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The Addison-Wesley Series in Finance
D e r i v a t i v e s IV a r k e t s '
Aladura Perstal
Copeland/Weston F/k'tpp')? and Colvorate Policy
Filancial
Dufey/Giddy Cses l International Ealdns
Marthinsen
Risk Takers: Uses J?7#Alpflle,
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Finallce: fnvd-nncn/-, Institlltions, and .#.f('l?7Jygd??7d?7,/
Eiteman/stonehillNoffett ultinational flf-undxx Finance
Gitman
Princles
Second Edition
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Principles t?./-lbncprrfcff'f?pt'lllcc -BriefEdition
Gj'tman/loehnk Fuldalnentals t////?,vdstf/prg GitmanNadura hztrodttctiol to Ffncnce HughesNacDonald International Banking: t-f?7# Cases Tcxt
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Derivatives
McDonald
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Megginson Colporate Fncncc Theol.y Melvin International ft//7dl?and Fztw3c: MshkinfEakins Financial A4't'I?-//J
and f?p-/fzfrfon.
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Moffett/stonehilliteman Fundalnentals of Finance
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RO B E RT L. Kellogg School of Management
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Editor-in-cbief: Denise Clinton Editor: Donna Battista Selkior Senior Project Mt7/vgw-l Mary Clare /lcEaving Developntel't Editon Marjorie Singer Anderson Senior Prodtttion Sltpervior: Nancy Fenton #lT./ Executtve A/ffp/zclla.gjjtlallagel ; Stephen Frail Deslkn Managen. Regina Hagen Kolenda Text Dcspicr? Rgip: Hpgen Kolenda : Rebecca Light Covel- Des i 4777 f/7rt7gc:Private Collectiort/Art for After Hours/superstock # blterior Covel Seniol'Manttfacturing fffyt'?'l Carol Melville Sltpplelllots Editon Marianne Groth Project Mf7/it7gt//?cn/'zElm Street Publishing Services, lnc. upo/isorl/g
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Copyright *2006 Pearson Education, lnc. reproduced, stored in a retrieval system, All rights reserved. No part of this publication may be mechanical, photocopying, recording, electronic, transmitted, in any form or by any means, or the of publisher. permission written otherwise, without the prior
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Library of Congress Cataloging-inublication McDonald, Robert L. (Robert Lynch), 19.54Derivatives markers, 2e / Robert L. McDonald. P. Cm. lncludes index. E ISBN 0-321-28030-X 1. Derivative securities. 1. Titlez 14G6024 A3 113946 2006 E 332 64'.5-dc21 E 0-321-28030-X ISBN 08 07 06 t)5 E
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CONTENTS
ElqT s
Exercise 57 Margins for Written Options Taxes
57
Quasi-trbitrage
4.3
Strategies 59 3.1 Basic lnsurance Strategies
59
Insuring a Long Position: Floors 59 Insuring a Short Position: Caps Selling Insurance 63
4.4 Golddiggers Revisited
Synthetic Forwards
3.2 3.3
3.4
-
Put-call Parity 68 Spreads apd Collars Bull and Bear Spreads Box Spreads 72 Ratio Spreads 73 Collrs 73
62
66
4.5
.
70 71
Speculating on Volatility
78
Straddles 78 ButyerfhrSpreads 81 Asymmetric Butterfly Spreads
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Introduction to Risk 91 Management 4 1 Basic ltisk Marfagement: The Producer's Perspective 91 92 a Minimum Price
a Forward Contract
Insurance: Guaranteeig with a Put Option 9) lnsuring by Selling a Ctl Adjusting the Amount of
'
9.5 96 tnsurance
Basic R-iskManagement: The Buyer's Perspective 98 Hedging with
a
Forward Contract
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Currency Contracts 154
Currency Prepaid Forward Currency Forward 156 Covered Interest Arbitrage
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127
Analogy 128 Pricing the Prepaid Forward by Discounted Present Value 129 Pricing the Prepaid Forward by Arbitrage 129 Pricing Prepaid Forwards with Dividends 131
.$.3 Forward Contracts on Stock
Uses of Index Futures
Asset Allocation 1.50 Cross-hedging with Index Futures
133
Creating a Synthetic Forward Contract 13' Synthetic Forwards in Market-Nlaking and Arbitrage 136 '
1.55
6.6 Carry Markets
Gold Futtzres
'
184
187
Seasonality: The Corn Forward Market 188 6.9 Natural Gas 191 6.10 011 194 6.11 Commodity Spreads 195 6.12 Hedging Seategies 196 6.8
Basis ltisk 197 Hedging Jet Fuel with Crude Oil Weather Derivatives 199 Cbapter 5';f77;772Iz?-.y 200 Readiug 201 f/rt/pcr Problelns 201
160 162
Chapter 7
199
lnterest Rate Forwards and
Futures 205 7.1 Bond Basics
,
205
Zero-coupon Bonds 206 Implied Forward Rates 208 Coupon Bonds 210 Zeros from Coupons 211 Interpreting the Coupon Rate 212 Continuously Compounded Yields 213
Chapter 6
'
181
Gold Investments 187 Evaluation of Gold Production
156
Commodity Forwards and Futures 169 6.1 Introduction to Commodity Fomvards 169 6.2 Equilibrium Pricing of Cornmodity Forwards 171 Nonstorability: Electricity 172 Pricing Commodity Forwards by Arbitrage: An Example 174
179
Storage Costs and Forward Prices 181 Storage Costs and the Lease Rate 182 The Convenince Yield 182
160
162 13?-olp/t?/pz.s Appotdix .%A:Taxes f//ot.f tbe Fortvard Price 166 and Appenix 5.B: Equating .Fo?-r&tz7Ws Flfp/rcs 166
176
Forward Prices and the Lease Rate
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5.5
ix
The Cpmmodit'y Lease Rate 178 The Lease Market for a Commodity 178
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Financial Forwards and 127 Futures 5.1 Alternative Ways to Buy a Stok 5.2 Prepaid Forward Contracts on Stock 128 Pricing the Prepaid Fonvard by
(7
with
.
Chapter
Chapter 4
Hedging
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142
Fumres Contracts
The S&P Futures Contract 143 and Marking 144 Margins to Market and Fotavard Comparing Futures Prices 146 Index Arbitrage in Practice: S&P Arbitrage 147 Quantolndex Contracts 149
108
FORWARDS, FUTURES,AND SWAPS 125
85 86
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Example: Another Equity-Linked Note 83 Cbapter
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5.4
Selling the Gain: Collars 108 Other Collar Strategies 112 Paylater Strategies 113 Selecting the Hedge Ratio 113 Cross-Hedging 114 QuanrityUncertainty 11 Cbapter Slfnc///t'Il3! 119 Reading 120 FJ/?W7cr f'?-t//p/t?a/.120
-
139
Does the Fotavard Price Predict the Future Price? 140 An Interpretation of the Forward Pricing Formula 141
Reasons to Hedge 103 Reasons Not to Hedge 106 Empirical Evidence on Hedging
lnsurance, Collars, and Other
Chapter 3
An Apparent Arbitrage and Resolution 175 Pencils Have a Positive Lease Rate
No-Arbitrage Bounds with Transaction Costs 138
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.
7.2
Forward Rate Agreements, Eurodollars, and Hedgng 214 Fonvard Rate Agreements 214 Synthetic FRAS 216 Eurodollar Futures 218 Interest Rate Strips and Stacks 223
CoxTElq'rs
k: Cox'rEx'rs Duratiop and Convexity
7.4
Duration 224 Duration Matching Convexity 228 Treasury-Bond and Trasury-Note Fumres 230
Repurchase Agreements Cbapter s'lf//i/lzf/r), 2.3.5Readilq 237 Furtber Pp-o/p/E7p7s 237 Price Appendix 7.A: Ilterest Rate and .8077:-/ Co/il/Ewlon. 241 Bonds 242 Bills 244
'
Chapter 8 swaps 247 8.1 An Example of a Commodity Swap
247
Physical Versus Financial Settlement 248 Why Is the Swap Price Not $20.50? 250 250 The Swap Counterparty The Market Value of a Swap 253
lnterest Rate Swaps
2.54
A Simple Interest Rate Swap 254 Pricing and the Swap Counterparty 255 Computing the Swap Rate in General 2.57 Th e Swap Curve 2.58 The Swap's Implicit Loan Balance 260 Deferred Swajs 261 Why Swap Interest Rates? 262 Amortizing and Accreting Swaps 263
Currency Swaps
264
Currency Swap Vormulas 267 Other Currency Swaps 267
268
Commodity Swaps
The Commodity Swap Price 268 and Swaps with Variable Quantity Price 269
8.5 8.6
Swaptions 271 Total Return Swaps Cbapter
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274
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Chapter 9
Parity and otheroption Relationships 281 9.1 Put-call Parit'y 281 Options on Stocts 283 Options on Currencies 286 Options on ljpnds 286 Generalized Parity and Exchange
Options
10.2
Two or More Binomial Periods
10.3 10.4 10.5
Put Options 328 American Options 329 Options on Other Assets
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Chapter 10
Binomial Option
Pricing: I 31 3 10.1 A Oneeriod
Binomial Tree
313
ing the Option Price 314 c The Binomial Solution 315 Arbitraging a Mispriced Option 318
A Two-periotl European Call 323 Many Binomial Periods 326
Chapter 11
Binomial
Pricing'. 11 343
1..1.4 11.5
330
Chapter 12
11.1 Understanding Early Exercise 343 11.2 Understandlg Risk-Neutral Pricing
11.3
3.46
The Risk-Neutral Probabtliv Pricing an Option tiing Real Probabiliries 347 The Binomial Tree and
Lognormality
Estimating Volatility 360 Stock; Paying Discrete Dividends 361 Nlodeling Discrete Dividends 361 Problems with the Discrete Dividend Tree 362 A Binomial Tree Using the Prepaid Forward 363 Cbapter &f77;D7tz?-), 36..$ Readlg 366 htrtber Problenls 366 Appendix 11.A: Priciltg Optiolls 1t,,9/7 Trtte Pl-obabilities J 69 Appeltdix 1 1.B: V/Jy Does Risk-Nelt 1n7/ Pric'g Fb?-/,:? 369 Utility-Based Valuation 369 Standard Discounted Cash Flow Risk-Neutral Pricing 371 Example 372 Why Risk-Neutral Pricing Works
323
option
The Black-scholes Formula 375 12.1 Introduction to the Black-scholes Formula 375 Call Options 375
346
12.2
Put Options 378 When Is the Black-scholes Formula Valid? 379 Applying the Formpla to Other
Assets
351
The Random Walk Model 3.51 Modeling Stock Prices as a Random Walk 3.52 Continuously Compouded Returns 353 The Standard Deviation of Returns 354 The Binomial Model 3.5.5 Lognormality and the Binomial Model 3.$5
M
Alternative Binomial Trees 3.$8 Is the Binomial Nlodel Realistic? 3.59
Option on a Stock Index 330 Options on Currencies 332 Options on Futures Contracts 332 Options on Commodities 334 Options on Bonds 335 Summary 336 Cbapter 51/771772t?ry 337 Fltrtber REutii/g 337 Pro/p/cais 338 Appendix 1 0.A: Taxes and Option Prccs 3#1
287
Options to ExchAng Stock 288 What Are Calls and Puts? 289 Currency Options 290 Comparing Options with Respect to Style, Maturity, and Strike 292 European Versus American Options 293 Maximum and Mipimtzm Option Prices 293 Early Exercise for American Options 294 Time to Expiration 297 Differenr strikePrices 299 Exercise and Moneyness 304 305 Cbaptel. Fffrl/Jc?Readiug 306 Prolllca/s 306 Appendix .9.42 Parity Soi/atfs JorAmerica'' Opfos.s J 10 Appendix 9.B: Algebraic Proojs of Strike-price Relations 31 1
Omptlt
272
A Graphical Interpretation of the Binomial Formula 319 Risk-Neutral Pricing 320 Constructing a Binomial Tree 321 Another One-period Example 322 Summary 322
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223
k
379
Options on Srocks with Discrete Dividends 380 Options on Currencies 381 Options on Futures 381
12.3
Option Greeks
382
Deflnition of the Greeks 382 Greek Measures for Portfolios optionElasticity 389
388
xii
XhwC
'r
Chapter 16
Problems Formldas for Appendix 14.A: zarcn.g Eotic Options 466 Asian Options Based on the Geometric Average 466 Compound Options 467 Infnitely Lived Exchange Option .,/62
12.4
12.5
Ptlrchased Call Option 396 Calendar Spreads 397 lmplied Volatility 400 Computing Implied Volatiliv 400 Using Implied Volatility 402
Perpetual American Options Brrier Present Values Perpetual Calls 404 Perpetu'l Puts 404
cbapter
13.5
403
403
z05
Fttrtber Reading 405 Problems 406 Append 12.AJ T/7cStak'dard No7'/af/l Distribtttiol' 409 Option Appendix 12.B: Formttlas Jt??Grqcis 410 Delta 410 Gamma 410 Theta 410 Vega 411 Rho 411 Psi 411
Market-Making and Delta-Hedging 41 3 13.1 What Do Market-Makers Do? 13.2 Market-Makef Risk 414
417
422
429
..
' '
rr
...-
14.4
ajaopiju/
Using Gamma to Better Approximate the Change in the Option Price 423
,tf
R.
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'o.jll
Other Bond Features 522 PutWarrants
Compensation Options
16.3
Whose Valuation? 5l5 Valuation lnputs 5l7 An Alternative Approach to Expensing Option Grants 528 Repricing of Compensation Options Reload Options 532 Level 3 Communications The Use of Collars in
Acquisitions
459
455
15.5
Cbapter F/frl/pcr Readq P?-o/p/c//is 543 ,slf//p/cfp--p
482
488
Strategies Votivatedby Tnx nnd Regulatory Considerations 490
Cbapter Fl/rf'/7crReading fl?-o/l/Ev/i.s 498
,9//77/72/?-)/
498 498
490 49.5
542 542
Chapter 17 17.1
486
Capital Gains Deferral' Tax-Deductitnle Equity
538
538
Options in Coupon Bonds 482 Options in Equity-lwinlcedNotes 483 Valuing and Structuring an Equity-Linked CD 483 Alternative Structures 485
Gold-Linked kotes 486 Notes with Embedded Options
523
The Northrop Grumman--f'lkv Merger
Bonds with Embedded Options
Golddiggers
!20
16.2
15.4 Engineered Solutions fr
Gap Options 457 Exchange Options 459 ,9//?277;t?-y
''$.';'.*
Zero-coupon Bonds 474 Coypon Bonds 475 Equity-t-inked Bonds 476 Commoditplnked Bonds 478 Currency-Linked Bonds 481
Compound Options 453 Compound Option Parity 454
European Exchange Options 461 Cbapter Reading' 462 .lhf)-#7c?.
;..
Finacial Ehgineering and Security Design 473 15.1 The Modigliani-Miller Theorem 473 15.2 Pricing and Designing Structured Notes 474
4.50
Stocks Options on Dividendaying with Compound Currency Hedging Options 456
14.5 14.6
.'
Chapter 15
Hedging Problem 44.5 Oprions on the Average 446 Comparing Asian Options 447 An Asian Solution for XYZ 448 Types of Barrier Options Currency Hedging 451
.'..'.N;>. J . (
FINANCIAL ENGINEERINGAND APPLICATIONS 4 71
Exotic Options: I 443 443 lntroduction Asian Optims 444
Barrier Optiops 449
a*
503
Debt qnd Equity as Options 503 Multiple Debt Issues 511 warrants512 co'nvertible Bonds 513 Callable Bonds 516 Bond Valuation Based on the Stock Price
PART FO
Chapter 14
14.3
.
corporate
Applications 503 16.1 Equity Debt, and Warrants
'
Market-Maha-ng as Instumnce 436 437 Sttmman' 438 C/pt7pf'c?. FJ/rl/JD438 Readq p/-o/p/tu/ps 438 Appotdix 13.A: 'Thy/op.Series 44j Approximations Appendix j3.B: Greeks j;; tbe Model 441
413
An Exampl of Delta-l-ledging for 2 D alrs 41# Interpreting the Prolit Calculation 418 Delta-l-ledging for Several Dgys 420 A Self-Financing Portfolio: The Stock Moves One (r 422 The Mathematics of
Delta-l-ledging
Analysis
The Blackcholes Argument 429 Delta-Hedging of American Options What Is the Advantage to Frequent Re-l-ledging? 431 Delta-Hedging in Practice 432 Gamma-Neutraliv 433 lnsurance 436 Mafket-Makers
'
13.4
The Blackcholes
XYZ'S
Option Risk in the Absence of Hedging 414 oi Delta and Gamma as Measures Exposure 416
Delta-Hedgzg
13.6
14.1 14.2
Chapter 13
13.3
424 Delta-Gamma Approximations Theta: Accounting for Time 425 Undersranding the Market-/laker's Prolit 427
395
Profit Diagrams Before Maturity
-$,,77,'77,,3,),
k
colq'rErq'rs
o h1 E rqT s
17.2
Real Options 547 Investment and the NPV Rule
548
Static NPV 548 The Correct Use of NPV 549 The Project as an Option 5.50 lnvestment under Uncertaint'y 551 A Simple DCF Problem 551 Valuing Derivatives on rhe Cash Flow 55l
Evaluating a Project with a z-eear Investment Horizon 54 Evaluating the Project with an lrnite lnvestment Horizon 558
Real Options in Practice
Lognormal Confdence Intervals 600 The Conditional Expected Price 602 The Black-scholes Formula 604 the Parameters of a Estmating 18..$
558
Peak-Load Electricity Generation Research and Development 563
559
Commodity Extraction as an ' Option 565
17.4
Lognormal Distribution How Are Asset Prices Distributed? 608
Single-Barrel Extraction under Certainty 565 Single-Barrel Extraction upder Uncertainty 569 Valuing an Infinite Oil Reserve Commodit'y Extraction with Shut-Down and Restart Options Permanent Shutting Down 574 Investment Whe Shutdown Is Possible 576 578 Restarting roduction Additional Options 578
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PART F
...
.
jk.j.;tryqt!(,q()..;;.) !rqqq4j,j;.z;.
. . . ..
.z
587
The Lognormal Distribution 593 A Lognormal Model of Stock Prices 595 Lognormal Probabilit'y Calculations 598 599
Computing Random Numbers
621
Simulating Correlated Stock 643 Prits
Monte Carlo Valuation
P?-o/J/cp/.
19.6
Ecient
Control Variate Method 630 Other Monte Carlo Methods 632 Valuation of American Options
21.1
646
21.2
Geometric Brownian Modpn
The Shqrpe Rqtio 659 The Risk-Neutral Procss It's Lemma 663
about
msk-Neutral Pricing
655
660
Functions of an It Process 663 Multivariate It's Lemma 66.5
Valuing a Claim on Stl 666 667 The Proceqs Followed by
The Black-scholes Equation Verifying the Formula for a
681
Derivative 683 The Black-scholes Equation and Equilibrium Returns 686 What If the Underlying Asset Is Not an Investment Asset? 688
Brownian Motion and It's
Lognormality 65.5 Relative Importance of the Drif't and Noise Terms 6.56 Correlated lt Processes 657 Multiplicaton Rules 658
20.4 20.5 20.6
The Black-scholes 679 Differential Equations and Valuation uder Certainty 679 The Valuation Equation 680 Bonds 680 Dividendaying stocks 681 The General Structure 681
Defnition of Brownian Motion 650 Properties of Bmwnian Motion 651 Arithmetic roWian Motion 6.5'3 The Ornstein-hlenbeck Process 6.54
20.3
675
Equation
Lemma 649 20.1 The Black-scholes Assunptin Stock Prices 649 Brownian Motion 650
624
Monte Carlo Valuation
Pl-ol7/tD//,s
Chapter 21
A pE'a dix 19.A: Fora7i//uz.JorGeometric Average Options 648
Chapter 20
XV
Fitrtbel. Reading
,9f?/7?/;f7?-),
Monte Carlo Vatuation of a European Call 62,5 Accuracy of Monte Carlo 626 Arithmetic Asian Option 627
19.5
Jumps 643
Generating 11 Correlated Lognormal 644 Random Vriables C/7f7p 645 ter Fltrtber Readiltg 645
Using Sums of Uniformly Distributed Random Varia es 622 Using the Inverse Cumulative Normal Distribution 622
Simulating a Sequence of Stock Prices 623
Converting p.NormalRandpm Variable to Standard Normal 590 Sums of Normal Random Variables 591
Probabilities
619
Simulating Lognormal Stock Prices 623
Chapter 18
18.4
Monte Carlp Valuation Computing the Optio Price as a Discounted Expected Value 617 Valuation with Risk-Neutral Probabilities 618 Valuation with True Probabilities
THEORY 585
18.2 18.3
605
Chapter 19 19.1
ADVANCEDPRICING
The Lognormal Distribution 587 18.1 The Normal Distribution
19.9
.
19.2 .
The Poisson Distribution 636 Simulatirlg Jumpswith the Poisson Distribution 639
,sfap/pfp-.)#
Cbapter .Flf?-#pcrReadinz 580 P/-ol7/cpo.$'580 Appendix 17.A: Calcttlation of Opfi??C$'# 583 Tinte to Drill a?: Oil 'T# Appendix 17.Bt T/pcSoltttion tf/ff/; Sbtttting Dorfal and Rcsfflrff/i,g 583 .,
19.7 19.8
Multiple
Histograms 608 Normal Probability Plots 609 6.13 Cbapter lhtrtber Reading 613 Prola/eaos 614 Appendix 18.A: T/lc Expectation ofa fwogaongt'l/Variable 615 a Ntllwffv Appca-x 18 rB: Casfnfco-n,g . .. ( Probabilit Plot 616
jiib
sifp.n//7f/rl,
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colql-Elq'rs
%.CONTENTS
xiv
21.4 21.5
690
Interpreting the Black-scholes Equation 690 The Backward Equation 691 Derivative Prices as Discounted Expected Cash Flows 692 Changing the Numeraire 693 Option Pricing When the Stok Price
Can
Jump
696
Merton's Solution for Diversifiable 697
Jumps I
Cbapter 5'J/777777f7r, 698 Fl/r#pcr Reading 698 .'I'h-olalEv/z,s 699 Appoldix 21 .A: Mltltivariate Black-scboles Analysis 700 Appendix 21.B: Proof ofpropositiol'
701
.'1
668 Proving the Propositior @ Specifc Examples 669 Q?7 670 Valuing a Claim on
630
.%l
633
20.8
Jumpsin the Stock Price Cbaptel. 67# .9/77777:f7?7,
672
Chapter 22 Exotic Options: 11 703 22.1 M-or-Nothing Options 703 Terminology 703 Cash-or-Nothing Options Asset-or-Nothing Options
704 706
21.1
k
xvi
Cox'rElq'rs CONTENTS
Ordinary Options and Gap Options Delta-l-ledging All-or-Nothing Options 707
706
All-or-Nothing Barrie Options
710
Cash-or-Nothing Barrier Options Asser-or-Noching Barrier Oprions Rebate Options 716
Barrier Options Quanto, 718
22.3 22.4
22.5
22.6
Variance and Volatility Swaps Pricing Volatility 759 xtending the Blackcholes
763
710 715
Currency-t-inlced Options
724
Foreign Euity Call Struck in Foreign Currency 728 Foreign Equity Call Struck in Domestic Currency 729 Fixed Exchange Rate Foreign Equity Call 730 Equity-Linked Foreign Exchange Call 731
Pricing
Measmement and Behvior ' Vol at ility 744
Value at lkisk for One Stock 815 817 VaR for Two or More Stcks Poitfolios for Nonlinear 819 VaR VaR for Bonds 826 Estimating Volatility 830 Bootstrapping Return Distributions 831 Issues with VaR 832 Alternative ltisk Measures 832 VaR and the ltisk-Neutral Distribution
Model 790 A Binomial Interest Rate Model
24.5
The Black-Derman--foy Model Verifying Yields 802 Verihring Volatilities 803 Constructing a Black-Derman--fby Tree 804 Pricing Examples 80.5
PART SlX
APPENDIXES 871
Appendix A
The Greek Alphabet
Continuous Compounding 875 B.1 The Language of lnterest Rates 875 B.2 The Logarithmic and Exponential Functions 876 Changing Interest Rates 877 Symmetry for lncreases and Decreases
878
837
Problellls
Appendix C C.1
Jensen'sInequality
881 Example: The Exponential Function 881 Example: The Price of a Call 882 Proof of Jensen's Inequality 884
C.2 C.3
26.2
Credit Risk 841 Default Concepts and Terminology 841 The Merton Default Model
26.3
Default at Maturity 843 Related Moels 845 and Default Experience Bond Ratings
Appendix D An Introduction to Visual Basic for Applications 885 D.1 Calculations without VBA 885 D.2 How to Learn VBA 886 D.3 Calculations with 'VBA 886
26.1
847
793
Using Ratings to Assess Bankruptcy Probability 847 Recovery Rates 8.50 Reduced Form Bankruptcy Models
796
26.4
Credit Instruments
873
Appendix B
83.5 Subadditive Iisk Measures Cbapter ,Slf?/pplzp-)?838 Rfr//pcr Reading 839 l''?-ol//tDns 839
Chapter 26
The Rendelman-Bartter Model 785 The Vasicek Model 786 The Cox-lngersoll-ltossModel 787 Comparing Vasicek and C1R 788 Bond Options, Caps, and the Black
Zero-coupon Bond Prices 794 Yields and Expected lnterest Rates Option Pricing 797
of
Time-varying Volatility: ARCH 747 The GARCH Model 751 Variation 755 Realized Quadratic
779
Equl'librium Short-Rate Bond Price Models 785
733
Historical Volatility 744 Exponentially Weiglzted Moving Average
746
Chapter 25 Value at Risk 25.1 Value at ltiik 813
The Belavior of Bonds and Inieresr' .. . . . '. . .. . . . .. Rates 780 A.n Impossibl l Prtcing Vdl 780 Etation for Bonds 781 An Equilibrium for Delta-Gamma Approxirtions Bonds 784
24.3
23.2
and
Chapter 24 Interest Rate Models 24.1 Market-Maling and Bond
Other Multivariate Options
Chapter 23 Volatility 741 23.1 lmplied Volatility 741
Model
Evidence 771 77.3 Cbapter &fn;?77'I).)? 773 htrtbel. Rct-ltfia,g Problel'ls 774 777 Appendix 23.4
727
Exchange Options 732 Options on the Best of Two Assets Basket Options 735 Cbaptel. &f7?;/?7t7?''y736 ltrtber Ref.ltfb/g 736 Problellls 737
lhfrl/Jcr
758
xvii
C/gtzplcrSltnul'an' 866 Flf/-#pcl-Readtg 867 f'lrol-l/c/as 867
sfp7/ntp'),808 Reading 808 f-b-o/pkcp/s809 Appendix 24.A: T/JC HeatbJtefrrow-Mb?-fo/zModel 811 Clmpter
757
Implied Volarility 764 Constant Elasticity of Variance 766 The Heston Model 768
JumpRisk
717
The Yen Perspective 720 The Dollar Perspective 721 A Binomial Model for the Dollar-Denominated Investor
-
Hedging and Pricing Volatility
%
852
853
Collateralized Debt Obligations 8.53 Credit Default Swaps and Related Structures 858 Pricing a Default Swap 862 CDS Indices 864
Problons
'
884
Creating a Simple Funcyion 886 A Simple Example of a Subroutine 888 Creating a Button to Invoke a Subroutine 888 Functions Can Call Functions 889 lllegal Function Names 889 Differences between Functions and Subroutines 890
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xviii
CONTENTS
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Using a Named Range to Read and Write Numbers from a Spreadsheet 891 Reading and Writing to CellsThat Are Not Named 892 Using the Cetls Functions to Read and Write to Cells 892 Reading from within a Function 893
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Creating a Binomial Tree 900 Other Kinds of Loops 901 D.9 Reading and Writing Arrays 901 rrays as Outputs 901 Arrays as lnputs 903 D.10 Miscellany 904 Getting Excel to Generate Macros for You
904 Using Multiple Modules 90.5 Recalculaon Speed 90.5 Debugging 906 Creating an Add-ln 906
Using Excel Functions 9om within VBA 893
Using 'VBA to Compute the Blackcholes Formula 894 The Object Browser 895
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envatives have moved to the center of modern corporate hnance, investments. and the mnnagement of financ ial institutions. They have also had a profound impact on other management functions such as business strategy, operations management, and marketina. A malor drawback. however. to makinz the power of derivatives accessible to students and practitioners alike has been the relatively high degree of mathematical sophistication required for understanding the underlying concepts and tools. With Robert McDonald s Derivatives Markets, we fnally have a derivatives text that is a wonderful blend of the economics and mathematics of delivatives plicing and easily accessible to A students and advanced undergraduates. It is a special pleasure for me to introduce this new edition, since I have long had the highest regard for the author's professional achievements and personal qualities. The book's orientation is neitheroverly sophisticated norwatered down, butrather ligor that creates an inherent iexibility for the structuring of a a mix o f intuition and derivatives course. The author begins with an introduction to forwards and futures and motivates the presentation with a discussion of their use in insurance and risk management. He looks in detail at forwards and futures on stocks, stock indices, currencies, interest rates, and swaps. His eeatment of options then follows logically from concepts developed in the eadier chapters. The heart of the text an extensive treatment showcases the author's of the binomial.option model and the Black-scholes equatio crystal-clear wliting and logicz development of concepts. Excellent chapters on financ ial engineering, security design, corporate applications, and real options follow and shed light on how the concepts can be applied to actual problems. The last third of the text orovides an advanced treatment of the most important concepts of delivatives discussed eadier. This part can be used by itself in an advanced derivatives course, or as a usef'ul reference in introductory courses. A rigorous development of the Black-scholes equation, exotic options, and interest rte models are . presente d us in g Brownian Motion and It's Lemma. Monte Carlo simulation methods are a l so discussed in detail. New chapters on volatility and credit risk provide a clea.r discussion of these fast-developing areas. D e rivatives concepts are now required for every advanced linarlce topic. Therefore, it is essential to introduce these concepts at an eady stage of MBA and undergraduate business o'r economics programs, and in a fashion that most students can understand. This text achieves tu s goa j jrj such an appealing, inviting way that students will actually enjoy theirjourney toward an understanding of derivatives. EotlAqoo
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llirty years ago the Black-scholes formula was nem and delivatives was an esoteric and specialized subject. Today, a baslc knowledge of derivatives is necessary to understand modern inance. For example, corporatlons routinely hedge and insure uslng denvatives finance activities with strucmred products and use den'vatives models in capital budgeting. This book will help you to understand the den'vative instruments that exist, how they are used, who sells them, how they are prlced, and how the tools and concepts are useful more broadly in snance. Den'vatives is necessarily an analytical subject, but 1 have tried throughout to emphasize intuition and to provlde a common sense way to think about the formulas. l do assume that a reader of thls book ah-eady understands basic fnancial concepts such as . Present value and elementary statistical concepts such as mean and standard deviation. Most of thebookshould thus be accessible to anyone who has smdied elementary fnance. For those who want to understand the sublect at a deeper level, the last part of the book and presents some Of the develops the. Black-scholes apploac jy to pjjcug (jesivatives .. . . standard mathematlcal tools used in option pncing, such as It s Lemma, There are also chapters dealing with applications: corporate applications, financial engiheering, and real options. In order to make the book accesslble to readers with widely varying backooounds . arld expenences, 1 use a tiered approach to the mathymatics. Chapters 1-9 emphasize present value calculations, and there is almost no calculus until Chapter 18. . . . using Excel spreadsheets on this book can be repllcated Most of the calculations in with the book. These allow CD-ROM that the comes you to expen'ment with the pricing models and bull'd your own spreadsheets. The spreadsheets on the CD-ROM contain option pricing functlons written in Visual Basic for Applications, the macro language ln Excel. You can easily incorporate these functions into your own spreadsheets. You can also exnmine and modif'y the Vlsual Baslc code for the functions. Appendlx D explains how to wn'te such f'unctions in Excel and documentation on the CD-ROM lists the option pncmg f.un dtions that come wit.h the book. Relevant built-in Excel t'unctionsare also mentloned tlurllghout the j)00 k .
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WHAT X11
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I N TH E SECO N D ED ITI O N
PREFACE
general introduction j third of the book. The srstconrse is a u to the 'Iirst two-thirds and last options, swaps, and structured products), t futures, products (principally to derivative The second course is for those wanting a markets in which they trade, and applications. and the ability to perform their own analysis. deeper understandilg of the pricing models basic statistics and have seen calculus, The advanced course assumes that students know option-pricing framework as fully as and from that point develops the Black-scholes scienMBA-level course will produce rocket lo-week possible. N0 one expects that a students cheating of derivatives and it would be tijts, but mathematics is the language to pretend otherwise. order than it occurs in the book, Yu may Want to cover the material in a different severa.l possible paths allow Qexible use of the material. 1 indicate so I wrote chapters to possible to hop around. For example, 1 through the material below. In many cases it i.s siniulation chapters on lognormality and Monte Carlo wrote the book expecting that the might be used in a first derivatives course. 1 introduc ihe basic building The book has five parts plus appendixes. Part Chapters 2 and 3 and call and put optins. blockj of' derivatives: forward contracts hedging and invetfent strategies. examine these basic instruments and some common risk manageent tools ad discusses why Chapter 4 illutrates theuse of derivatives as chapters focus on undefstanding the firms might care about risk management. These pricing. contracts and strategies, but not on and swaps contracts. 1:! these Part 2 considers the pricing of forward, futures, pre-speified pzce, at a fture day. The contracts, you are obligated to buy an asset at a Chapter and how is it detrinined? Iain question is: What is the pre-specified price, commodities, assets, Chapter 6 discusses 5 examines forwards and futures on inancial forward contracts. Chapter 8 shows how rate and Chapter 7 looks at bond and interest from forward prices. swap prices can be deduced intuition about options prior to Part 3 studies option pricing. Chapter 9 develops 10 and 11 cover biomial option delving into the mechanics of option pricing. Chapters formula and option Greeks. Chapter 13 pricing and Chapter 12, the Black-scholes used by market-makefs when managing explains delta-hedging, which is the technique hedging relates to pricing. Clppter 14 looks the risk of an option position, and how compound including Asian options, barrier ojtins, at a few important exotic options, options, and exchange options. chapters are applied in Part 4. Chapter 15 The techniques and formulas in earlier from the which is the creation of new financial products covers fnancial engineering, compensation and equity pricing, derivatives building btcscks in earlier chapters. Debt 17 studies real optins-the 16. Chapter in covered Chapter options, and mergers are valuation and management of physical investapplication of derivatives models to the lnents.
hedging in depth. The material in this part Finally, Part 5 explores pricing and assumptions underlying the standard derivatives explains in more detail the structure and model and shows how the Black-scholes models. Clapter l 8 covers the Iognormal discusses Monte Carlo valuation, a powerful formula is an expected value. Chapter 19 20 explains what lt means to say that and commonly used pricing technique. Chapter
%
XZii
stock prices follow a diffusion process, and also covers lt's Lemma, which is a key result in the study of derivatives. (At this point you will discover that It's Lemma has already been developed intuitively in Chapter l3, using a simple numerical example.) Chapterzl delives the Black-scholes partialdifferential equation (PDE).Although the Black-scholes fonllltlais famous, the Black-scholes eqttatiol, discussed in this chapter, is the more profound result. Chapter 22 covers exotic options in more detail than Chapter 14, including digital barrier options and quantos. Chapter 23 discusses volatility estimation and stochastic volatility pricing models. Chapter 24 shows how and binomial analysis apply to bonds and interest rate delivatives. the Blaclscholes Chapter 25 covers value-at-risk, and Chapter 26 discusses the burgeoning market in credit products. WHAT
IS NEW IN THE SECOND
EDITION
rfhere are two new chapters in this edition, covering volatility and credit risk: * Chapter 23 covers empirical volatility models, such as GARCH and realized volatility',financial instruments that can be used to hedge volatility, such as variance swaps', and pricing models that incorporate jumps and stochastic volatility, such as the Hestop model. . Chapter 26 covers stnlctural models of bankruptcy risk (the Merton modell; tranched structtlres such as collateralized debt obligations', credit default swaps and credit indexes. throughout the book. Among the
There are numerous changes and new examples changes are the following: important more
. An expanded discussion of bond convexity * An expanded treatment of computing hedge ratios * An' expanded treatment of convertible and callable bonds * Discussion of the new option expensing expensing proposal
rules in
FAS 123R and the Bulow-shoven
. Discussion of a variable prepaid forward on Disney stock issue by Roy Disney . ln-depth discussion of a mandatorily convertible bond issued by Marshall & Ilsley, including pricing and structuring . The
use of sirulation
to price American options
. Additional discussion of implied volatility . Enhanced discussion of te link between discounted cash flow valuation and riskneutral valuation
.
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xxiv
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optional. The advanced course assumes student.s havq already 5.1, 5.2, 7.1, and Appendix B are recommended background.for aII introductory courses.
-
taken a basic course. Sections 1
THE MATERIAL
.4,
the material in this book. The matelial is There are potentially many ways to cover mathematical dificulty, which means that generally presented in order of increasing For example, tixedincome is related-material is sometimes split across distant chapters. exotic options in Chapters 14 and 22. Each of these covered in Chapters 7 and 24, and neighboring chapters. As an illustration of one way to use chapters is at the level of the chapters I skip in the courses 1 teach (withinthe the book, here is the material 1 cover constraints): some specisc topics due to time 15.42t5.5, 16, 1-6, 7.1, 8-10. 11.1-11.2, 12, 13.1-13.3, 14, * Ineoductory course: '
13, 18-22, 7, 8, 15.1-15.3, 23, 24, 25, 26. * Advanced course'. possible sets of chapters to use in courses that The table on page xxv outlines some sections of the book that provide background have different emphases. There are a few short-sales (Section 1.4), conreader should understand. These include on topics every (Sections 5.1 and 5.2), contracts prepyid fotward tinuous compounding (Appendix B), 7.1). (Section an'd zero-coupon bonds and implied forward rates
8. Swaps 9. Parity and Other Option Relationships l0. Binomial Option Pricing: l l 1. Binomial Option Pficing: 11 12. The Black-scholes Formula 13. Market-Making and Delta-l-ledgng 14. Exotic Options: l
A
NOTE
ON EXAMPLES
assist you this book display intermediate steps to Many of the numerical examples in it will also be possible for you to crein following the calculations. ln most cases basic assumptions. and compute the same answers starting from the ate a spreqdsheet generally rounded to three or four decimal However, numbers displayed in the text are signifcant digits. This creates a points, while spreadsheet calculations have many morewould obtain using spreadsheet, a those you dilemma: Should results in the book match equations? displayed obtain by computing tle or thoje you would will provide tlze results the numerical examples in the book rule, general As a spreadsheet. The displayed entering the equations directly in a will be you would obtain by logic of a calculatiqn, but a spreadsheet caleulations will help you follow the helpful in reproducing the linal rsult.
SUPPLEMENTS
accompanies materials for both instructors and smdents A robust package of ancillary the text.
15. Financial Engineering *
l6. Corporate Applications l7. Real Options 18. The Lognormal Distribution ..
l9. Monte Carlo Yaluation 20. Brownian Motion and It's Lemma 21 The Black-scholes Equation .
22. Exotic Options: FI 23. Volatility 24. Interest Rate Models 25. Value at Iisk 26. Credit lkisk
lnstructor's
Resomces
Forinstructors, an extensive set of online tools is available for download from the catalog P age for Derivatives Markets at www.aw-bc.com/finance.
i
AcxxowuEocuEx'rs
xxvi
% PREFACE Mark Cassano, University of CalAn online lnstructor's Solutions Manual by solutions to Ohio State University, contains complete gary, and Rdiger Fahlenbrach, problems. selected solutions spreadsheet to all end-of-chapter problems in the text and Indianapolis, features of University Will, The online Test Bank by Matthew W. and questions, short-answer five approximately ten to Vteenmultiple-choice qtlestions, for each chapter of the book. one longer essay question electronic formats, includingWindows The Test Bank is available in both print and Test Bank are and Microsoft Word files. The TestGen and r Macintosh TestGelt fles available online at hup://www.aw-bc-com/irc. Cao, Pennsylvania State UniOnline Powerpoint slides, developed by Charles Emm, Georgia State Uniand Ekaterina versity', Ufuk Ince, Unikersity of Washington; Copies of the slidej 4an the book. selected from art versity, provide lcture outlines and facilitate note taking duling class. be downsized and distributed to students to the computerized Test Bank filej (Testcontains The lnstructors Resource Disk the Test Bank liles (Word) and Powerpoint Gen), the Instructor Manual files (Word), 'liles.
Student Resources A printed Solutions ilanual by
Mark Cassano, University qf Calgary and Riidiger all the ven-numbered problems Fahlenbrach, Ohio State University, provides answers to in the textbook. Solutions, by Riidiger Fahlenbrach, New to this edition, Practice Problms and and worked-out solutions for each problems additional contains
Ohio State University, chapter of the textbook. functions in Excel are included on Spreadsheets with user-defined option pricing functions are written in VBA, with Excel the book. These a CD-ROM packaged with Visual Basic editor built into Excel. These the code accessible and modifable via the the book's Web site. spreadsheets and any updates are also posted on
ACKNOWLEDGMENTS book in 1994 by asking that the Kellogg Kellogg student Tejinder Singh catalyzed the derivatives course. Kathleen Hagerty and I Finance Department offer an advanced notes (developedwith Kathleen's initially co-taughtthat course and my part of the course third of this book. help and feedback) evolved into the last edition, I received invaluable assistance from kiidiger In preparing the second material with a cdtical of Fahlenbrach, Ohio State University, Fho read much the new other mistakes and offered valuable suggestions. Numerous eye, and who both caught the first edition. Colleagues in sttldents, colleagues. and readers provided comments on with their time include Torben Anthe Kellogg hnance department who were generous Deborah Lucas, Mitchell Petersen, Ernst dersen, Kathleen Hagerty, Ravi Jagannathan, Many Kellogg MBA and PIA.D.stuSchaumburg, Costis Skiadas, and David Stowell. Arne Staal, Caroline Sasseville, and Alex dents helped, but I want to especially thank
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xxvii
Wolf. Others who reviewed new material include David Bates, University of Iowa; Luca Benzoni, University of Minnesota; Milhail Chernov, Columbia University', and Darrell Dufie, Stanford University. Mark Schroder, Michigan State University, kindly provided code to calculate the non-central chi-squltred distribution, and Kellogg student Scott Freemon implemented this code in VBA. A special note of thanks goes to David Hait, president of OptionMetrics, for permission to include options data on the CD-ROM. ' I also received help and comments from George Allayanis, University of Virginia; Jeremy Bulow, Stanford University', Raul Guerrero, Dynamic Decisions', Darrell Karolyi, Compensation Strategies, lnc.; C. F. Lee, Rutgers University; David Nachman, University of Georgia', Anil Shivdasani, University of Nol'th Carolinai. and Nicholas Wonder, Western Washington University. I would like to particularly thank those who provided valuable feedback for the second edition, including Turan Bali, Baruch College, City University of New York; Philip Bond, Wharton School, University of Pennsylvania; Michael Brandt Duke University',Chades Cao, Pennsylvaniastate University',Bruce Grundy, Melbourne Business School, Australia; Shantaram Hegde, University of Connecticut', Frank Leiber, Bell Atlantic', Ehud Ronn, University of Texas, Austin; Nejat Seyhun, University of Michigan', John Stanseld, University of Missouri, Columbia', Christopher Stivers, University of Georgia', Joel Vandeni Dartmouth College; and Guofu Zhou, Washington University, St. Louis. I would be remiss not to aclcnowledge those who assisted with the srst edition, including Tom Arnold, Louisiana State University', David Bates, University of Iowa', Luca Benzoni, University of Minnesota', Mark Broadie, Columbia University', Mark A. Cassano, University of Calgay; George M. Constantinides, University of Chicago',Kent Daniel, Northwestern University', Jan Eberly, Northwestern University; Virginia France, University of Illinois', Steven Freund, Suffolk University', Rob Gertner, University of Chicago; Kathleen Hagerty, Northwestern University', David Haushalter, University of Oregon; James E. Hodder, University of Wisconsin-Madison; Ravi Jagannathan, Northwestern University', Avraham Kamara, University of Washington', Kenneth Kavajecz, Wharto:i School, University of Pennsylvania', Arvind Krishnamurthy, Northwestern University; Dennis Lasser, State University of New York at Binghamton', Cornelis A. Los, Kent State University', Deborah Lucas, Northwestern University', Alan Marcus, Boston College', Mitchell Petersen, Northwestern University', Todd Pulvino, Northwestern University', Ernst Schaumburg, Northwestern University', Eduardo Schwartz, University of California-l-os Angeles; David Shimko, Risk Capital Management Pqtners, lnc.; Anil Shivdasani, University of North Carolina-chapel Hill; Costis Sldadas, Northwestern University; Donald Smith, Boston University; David Stowell, Northwestern University', Alex Triantis, University of Maryland; and Zhenyu Wang, Yale University. The following served as software reviewers'. James Bennett, University of Massachusetts-Boston; Gordon H. Dash, University of Rhode Island', Adam Schwartz, University of Mississippi; and Robert E. Whaley, Duke University. Special thanks are due to George Constantinides, Jennie France, Kathleen Hagerty, Ken Kavajecz, Alan Marcus. Costis Skiadas, and Alex Triantis for their willingness to
'
AcltNowLEocu
xxviii %.P REFACE multiple times and for class-testing. Mark read and comment upon some of the material which I used both to compute the Broadie generously provided his pricing software, calculations. Heston model and to double-check my own Will, and Charles Cao for I thank Rdigey Fahlenbrach, Mark Cassano, Matt In addition, Riidiger book. for this their excellent work on the ancillary materials and KlishNovikov, Dnaitry Fahlenbrach, Paskalis Glabadanidis, Jeremy Graveline. Kaplin and Andy book for the checkers namurthy Subramanian served as accuracy jrovided programming assistance. Burghardt of Can-Futures, Andy Among practitioners who helped, 1 thank Galen Jacobson of the International Alex of Moor of E1 Paso Corporation, Brice Hill lntel, Tradelink, L.L.C. of Securities Exchange, and Blair Wellensiek intellecmal debts. From the many, Iwant With any bok, there are many long-term several classes from Robert Merton at tale good fortune to to single out two. I had the book is deeply in his debt, and M1T while 1 was a graduate student. Every dedvatives 1970s the are as essential today as they this one is no eycepton. His classic papers from with Dan Siegel, with worling amount learned an enormous were 30 years ago. I also 1991 death in at the age of 35 was Dan's options. whom I wrote several papers on real personally. profession, as well as to me a great loss to the made it clearfrom the outset The editolial and production team atAddison-Wesley lucky to have the project overI that their goal was to produce a high-quality book. was Donna Battista. Project Edity, Finance and tireless seen by Addison Wesley's talented myriad details and offered exof track expertly kept Manager Mary Clare McEwing Editor Mmjorie Singer Development board. sounding' cellent advice when l needed a significantly. Promanuscript improving the suggestions, Anderson offered innumerable manuscript into a physical marshalled forces to t'ul'n duction Supervisor, Nancy Fenton Services Publishing E1mStreet at excellent teams book. Among those forces were the edition, of the design first the compliments on and Techsetters. I received numerous second. Kudos are due to Gina Kolenda which has been carried tllrough ably into the and cover design. and Rebecca Light for their creativity in text tried hard to minimize errors, including have The Addison-Wesley team and 1 of course, 1 alone bear Nevertheless, above. noted the use of the accuracy checkers will be available at updates software and responsibility for remaining errors. Errata find do errors so we can if lnow you 1et Please us www.aw-bc.com/mcdonald. update the list. revision using Gnu Emacs and MITeX, 1 produced the original manuscript and authors. l am deeply patef'ul to the worldtools for extraordinarily powerful and robust this software. wide community that produces and supports Through both editions 1 thanks go to my family. My deepest and most heartfelt and tolerance. This book is have relied heavily on their understanding, love, support, children Claire, David, and Henly dedicated to my wife, Irene Freeman, and 111.M '
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xxix
Robert L. McDoltald is '?3W?llj Ndp?l?/lcr. Distillgltished Professor tTFf/lt-I?gcc atNorthJs/c,re?-/l University :5'Kellogg School ofManagemellt, lWpc?-dhe has rt-Illg/?/' since 1984. He is co-Editor of the Review of Financial Smdies altd llas !7c77Associate Editor of Analysis, Management #?c Journal of Finance, Journal of Financial and Quantitative Science, and tl/'/ze/--jt?l/?-??t'l/.. He tu a fh ill fct/nt//??fc..-t???? the U/?ve?-.W/yofNorth Carolina at Chapel Hill and a PIt.D. ?? Economicshnnl MlT
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and is pejorative. Suppose your family grows corn automatically think the term Yo provides insurance'. to mi11into cornmeal. The bet your friend's family buys corn your income. Your supplements this price; low sells for a earn $1 if your family's corn family the high cost of offsets buys is expensive; this friend earns $1 if the corn his unfavorable outcomes. The the bet hedges you both against corn. Viewed in this light, both of you. contcact has reduced risk for price of of contract simply to speculate on the Investors could also use this kind /74)/ the is It point: is key contract is not insurance. And that a llt7l it is corn. ln this case the ltsed, and JW/t?uses it, lt'Il detennines JW7cle?-olitself bltt Fitnj?it is ct?pl/?-t'7J everything. risk-redttcillg. Context is if you are new to the subject the impliAlthougl w've just defned a delivative, be' obvious dght away. You will come to a cations of the desnition will probably not through te book, smdying different deeper understanding of derivatives as we progress economics. products and their underlying lvbet''
Uses of Derivatives
derivatives? Here are some motives: What are reasons someone might use
and other users to reduce companies Derivatives are a tool for Risk manasement seller this in a simple wa#: The fnrmer-a risks. The corn example above illustrates low. contrac' t which makes a payment when'the plice of corn is of corn--enters into a h farmer who we therefore ty is hdging. This contract reduces the risk of loss for t. e forbiddingly complex, but many deriktives are It is common to think of derivatives'as Automobile is a derivative, for example. simple and familiar. Every form of insurance around a have an ccident. If you wrap fourcar insurance is a bet on whether you will valuable', if the car remains intact, it is not. tree, your insurance is in investment vehicles. As you will see later speculation Derivatives can serve as make bets that m'e highly leveraged (thatis, the book, derivatives can provide a way to of maling be large relative to the init'ifll cost the potential gain or loss on the bet can that the S&P bet example, if want to you specic view. For the bet) and tailored to a dedvatives today, can be from 1300 and 1400 one year ,
500 stock index will be between constructed to let you do that.
effect Sometimes derivatives provide a lower-cost way to transaction costs wish fund mut'ual of may a transaction. For example, the manager other a particular financial and paying brokers this entails paying fees to to sell stocks and buy bonds. Doing which we will discuss later. It is possible to spread, bid-ask the trading costs, such as economic effect as if stocks had acmally trade derivatives instead and achieve the same transaction the delikative might result in lower been sold and replaced by bonds. Using and buying bonds. costs than acttlally selling stocks regulatory restrictions, lt is sometimes possible to circumvent o' Regulatory arbitrage Derivatives are ften used, for examby trading derivatives. taxes, and accounting rules eliminate the risk of holding and of stock (receivecash ple, to achieve the economic sale
Reduced
Is A DERIVATIVE?
%
the stock) while still maintaining physical possession of the stock. This transaction may allow the owner to defer taxes on the sale of the stock, or retain voting lights, without the risk of holding the stock. Thse are common reasons for using derivatives. The general point is that derivatives provide an alternative to a simple sale or purchase, and thus increase the range of possibilities for an investor or manager seeldng to accomplish some goal.
Perspedtives on Derivatives How you think about derivatives depends on who you are. ln this book We will think about tluee distinct perspectives on derivatives: End-users are the coporations, investment managers, and perspective derivative enterinto who contracts forte reasons listed in the previous section'. investors speculate, reduce risk, costs, or avoid a nlle or regulation. End-users have a to manage riskreduction) and goal (forexample, care about how aderivative helps to meetthat goal.
The end-user
The market-maker
perspective
Market-makers are intermediaries, traders who will.
buy derivatives from customers who wish to sell, and sell derivatives to customers who wish to buy. ln order to make money, market-makers charge a spread: They ate like buy at a low price and sell at a high price. In tls rejpct markt-ralts Marketprice sell the retail price. wholesale and the low who higher at buy at grocers makers are also like grocers in th a t their inventory reqects customer demands rather than their own preferences: As lng aj shoppers buy paper towels, the grocer doesn't care whether they buy the decorative or super-absorbent style. Mter dealing with customers, market-makers are left with whatever position results from acommodating customer demands. Market-makers typically hedge this lisk and thus are deeply concerned about the mathematical details of pricing and hedging. The economic
observer
Finally, we can look at the use of derivatives, the activities of
the marlet-makers, the organization of the markets, the logic of the pricing models, and try to make sense of everything. This is the activity of the economic observer. Regulators must often don their economic obselwerhats when deciding whether and how to regulate a certain activity or market participant. These three perspectives are intertwined throughout the book, but as a general
point, in the erly chapters te book emphasizes the end-user perspective. In the late chapters, the book emphasizes the market-maker perspective. At all times, however, the economic observer is interested in mnking sense of everything.
Financial Engineering and Security Design the major idea-is that it is generally One of the major ideas in derivatives-perhaps possible to create a given payoffin multiple ways. The construction of a iven financial ngineerig. The fact that product from other products is sometimes called linancil
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TO DERIVATIVES
need to hedge their implications. First, since market-makers this is possible has several 'I'he marketworks. market-mnking understanding how positions, this idea is central in that pays position ofsetting and ten creates an maker sells a contract to an end-user, posidon. hedged This creates a laimif it is necessary to pay the customer. how it can contract can be replicated often suggests given Second, the idea that a risk, initial the change dials effect, to tarft market-maker can, in be customized. The These changes pennit the cteation derivative. of paymentcharacteristics a premium, and given situation. of a product that is more appropziate for a about a given derivative by realizing Thiyd,itis often possible to improve inmition already understand. that it is equivalent to something we arbiage create a payoff, the regulatory Finally, because there are multiple ways to code, or in stdp. Dijtinctions existing in the tax discussed above can be dificult to is since a particular secttrity or derivative that regulations, may not be enforceable, the replaced by one that is tzeated differently but has regulated or taxed may be easily same economic prosle. that derivative products can generally be A theme running throughout the book is constnlcted from oter products.
MARKETS
sayin: that the Dow Jones lndustrialAverage has gone up We take for granted headlines arld interest rates have risen. But why 100 points, the dollar has fallen against the yen, index (such zise and fall of a pnrticular snancial do we care about these things? ls the and winners trak lndustrial Average) simply a way to keep score, to as the Dow Jones where watching sports; we watching the stock market like losers in the economy? ls of sound atld f'ury,but f'ull by told journalists, tale root for certain players and teams-a signifying nothing? often underappreciated, impact on Financial markets in fact have an enormous, consider tll role of fnancial markets we will everyday life. To help us understand the and both children 2.3 and Sarah Average have Average family, living in' Mytown. Joe income pays for employer in Mytown. work for the XYZ Co., the dominant lef4 is over goes cloting, and medical care. What their mortgage, trarlsportation, food, retirement. children's college tuition and their own toward savings earmarked for their of the markets and dezivatives play in the lives What role do global finarlcial 'rheir
Averages?
Financial Matkets and the Averages
affect their of the ways in which financiql markets The Averages are largely unaware lives. Here are a few: rieed for money to finance employer, XYZ Co., has an ongoing * The Average's dependnt on tll local bnnk forfunds because operations and investments. Itis not stocks and bondj in global rarkets. it can raise the money it needs by issuing '
MARKETS
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lisks. In addition to having property and * XYZ Co. insures itself against certain for its buildings, it derivatives markets to protect global casualty insurance uses against adverse interest commodity price changes. By and itself rate, currency, risks, able into bnnknlptcy, XYZ is less and less being likely to go to manage these unemployment. throw the Averages into likely to
* The Averages invest in mut'ual f'unds. As a result they pay lower transaction costs than if they tried to achieve comparable diversifcationby buying individual stocks. * Since both Averages work at XYZ, they run the risk that if XYZ does fall on hard times they will lose theirjobs. The mutual funds in which they invest own stocks in a broad array of companies, ensuring that the failure of any ope company will not wipe out their savings. * 'I'he Averages live in an area susceptible to tornadoes and insure their home. If theirinstlrance company were completely locak itcould notoffertornado insurance because one disaster would leave it unable to pay claims. By selling tornado risk in global markets, the inslzrance company can in effect pool Anytown tornado risk with Japan earthquake lisk and Flolida hurticane risk. This pooling makes instlrance available at lower rates. *
1.2 Tt4E ROLE OF FJNANCIAL
F1NANCIAL
Averages borrowed money f'rom Anytown bnnk to buy their house. The bnnk sold the mortgage to other investors, freeing itself from interest rate and default risk associated with the mortgage, leaving that to others. Because the risk of their mortgage is borne by those willing to pay the highest price for it, the Averages get te loWestpossible mortgage rate.
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In all of these examples, pnrlcular financial functions and rijks have been split up and parceled out to others. A bank that sells a molgage does not have to bear the risk of the mortgage. An insurance company does not bear a1l the Iisk pf a disaster. ltisk-sharing is one of the most important f'unctions of fnancial markets.
Risk-sharing Risk is an inevitable part of our lives and all economic activity. As we've seen in the example of theAverages, tinarlcialmarkets enable the fipgncial losses from at least some of these risks to be shared. Risk arises from natural events, such as eartquakes, Qoods, and hurricanes, and from unnatural events such as wars and polhical conQicts. Drought and pestilence destroy agriculttlre every yearin some part of the world. Some economies boom as oters falter. On a more personal scale, people are born, die, retire, find jobs, losejobs, marly divorce, and become ill. In the face of this risk, it seems namral to have arrangements where the lucly share with the unlucky. Risk-sharing occurs informally in families and communities. The insurance market makes formal risk-sharing possible. Buyers pay a premium to obtain various linds of insurance, such as homeowner's insurance. Total collected premiums are then available to help those whose houses bul'n down. The lucky, meanwhile, did not need inslzrance and have lost their premium. The market makes it possible for the lucky to help the unlucky.
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and interest ln the business world, changes in commodity prices, exchange rates, dollar becomes lf down. the burning house rates can be the financial equivalent of a It makes others and hurt. companies helped are are expensive relative to the yen, some risk, this exehange that the companies enabling so to sense for there to be a mechanism lucky can, in effect, help the unlucky lisk. Consider an insurance company that provides Even insurers need to share A large earthquake could generate clnims residents. earthquake insurance for California companies sufxcient to bankrupt a stand-alone insurance company. Thus, insurance claims. aainst larje ???c?'ker insuranc reinsurrs, buy, from to often use the reilsurance Iisks, risks enablinjinsurance bcome more t thereby Reinsurers pool diferentkinds of widely held. catastopliebbnds-bonds ln some cases, reinsurers fnmhersharerisks by issuing earthquake, that the issuer need not repay if there is a specifed event, such as a larg tisk earthquake willing accept Bondholders can to causing large insumnce claims. is if bond tlle there no interest payments on buy these bonds. in exchange for greater those exactly by risk erthquake be borne to allows earthjuake. An eazthquake bond investors who wish to bear it. Although there are mechanisms for sharing many kinds of risks, some have argued lisk-sharing is possible and desirable. Tlze ecolmrnist Robert that significantly more creation of entirely new markets for risk-sharing, the envisions 2003) Shiller tshiller, risks associated Fitfl house prices), incometrade including home equity insprance (to be need fully repaid ifwages decline in a particular that loans not linked loans (personal with occupation), and macro insurance (contracts payments linked tp ntional incomes). inciusive markets for While these markets do not yet exist. there is a trend toward more created an risk transfer. #or example, Goldman Sachs and Deutsche Bnnk httve recen with based claims buy possible payouts wlzich is to it market derivatives'' in teconomic economic statistics. The box on page 7 discusses this market. on You rnight be wondeling what this discussion has to do with the notions of diversiable and nondiversifable risk familiar from portfolio theory. ltisk is diversiliable risk if iy is unrelated to other risks. The risk that a lightning strilce will cause a factory idiosyncratic and hence diversisable. tf jny ikestors to burn down, for exnmple, lisk, this it has no signiticant effet on anyope. Rijk tat dpes not piece of share a small nondiversilia b le risk. % e ri sk of a stock spread vanish when across many investors is nondiversifable. example, is market crash, for Financial markets in theory setwe two purposes. Markets permit cliverslfable risk By deEnition, diversifiable risk vanishes when to be widely shared. This is ecient: markets permit nondiverstfiale risk, time, At shared. te same it is widely snancial which does not vanish when shared, to be held by those post willing to hold it. Thus, the ctmcdpf- and ??yl?-kcl' discttsed i'n this the l?7#twnc?7M/ ecoltomic idea ll?p#cl-fl/iWq p'??d/f. p-l/c-,/lt7?-ii/g mechallsms evelyone. book is 1/2:71the existelzce of .is
1.3 DERIVATIVES
IN PRACTICE
Derivatives use and the variety of derivatives have grown over the last 30 years.
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A futures exchange is an orgaized and regulated marketplace for ading futures contracts, a lind of derivative. Figure 1.4 depicts f'utures contract volume for the three largest U.S. f'utures exchanges over the last 30 years. Clearly, the usage of futures contracts has grown signifcantly. Exchanges in other countries have generally experienced comparable or greger growth, ln 2002, Eurex, an electzonic exchange headquartered in Frankturt, Germany, traed 528 million contractsj the largest trading voltlme of any futures exchange in the world.
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1.4 BUYING AND SHORT-SELLING FINAKCIAL
NASDAQ100 index
Namral gas
Eurodollars
Heating oil
Gasoline Gold
Wheat
Nikkei 225 Pork bellies Heating and cooling degree-days
Oats
Japanese yen
Electricity
Corn Soybeans
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ASSETS
Throughout this book we will tnlk about buying and selling-and short-selling-assets such as stocks. These basic transactions are so important that it is Worthdescribing the details. First, it is important to understand the costs associated with buying and selling. Secopd, a very important idea used tlzropghout the bpok is that of short-sales. The concept of short-selling should be inmitive-a short-sale is just the opposite of a purchase-but for almost everyone it is hard to grasp at first. Even if you are fnmiliar with short sales, you should spend a few minutes reading this section.
Buying an Asst
ln recent years the U.S. Securities andExchange Comnzission(SEC), FinancialAccounting Standards Board (FASB), and the lnernational Accounting Standard Board (IASB) have increased the requirements for corporations to report on their use of derivatives. Nevezlheless, surprisingly little is known about how companies acmally use delivatives lisk. will be The basic sategies companies use are wemunderstood-and to manage perceived risk what of example, frction for described in this book-but it is not lcnown, do We companies the all frequently aggregate. in is hedged by a given company, or by hedging. hedging for rationale specific not or not lnow a company's We would expect the use of derivaives to vary by type of ftrm. For exnmple, financial firms, such as banks, are lzighly regulated and have capital requirements. They may have assets and liabilities in different currencies, with different maturities, and with
Suppose you want to buy 100 shares of XYZ stock. This seems simple: ((fthe stock price is $50, 100 shares will cost $50 x 100 $5000. However, this calculation ignores ansaction costs. First, there is a commission, which is a eansaction fee you pay your broker. A commission for te above order could be $15, or 0:3% of the purchase price. price'' is, surplisingly, imprecise. There are in fact two Second, the term prites, a price at which you can buy, and a price at which you can sell. The price at which you can buy is called te offer price or ask price, and the price at which youcan sell is called the bid price. Where do these terms come from? To buy stock, you can pick up the phone and call a broker. If the stock is not too obscure and your order is not too large, your purchase will probably be completed in a matter of seconds. Have you ever wondered where the stock comes from that you have just bought? lt is possible that at the exact same moment, another customer called the broker and put in an order to sell. More likely, howekr, a market-maker sold you the stock. Market-makers do what their nnme implies: They make markets. If you want to buy, they sell, and if you wany to sell, they buy. ln order to enrn a livin'g, market-makers sell for a high price and buy for a 1ow price. Ifyou deal with a rparket-maler, therefore, you buy for a high price and sell for a 1ow price. This difference between the price at which you can buy and te price at which you can sell is called the bid-ask spread-s In practice the bidrask spread on the stock you are buying may be $49.75to $50. Tllis means that you can buy for $50/shareand sell for $49.75/+%e. If you were to buy
lx'he table lists only a fraction of the contracts traded at these exchanges. For example, in June 2004, the Chicago Mercantile Exchange Web page listed futures contracts on over 70 different underlying assets ranging from butler to the weather in Amsterdam.
51fyou think a bid-ask spread is unreasonable, mskwhat a world without dealers would be like. Every buyer would have to lind a seller, and vice versa. The search would be costly and take ime. Dealers, because they maintain inventoly offer an immediate transaction, a service called imnediacy.
Table 1.1 provides examples of futtlres conacts traded aythese exchangesx Much commercii delivatives trading occufs in the over-the-counter market, where buyefs bnnks and dealrs rather than on an exchange. It is dicult ' gnd sellers transact with volume. However, in some markets, such as obtain statistics over-the-counter for to market is signiscantly larger than the over-the-counter the clear that currencies, it is market. exchange-traded
How Are Derivatives Used?
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INTRODUCTION
To
BUYING
DERIVATIVES
commission twice, and you would pay the immediately and then sell. you would pay the bid-ask spread. difference between the Apparent rapid price :uctuations can occur because of the the bounce.'' bid is $49.75and the ask is If called bid-ask This is bid and ask prices. stock was last aded $50, a series of buy arid sell orders will cause the ptice at which the price has not changed, however, because to move between $49.75and $50. The changed. the bid and ask have not t'tnle''
commission Suppose XYZ is bid at $49.75and offered at $50, and the Example 1 100) + $15 $5015. lf you is $15. lf you buy 100 shares of the stock you pay ($50x Your round immediately sell them again, you receive ($49.75x 100) $15 $4960k receive from and what what you you pay tlip transactin cost--the difference between $51. k the bid and ask ptices-is $5015 $4960 a sle, not counting changes in
AND SHORT-SELLING
FINANCIAL
ASSETS
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There are at least tluee reasons to short-sell:
1. Speculation: A short-sale, considered by itself, makes money if the price of the stock goes down. The idea is to hrst sell higi and then buy low. (With a long position, the idea is to first buy low and then sell high.) 2. Financing: A short-sale is a way to borrow money, and it is frequently used as a form of hnancing. This is very common in th bond market, for example. 3. Hdging: You can undertake a short-sale to offset the risk of owning the stock or a delivative on the stock. This is frequently done by market-makers and traders.
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arld come f-rom. Incidentally, this discussion reveals where the terms like it sounds bid price backward. 'I'he terminology is that the You might at lirst think price at should be what you pay. lt is in fact what the market-nlaker pays; hence it is th what sell it is will forvhence market-maker what the offer is plice which you sell. The reflects the perspective of the market-maler. you have to pay. The terminology happens to your shares after you buy them? Generally they One last point: What carefully, ' are held by your broker. If you read the ffne print on your brokerage coneact shares to another investor. Why would your broker typically has the right to lend your topic, want to bon'ow your shares? The answer to that brings us to the next 'Ibid''
Efask''
anyone short-sales. associAlthough we have focused here on shares of stock, there are similar issues ated with buying any asset.
Shortlling exnmple, When we buy something, we are said to have a long position in that thing. For sell if we buy the stock of XYZ, we pay cash and receive the stock. Some time later, we that is lending, in the sense we pay money te stock and receive cash. This transaction receive may not be of The return we rate back the future. in receive and today money stock price lnown in advance (ifthe stock price goes up a lot, we get a high remrn; ifthe nonetheless. of loan but it is a ldnd goes down, we get a negative ret-urn), entails The opposite of a long position is a short position. A short-sale of X'YZ cash. time later, receiving Some we the borrowing shares of XYZ and then selling them, the lender. A short-sie can it it, and to ret'urn cash paying for stock, the back XYZ buy ft'om a be viewed, then, as just a way of borrowing money. When you borrow money advance. of in bank, you receive money today and repay it later, paying a rate interest set the This is also what happens with a short-sale, exeept that you don't necessarily know : ;
rate you pay to borrow.
These reasons are not mumally exclusive. For example, a market-maler might short-sale to simultaneously hedge and snance a position. use a Because short-sales cgn seem confusing, here is a detailed example that illusates how short-sales work. Example: short-sellingwine There are markets for many collectible items, such as :ne wines. Suppose there is a wine from a particular vintage and producer that you believe to be overpriced. Perhaps you expect a forthcoming review of the wine to be negative, or perhaps you believe that wines soon to be released will be of extraordinary quality, driving down prices of existing wines. Whatver the reason, you tlzink the price will fall. How could you speculate based on this beliep If you believe prices will rise, you would buy the wine on the market and plan to sell after the plice lises. However, if you believe prices will fall, you Would do the opposite: Sell today (at the high price) ad buy sometime later (at the 1owplice). How can you accomplish tlzis? In order to sell today, you must first obtain wine to sell. You can do this by borrowing a case from a collector. 'I'he collector, of course, will want a promise ihatyou will return the wine at some point', suppose you agree to returfl it one week later. Having reached agreemet, you borrow the wine and then sell it at the market plice. After one' week, you acquire a replacement case on the mtrket, then ret'ul'n it to the collector from whom you originally borrowed the wine. If te price has fallen, you will have bought the replacement wine for less than the price at which you sold the oliginal, so you make money. lf the price has risen, you have lost money. Either way, you havejust completed a short-sale of wine. The act of buying replacement wine and remnng it to the lender is said to be closing or covering the short position. Note th a t short-selling is a way to borrow money. Initially, you receive money selling the wine. A wek later you paid the money bak (youhad to buy a teplacefrom ment case to return to the lender). The rate ot interest you paid was 1c)wif the price of the replacement case was lom and high if the price of the replttcement case was high. This example is obviously simplifed. We have assumd several points:
* lt is easy to find a lender of wine. * lt is easy to buy, at a fair price, satisfactory wine to remrn to the lender: The wine you buy after one week is a pelect substimte for the wine you borrowed.
BUyl NG AN D SH O RT-SELLI NG
TO DERIVATIVES
k. INTRODUCTION
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The preceding examples were simple illustrations of the mechanics and economics of short-selling, and they demonstrate the ideas you will need to understand our discussions of derik.atives. It turns out, however, that some of the complexities we sldpped over are easy to understand and are important in practice. In this section we use the wine example to illustrate some of these practical issues.
credit rislt As the short-seller, you have an obligation to the lender to ret'urn the wine. The lender fears that you will renege on this obligation. This concern can be addressed with collateral: After you sell the wine, the lender can hold the money you received from the lender keeps the money selling the wine. You have an obligation to return the wine; ' . don't. that the in event you Holding on to the money will help the lender feel inofe scqre, but after thinking the matter over, the lender will likely want more from you thn just the current value of the wine. Suppose you borrow $5000woft of Wine. Whai happens, the lender will think, if the price of that particular wine lises to $6000one Week lat/ This is a $1000 losson your short-sale. In order to returu the wl, you Fill hve to pay $6000for wine youjustsold for $5000. Perhaps you cannot afford the xtra $1000and you will fail to rettlrnthe borrowed wine. The lender, thinking ahead, will be worried at the outset abou thispossibility and will ask you to prov tde lnole than the $5000th wie is worth, say extra $1000. Ius exea amount is called a haircvt, and serkes p proiect tlae lender against your failure to rettlrn the wie when the price lises.o In practice, short-sellers must have funds--called capital-ao be able to pay haircuts. I'he amount f capiti Places a limit on their ability tp short-sell. ''''
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Now consider a short-sale of stock. As with the previExample: short-selling stoclt selt iq receiving cash Fhp you short-sell stock you borrow the stock and ous eyopl, it the original today. At some f'uture tiate you buy the stoc k in the market and rettll'n to of te stock you shorttoday, equal to the market value owner. You have cash coming in market then-current its sell. ln the f'uture, you repay the bonowing by buying the asset at short-selling of a.loan. Thus, price and remrning the asset-this is like the repayment interest rate you pay is not that the except borrowing money, equivalent to a stock is stock plice. The rate of known in advance. Rather, it is detennined by the change in the security falls in price. In effect, if the interest is high ifthe security rises in price and 1ow which you borrow. With a shol-t-sale,you te rate of return on the security is the rate at rather than the buyer of a security. are like the issuer of a security 1.2 depicts the cash Suppose you want to Short-sell IBM stock for 90 days. Table shprt-setler must in t'urn dividends, the flows. Observe in particular that if the share pays wtne! This with alise did not make dividend payments to the share-lender. This issue and dividnd ordinary payment, dividend payment is taxed to the recipient, just like an it is tax-deductible to the short-seller. opposite of the cash flows Notice that the cash flows in Table 1.2 are exactly the l/2c opposite of bttying. literally from purchasing the stock. Thus, short-selling is
The Lease Rate of an Asset make payments to We have seen tat when you bonow an asset it may be necessary to stock are an exnmple of this. We will refer to the the lender. Dividends on of the asset. This concept will arise payment required by the lender as the lease rate discussions of frequently, and, as we will see, provides a unifying concept for oqr layer derivatives. under some circumstances The wine example did not have a lease payment. But Wine does not pay an explicit it might be necessary to make a payment to borrow wine. seeing bottles in the enjoys dividend but does pay an implicit dividend if the owner 'short-sold
15
ltisk and Scarcity in Shortelling
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scarcity As the short-seller, do you need to won'y about the short-sale proceeds? The lender is going to have $6000 of your money. Most of this, however, simply reqects your obligation, and we could ask a trustworthy third party, such as a bank, to hold the money so the lender cnnnot abscond with it. However, when you remrfl 4he wine, you fnrd?'dJl. This raises te question: What rate are going to want your money back, pltts you? the will the lender Over of interest pay course of the short-jale, the lender can en'rning. 6%. The lender could offer to pay yo 4% on the funds, invest your money, say, difference the 6% ealmed on the money and the 2% between keep thinking to qs a fee What happens the lender and paid if borrower negotiate? the 4% to you. point: The of is Here the interesting interest the lender pays on the cpllateral rate people how going depend want to borrow wine from the particular vintage is to on many
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f'Note that the lender is not concerned about your failure to perform when the price goes down because the lender has the money!
16
%.llq-rRoouc-rlohl
To
PROBLEMS
DERIVATIVES
practical matter, it may not be and prbducer, and how many are willing to lend it! As a wine, the lender will offer borrowed high demand for easy to 5nd a lender. lf there is willing for being to lend something that essentiily enrning a fee a low rate of interest, the lender might onelude the wine, is scarce. However, if no one else wants to borrow close to the market interesi of offer ' th an n othing and you a rate that a small fee is better rate.
different markets, the repo The rate paid on collateral is called different things in market. Whatever it is called, stock the in rate in bond markets and the short rebate another cost to your is of interest market rate the difference between this rate and the short-sie.
k
Jorion (1995) exnmines in detail one famous tderivatives disaster'': Orange County in Califonzia. Benzstein (1992) presents a histoly of the development of 5nancial markets, and Bernstein (1996)discusses the concept of risk measurement and how it evolved over the last 800 years. Miller (1.986)discusses origins of past Enancial innovation, while Merton (1999)and Shiller (2003)provide a stimulating look at posmarkets. Froot and O'Connell (1999)and Froot jiblelrl/rd developments in snancial (2001) examine the market for catastrophe reinsutance. D'Avolio (2001)explains the economics and practices associated with short-sales. Finally, Lewis (1989)is a classic, funny, insider's account of investment bnnking, offering a different (to say the least) perspective on the mechanics of global risk-sharing.
PROBLEMS the pric of plpthing Derivati.vesare financial ins%ments with apayoffdeterminedby lisk nd i ieduce speculation, management and else. They can be used as a tool for transaction costs or avoid tues and regulation. risk-sharing. One important f'unction of financial markets is to facilitate optimal d ith cpincid w an increase in The growth of derivatives markets over the last 50 years has shck, the ablidpment 1973 oil the zisks evident in variou markets. Events such as the created have a ew role of fixed exchange rates, and the deregulation of energy markets for dedvatives. ther cash) Ashort-sale entails borrowing asecurity. selling it, maki dikid:d (or coneepttlally the short-sale is payments io the security lender, and then retunng it. A of financing, used for speculation as a form opposite of a purchase. Short-sales can be details f short-selling in practic arl be undrstood of the hedge. Many o or as a way to short-seller and scarcity of shares ihgtcan be bkrowed. as a response to credit risk o f the lnderss' liig with Short-sellers typically leave the short-sale proceeds on deposit wit.h ,
1.1. Heating degree-day and cooling degree-day futtlres contracts make payments based on whether the temperature is abnormally hot or cold. Explain why the following businesses might be interested in such a contract: a. Soft-drink manufacmrers. b. Ski-resort operators. c. Electric utilities. d. Amusement park operators. 1.2. Suppose the businesses in the previous problem use f'umres contracts to hedge their temperamre-related risk. Who do you think might accept the opposite lisk? 1.3. ABC stock has a bid price of $40.95and an ask plice of $41.05.Assume there is a $20 brokerage commission. a. What amount will you pay to buy 100 shares? b. What amount will you receive for selling 100 shares? c. Suppose you buy 100 shares, then immediately sell 100 shares with the bid and ask prices being the same in both cases. What is your round-trip transaction cost?
collateral is called the short additional capital called a haircut. The rate paid on this the interest rate. rebate, and is less t11:,11
FURTHER
READING
this chapter. HowThe rest of this book provides an elaboration of themes discussed in 3, and4induce 2, discussion. Chapters the related directly to ever, certain chapters are derivatives, and show how forward and option conacts, which are the basic contracts in derivatives marketdetailhow in discusses 13 Chapter they are usedin risknanajement. derivatives can be combined makers manage their risk, and Chapter 15 explains how products. with instruments such as bonds to create customized risk-management sites for TheWeb thatlistthirconeacts. sites haveWeb The derivatives exchanges (Chicago Board of Trade), wwmcme. the exchanges in Figure 1.4 are www.cbot.com (New York Mercantile and Exchange), www.nymex.com com (Chicago Mercantile www.gs.com/econderivs. discussed at Economic derivatives are Exchange).
1.4.
Repeat the previous problem supposing that the brokerage fee is quoted as 0.3% 0 f the bid Or ask price. '
1.5. Suppose a security has a bid price of $1t0 and an ask price of $100.12.At what price can the market-maker purchase a security? At what price can a marketmaker sell a security? What is the spread in dollar terms when 100 shares are traded? '
1.6. Suppose you short-sell 300 shares f XYZ stock at $30.19 with a commission charge of 0.5%. Supposing you pay commission charges for purchasing the seculity to cover the short-sale, how much proht have you made if you close the short-sale at a plice of $29.87?
18
k. IN-rRoouc-rlolq
To
DERIVATIVES
1.7. Suppose you desire to short-sell 400 shares of JK.I stock, which has a bid price of $25.12and an ask price of $25.31.You cover the short position 180 days later when the bid price is $22.87and tle ask price is $23.06. a. Taldng into account only the bid and ask prices and interest), what proft did you earn?
(ignoringcommissions
b. Suppose that there is a 0.3% commission to engage in the short-sale (this is the commission to sell the stoek) and a 0.3% commission to close the short-sale (this is the commission to buy the stock bck). How do these commissions change the prct ilz the previous answer? paid nothing c. Suppose the 6-month interest rate is 3% and that you are much interest do you lose dpling the 6 proceeds. How short-sale the on months in which you have the short position? '
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1.8. When you open a brokerage account, you typically sign an agreement giving the b.rokerthe light to lend your shares without notifying or compesating you. Why do brokers want you to sign this agreement? 1.9. Suppose a stock pays a quarterly dividend of $3. You plan to hold a short position in the stock across the dividend ex-date. What is your obligation on that date? If of the you Jtre a taxable investor, what would you guess is the tax consequence deductible'?) dividnd the be would tax to particulm (In expect payment? you Suppose the company announces instead that the dividend is $5. Should you care that the dividend is different fromavhat you expected? 1.10. Short interest is a measure of the aggregate short positions on a stock. Check an service for the short interest on several stocks online brokerage or other snancial of your choice. Can you guess which stocks have high short imerest and wlzich have low? Is it theoretically possible for short interest to exceed 100% of shares outstanding? 1.11. Suppose that you go to a bank and borrow $100. You promise to repay the loan in 90 days for $102. Explain tls transaction using the terminology of short-sales. 1.12. Suppose your bnnk's loan officer tells you that if you tale out a mortgage (i.e., than you borrow money to buy a house) you will be permitted to borrow no more terminology of using ansaction the this Desclibe 80% of the value of the house. short-sales.
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chapter introduces the basic derivatives contracts: forward contrats, call options, and put options. T hese f'un damental contracts are widely used and serve as building blocks for more complicated dedvatives tht we discuss in later chapters. We explain here how the contiacts work and how to think about their risk. We also introduce an exemely important tool for analyzing derivatives positions-namely, payoff and proft diagrnms. The terminology and concepts introduced in this chapter are fundamental and W ill be used throughout this btjok. ,
2.1 FORWARD
CONTRACTS
Suppose you wish to buy a share of stock. Doing so entails at least three separate steps: (1) setting the price to be paid, (2)transfrring cash from the buyer to the seller, and (3) 'ansferring the share from the seller to the buyer. With an outright purchase of stock, :11three occur simultaneously. However, as a logical matter, a price could be set today and the ansfer of shares and cash would occur at a specifed date in the f'uture. This is in fact the definition of a forward contract: lt sets today the terms at which you buy or sell an asset or commodity at a specitic time in the future. A fomard contract does the following:
* Species the quantity and exact type of the gsjet or commodity the seller must deliver. * Speciss delivery logistics, such as time, date, and place. @ Specifies the price the buyer will pay at the time of delivery. * Obligates the seller to sell and the buyer to buy,subjet t the above spciscations. 'rhe
The time at which the contract settles is called the expiration date. asset or commodity on which the forward contract is based is called the underlying asset. Apart from commissions and bid-ask spread: (see Section 1.4), a forward contract requires no initial payment oc prerniumk The conact'ual forward price simply represents the price at which consenting adults agree today to transact in the f'uture, at which time the buyer pays the seller the forward price and the seller delivers the asset. Futures contracts are similar to fomard contracts in that they create an obligation to buy or sell at a predetermined price at a future date. We will discuss the instimtional and pricing differences between forwards and futures in Chapter 5k For the time being, think of them as interchangeable.
21
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DJ lndustrial Avefage ((2n-$11x Index 9235 4>2 9953 19085 9951 1162 113 15 57 s'lt 19072 319 116 19575 8442 159 125 nacvc1 149; 1:08 lnt 4,6 Est vcl !n n 911;Qpan i6. /5/4, +123J712 p?l: B'l19193-132 99533 cnse n ldz Mini Dl Industfial Average (T)-$5 x laex ,9
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Mini S&P 500 (r.MD4%x index
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S&P Midcap 400(D-$52 x kdp.x 575.1 87.($ F73.10 +17.15 61.552 598.79 B,Mq sept 56 Et vcl495:vQl l/ln 7p; Qpen Int 13,M +.63. I(k prl; l1l54.852l.c 511Jk f-lH 571.02, Nasdaq 10 (r.ME)-$1(11 x Index 79,921 lzs BD5P 8995 r59 :1.55599Bz Jept l'Wp Qpen 1nt7431J9 Estvn1144R vnI
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source: J.Ptz//Street Jounal, July 28, 2004, p. C-16.
Figure 2.1 shows f'uttlres price listings from the JPcl!/Street Jt?lf?p7tz/for futtzres lndustrial Average (DJ 30) contracts on several stock indices, including the Dow Jones indices are the upderlying assets for . and the Standard and Poor's 500 (S&P 500). these examples the contracts. (Astpck index is the average price of a group of stocks. In 'I'he stock.) hrst column the price of just one we work with tls group price rather tan the plice at show that follow of the listing gives the expiration month. The columns settlement and day, duling the the the beginning of the day (theopen), the high and 1ow
.
The Payoff on a Forward Contract Every forward conact has bot.h a buyer and a seller. The term long is used to describe the buyer and short ij used to describe the seller. Generally, a long position is one that makes money when the price goes up and a short is one that makes money when the price goes down. Because the long has agreed to buy at the fixed fonvard price, a long position profits i.fpdces rise. The payoff to a contract is the value of th position at expiration. The payoff to a long fonvard conact is Payoff to long fomard
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plice, which reQects the last transactions of the day. and 1ow The listing also gves the price change from the preyious day, the high of number the which measures during the life of the futures contract, and open interest, seller, and both buyer a a a contracts outstanding. (Since each trade of a contract has buyer-sellerpair counts as one contract.) Finally, thehead of the listing tells us where the Mercantile Exchange and contracts trade (theChicago Board of Trade (CBTJ Chicago is 500 the S&P $250 times the index (CME1), and the size of the contract, which for detail 5. There are Chapter in in more value. We will discuss such f'utures contracts 2.1, both Figure those in than eontracts many more exchange-traded stock index f'uttlres in the United States and amund the world. September and December The priee quotes in Figure 2.1 are from July. July in for purchase of the index plices for tlze two conacts are therefore prices set 500 f'utures pliee is $1092.50and in later months. For example, the September S&P 'rhe
23
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DI-AIG Commodlty Indextnmg x inlpx 125.9 419.1 2,886 455.9 r. Est k;t Q;vQIMQn (); ())Qr1 lflt 2:26, tm:1 (lnse14.1.311$. 1.0111.9242 ldx )rl: Hl1M.7.13: x index S&P 500 Indexto-sa 50pt 10231p 199692 1 2329 199250 9 116929 7819 575,917 9e( 129352 19559 1,932' 199299 97A2 ll 21 72199 12,717 st kol 39763;v:l McR .t37876)Q)Cn1nt5.z8912+0.7l. ldx p#) Hl19.F2 : 18l.7l Oere 1021.83,
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the December price is $1092.90.1By contrast, the current S&P index price that day is $1094.44. This is the spot price for the index-the market price for immediate delivery of the index. There are many more exchange-traded f'umres contracts than just those listed in Figur 2.1. As we will see in Chapters 5, 6, and 7, there are also f'utures conacts on interest rates and commodities. Fumres are widely used in lisk management and as an alterative way to invest in the underlying asset. Agricultural futures (suchas corn and box on page 24 soybeqs) can be used by fnrmers and others to hedge crop prices. discusses an unsuccessftll proposal for a new f'utures contract that was in the news in 2003. We will discuss in Chapter 5 how forward and fumres prices are determined and details about how fumres contracts work. 1.11this chapter we take prices as given more and exnmine proft and loss on a fonvard contract. We will also see how a position in a forward conact is similar to azld different from altemative investments, such as a direct investment in the underlying index.
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these calculations, consider a fonvard conact on a hypothetical stock index. Suppos: the non-dividend-paying S&R (pecial and Rich''l 500 index has a current plice of $1000 and the 6-month fonvard price is $1020.2The holder of a long position in the S&R fonvard conact is obligated to pay $1020in 6 zonths for one unit
To illusate
l'rhe use and nonuse of dollar signs for fumres prices can be confusing. Many futures prices, in particular those forindex ftltures, are in practice quoted without dollar signs, and multiplied by a dollar amount to determine the value of the contract. In this and the next several chapters, we will depart from this convenuon and use dollarsigns for index fumres prices. When we discuss the S&P 500 index fumres coneact in Chapter 5, however, we will follow practice and omit tlle dollar sign. 2We use a hypothetical stock index the S&R in order to avoid complications dends. We discuss dividends and real stock indices in Chapter 5.
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$ Ah1 INTRODUCTION
24
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We can graph the information in Table 2.1 to show the payoffin 6 moths on the forward contract as a function of the index Figure 2k2graphs the long and short positions, with the index price at the expiration of the forward contract on the ho/zontal axis and payoff on the vertical axis, As you would expect, the two positions have a zero payoff when the index pdce in 6 months equals the forward price of $1020. The graph fr the short forward is a mwr image (aboutthe x-axis) of the graph for the long fonvard. For a given value of the index, the payoff to the short is exactly the opposite of the payoff to the long. In other words, the gain to one party is the loss to the other. This kind of graph is widely used because it summarizes the risk of the position glance. at a
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26
AN INTRODUCTION
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TO FORWARDS
AN D OPTlON5
FORwARD 1/11154!! : 74: ($';#' ;k' 1* itf ))fl: q' i'(2;157(42,6(J7/j* F7;TT7T:ITI:)Y (r' j''(q' 7771:77:7:7744:7:! !.(;! p.j'(q' t')'j;'. #' (6* (f' q)' /'t:' ((' (i'lrj (' @' i ( (l !! ( (;'t' (' q T'lqq:qi j. (!(j1 . .!!! ';( iEEi?E ')q');; j'yjf jjf k:kjkjjjj;'.. j'r jqf .yy' ),f r'. )'. j'yf y'. tyt'. y' r' . (yj )' (('p)yj'y,yyj;j));. . .' 1(;'(i.i'(:(:( . .q..' . ..ir.'...(.(! tj'. t'jyy'. 'j.y billl.' jjjr.i.yk ., q!!ii::::kr..... :t!:(;i!,-... .IltitiiL ; tk ' :ki - ;,p; ) ' ,.;E ;. q. jy; ; ) llli/t); y ( . . LcL,jL.-.--.-j-j.;;-.L-L-jkL;LLj.,.;j--.L-L..kj,..q,;.j-L;-.k;.; j j ltyyy-y--jjj-iyy ))r));j. tr . j j. ' . ( .. .l !.!rE!.Ji..qi......:):.?.-...,.s..........................-.... ...r.........-..y.-y-y--.u).),s. . .),:jygj,.... . . 1!1) iitELlt.-y,; d::z)p ............E Sllk.:j;;-. :@1;; ?:Iq!I,(E (;). . rjgy,--' ;'f' ',F''. :'C' )' ??' ;''-'-':'kf t'(' )' 1* #'
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Comparson
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after 6 months of a Iong positon in the s&R index versus forward contract in the S&R index.
150 1O0
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forward shortforward
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The S&R forward contract is a way to acquire the index by paying $1020after 6 months. 'An alternative way to acquire the index is to purchase it outright at time 0, paying $1000. Is there any advantage to using the forward contract to buy the index, as opposed to purchasing it outright? lf we buy the S&R index today, it costs us $1000. The value of the position in 6 months is the value of the S&R index. The payoffto a long position in the physical S&R index is graphed in Figure 2.3. For comparison, the payoffto the long forward position, is graphed as well. Note that the axes have different scales in Figures 2.3 and 2.2. To ee how Figure 2.3 i.s constnlcted, suppose the S&R index price is $0 after 6 months. tTl1is isjusi a thought experiment for te purpose of constructing the graph, but fims in if you would like to be concrete, imagine that the S&R index contained ltret the year 2000 which would be bnnkrnpt in 2001.) If the ipdex price is $0, the physical For index will be wprth $0; hence we plot a 0 on tlze y-axis against 0 on te I-axis. :11other prices of the S&R index, the payoff equals the value of the S&R index. For example, ifwe own the index mzd te price in 6 months is $750,the value of the position If the index price in 6 months is
-1500
950 1000 1050 1100 1150 1200 S&R lndex Price ($)
Comparing a Forward and Outright Purchase
is $750.
$0, the payof'f to the forward conact,
using
equation (2.1),is =
0
-
$1020
-$1020
=
lf instead the index price is $1020.the long index position will be worth $1020and the fomard contract will be worth $0.
-1020
jiza 0
500
1000 1500 S&R Index Price
2000
2500
(s)
Wit.h both posidons, we own th index ftr 6 mpnths. What the sgpredoes not reqect, however, is the different initial investmepts required for the twp positions. With the cash index, we west $1000initially and then we opn the index. With the fol-ward The conact, we invest $0 initially and $1020 after 6 months; then we own te payoffgraph tells us how much mony we Enancing of the two positions is different. end up with after 6 months, butdoes npt acctwntfor the initial $1000 westpenywiih the outright purchase. Figtlre 2.3 is accurate, but it does not answer our quesiion-namely, whether there is an advantage to either a fonvard purchase or an outright purchase. Both posidns give us ownership of te S&R index after 6 months. We can compare them faidy if we equate the nmounts iniially invested and then account for interesi enrned over the 6 mothsk We can do this in eiter of two equivalent ways: 'lkeasury in zero-couppn bpnds (forexample, bills) along with the 1. Irkvest $1t0 forward 'coniacts '' in which case each position initlallv costs $1006at time 0.
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Borrow to buy tlle physici $0 at time 0.
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Suppose the 6cmonth interest rate is 2%. With alternative 1, we pay $1000today. $1020. At that point, bopd ij Frt.h $1000 x 1.02 6 months ihe kro-cofo bonct proceeds to pgy the frward price of $101t. We then own the index. we use the The net effect is that we pay $1000inially an (i own the index after 6 months, just as if we bought tlle index outright. hwesting $1000and at the same time entering a long forwardconeact mimics the effect of buying the index outright.
Aer Payoffto long forward
..+.
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Cox-rRAc'rs
=
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28
%.Ah1 IxTRooucTloN
'ro
FORWARDS
FORwARD
AND OPTIONS
cox'rRAc'rs
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29
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With alternative 2, we borrow $1000to buy the index, which costs $1000. Hence months we owe $1000plus interest. At make no net cash payment at time 0. After 6 we $1020for the borrowed money. Th ne# effecy ij that time we repay $1000 x 1.02 also osvn te index. that we invest nothing initially, and after six months pay $1020.We into a lone forward Borrowine to buv the stock terefore mimics the effect of enterine =
contract.
We conclude that when the index pays no dividends, the cmly difference betaveen of a payment that will the forward contract and the cash index investment is the timing using the interest be made for certain. Therefore, we can compare the t'wo positions by conclude that the forward example, we rate.to shift the timing of payments. In the above in the timing of contract and the cash index are equivalent investments, differing only other. the cash flows. Neither form of investing has an gdvantage over the This analysis suggests a way to systematically compare positions that require differentinitial investments. We can assume thatweborrow any requiredinitial payment. At expiration, we receive the payofffrom the conuct, and repay any borrowed amounts. differing Wewill call this the net payoff orprofit. Because this calculation accounts for than rather payoff profit primarily use initial investments in a simple fashion, we will diagrams and prost the book/ same are Note that the payoff diapams throughout the initial investment. requires it no because conact forward for a point in To summazize, a payoff diagram graphs te cash value of a position at a in of investment value the time. A profit diagram subtracts from the payoff the future
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2500 2000 lsto
$1020
1000 500
+ $1000 bond
-*
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Long forward
-1-
0
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Fonkafd
+
bond
1020
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--
504
0
1000 1500 S&R Index Price
2t)0
500
($)
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the position. Tls discussion raises a question: Given our assumptions, should we really expect value of the index? The answer in the forward plice to equal $1020,which is the f'uture until explanation Chapter 5. this case is yes, but we defer a detailed
Zero-coupon Bonds in Payoff and Proft Diagrams purchase The preceding discussion showed that the long forward conact and outright value time take o money of the physical S&R index are essentially the same once we and into account. Buying the phlsical index is like elgering into the forward coneact point thij same simultaneously investing $1000 in a zero-coupon bond. We can see graphically by using a payoff diagram where we include a zero-coupon bod. Suppose we enter into a long S&R index fomard position, and at the same time tThis was purchase a $1000 zero-coupon bond. which will pay $1020after 6 months.
by holding the index paid a dividend in tis example, then we would receive the dividend 5 how to will Chapter We in contract. the see forward entered into when we physical index, but not '; take dividends into account in this comparison. 4The term protit'' is defined vmiously by accountants and economists. Al1of our proht calculations with another, not computing profit in any absolute sense. are for the purpose of conlparing one position
31fthe
alternative 1 in the previous section.) Algebraically, the payoff to the fonvard ptus the bond is Forward + bond
=
Spot prie
at expiration
Forward payoff
-
$1020 + $1020 Bond payoff
= Spot price at expiration rl'hisis the same as te payof to westing in the physical index. 'l'he paypff diagram for this position is an easy modifcation of Figure 2.3. We simply add a line representing the value of the bond after 6 months ($1000x 1.02 $1020), and then add the bond payof to the fonvard payoff. This is graphed in Figure 2.4. fomard plus bond looks exactly like the physical index in Figure 2.3. What is the proft diagram coaesponding to this payoff diagram? For the forward contzct, prtt is the same as the payoffbecause there is no initial investmeni. Profit for thefomardjlus bond is obtained by subeacting thefuture value of the irtitial investment. The inidal investment was the cost of the bond, $1000. 1ts f'umre value is, by desnition, $1020, the value of tlae bond after 6 months. Thus, the prost diagram for a fonvard conact plus a bond is obtained by ignoring the bond! Put differently, adding a bond to a posi:on leaves a prost diagrgm unaffected. i Depending on the context, itcanbe helpf'ul to clraw eitherpayofforprotdiapams. Bonds can be used to shift payoff diapams vel-tically, but do not change the prost calculation. >
=
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30
AN INTRODUCTION
TO FORWARDS
CALL
AN D OPTIONS
'l'he foregoing discussion assumed that at expiration of the forward eontact, the contract called for the seller party short the forward contract) to deliver the cash S&R indek
tthe
buyer ttheparty long the forward contact). However, a physical transactio in a broad stock index will likely have signi:cant eansactioncosts. An alternative settlement lnstead of requiring delivefy of the procedure t.ha t is widely used is cash settlement. parties make a net cajh actual in dex the forward contract settles hnancially. The two occurred ad bpt.ilparties payment, which yields the same cash flow as if delivery had example. with this an )'. had then closed out their positions. We can illustrate
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contracts are transacted directly between two pnrties means that each counterparty bears the credit risk of the othens Credit risk is an important problym with a1l derivatives, but it is also quite complicated. Credit checks of counterpalties and credit protetions such as collateral and bnnlcletters of credit are commonly employed to guard against losses from counterparty default.
Cash Settlement Versus Delivery
to the
Op-rlolqs
,
,
2.2 CALL OPTIONS
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We have seen that afomard contract obligates the buyer ttheholder of tle long position) to pay the forward price at expiration, even if the value of te underlying asset at expirathe forward price. Because losses are possible with a forward contract, tion is less t.11:.11 wonder: is namral it Could there be a contract where the buyer has the right to walk to away ffom the deal? 'I'he answer is ys; a call option is a contract where te buyer has the right to buy, but not the obligation to buy. Here is an exapple illustrating how a call option works at
Suppose that the S&R index at expiration is $1040. ecause ie q Example 2.2 !)! forward priee is $1020, the long position has a payoff of $20. Similady, the short yjtj positipn loses $20. With cash settlemnt, the shol't simply pays $20 to the long, with tl .4)'q) . . ansaction costs. lt is as if the long paid ljt no trnsfer of the physical asset, and hence no ansaction $1020,acnuired the in4ez worth $1040,and ten immediately soldit with no
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lfthe S&Rindex price atexpiration hadinsteadbeen $960,thelongposition would and the short would have a payofof $60. Cash settlement in this 7 have a payoff of tlj k tij case entails the long paying $60 to the short.
Suppose tlwt the all buyer ages to pay $1020for the S&R index in r Example 2.3 11 months but obligated to do so. 6 is buyer has purchase j a ca jj option.) If in 6 not (jt t'1*he tl mont: the S&R ptice is $1100,the buyer will pay $1020and receive the index. This is ,).) 'l' a payoff f $80 per unit of the index. If the S&R price is $900,the byer walks away.
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Cash setllement is feasible only when there is an accepted reference plice upon which the settlement can be based. Cash settlement is not limited to forward contractsdelively virtually any financial conact can be settled using cash rather t.1::.11
Now thirlk about this transaction from the seller's point of view. The buyer is in coneol of the ofdon, deiding when to buy the index by paying $1020.Thus, the lights of the oydon buyer are obligations for the option seller.
Credit Risk crdit risk, arly derivatives coneactvhas Any forward or futures contract-indeed, who fails to make counterparty owes money which means there is a possibility that the the price and lixed price spot a payment. J.f you agree to sell the index in one yea.r at a the obligated buy to the is countepmy t'urns out to be lower than the forward price, will for the that counterparty risk some face the index for more than it is worth. You the index. Similarly the counterparty fades the for forward price fail te pay to reason risk that you will not f'ulNllthe conact if the spot price in one year turns out to be higher than the forward price. minimize With exchange-traded contracts, the exchange goes to g'reat lengths to ultimate counterparty and being the participants a1l collateral of this risk by requiring in a11 ansactions. We will discuss credit risk and collateral in more detail when we discuss f'umres contracts in Chapter 5. With over-the-counter contracts, the fact that the
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T Example 2.4 If in 6 monts the S&R pri ij $1100,the seller will receive $1020 ((i !' . lf the S&R ).) and gike up all index worth mote, for a loss of $80 per unit of te index. ll rice the will less the seller obligation. is has buyer than Thus, $1020, buy, not so )ll P no ')) at expiration, the sellr will have @payoff whih is zero (if te S&R price is less than k1 $1020)pr nega tive (ifthe &R pric is jreaterflaan $t02O). k E @
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5Of course, credit risk als exist.s in exchange-traded contracts. The specihc details of how exchanges are strucmred to minimize credit risk is a complicated and fascinating subject (see Edwards and Ma (1992), ch. 3, for details). In practice, exchanges are regarded by pal-ticiplmts msgood credit rislcs.
i
32
%AN
INTRODUCTION
To
FORWARDS
CALL
AND OPTIONS
Does it seem as ifsomething is wrong here? Because the buyercan decide whether make money at expiration. This situation suggests tat the seller to buy, the seller cannot tfblibed'' to enter into the contract in the flrst place. At the time the must, in effect, be buyer and seller agree to the contract, the buyer must pay the seller an initial priee, or prenlillm. This initial payment compensates the seller for being at a disadvantage at expiration. Contrast this with a forward coneact, for wlich the initial premium is zero.
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Option Terminology
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He4e are some key terms used to describe options: Strikeprice: The strike price, orexercise price, of a call option is what the buyer strike price was $1020. The strike pays for the asset. In the exnmple above, the price can be set at any value.
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Source: Wl? Jrrecr Journal, July 28, 2004, p. C-15-
Exercise: Th exercise of a call option is the act of paying the strike price to receive tlle asset. 1.rlExample 2.3, the byr decided after 6 monthj whether to is, whether t pay $1020tttzetcike plice) to ieive the exercise the option-tat S&R index.
practice. While most exchange-traded options are American, the options in Figure 2.5 d are European. Later in the book we will discuss American options in more etail.
Expiration: The expiration of the option is the date by which the option must either be exercised or it becomes worthless. The opdon in Exaple 2.3 had an expiration of 6 months.
Payoff and Prolh for a Pttrchased Call Option
.
We can graph call options as we did forward conacts. 'I'he buyer is not obligated to buy the index. and hence will only eyercise the option if the payoffis greater t.1)a11 zero. 'Fhe algebraic expression for the payoff to a purchased call is therefore
Exercise style: The exercise style of te option govet'ns the time atwllich exercise could occur only at expiration. Such an can occur. In the above example, exercise option. European-stle be tf the buyer has the tight to exercise option is said to a at arly time during te life of te option, it is an American-style option. lfte buyer butnotforthe entirelife of the option, the can only exercise dllring specifiedperiods, by option. (The Bermudan-sle terms 'European'' apd is option a Berfpudan and Americgn, geography. Etlropean, with nothing do the way, have to options are bought and sold worldwide.)
Purchased call payoff
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The expression maxlc, l?qmeans take the greaterof the two values a and b. (Spreadsheets contain a max function, io it is egsy to compute option payoffs in a spreadsheet.)
Hytmerican,''
To summarize, a Etlropean call opion gives the owner of the call the right, but not the obligation, to buy the underlying asset on the expiration date by paying the sfrike price. The option described in Exnmples 2.3 and 2.4 is a 6month frlrcwctw-./-yfeS&R call with a strike price of$1020. The buyer of the call can also be descried as having a long position in the call. Figure 2.5 presents a small portion of the opdon price listings for the S&P 500 Index option traded at the Chicago Board Options Exchange. Each roW represents a or a P to denote call or different option, with tlle expiration mont. strike price, a the last trade of the day, premium day, that at the put, the number of coneacts traded wit futures, As and every opion trade open interest. the change from the previous day, buyer-seller pairs. of number the interest measurrs requires a buyer and a seller, so open option. buying of mechanics the of an 'Fhe box below discusses some For the time being, we will discuss European-style options exclusively. We do this because European options are the simplest to discuss and are also quite common in
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As discussed before, thepayoffdoes nottake account of the initial cost of acquiring For a purchased option, the premium is paid at the time the opion is
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AND OPTIONS
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6It is not important at this point how we compute this price, but if you wish to replicate the option premiums, they are computed using the Black-scholes formula, which we discuss in Chapter l2. Usng the BSCal1 spreadsheet function accompanying this book, the call price is computed as BSCal1(1000, 1000, 0.3, 2 x 1n(1.02), 0.5, 0) = 93.81
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36
AN INTRODUCTION
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37
If the index rises, the fonvard conact is more proftable than the option because it does not entail paying a premium. lf the index falls suciently, however, the option is more proftable because the most the option buyer loses is the f'uture value of the premium. This difference suggests that we can think of the call option as an insured position in the index. Insurance protects against losses, and the call pption does te same. Canying the analogy a bit further, we can think of the option premium as, in part, re:ecting the cost of that insurance. The fonvard, Which is free, has no such insurance, those on te call. and potentially has losses larger t.11a11 This discussion highlights the important point that there are always trade-offs in folavard contract outperforms the call if the index rises and selecting a position. undemerfonns the call if the index falls sufliciently. When a11contacts re fairly priced, you will not find a contract that has higher profits for a11possible index market prices.
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.
.
.
'- -. -
-.
.
. -
.
-
-
call:
-
wojjt ($)
-
.
-
-
-
Profit at expiraton for purchase of 6-month S&R ndex call with strike price of $1000 versus profit on long S&R index forward
200 15O
Written call payoff
=
Written call proft
=
-
-
max(0, spot price at expiration
-
maxgo, spot price at expiration
-
1O0 Index price
50
position.
=
--'u'
qn
Profit
strike pricej
+ future value of option premium
$1020
-*95.68
=
lndex Price
100
=
$1000
r'
p Example A.7 Consider a looo-strike call option on the S&R index with 6 months jl. expiration. At the 17me the option is written, the option seller receives the premium @to 1! . tl of $93.81. 1'1 SuPP ose the index in 6 months is $1100. It is worthwhile for the option buyer to )) ) pay the $1000strike price to acquire the index worth $1100.I'hus, te option writer will (t@ . . sell worth the stlike price of $1000. Using the index, equation (2.5), $1100,for ) have to l'1 written call payoff is li) the yty i lt maxp, $1100 $1000) )1.5.) . j.' jtt prernium has enrned 2% interest for 6 monts alzd is now worth $95.68. Prolit for tt. the written call is 6!. ''
Purchased call + Long fonvard
-150
+
-200 -250
800
(2.6)
This exnmple illustrates the option writer's payoffand prost. Just as a call buyer is long in the call, the call seller has a short position in te call.
0 .
strike price)
850
900
950 1000 1050 1100 1150 1200 S&R Index Price ($)
.
p
.
,
-$100.
The last column in Table 2.2 computes the call profit at different index values. Because a purchased call and a forward conlact are both ways to buy the index, it is interesting to contrast the two. 'Fhus, Figure 2.7 plots the profit on both a purchased call and a long forward contract. Note that prot and payoff diagrams for arl option differ by the f'uture value of the premium, whereas for a forward conlact they are the same.
-
=
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=:
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%.AN
38
INTRODUCTION
To
FORWARDS
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[email protected])t.: j.'/..:r-.r.C-rtht..:..-.-....'i.!. jjjjy!...,: )(yjyjjy. -yyyrjyjj, yj .IiisEl .J) ()(t(:yjj. y.g.y.y.;(y lilb?sLbbqkb?.s6'iL-l yjt;. j qllii-glllikkkr.. . .-..i-G-.-....-'.'..'.....':..(22. qrr i.).-.. .,,--L-. .' -' ';' -'. jgjy)f 'JJY F' j':(';q' y;'yyyy'.,yyjjjyy. ('
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A put must have a premium for the snme reason a call has a premium. The buyer of te put conols exercise', hence the seller of the put will never have a positive payoff at expiration. A premium paid by the put buyer at the time te option is purchased compensates the put seller for this no-win position. It is hnportant to be crystal clear about the use of the tenns and in the above exnmple, because tere is potential for confusion. The buyer of the put owns of r/2cpttt is a a contract giving the right to sell the index at a set price.Thus, the @puyer seller of the indexl Similarly, the seller of the put is obligated to btty the index, should the put buyer decide to sell. Thus, the buyer of the put is potentially a seller of the index, and the seller of the put is potentially a buyer of the index. (1fthinking through these transactions isn't automadc for you now, don't wony It will become second nature as tI'IiIIIC about options.) you cmtinue to Other terminology for aput option is the snme as for a call pption, with the obvious lnparticular, ihe sttikepriceis te ajreed-upon selling becomes change tat price ($1020in Example 2.8), exercising the option means selling the underlying asset in exchange for the sfrike price, and the expiration date is tat on which you must exercise the option or it is valueless. As wit.h call options, th e are European, American, and
E: q:
i
.
..+.
250
d-month S&R call with strike o f 100Uversus profit for short S&R forward.
+
call written frward Short
'buyer''
E
200
150
Prof'it
595.68
=
Index price
100
=
$1000
so 0 -50
Index
#rice
=
10 )()
'
-100
tbuy''
-,150
-200 800
850
900
950 1000 1050 1100 1150 1200 S&R lndex Pdce (5)
1)'The
-
maxp,
$95.68.
%
Put option payoff
Figure 2.8 depicts a graph of the option writer's profit, graphed against a short forward contract. Note that it is the mirror image of te call buyer's prost ilzFigure 2.7.
2.3 PUT OPTIONS
=
maxlo, stlike price
'
'
''
'
Suppose that the seller agrees tovjell the S&R index for $1020 in 6 but is not obligated to do so. lrfheseller has purchased a put option.) tf in 6 ).ttymonths gnd walk away. If ')' months the S&R price is $1100,the seller will not sell for $1020 will kvill sell will and at for $1020 enrn $120 that tlme. l(' the S&R price is $900,the seller
spot plice at expirationl
-
(2.7)
t')' () Example 2.9 Cnsider a put option 1171 and a stnke price of $1000.
on the S&R indx with 6 monthsto expiration '
!1 ll Suppose the index in 6 months is $1100. lt is not worthwhile to sell te index i! i worth $1100for the $1000stlike price. Using equation (2.7),the put payoff is .j j.j
E)-y!!
.
.
'
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.(.. '(
llj
@!
maxlo,
(
.tiy ) lf the index were 900 at epiration, lrtj;t lj payoff is then
)) k'
maxp,
$1000 $1100) $0 =.z
-
it is worthwltile .
selling the index for
$1000. The
.
$1000 $900) -
=
$100
%.
..''
,
.
t
ti!
k.
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The put buyer has a long position in the put. Here is an example.
-'
We introduced a call option by compnring it to a fonvard coneact in which the buyer need not buy the underlying asset if it is worth less than the agreed-to purchase price. walk Perhaps you wondered if there could also be a contract in which the seller ould option is a is put A sell. The answer ys. away if it is not in his or her interest to eample is obligation. Here arl contract where the sellr has the right to sell, but not the worlcs. illustrate how option a put to
r Example 2.8 ).lt
-'
'
'
rf'he
=
option writer keeps the premium, for a protit after 6 months of
'.'
We now see how to compute payoffandprostfor a plzrchased put option. The put option buyr gives the put buyer the right to sell the underlying asset for te stlike price. does this only if the asset is less valuable t-1:a1:the stlike plice. Thus, the payoff on the put op on is
$900 $1000) $0. -
'
Payoff and Proft for a Purchased Put Option
.
)i) ).')
'tsell.
tlseller''
Bermudan put options.
)'@I. tlze index is 900 at expiration, it is not worthwhile for the option buyr to pay the t,.). t)) $1000 strlke plice to buy the index wocth $900. The payof'f is then )'1
11
op'rloxs k.
PuT
AND OPTIONS
As with the call, the payoffdoes not take account of the initial cost of acquing the position. At the time the option is acquired, the put buyer pays the option premium to the put seller; we need to account for this in computing prost. Jfwe borrow the premium
k
40
AN Ilq-rRooucTloN
To
FORWARDS
.jji:)' .'-' .k11j;2j;j;..'jj;::i,j.: jj;:!,;.d qlllr..lll'.rll''.
-' '-Eky' ' ' .Ijj:,.,,kkt,k;'. ;':';.' E '.y' r'ii illj'lf lr:' :111:)* (Ijjjjjy'.I tt;f :'k' f' ,'.
=
maxlo, strike price
'.'
spot price at expiration)
The following example illustrates the computation of prost on the put.
.' r'
Use the same option as in Example 2.9, and suppose thatthe lisk-free ratq is 2% over 6 months. Assume that the premium for this put is $74.20. The f'uture -) ( value of the put premium is $74.20 x 1.02 = $75.68. t'j If the S&R ipdex price at expiration is $1100, the put buyr will not xetcise the )y Clt )q OPtion. Usig etation (2.8),profit is
1j Example 2.19
E E
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value of option premim - futtlre
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EE
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lql:E
Purchased put profit
ti!
i( .y jg(
.
g.. reoecng the loss of the frenliup. .. ' expiration, .. the put buyer exercises the put, tt lf the indx is $900at tt t) for $1000. Prot is then ,lrj y(jj 1) maxgo, $1000 $900) $75.68 $24.32 jty ll ' ltt . . 11 (E .(
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850 900 950 1000
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-$75.68
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$124.32
150
-75.68
74.32
100
-75.68
24.32
1150 1200
.
,,
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.
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50
1050 1100
' index llipg tfke
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$800
=
-
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-$75.68
$1000 $11001 $75.68
.
.
.
maxlo,
.
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41
jjj;;illifd t'(IrIFIII'iITI:'IIICY:jIIEiSCIIPi7I',IjjjjIJqjjjjjjjqjIjjjj'iI .jjjr' j;;,,k,;'. @' j:' 1* (EY !!' ?' )'. :'(!j' yrf ;!ii1;;' Fi ( ( !'; ! i'))' (1* qk' )' (' f' ;q' ' (47!!* .IiiEjIi'. iIIl)!iik:i1IL)'t1lIi):iI1)'!Iik:iIlp)'. 'tii'j pr:s ?p8! q:!7..i,t:j,.L'. r' .iIrq1i;;'i(E!)! i;lil! ,1J:12 t'(i'll'ijiii' ljjjjjy'-i r' )' rlf i)'. (' j'jjf q' jjyj'.' jy.jjlj--lf (qiklj))i''i''. L;', ');q':.ryjjjj.; (jj,';:'jjjjjjjjj;jtp(. )'' qrf (!Ir!!iik,-ri:iII(()'. (jf )'('y' q.L', . ' .'' 7:. (' !. . :il. (( tfyjlflljj'jyyjy' .. t'q')j))'(11/9:211)y)k1.)*. kr'q #.jy(jtr r:.).) ty( ( (j( . q. j )y(g. jy . j))yj:. g yj)y)j. . j yyyjtt,tjky : ..ELy(( )tt.....(......-.....-..,).k......:-..):.....-j ..i:s. . :7lIti.. ..stkk. . jjjjjy,;-;; . .,,,,,.,.$,..;,.':','(bL.(..,,.,.,.,).L.L,.,.,),(..LL).,,,.,j j j. .y .- 1F:. 1). . d::)l I.., rl.::),l,::), 1,. ..l).. !ilse d::.l 4:2)1 d::), it ahl-te 1..dt!skI..,..d 1#., rrt .. k:.i,---kk1q.t. . n t s 1..d::,pm a I::)h u I..,::r e t!ik!!q;t: s,t:,7.i1-:: put option with a future value of premium of $75.68.
nmount, we must pay 6 months' interest. The option prost is computed as -
op-rloxs k
PuT
AND OPTIONS
-t5.68
-25.68,
-75.68
-75.68
-75.68
-75.68
-75.68
-75.68
-75.68
-75.68
-75.68
-75.68
-
-
-
=
reqecting the payment of premium.
y' F' F!'' 'ts;' EE E ! EE i: E qf ft' !' $!92 .iEE.)!Ej.i.i f:. i('E qlq:pp' )@)75)* 1* 14q11)*,t11 )!14r*/* i'i)rq' lisjgg'ttl''tq.)i q' ltpiiq '(q . E(f( iE'EE71(ICIE F' !f p1' qf'lpfilftppiqtllfeliltqt'ylllljtq'..r;slf )' t'j';@' lqrlq' . iq i;q Ei ! )kl rs!r' (pr (r ' rv?r;pi,)#-i..EE'.i!..i!-E-E;TE)E).-.-.r,):.ityi.). '(i EEti . . . . .. .. (.. !:41)7r11 ..'' . . ' . syrtE-t;IjEjEt..).ijt.gEEt.i,.q'.j.y).;)(yt.y-;y-.riy.tjr;:y(r.i.)j.
qjlljyyrf )qt@t7... -'--!(;).,!!;;''!--.--....,-;#:.T'k).,..p,,kt'. 1l:2. ). . .4))( ;.yysyjjjt-j.sjy.j. .).rrrt,t)!l1Fi').-..-.. .11!141,:. -l..,...-E,.G.,-----.'..'..,.'.'.?b.E-y:')-.:-t--..-.'.2.'..'.,'-..-.-. i:s,t-.-#?.:.-:-,.-...:.;.r.t,p...-)
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Table 2.3 computes the payoff and prot on a purchased put for a range of index values at expiration. Whereas call profit increases as the value of the underlying asset creases, put proEt increases as the value of the underlying asst decreases. Bcause a put is a way to sell an asset, we can compare it to a short forward position, which is a mandatoly sale. Figure 2.9 graphs prost from the purchased put described in Table 2.3 against the profit on a short fonvard. We can see from the graph that if the S&R index goes down, the short forward, which has no premium, has higher prot than the purchased put. tf the index goes up suciently, the put outperforms the short forward. As with the call, the put is like an insured forward contract. With the put, losses are limited should the index go up. With the short forward, losses are potentially unlimited.
--
,
[email protected]'.Profit purchased
(.... .
-h-.
...s........
.-.............,.,,,;;....
q
.;,
.
yytyt..jy)ty.j-yyy; .
..,.
.
on a S&R index put with strike price of $1000 versus a short S&R index forward.
jtyg;j yjjytjgjtj2r4y; kjgy jjjy;j g;j .jyy
-.s
250 200
y(s
=
Index price
50
=
:1020 k.
0 -50 -100
Profit Index price
=
-$75.68
=
$1000
-150
Payoff and Proft for a Written Put Option
''This price is computed nsing the Black-scholes formula for the pfice of a put: BSPut(1O00, 1000. 0.3, 2 x 1n(1.02), 0.5, 0) 74.20. We will discuss this formula in Chapter 12.
purchased put short forward
150
-200
Now we examine the put from the perspective of the put writer. The put writer is the counterparty to the buyer. Thus, when the contract is written, the put writer receives the
+
.
800
850
900
950 1000 1050 1100 1150 1200 S&R lndex Price ($)
prenziup. At expiration, ifthe put buyer elects to sell the underlying asset, the put writer must buy it. 'I'he payoff and profit for a written put are the opposite of those for the purchased put: Written put payoff
=
=
maxlo, stlike price
-
spot price at expiration)
(2.9)
%
42
AN INTRODUCTION
Written put prost
TO FORWARDS =
-
AN D OPTIONS
maxgo, strike price
spot price at expiration)
-
The RMoneyness'' of
+ future value of option premium
Consider a looo-sttike put option on the S&R index with 6 months Example 2.11 expiration. At the time the option is written, the put writer rceives the premium of to $74.20. Suppose the index in 6 months is $1100.The put buyer will not exercise the put. Thus, the >utwliter keeps the premium, plus 6 monts' intertst, for a payoff of 0 and profit of $75.68. l.f the index is $900in 6 months, the put owner will exercise, selling the index for $1000. Thus, the option writer will have to pay $1000for an index worth $900. Using equation (2.9),the written put payoff is maxlo,
$1000 $900) -
2.4 SUMMARY OF FORWARD
$75.68. Profit for
AND OPTION
Figure 2.10 graphs the prost diagram for a written put. As you would expet, it is the mirror image of the purchased put. -' ,'-,----' -r' -' )' 'k.'. q'. ()l::'d Cliilii.d #' y' )' );' r' y' ('
j'. ;'.-' (iI(E::-,' jy'. !!riI:::i;ik-'-. y'. jjyjd y' ,jj:L)jjj,j'k. .'i1I,iiiq'..' r-' rl'p,l 1IpIl'..' j'y' (' )' ild y;d ;'lyyyy'rd Tqq!tqqd ld t--);;'. ..!iiik:.iir'.. .',''j' -t'jj --' -' --k(:-' -d --...-----:---.------..--rt'-----:t' 'l:::::,-d
''-.' .'.'..'.'t;ki'. .''
EiEEEEEi(((:E'('qjEl('((fCf,((?i$t,'E(: .(E):!E)7E iE EIEIEETIEEFE:EEI: Ey 'f;iijlltfttllptlllpl)i) qTUN:rq('iFC(EE'@lq4(7(.'T .y . E E E-(.!. q . j ... g..(y.yyyy .(.yyyjjy;yy;jjjjjj.. . . jyyjjjjtjjjjjjyj gj y tty -irlhkilrkili-.-. ytg j!.jyyy.y.- yjjyyyyy.y.yj.gy.gy.;yyg.,gk-.'y-.-jj.-.y.jy(g;-ygyjy. ; . k.-.. . . EE ;i jr, ..)... :: r --. ..( -. -. j- .-:....; q -. 'rl... . .. (E
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-
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Written S&R index put option with strike of .$1000 versus a Iong S&R index forward Contract.
-.
'.
. . (ytyj(g.j.;j .1i1ii:i1i,i.. iliijii!:l ' . yyittyyk,y .)- . . -
-.
POSITIONS
We have now exnmined six different positions: Short and long fomards, and purchased and written calls arld puts. We can categodze these positions in at least two ways. One way is their potential for gain and loss. Table 2.4 summarizes the maximum possible gain and loss at maprity for fonvards and European options. Another way to categorize the positipns is by whether the positiops reprsent buying or selling the underlying asset. Those that represent buying are fundarpntally long wit respect to the underlying asset, while those that represent selling are f'undamentally short with respect to the underlying asset.
%
-7$24,32
=
yrty . . . . ..... qq. y. jjtrtjjj jjjqjtjllilissEl'i SkqLqjjqL ;y y. 8).(jIIEI. y
Option
nn
-$100
=
The premium has earned 2% interest for 6 months, and is now worth the written put is therefore
-$100 + $75.68
43
Options are often desclibed by their degree of moleypess. This term desclibes whether the option payoffwould be positive if the option were exercised immediately. (The tenn is used to describe both American and European options even though European options cannotbe exercised until expiration.) An in-the-money option is one wlzich would have a positive payoff (butnot necessnrily positive profit) if exercised immediately. A call the asset p rice and a put with a strike price greater tan te with a strike price less t.11a11 prfce bpth in-the-money. asset are option is one tat would have a negative payoffif exercised A.n out-of-th-money immediately. A call with a strike price greater than the asset price and a put with a strike price less than the asset price are bot.h put-of-the-money. An at-the-money option is one for which ihe strike price is approximately equal the asset price. to
The put seller has a short position in the put.
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$
5U M MARY OF FO RWARD AN D OPTI ON POSlTl ONS
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-200 800
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44
AN INTRODUCTION
TO FORWARDS
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ora-rloxsARE
AN D OPTIONS .
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-150
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850
900
950 1000 1050 110 S&R lnde.x Price ($)
-150 -200
1150 1700
Long Positions The following positions are long in the.sense that there are circumstances in which tey ' represent either a right or an obligation to bkt' the underlying asset:
Long forward: A n o bligation to buy at a ftxed price.
-
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950 1000 1050 1100 1150 1200 S&R Ipdex Price ($)
Figtlre 2.12 compares these three positions. Note that the written calli short when the asset price is greater than the strike price, and the purchased put is shol't When the the strike ptice. All frce ofthese Jm-f/tm. benehth'omfalling asset price is less t.11:11 ices. #?
call: The right to buy at a fxed price if it is advantageous to do so.
Written put: An obligaion of the put writer to buy the underlying asset at a fixed plice if it is advantageous to te option buyer to sell at that plice. tReca.ll tat the opion lfydr decides wheter gr not to exercise.) Figure 2.11 compares these three positions. Note tat the purchased call is long putis log when the asassetprice is greaterthan the strikeprice, andtewlitten the when risingprices. lrcd ofthesepositions svike price. l7c?7c/r.#'t???l price is less than AIl te set
2.5 OPTIONS
ARE INSURANCE i
J.n many investment strategies using options, we will see that options serve as insurance against a loss. ln what sense are options the same as insurance? In this section we answer this questipn by cpnsidering homeowner's insurance. You will see that options are literlly insurace, and instlrance is an bption. A homeowner's insurance policy promises tat in the event of damage to your house, te zsurance company will compensate you for at least part of the damage. greater the damage, the more tlle insurance company wi.ll pay. Your insurance policy thus drives its value from the value of your house: It is a derivative. 'l'he
Short Positions The following positions are short i.n the sense that there are circumstances in which they represent either a right or an obligation to sell the underlyixng asset:
Short forward: Arl obligation to sell at a fixed price. Written call: An obligation of the call writer to sell the underlying asst at a lixed price if it is advantageous to the option holder to buy at that price (recallthat the ' option bktyer decides Fhether to exercise). Purchased
.
put:
The right to sell at a fixed price'if it is advantageous to do so.
Homeowner's Insttranc ls a Put Option To demonsh->t how hoineowner's instlrance acts as a put option, suppose that you own a house that costs $200,000to build. To make ttils example as simple as possible, we assume tat physical damage is the only thing tat can affect the market value of the
house.
'
%.AN
46
llqTRoouc'rlolq
To
AND OPTIONS
FORwARDS
op-rlorqsARE
Let's say you buy a $15,000insurance policy to compensate you for damage to the house. Like most policies, this has a deductible, meaning that there is an amount of damage for which you are obligated to pay before the insurance company pays anythipg. Suppose the deductible is $25,000.lf the house suffers $4000damage from q stonp, you pay for a11repairs yourself. lf the house suffers $45,000in damage from a storw, you pay $25,000 and the insurance company pays the remaining $20,000.Once damage occurs beyond the amount of the deductible, the insurance company pays for all furtherdnmage, up to $175,000. (Why $175,0002Because the house can be rebuilt for $700,000,mzd you pay $25,000of that-the deductible-yourself.) Let's graph the prost to you for tllis insurance policy. Put on the vertical axis profit the on the insurance policy-the payoff less the insurance premium-and on te horizontal axis, the value of the house. .1f the house is undamaged (the house vtlue is $200,000) the payoff is zero, and prost is the loss from the unused insurance premium, $15,000. lf the house suffers $50,000damage, the insurance payoff is $50,000less te $25,000 deductibl, or $25,000. The profit is $25.000 $15,000 = $10,000. tf the house is completely destroyed, the policy pays $175,000,and your proft is $160,000. Figure 2.13 graphs the proft on the insurance policy. Remarkably, the insurance policy in Figure 2.13 has the same shape as the put option in Figure 2.9. An S&R put is insurance against a fall in the price of the S&R index, just as homeowner's insurance insures against a fall in the price of the house. h'surance ctwnptwzd. are f?)the bttsiness of I$?r9?l.g put optiolls! The $15,000insurance premium is like the premium of a put, and the $175,000level at which insurance begins to male payments is like the strike price on a put.
INSURANCE
%.
47
'l'he idea that a put option is insurance also helps us u'nderstand what makes a put option cheap or expensive. Two important factors are the riskiness of the underlying of te deductible. Just as with insurance, options will be m6re asset and te Jzmount expensive when the upderlying asset is rislier. Also, the option, lile insurance, will be less expensive as te (ie du ctible gets larger (for the put option, this means lowering the sike price). You have probably recognized that there are some practical differences between a financi put option and homeowner's insurance. One important difference is that the S&R put pays off no matter why the index price declines. Homeowner's insurance, on ln the other hand, pays off only if the house declines in value for specified resons. decline real is covered typicl in by homeowner's particular, a simple estate prices not instlrance policies. We avoided this omplication by assuming at te outset that only dnmage could affect te value of the houe.
-
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$160,000
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Gain on insurance due to damage
100,000
.
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lf we accept that insurance and put options are the same thing, how do we reconcile this with the common idea that buying insurance is prudent and buying put options is lisly?
The risk of a derivadve or any other asset or scurity can only be evaluated in the context. Figure 2.13 depicts the tisk of an insurapce contract without ctpnfcrnr This would be like owning insurance on your neighbor's house.. risk ofthe nwretfa.zf. because you would buy the insurance policy, and you would lose It would be entire investment if there were no insurance claim.B we do not normally thirlk of your Iike this, but it itluseates the point that insurance policy is put option a on an insurance the insured asset. In the snme way, Figure 2.9 depicts te risk of a put option without considering the risk of any other positions an investor might be holding. In contrast to homeowner's instlrance, many investors do ownput opions without owning the undertying asset. 'I'his is why opdons have a reputaion for being risky while homeowner's insurance does not. With stock optiops it is possible to own the insurance without the asset. Of course, many investors who own putoptions also own the stock. Forthese investors, theriskis like that of insurance, which we normally think of as risk-reducing rather than lisk-increqsing. T'risly''
Call Options Al.e Also lnsurance Loss of $15,000 premium if hous is uhdamaged
50,000
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But l Thought Insurance Is Prudent and Put Options Are ltislty
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C:.II options can also be insurance. Whereas a put option is inslzrance for an asset we already own, a call option is insurance for an asset we plan to own in the f'uture. Put dilerently, a put op tion is insurance for a long position while a call option is insurance for a short position.
-
100,000 150,009 200,000 House Price (5)
250,000
8Of course, in real life no insurance company will sell you insurance on your neighbor's house. The reson is thatyou will thenbe temptedto causedamagein orderto makeyourpolicy valuable. Insurance hazard.'' companies call this ''moral
i
k. AN
48
EXAM PLE: EQU ITY-LI N KED CDS
AND OPTIONS
TO FORWARDS
INTRODUCTION
''''''
* Is the CD fairly priced? * How can we decompose the product in terms f options and bonds? @How does the issuing bnnlc hedge the zisk associated with issuing the product?
''''''
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CDS
EQUITY-LINKED
@How does the issuing bnnk make a profh?
Although options and forwards are important in and of themselves, they are also comFor used as buildia blocks in the construction of new fnancial instrriij. mo-nly otfr piodts tht alloWivestors cpmpanies westmt example,bnnks and. insuranc ' . . ' . . prvide if the rnarket and stock ris beest idex i a guarazired retr tat from a a to rkd iikj t tools notd cDs can rever e-en gineer'' such equitplinkd '''''''''
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A simple 5 1/2 year CD with a retllrn llnked to the S&P 500 might have the following stnlcture: At maturity, the CD is guaranteed to repay the invested amount, plus 70% of the simple appreciation in the S&P 500 over that time.lo We can ask several quesions about the CD:
Return to the earlier example of the S&R index. Suppose that the current price of S&R index is $1000 and that we plan to buy the index in the future. If we buy arl the S&R call option wit a strike price of $1000, this gives us the zight to buy S&R for a By buying a call, we have bought insurance against an mnximum cost of $1000/share. prie. the in increase
2.6 EXAMPLE:
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To understand this product, suppose the S&pirldex is 1300 initially nd arl investor invests $10,000.Ifthe index is below 1300 after 5.5 years, the CD remrns to the investor the oliginal $10,000investment. If the index is above 1300 after 5.5 years, the investor receives $10,000plus 70% of the percentage gain on the index. For example, if the index is 2200, the westor rceives
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50
EXAM PLE: EQUITY-LI
AN D OPTIONS
TO FORWARDS
AN INTRODUCTION
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Economics of the CD Now we are in a posiion to understand te economics of this product. TIAiIIIC about what happens if te index is below 1300 at expiration. We pay $10,000and we receive $10,000 back, plus an option. we have forgone interest on $10,000in exchange for the possibility of receiving 70% of the gains on the S&P. Suppose that the effective rfhus,
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742
y Equity-lainlced CDs?
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51
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annual interest rate is 6%.,after 5 1/2 years, the buyer has lost interest of
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With reverse-engineedng, we see that an investor could create the equivalent of an equity-linked CD by buying a zero-coupon bond >nd 0.7 call options. Wy, then, do products like this exist? Consider what must be done to replicate the payoff. If a retail investor were insure to an index westment using options, the investor would have to learn about options, decide what maturity, stlike price, and quantity to buy, and pay transaction costs. Exchange-trade d opt ions have at most 3 years to mamrity, so obtaining longerterm protection requires rolling over the pitipn at plp pnt. An equitplinked CD provides a prepackaged soluiton. It may provide a pattern of market exposure tht many investors could not otherwise obtain at such 1ow transaction
.
costs.
The idea that a prepackaged deal may be attractive should be falrtilir to you. bpilding blocks, as it wer-and they Supermarkets sell whole heads of letmce-salad also sell, at a premium price, leytuce already washed, torn into bite-sized piecs, and mixed as a salad. The transaction cost of salad preparation leads some consumers to prefer the prepackaged salads. What does the :nancial institmion get oui of this? Just as the supermarket earns prost on prepackaged salads, the issuing bank wants to earn a transaction fee on the CD. When it sells a CD, the issuing balzk bonows money tthezero-coupon bond portion of the CD) and receives the premium for wliting a call option. The cost of the CD to the bank is te cost of the zero-coupon bond plus the cost of the call option. Obviously the bank would not issue the equity-lirtked CD in the srstplace unless it was less expensive than Zternative ways to attract deposits, such as standard CDs. The equity-linked CD is risk'y because the bnnk has written a call, but th bank can manage this risk in several ways, one of which ij to purchase call options from a dealer to offset the risk of having written calls. Using data from the early 1990s, Baubonij et a1. (1993) estimated that issuers of equity-linked CDs earned about 3.5% of the value of the CD as a fee, with l about 1% as the transaction cost of hedging the wlitten call.1
l IA back-of-the-envelope calculation in Chapter 15 suggests the issuer fees for this product are in the neighborhood of 4% to 5%.
k
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52
AN INTRODUCTION
FURTHER
AND OPTIONS
FORWARDS
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Forward conacts and put and call options are th basic derivative instruments that can be used directly and that selwe as building blocks for other instruments. A long fomard contract represents an obligation t buy tlle uderlying asset at a lixed price, a call option gives its owner the right (but not the obligatin) to buy the undyrlying asset at a sxed'price, and a put option gives its owner the light (but not the obligation) to sell the underlying asset at a fxed pdce. Payoff and profit diagrams re commonly used tools for evtluatinj the risk of ihse coneacts. Payoff diagrams show the gross value of a position at expiration, and profit diagrams subtract from the payoff the future value of the cost of the position. Table 2.6 summarizes the characteristics of fonvards, calls, and puts, showing which are long or short with respect to the underlyig asset. The table describes the
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strategy asspciated with each: Forward conacys guarantee a price, purchased options are insurance, apd written ptions are selling insurance. Figure 2.15 provides a grapllical summaiy o f these posidons. Optios can als be viewed as insurnce. A lut option gives the owner the right sell if the price declines, just as insurance gives the insured the right to sell (put) a to damaged asset to the insurance company.
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READING
READING
We use the concpts introduced in this chapterthroughoutthe restof tiis book. Chapter 3 f basic option stratygie! which are Wilely used in practice, including Presents a'nmbef nd Qoors. collars, Chapter 4 presents the use of options in risk managerent. caps, A more general question raised implicitly in this chapter is how the prices of fomards and options are detenmined. Chapter 5 cpvers financial forwards and f'utures in dtail, qnd Chpptr 10 inoduces the basic ideas underlying option pricing. Brokefj iutinely su/tlly optios customefs with an introductory pamphlet optin titled Chal tacteristics :7l7# Risks of Standardized yptjsn'. vus js availabout You can also obtain current option prices able online from http://www.cboe.com. CBOE'S sites such and various brokerage sites. Web from as the
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54
%
AN INTRODUCTION
PROBLEMS
AND OPTIONS
TO FORWARDS
The notion that options are insurance has been applied in practice. Sharpe (1976), for example, analyzed optimal pension f'unding policy tnking into account pension insurance provided by the Pension Benest Guaranty Corporation. Merton (1977a)obseryd thatbnnk depositinsurance and in fact any loan guarantee can be modeled as aput option. Baubonis et al. (1993)discuss equitplinked CDs.
PROBLEMS In the following problems, i.f the yields 1 + ?' after one year.
4teffective
annual interest rate'' is r, a
$1 investment
2.1. Suppos XYZ stock has a price of $50 and pays no dividends. The effective annual interest rate is 10%. Draw ayoff and proft diagrnms for a long posidon in the stock. Verify that profit is 0 at a price in one year of $55. 2 2 Using ihe sa'me information as the previous question, draw payoff and prost diagTnms for a short position in the stock. Venf'y' that prot is 0 at a price in one ' of $55. year 2.3. What position is the opposite of a plzrchpsed call? 'Fhe opposite of a purchased P ut? a. Suppose you enter into a long 6-month fo-ward position at a forward Price of $50. What is the payoff in 6 monts for prices of $40,$45,$50. $55, and $60? -
any advantage to investing
Why?
c. Suppose XYZ paid a dividend of $2 pr year and everything else stayed the same. Now is there any advantage to investing in the stock or te forward contract? Why? 2.8. Suppose XYZ stockpays no dividends and has a currentprice of $50. The forward pfi.ce for delivery in one year is $53. Jf there is no advantage to buying either the stock or the forward contract, what is the l-year effective interest rate? 2.9. An offmarket forward contract is a forward where either you hake to pay a premium or you receive a premium for entering into the contract. (With a standard forward contract, the premium is ero.) Suppse the effective annal interest rate ' is 10% and the S&R index is 1000. Consider l-year forward contracts. a. Verlf'y'that if the fonvtrd price is $1100,the jroEt diagrams for the index and the l-yellr fonvard are the same. b. Suppose you are offered a long forward contract at a forward price of $1200. How much would you need to be paid to enter into this contract? c. Suppose you are offered a long forward contract at $1000. What would you be willing to pay to enter into this forward contract? 2.10. For Figure 2.7, verify the following'. a. The S&R index price atwhichthe call option diagram intersects thex-axis is $1095.68.
c. Comparig the payoffs of part () and (b), which onact more expensive (i.e.,the long call or long fonvard)? Why?
b. The S&R index price at which te the same proft is $924.32.
should be
b. Suppose you buy a 6-month put option with a strike price of $50. What is the payoff in 6 months at the sate prices for te underlying asset? which coneact c. Comparing the payoffs of palrts (a) and (b), more e kP e j ive (i.e., the long pt or short forwrdj'i Why?
should be
.
2.6. A default-free zero-coupon bond costs $91 and will pay $100 at maturity in 1 year. What is the.effective annual interest rate? What is the payoff diagram for
thebond?
'
The proht diagram?
Suppose XYZ stockpays no dividnds and Vs a currentjrie of $50. Thefomard annual interest Plice for delivery in 1 year is $55. Suppse the l-yeaf effctik rate is 10%.
a. Graph the payoffandprostdiagrmns with a fonvard price of $55.
for afolward contract on XYZ stock
'
55
in the stock or the forward contract?
b. Suppose you buy a 6-month catl option with a stlike plice of $50. What is the payoff in 6 monts at te snme prices for the underlying asset?
a. Suppose you enter into a short 6-montll forward position at a forward price of $50. What is the payoffin 6 months for prices of $40,$45,$50, $55, and $60?
2.7.
b. Is there
k
call option and
forward contrat have
2.11. For Figure 2.9, velify the following'. a. The S&R index price at which theput option diagram intersects thex-axis is $924.32. b. The S&R index price at which the put option and forward contract have the same prost is $1095.68. 2.12. For each entry in Table 2.4, explain the gain or loss occurs.
circumstances
in whih the maximum
2.13. Suppose the stock price is $40 and the effective anual interest rate is 8%. a. Draw on a single graph payoff and prost diagraps for the following options:
$9.12. (ii) 4o-strike call with a premium of $6.22. (iii) 45-strike call with a premium of $4.08. (i) 35-strike
call with a premium of
$
AN #NTRODUCTION
To
FORWARDS
AND OPTIONS
APPEN D IX
b. Consider your payoff diagram with all tlzree options graphed together. lnmitively, why shotzld the option premiumdecrease withthe strikeprice?
2.14. Suppose the stock price is $40and the effective annual interest rate is 8%. Draw payoff and profit diagrams for the following options: a. 35-stlike put with a premium of $1.53. b. zm-strike put with a premium of $3.26. c. 45-strike put with a premium of $5.75. Consider your payoffdiagram wit :11 three options graphed together. Inmitively, why should the option premium increase with the strike price? 2.15. The profit calculation in the chapter assppes thai yolz bpn'owat ktftxed interest rateto finance investments. An alterntive way to bon'w is to Short-sell Whatcomplications would arise in alculating prot if you tinanceda $1000S&R index investment by shorting IBM stock, rathr than by borrowing $1000? 2.16. Conseuct a spreadsheet that penrnts you to compute payoff and proht for a short . and long stock, a short and long fonvard, and purchased and wlitten puts and callst The spradsheet should 1et you specify the tock pdce, forward price, interest rate, option strikes, and option premiums. Use the spreadsheet's mlx f'unction to compute option payoffs. 'stock.
.
'
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'
2.A:
MO RE ON BUYI NG A STOCK
$
OPTI ON
In June 2003, lomega Corporation declared a $5 dividend, payable on October 1. At the time of the declaration, Iomega's share price was $11.40.Since the dividend was 44% of the share price, the OCC reduced a11Iomega ption strike prices by $5, effective October 2.12 When we discuss option pricing, we will see that it is necesjary to take dividends into account when pricing an option.
Exercise Some options, for example those that are cash-settled, are automatically exercised at maturity; the option owner need not take any action. Suppose you own a traded option that is not cash-settled and not automatically exercised. ln this case you must provide exercise instructins plior to the broket'j deadlin. If you fail to do so, the option Will ekpire worthled:. When you exercise the option, you genefally pay a commission. If you do not wish to oWn the jtock, exercising th option would require that you pay a commission to exercise and then a cdmmission to sell the shares. lt might be preferabl to sell the option instead of exercising it. If you do wish to own the underlying asset, you can exercis the option. The option writer who is obligated to f411:11the option exercise (deliveringthe shares for a call or buying the shares for a put) is said to have been assiglied. Assignmet can involve paying a commission American-style options can be exercised prior to expiration. lf you own an option
AppENolx A
2.A: MORE OPTION
s'rocx
oN BUVING
The box on page 34 discusses buying options. are at least four practical issues tllat an option buyer should be aware of: Divideds, exefise, margins, and taxes. In this section we will focus on retail westors and xchange-eaded stock options. Be aware that specisc nlles regarding margins and tus change frequently. This section is intendd to help you identify issues and is not intended as substimte for professional brokerage, accounting, or legal advice. rfhere
Dividends The owner of a standard call option has the zight to buy fixed number of shares of stock at a fixed price, btlt has no light to receive dividends a paid on the underlying stock over tle life of the option. When a stock pays a dividend, the stock price declines by approximately the amount of the dividend. This decline ttle price lowrs the ret'unz in the owner of call option. to a F0r exehange-traded options in the United States, there is typically no adjustment in the terms Of the option if te stock pays an dividend (onethat is typical for the stock). However, if the stock pays an unusual dividend, then oficials at the Options Clealing Corporation (OCC) decide whether or not to make mzadjustment. 'Eordinary''
and fail to exercise when you should, you will lose money relative to foll6wing th optimal exercise strategy. If you write the option, ad it is exercised (you are assigned), you will be required to sell the stock (if you sold a call) or buy the stock (if you sold a put). Therefore, if you buy or sell an Amelican option, you need to understand the circumstances under which exercise might be optimal. Dividends are one factor that can afect the exercise decision. We discuss early exercise in Chapters 9 and 11. E
M argmsfor Written Options '
Purchased options for which you f'ully pay require no margin, as there is no counterparty
fisk. Wit.h written option positions, however, you can incuralarge loss if the stock moves against you. When you write an option, therefore, you are required to post collateral to insure against the possibility that you will default. This collateral is called margin. Margin nlles are beyond the scope of this book and change over tile. Moreover, different option positions have different margin rules. Both brokers and exchanges can provide information about current margin requirements.
lzlkeducing the strike price by the amount of the dividend leaves call holders worse off, albeit betteroff than if no adjustment had ben made. If S is the cum-dividend stock prce and S D the ex-dividend stock prie, Merton (1973b,p. 152) shows that to leave the value of a call position unchanged, it is D) 1 necessary to reduce the strike price by the factor CS D)/S, and give the option holder S/S additional options. An option with a value protected against dividends is said to be payout-protected. -
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%
AN INTRODUCTION
To
FORWARDS
AND OPTlON5
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Taxes
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rules for derivatives in general can be complicated, and they change frequently the tax law changes. The taxation of simple option as transactions is straiglafbnvard. ,If you purchase a call option stock and then sell it gain or or loss on the position is treated like gain or loss on a stock, and accorded long-term or short-term capital gains treatment depending on the length of time for which the position has been eld. I.f you purchase a call option and then exercise it, the cost bass of the resulting stock position is the exercise plice plus the option premium plus commissions. The holding period for the resulting stock position begins the day after tlae option is exercised. The time the option is held does not contribute to the holding period. The rules become more inicate when forwards and options are held in tandem with the underlying asset. The for this complexity reasons TM laws in m'e not hard to undzstand. he United statesaccord dierent tax different kinds of inome. treatment to The t?zx code views interest income, dividend income, and capital gains income as ydistinctand subject to different tax rules. Futures also have special ruls. Hbwever using derivatives, one kind of income can be turned into another. We saw in tltis chapter, ' Or example, that buying zero-coupon bonds and a fomard cpntract mimics a stock investment.
One category of special rules governs a constructive sale. tf you own stock, entering into certain option or forwtrd positions can trigger a constructive sale, meaning that even if you continue to own the stock, for tax purposes you are deemed to have sold it at the time you enter into the fonvard option or positions. By shorting a forward against the stock, for example, the stock position is transformed into a bqpd position. When you have no risk stemming from stock ownership, tax law deems you to po longer be an owner. The so-called straddle rules are tax Iules intended to control the recognition of losses for ttx purposes when there are offsetting risks as with constl-uctive sales. positions often arise when investors are undertaking tax arbitrage, which is whysuch tlae positions are accorded special treatment. A stock owned together with a put fs a tax straddle.l3 Generally, the straddle rules prevent loss recognition on only a part of the entire position. A straddle for tax purposes is not the same tlng as an optiop straddle, discussed in Chapter 3.
It is probably obvious to you that if you are tuable and transact in options, and especially if you have both stock and offsetting option positions, you should be pmpared to seek professional tax advice.
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Table 3.2 presents the payoff and proft for a short position in the index coupled with a purchased call option. Because we short the index, we earn interest on the short proceeds less the cot of the call option, giving as the f'umre value pf the cost. of The payoff and profit diagrams 3.3 columns graphs the Figure Table 3.2. with insured index position in Figure 3.1, we of purchased As the put. resemble those a Fiyh ash flows. payoj. in caref'ul in dealing The panel (c) of Figgre 3.3 is have yo be this coupled with ln purchased borrowing. like that of a put case, the payoff diagram call equivalent that from buying a put and the and buying is index to for shorting a diagrams of valtle borrowing the present are unaffected $1000($980.39).Sinc ptotit is exactly profit diagram the in panel however, the by borowing, same as that for a (d) with this comparing panel Figure 2.9. Not S&R index by You put. purchased can see (d) position look like has the insured it loss the short only does smne as a purchased a ptlt, of the price is bove which is value fumre tlle $74.201put $1000: $75.68, put if the -$924.32
premium.
Insuring a Short Positih:
Caps
If we have a short position in the S&R index, we experience a loss when the index rises. We can insure a shott positiop by purchasing a call option to protect against higher a price of repurchasing tlie index.l Buying a call option is al.socalled a cap.
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Keep in rnind that if yo have an obligation to buy the index in the ftlttlre but the price is not fixed, thenyou have an inlplicit short position (if the price goes up, you will have to pay more). A call is insurance for both explicit and implicit short-sellers.
Selling Insttrance We can expect that some investors want to purchase insurance. However, for every insurance buyer there must be an insurance seller. ln this section we exnmine strategies in which investors sell insurance. It is possible, of course, for an investor to simply sell calls and puts. Often, however, investors alsp have a position in the asset when they sell insurance. Writing an opt ton w h n there is a corresponding long position in the underlying asset is called covered writing, option overwriting, or selling a covered call. A11 three tenns mean
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64
COLLARS,
AN D O-rH ER ARATEGI
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(1(y,, Short S&R Index
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shows the payoff diajram for a purchpsd index call with a strike price of $1000 (column2 in
Table 3.2). Panel (c) shows the combined payoff diagram for the short indx and long call (column3 in Table 3.2). Panel (d) shows the combined profit diagram for the short index and Iong call, obtaned by adding $924.32to the payoff diagramin panel () (coItlmn 5 tnTable 3.2).
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covered puts A covered put is achieved by writing a put against a short position on the index. The written put obligates you to buy tlze index-for a loss-if it goes down'
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which is the future value of the prernium received from writing a looo-strike put. 'I'he proft from writing the looo-strike call is colputed in Table 3.3 and graphed in Figure 3.4. If the index falls, we lose money on the index but the option premium partially offsets te loss. If the index rises above the stlike price, the written option loses money, negating gains on the index. Comparing Table 3.3 with Table 2.3, we can see that writing the covered call generates exactly the same proht as selling a put. Why would anyone write a covered call? Suppose you have the view that the index is unlikely to move either up or doWn. l'l71isis sometimes calle j a neutr al', market view.) If in fact the index does not move and you have wlitten a call, then you keep the premium. lf you are wrong and the stock appreciates, you forgo gains you would have had if you did not write the call.
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65
Profit on Written Call
Proht on S&R Index -1000
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Because the covered call looks like a written put, the muimum prot will be the prost is same as with a written put. Suppose the index is $1100at expiration. The
Long S&R Call 2000
ARATEGI
selling the call. A payoff with limited profit for price increases and potentially large losses for price decreases sounds like a written put.
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essentially the same thing.z 1.rlcontrast, naked Avriting occurs when the writer of an option does noi hay a position in the asset. In addition to the covered writing strategies we will discuss here, there are other insurance-selling sategies, spch as delta-hedging, which are less tisky than paked writing and are used in practice by market-makers. We will discuss thse other strategies later in the book, pmicularly in Chapter 13. call writing f.fwe own the S&R index and simultaneously sell a call option, written have covered call. A covered call will have limitd prostability if the index a we because option writer is obligated to sell the index for the sfrike price. increases, an Should the index decrease, the loss on the index is offset by the prernium enrned f'rom Covered
z'Ikchnically,
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Short S&R Put
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payoff
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(b)
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($)
combmedPayoff
1000
($) Combined
Profit
2(00
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$95.68
1000
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'
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00 o
1500 200 S&R Ipdey at Expiration
0
500 000 150 2000 $&R Index at Expiradon
(d)
(c)
(d)
It is possible to mirnic a long forward position on an asset by buying a call and selling a Put with each option having the same strike price and tilr to expiration. For exnmple, we could buy the looo-strike S&R call and sell the looo-strile S&R put, each with 6 months to expiration. ln 6 months we will be obliged to pay $1000 to buy the index, just as if we had entered into a forward contract. For example, suppose the index in 6 months is at 900. We will not exercise the call, but we have written a put. The put buyer will exercise the right to sell the index
.
=
500 1000 1500 2000 S&R ladex at Expiradon
FORWARDS
500 1000 1500 2000 S&R Index at Expiration
(a)
-1000 0
'.--'-''''---
-1000
-1000
(c) is the
Panel
gjjygjjy jtgg;j jtg yyyj.yyyy;j jjjyjj tjjyy (;j
-.. - - .. ... .
-1000
in price. Thus, for index plices below the trike pric, the loss on the written put offsets the short stock. Fr index prices above the strike price, you lose on the short stock. A'position where you have a constant payoffelow the strike and increasing losses above the stnke sounds like a written call. ln fact, shorting the index and writing a put Produces a prost diag'ramthat is exactly the same as for a written cplt. Figure 3.5 shows this graphically, and Problem 3.2 asks you to verify this by constructing a payoff table.
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Payoff and profii diagrams for writing a covered 5&R call. Panel (a) is the payoff to a jl rjj; j; k k(j) (lsiticjrj. Panel (b) is the payoff to a short S&R call with strike price of 1000. Panel (Q'isthe combined payotf for the 5&R indx and written call. Panel (d) is the combined profit, obtained b# subtacting - ($1O00 593.809) x 1 $924.32 from the payof in ppnel (c). .02
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for $1000;therefore w af obligated to buy the index at $1000. If instead te index is at $110t, the pt is not kercised, but we exrcise the call, byig the index for $1000. Thus, whether the index lises or falls, when the options expire we buy the index for the strike price of the options, $1000. The purchased call, written put, and combined positions are shown in Figure 3.6. The purchase of a call and sale of a put creates a synthetic long fo>ard contract, wlzich has two mipor differences from the act'ual fonvard: 1. The forward contract has a zero premium, while the synthetic forward requires that
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Equating the costs of the alternative ways to buy the index at time t gives us
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15O
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so -50 -100
Ptlrchased call + Written put Combined positon
H-
150 -200
-
Puttf,
F)
=
PV(F(),w
-
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In words, the presentvalue of the bargain elementfrombuying theindex atthe strikeprice (the right-hand side of equation (3.1)) must be offset by the initial net option premium (the left-hand side of equation (3.1)1.Equation (3.1) is lnown as put-call parity, and one of the most important relations in options.
slozs N
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=
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Example 3.1 As an example of equation (3.1),consider buying the 6-month 10001)l@p ' S&R call for a premium of $93.809and selling the 6-month looo-strike put for ).1stnke )))) premium of $74.201. These transactions create a synthetic forward permitting us to C a '!'1 buy the index in 6 months for $1000. Because the actual forward price is $1020,this synthetic forward permits us to buy the index at a bargain of $20, the present value of !q.g . )ll which is $20/ 1.02 $19.61. The difference in option premiums must therefore be ). .'tq In fact, $93.809 $74.201 $19.61.This result is exactly what we would get )t.,$19.61. .'
(E
-250
800
850
950 1000 1050 1100 150 S&R Index Pdce ()
900
1200
i Et
(.'E
=
=
-
expensive t.11a:1 the put. ln fact, in Figure 3.6, tlzeimplicit cost of te synthetic forwardthe price at which the profit on the ombined call-ppt position is zero-is $1020,which is the S&R forward price. Sirnilarly. it we set the sike price high, we are obligated to buy the index at a high price relative to the forward price. To offset the exta cost of acquiring the index using the high stdke options, it makes sense that we would receive payment initially. This would occur if the put that we sell is more expensive than the call we buy. Finally, if we set the strike price equal to the forward price, tlen to mimic the forward th: irlitial yremim ntust tualzero. I this ase, put nd call prmiums must
jipwith
t
equation
(3.1):
$93.809 $74.201 PV($1020
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-
be equalr
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A forward contract for which the premium is not zero is sometimes called an offmarket forward. This trminology arises since a true forward by desniitpn has a zero premium. Therefore, a forward contract with a nonzero premium must have a forward market (forward)price.'' Unless the stlikeplice equals te folavard price which is price, buying a call and selling a put creates an off-market forward. Iloffthe
We have seen earlier that buying the index and positions prost buying a put generates the same as buying a call. Similarly, selling a covered call calll selling generates the same profit as selling a put. Equation (buying the index and a (3.1) explains why tis happens. Consider buying the index and buying a put, as in Section 3.1.. Recall tat, in this and the index price equal to $1000. example, we have the forward price equal to $1020 equals of forward price the index price. Rewriting equation the Thus, the present value (3.1) gives Equivlente
Put-call Prity . .
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We can summarize this argumet by saying that the ndr cost t/-lllyf??' the f?7#c-'t: ttsing options ??2II'f eqttal (he n'f cost ofbttying #?dinde,x ldWlg afonvalzl contract. If at time 0 we en terinto a lonjforward positton expiring at iime F, Fe obligate ourselves to buying the index at the forward price, F(),w.'I'he present value of buying the index in the ftlture is just the present value f the forward price, PV(F0,r). q lfinstead we buy a call and sell a put today to guarantee tlle purchase price for the index in the f'utwe, the present valpe of the cost is the net option premium for buying the call and setng the put, CalltA-. FJ Puttff. F. plus the present value of the strike price, PV(#). trfhenotations Calltf, 7') and Puttff, r) denote the premiums of options wit.h stdke price K and wit t periods until expiration.) ,
-
'
of diferent
PV(Fo,w) + Puttf,
F)
=
$1000 + $74.201
=
Calltff, F) + PV(#)
$93.809+ $980.39
That is, buying the index and buying the put cost the same, and generate the same payoff, as buying the call arld buying a zero-coupon bond costing PV(#). (Recall from Section 2.1 that a bond does not affect protit.)
% Ilqsu RANCE,
70
AlqD OTH ER ARATEGI
COLLARS,
Similarly, in the case of PV(Fo,w)
writing
F)
=
PV(#)
That is, writing a covered call has the same proft as lending PV(#) Equation (3.1)provides a tool for constructing equivalent positions.
and selling a put.
In deriving equation (3.1), and in some earlier discussions, we relied the that if idea two differept investments generate the same payoff, they mgst have on the same cost. This commonsensical idea is one of 4he most important in the book. If eqpation (3.1)did not hold, there would be both low-cost and high-cost ways to acquire the index at time F. We could simultaneously buy te index at 1ow cost and sell the index athigh cost. This transaction has no risk (sincewe both buy and sell the index) and generates a positive cash flow (becauseof the difference in costs). Taking adkantage of such an opporlnit.y is called arbiage, and the idea tat prices should not pelmnit pricing.'' We implicitly illustrated this idea earlier in arbiage is called'tno-arbiage showing how owning the index and buying a put has the same profit as a call, etc. No-arbitrage pricing will be a major theme in Chapter 5 and beyond. a No arbitrage
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You might ask: Is there a lower-cost way to speculate tat the stock price will rise, still has the insurance implicit in the call? The answer is yes: You can lower the that cost of your strategy if you are willing to reduce your prot should the stock appreciate. You can do tlzis by selling a call at a higher strike price. 'T'he owner of this second call buys appreeiation above the higher strike price and pays you a premium. You achieve a lower cost by giving up some portion of prct. A position in which you buy a call :nd sell an othemise identical call with a higher stn'ke price is an example of a bull spread. Bull spreads can also be constructed using puts. Perhaps surprisingly, you can achieve the same result either by buying a low-strike call and selling a high-strike call, or by buying a low-stlike put and selling a high-strike put. Spreads constructedwith eithercalls orputs are sometimes calledverticalspreads. The terminology stems from the way option prices are typically presented, with strikes arrayed vertically (as in Table 3.4).
There are many well-lnown, commonly used strategies tat combine two or more options. ln this section we discuss some pf these strategies and explain the motivation for using them. The underlying theme in this section is that there are always trade-offs in designing a position: lt is always possible to lower the cost of a position by reducing its payoff. Thus there are many variations on each particular strategy. All the examples in this section will use the set of option plices in Table 3.4. We will assume the continuously compounded interest rate is 8%. ';'7* j'f)' ?' T' 'SX kjjj(r)'. j'rjf )' f' @' ;'(' )' j'g' 1* ;r' qltqqqr;);f y' yjjltatjjljijiiz'k'k. tjjzyt;jjrgr'-d r' y'. j': t;f q'lrf j'k)f (tf ytf tt'':'p' ' E' i ! : (q55:'7* j'y' '')'()' ('. jjj'yy (': (jf )' E i jEE ' E E 'E '' E'' E EE'E'' ' : '. (('E..(. E..E(.iE;.q.(.E(.q. ('!:'.lj.(! (! ; (q'(... .ti(i.ij'ir( ; ;1:.j(.'(' (..('(.(j.((?'iE'EE.(.( @ . (.!.((!'( .q 1* .(( Eyrj,j . ! .y i. E! E i .EE . E E.E. i. E E E -L,)q.;-;L-t3qq'.;.. i. !t.EIE.-.t-.I;ftii-.... E. .E .rt-' E .E E . . g .E .. .( ( y y . , r r . .. . q r . j y . . j g j ! ( j . j j y . g . j ;y. . ....y.y.y..jryr. yj gj. yj.;j j ;j..jy,;r)r-))j. y.. .j.kj.yj. .;Igtii;l j. yyy .i:k(..... . .))t!lkk IjEjjL...?.:)t-.j.y.-j . .j ..
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3.3 SPREADS AND COLLARS
'
COLLARS
An option spread is a position consisting of only calls or only puts, in which some options are purchased and some written. Spreads ar a common strategy. ln this section we defne some typical spread strategies and explain why you might use a spread. Suppose you believe a stock will pppreciate. Let's compare two Fays to speculate belief: entering into along forward contract orbuying a call option with the strike this on equal price to the forward price. The forward contract has a zero premium and the call has a positivepremium. A difference in payoffs explains the difference in premiums. J.f the stock price at expiration is greater than the forward plice, the forward contract and call have the same payof'f. If the stock price is less tan the forward price, however, the forward conact has a loss and the call is worth zero. Put-call parity tes us that the call is equivalent to the fonvard contract plus a put option. Thus, the call premium equals the forward the cost of the put, which is insurance against the stock price being less t11:,11
Puttff, F)
-
Alqo
Bull and Bear Spreads
a covered call, we have
Calltf,
-
SPREADS
Es
(. ..
0.44
1.99
5.08
SAnother way to express the plinciple of no arbitrage is using profit diagrams. Given two profit diagrams, there is an arbtrage opportunity if one diagram is everywhere above the other.
-'.
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r Exam' pIe 3.2 To see how a bull spread arises, suppose we want to speculate on the 1i1 # stoc k p rice increasing. Consider buying a 4o-strike call with 3 months to expiration. lil
jt) From Table 3.4, the premium for this eal1 is $2.78. We can reduce the cost of the )' Position-and also the potential profit-by selling the zz-strike call. ).1 l'1 An easy way to construct the graph for this position is to emlate a spreadsheet: :;( l)( For each plice, compute the prot of each option position and add up the profits for the individual positions. lt is worth worling through one example in detail to see how this ',! ;( . . is done. . t') The initial net cost of the two options is $2.78 $.97 $1.81. With 3 months (!r'l ) . t) interest, the total cost at expiration is $1.81 x (1.0833)0.25 $1.85.Table 3.5 computes 'i) the cash flow a t expiration for both options and computes profit on the position by .,! .# )ij subtracting the future value of the net premium. lt Figure 3.7 graphs the position in Table 3.5. You should vel'ify that if you buy the )) %. f 4o-strike put and sell the 45-strike put, you obtain exactly the same graph. 'i: :
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AN D OTH ER ARATEGI
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The opposite of a bull spread is a bear spread. Using the options from the gbove example, we could create a bear spread by selling tle 4o-strike call and buying the 45-strike call. The prost diagram would be exactly the opposit of Figure j.7.
Box Spreads A box spread is accomplished by using options to create a synthetic long fonvard at one price and a synthetic short fonvard at a different price. This sategy guarantees a
A ratio spread is constnlcted by buying ?n calls at one strile and selling 11 calls at a different stdke, with a11options having the same time to mat-urity and same underlying asset. Ratit? spreads can also be constructed with puts. You are asked to constnlct ratio spreads in pfoblm 3.15. Also, a ratio spread constructed by buying a low-stn'ke call and selling'two higher-strike calls is one of the positions depicted in the chapter summal'y in Figure 3.17. Since ratio spreads involve buying and selling unequal numbers of options, it is signicanc of this may not possible to construct ratio spreads with zero premium. be obvious to you now, but we will see in Chapter 4 tat by using ratio spreads we can construct paylater strategies: insurance that costs nothing if te insurance is not needed. The trade-off to this, as you might guess, is that the insurance is more costly if it is 'he
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Collars A collar is the purchase of a put option and the sale of a call option with a higher strike plice, with both options having the same underlying asset andhaving the same expiratio date. lf the position is reversed (saleof a put and purchase of a calll, the collaris written. The collar width is the difference between the call and put strikes.
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The collar depicted in Table 3.6 entails paying a net prezium of zero-tost $1.02: $1.99 for the purchased put, gainst $0.97 for te written call. It is possible to find strile prices for the put and call such that the two premiums exactly offset one another. This position is called a zero-cost collar. To illustrate a zero-cost collar, suppose you buy te stock and buy the 4o-stn'ke put thathas a premium of $1.99.Trial and error reveals that a call with a strike of $41.72 ao has a premium of $1.99kX'hus, you can buy a 4o-strike put and sell a 41.72-st.1.111e call without paying any premium. 'T'he result is depicted in Figure 3.9. At expiration, the collar exposes you to stock prie movements between $40 and $41.72,coupled with k downmde protection below. $40. You pay for this protection by giving up gains should the stock move above $41.72; For any given stock tllere is an intinite number of zero-cost collars. One way to is to lirst pick the desired put sfrike below te forward price. It is then possible this see find to a stlike above the fonvard price such that a call has the same premium. collars
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If you have a short position in the stock, you can collar te position by buying a call for insurance and selling an out-of-the-money put to pnrtially ftlnd the call purehase. The result looks like a bear spread.
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One aspect of the zero-cost collar that may seeril puzzling is that you can snance te purchase of an at-the-money put by setling m4out-of-tlie-money call. ln the above example, wit.h the stock at $40, you were able to costlesdly buy a zlo-strike put by also selling a 41.72-s11-1e call. This makes it seem as if you have free insurance with some possibility of gain. Even if you are puzzled by this, you probably realize that 'Tree'' insurance is not possible, and something must be wrong with this way of thinking about the position. collars
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d $41 72 of collars, considerqn executive who ownj a large To illustrate he us adpcing using executives stock. Such frequently hedgetheir stockpositio, positionin compmw tvt mattllitv yeis exnmple, several for ctgarj witll Sup/os? Vicrosoft cost to zerohak a price of $3t/shar and >r executive wishes to edge 1 rnillion shares. If the ekecutive buys a 3o-strik put with 3 years to paturity, what 3-ye@reall will have the eftectike annual sk-free rate of 6%, a zero dtvidend snme premium? Assuming an 3o-strike Blpck-scholes price and 40% voladlity, fr is the $5.298 a put with yield, a with a solver), option numerical call maturity. and trial Using 3 years to a en'or (or a highly agnin, ollar the premium. of Once zerp-cost has tll snme strike $47.39 seems Itwevr, thif cpmparispn doe not take into account financing cost. asymmetri. . J . . stock in three years for $30/sharewill in fact have lost three years executive selllng of interest: worth $30 x ((1.06)3 1) $5.73. an
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cost i.e., the ftlttlre value of the stock price. In the glrday example above, this would require that you set te put stdke equal $40.808, to which gives a premium of $2.39k) The call premium at this strike is also $2.39!Ifyolt p'y to ??.If?'d against aII Iosses tp?) the stock, fncfllfll.,g interest, J'/?e??. a zero-cost collar 7itz-zero w/#/'. ' This is an implication of put-call patity, equation (3.1).lt trns out tfiat is also the theoyetical folward price. $40.808 lf we set the strike equal to the forward price, the call premium equals the put premium. ,
3.4 SPECULATING ON VOLATILITY The positions we have just considered are a11directional: A bull spread or a collar is a bet that te price of the underlying asset will increase. Options can also be used create pojitions rhatare nondirectional with respect to the underlying asset. With toa nondirectional position, the holder does not care whether the stock goes up or down, but on1# how much it moves. We now examine stradles, strangles, and buttertly spreads, whih are examples of nondirectional speculations.
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strangle The disadvantage of straddle is the high premium a cost. To reduce tlz premium, you can buy out-of-the-money optiond ratherthan at-the-inopey pptio. Such a position is called a strangle. For examjle, consider buying a 35-stlike put and 45stlike call, for a total premium of a $1.41.with a f'uttlrevalue of $1.44. Wese reduce your muimum loss if the options transactions expire with the stock near $40, but they also increase the stock-price move required for a pro:t.
Figure 3.11 shows the 4o-strike seaddle graphed against the 35-45 strangle. Tlzis comparijon illuseates a key pointp ln comparing Any two faidy priced option positions, t.here w ill always be aregion where each outperfonus the other. lndeed, this is necesiary to have a fairly priced position. J.nFigure 3.11, the sangle outpedbrms the saddle roughly when t stpck prie at expiration is between $36.57and $43.43. Obviously, there is a much broder range in which the staddle outperforms the sangle. How can vou decide which is j.jy e jiet. ter investment? The answer is that unless yo tt have particular view on the stock s performance, you cnnnot say that one position is preferable to the other. An option pricing model implicitly evaluates the likelihood that one strategy will outperform the other, and it computes option prices so that the two strtegis aie equivalently fair deals. An investor might have a preference for one sategy over the other due to subjective probabilities that differ f'rom the market's.
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Written straddle What if an westorbelieves that volatility is lokver than te market's assessment? Because a purchased saddle is a bet that volatility is high (relativeto te mlket's assessment), ?. m-ittn straddl-selling a call and put wit.h the same sttike price and time to expiration-is a bet tat volatility is 1ow (relativeto the market's L' assessment). Figure 3.12 depicts a written straddle, which is exactly the opposite of Figure 3.10, the purchased saddle. The written straddle is most prostable if the stock plice is $40 at expiration, and in this sense it is a bet on low volatility. What is strilting about Figure 3.12, however, is the potential for loss. A large change in the stock plice in either direction leads to a large, potentially unlimited, loss.
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% INSURANCE,
80
COLLARS,
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S PECU
STRATEGIES
AND OTHER
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.
20
The straddle writer can insure against large losses on the sladdle by buying options io protect against losses on both the upside and downside. Buying an out-of-the-money put provides insurance on the downside, protecting against losses on the at-the-mony written put. Buying an out-of-te-money call provides insurance on the upside, protecting against losses on the written at-te-money call. . Figure 3.13 displays the saddle written at a strike price of $40, iong Fith the of options to safeguard the position: A 35-strike put and a 45-strike cil. The net combining tese tlu'ee seategies is an insured written saddle, which is called abuiierfly spread, graphed in Figure 3.14. lt can be thought of as a wlitten stradtle for the timid (or for the pnldentl). Comparing the butterlly spread to the written straddle (Figure 3.14), we see tat the butterfly spread has a lower maximum pro:t (dueto the cost of insurance) if the stock at expiration is close to $40, and a higher proft if there is a lrge move in the stock price, in which case the instlrance becomes valuable. spread we gain imWe will see in Chapter 9 that by understanding the buttey portant insights into option prices. Also, the butterlly spread can be created in a variety of ways: solely with calls, solely with puts, or by using the stock and a combination spread in Figure 3.14 of calls andputs.s You are asked to verif'y this in Problem 3.18.
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($)
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25
30
35 40 45 XYZ Stock Price
50
55
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($)
lt might occur to you that an investor wishing to bet that volatility will be low could write a saddle and acquire insurance against extreme negative outcomes. That inttlition is correct and leads to our next sategy.
-20
20
25
30
35 40 45 XYZ Stock Price
50
55
60
()
snchnically, a true butterlly spread is created solely with calls or solely with puts. A butterlly spread butterfly.'' created by selling a straddle and buying a strangle is called an t'iron
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50
55
-10
60
20
25
30
35 40 45 XYZ Stock Price
($)
huying a 35-40 bull spread and a 40-45 bear
-
50
55
60
($)
In general, consider the strike prices ffl, Kz, and Ks, where #1 Define so that #3 Kz -
= Ks
Examine Figure 3.15. It looks like a butterfly spread except that it is asymmezc: The ' . peak is closer to the high strlke than to the 1ow strike. This picttlre wgs created by buying two 35-stlike calls, selling ten 43-strike calls (witha premium of $1.525,using and buying eight 45-stHke calls. The position. is like a the jiuljtin iri Tble butterfly in that it earns a proft if the stock stays within a small range, and the loss is the same for high and low stoclt prices. However, the prolit diagrnm is now tilted to the right, rather tan being symmetric. Suppose you lnew that you wanted a position that looks like Figure 3.15. How would you lnow how malzy options to buy and sell to construct this position? In order to obtain this position, the s'trikes cleady have to be at 35)43, and 45. The total distance of the way from 35 to 45. In between 31 and 45 ls 10. Te num ber 43is 8t% (= 43-3/ jo ) fact, we can wlite 43 as '''
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Asymmetric Butterdy Spreads
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-
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Kz
=
ffl +
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0.2, as in the above example. Forexample, if #1 = 35, Kz = 43, and Ks = 45, then for evel'y Kz call we write, we buy #1 In order to constrtlct an asymmetric butter, K5 calls. calls arfd 1 You should venf'y' tat if you buy two 35-st1ike puts, sell ten 43-srrike puts, and buy eight 45-strike puts, you duplicate the profit diagram in Figure 3.15. =
-
3.5 EXAMPLE:
ANOTHER
EQUITY-LINKED
NOTE
lion by issuing In Jgly 2004, Marshall & Ilsley Corp. (ickersymbol h41) raised $400kml* 2007.6 maturity Instead of maling matllring in August effecdvely bonds payment a irl cash, the bonds pay the holder in shares of Marshall & llsley's own stock. A bond that, under some circumstances, pays the holder in stock instead of cash is called a
6ln'orsimplicity we will refer here to the Marlhall & llsley security as a Chapter 15, it was actually a bond plus a fonvard contract.
''bond-''
As we will dscuss in
'
% INsuRANcE,
84
COLLARS,
CHAPTER
STRATEGIES
AND OTHER
%.
85
stock. Both graphs ignore dividends and other distributions. Based solely on compadng the mattlrity payoffs, investing in the bond is inferior to investing in the common stock. on the stock (an However, distributions on tlae bond (6.5% armually) are greater t.11:.11 analysis it is impossible valuation withoutperforming a annui dividend of about 2917);so Accounting indifferent. other, the or be to say Whetherinvestors should prefer one or bond to lead the upward and payoff lines shif't would the for the distributions would bond the outyerforming with stock prices, stock the at 1ow outperfonn 0.6699 shares of
Marshall & Ilsley note always settles in stock; hence it is called copvertible bond. bond. We will discuss this particular bond more in Chapter mandatorily convertible a interestinM but bond is 16the at this point because the payoff strucmre resemblej a rl'he
CO
SUMMARY
.
The bond pays an annual 6.5% coupon and at maturity makes paymets in shares, the numberof shares dependentupon the Erm's stockprice. 'I'he specifc terms pf the with mamrity payment are in Table 3.7. To interpret this payoff, note that when the Marshall & Ilsley stock price at maturity is between $37.32and $46.28,the payoff is a varying number of shares, selected so the bond is worth $25(e.g.,0.5402 x $46.28= $25). Figure 3.16 graphs the mamrity payoff of the bond as a function of the N.!Istock in price three years, against the payoff of owning 0.6699 shares of Marshall & llsley
at high pfices. The Marshall & llsley share price was $37.32on the day the bond was issued. The bond was designed so tat, at issue, it would sell for $25, which is the same price as 0.6699 shares. The bond was also designed so that bondholders would forgo 24% of the appreciation on the stock above $37.32. We have $37.32 x 1.24 $4,6.28,and 25/46.28 0.5402. Tlzis accounts forthe flatrangein thepayoffand the numberof shares exchanged above a share plice of $46.28. Finally, in order for the bond to underperform the stock above $37.32,yet to sell at the share price, it is necessary to compensate bondholders with additional payments. The 6.5% coupon accomplishes this. How would we pzice te bond? The p'aph in Figure 3.16 should remind you of graph is $y (1) owning 0.6699 shares a written collar. One way to construct the same with a strike price of $37.32,and calls selling 0.6699 dividends), (2) of stock (ignoring algebraic expression for the price strike of $46.28. (3) buying 0.5402 calls with a payoff is .
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1/1.2i high Note that the bond holder is implicitly selling a low strike call and buying divialso forgoes premium. bondholder option owed 'Fhe strike calls, and is therefore effectively wllich distribution, is the explain 6.5% dends on the stock. twf)factors the dividend plus the amortized option premium. By plicing these options and valuing the 6.5% distribution, we can arrive at a fair price for the bond. 'Fhis discussion leaves unanswered the question of why Marshall & Ilsley would issue such a bond. We have also not discussed a1l the details of the bond's strutttlre. We return to these issues in Chapter 15.
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Puts are insurance against aplice decline and calls are insurance agaibst a price increase. Combining a long or short position in the asset with an offsetting position in options (for example, a long position in tlae asset is coupled either with a purchased put or written call) leads to the various possible positions and their equivalents in Table 3.8.
k
86
INSURANCE,
AND OTH ER ARATEGI
COLLARS,
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(floor)
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=
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-
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Collar
Stock Price
.
Stock Price
Profit
Profit
Buying a call and selling a put wit the same strike price and time to eypiration creates an obligation to buy the asset at expiration by paying the strike price. This is a synthe zic forwarct. A synthetic forward must have the same cost in present val terms as a trtle forward. This observation leads to equatin (3.1): -
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(3.1)
Skaddle
Strarfgle
stockPrice
jtock Price juoyky
Profit
explains the difference in call and put premiThis relationsllip, called put-call rf7p#@, othemise options. lt is for identical one of the most important relationships in ums
Butterfly
Rat'io Spread
Stock Phce
stockPrice
derivatives.
There are numerous sategies that permit speculating on the direction of the stock the size of stock plice moves (volatility). Some of tese positions are summarized on or graphically in Figure 3.17. We also categorize in Table 3.9 various strategies according to whether they reqect bullish or beatish views on the stock plice direction or volatility.? Netscape PEPS were equivalent to a bond coupled with an option spread, illustrating that the tools in this chapter have applicability eyond speculative investing.
.'
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FURTHER
READING
In Chapter 4 we will see how flrms can use these stategies to manage zisk. We will further explore put-call parity in Chapter 9, in which we also will use bull, bear, and butterlly spreads to say more about what it means for an option to be fairly priced. Put-call parity was flrst demonsated in Stoll (1969).Merton (1973a)corrected the original analysis for the case of Amelican options, for which, because of early exercise, parity need not hold. Ronn and Ronn (1989)provide a detailed examinadon of price bounds and returns on box spreads. There are numetous practitioner books on option trading sategies. A classic practitioner reference is McMillan (1992).
7erable3.9 was suggested by David Shimko.
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No price view
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Price will increase
Buy calls
PROBLEMS
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3.1. Suppose that you buy the S&Rindex for $1000,buy a looo-strike put, and borrow $980.39. Pelbrm a payoffand prost calculation mirnicking Table 3.1. Graph the resulting payoff and prost diagrams for the combined position.
.
88
%.IlqsuRAxcE,
COLLARS,
AN D OTH ER ARATEGI
PROBLEMS
Es
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following 6-??7o?7r/7J5?a$?'J?-#>?-cc /c
assume
is $1020,t7?/#
6month
the cf/c/ve
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rate is 2%, the vS&k S&R options wr 6 ??7t???r-
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3.12. Suppose you invest in the S&R index for $1000,buy a 950-stI4ke llW-strike call. Draw a prost diagram for zero-cost collar?
to expiration: Strike $950 1000 1020 1050 1107
Call
Put
$120.405
$51.777
93.809 84.470
74.201
71.802 51.873
101.214 137.167
a. loso-strike S&R straddle. b. Written 950-strike S&R straddle.
84.470
. ..
.
.
.
S&R straddle.
3.14. Suppose you buy a 950-strike S&R call, sell a looo-strile :-rike S&R put, and buy a looo-strike S&R put. a. Venfy' that there is no S&R price risk in this b. 'What is the irtitial ost of the position?
. '
mont
-
diagram by borrowing
'
s.
c. Consider buying n 950-stlike calls and selling ??7 1050-sike calls so that the premium of the position is zero. Considering your analysis in (a)and (b), what can you say about /1/??1? What exact ratio gives you a zero premium?
3.7. Vel'ify that you earn the same proft and payoff by (a) shorting the S&R index for $1000and (b)selling a loso-stn'ke S&R call, buying a 1050-stlike put, and borrowing $1029.41.
3.16. J.nthe previous problem we saw that a ratio spread can have zero irliti premium. Can a bull spread or bear spread have zero irlitial premium? A butterlly spread? Why Or Why not?
3.8. Suppose the premium on a 6-month S&R call is $109.20and the premium on a put wit the same strike price is $60.18.What is the jtrile price? .
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3.9. Construct payoff and profit diagrams for the purchase of a 950-strike S&R call and sale of a looo-strike S&R call. Verify that you obtain exactly the samrm/r diagram for the purchase of a 950-strike S&R put and sale of a looo-strike S&R put. What is the difference in the payoff diagrams for the call and put spreads?
l
u
a. Buy 950-strike call, sell two 1050-strike calls. b. Buy two 950-stn'ke ealls, sell three loso-sfn'ke calls.
3.6. Verify that you enrn the same prolit and paypff by (a) buying the S&R index for $1000 and (b) buying a 950-strike S&R call, selling a 950-stlike S&R put, and lending $931.37.
3.10. Constnzct payoff and proft diagrams for the purchase of a 1050-stlike S&R call and sale of a 950-strike S&R call. Vel'ify that you obtain exactly the same proht diagram for the purchase of a loso-str1'ke S&R put aad sale of a 950-strike S&R put. What is the difference in the initial cost of these positions?
'ansaction.
3.15. Compute profit diagrams for the following ratio spreadj:
$1029.41and buying a 1050-strike put.
Why is there a difference?
S&R call, sell a 950-
c. What is the value of 4heposition after 6 months? d. Verify that the implicit interest rate in these cash :ows is 2% over 6
3.5. Suppose you s ort theS&R index for$1000 andbuy a loso-sfn'ke call. Construct diagrams forthis position. Verify thatyou ob,tain the same payoff payoffandproqt andprofit
.
c. Simultaneous purchase of a 1050-strike straddle and sale of a 950-strike
.
'
put, and sell a How position. close is this to a this
3.13. Draw profit diagrams for the following positions:
3.3. Suppose you buy the S&R index for $1000and buy a 950-strike put. Construct 'payoff and prost diagrals for this position. Verify that you obtain the same payoff nd prost diagram by inves ting $931 37 in zero-coupon bonds and buyig a 950-strike call. 3.4. Suppose you short the S&R index for $1000and buy a 950-strike call. Constnlct payoffand prostdiagrams for this position. Vel'ify tat you obtain the sam payoff and profit diagram by borrowing $931.37and buying a 950-strike put. .
89
3.11. Suppose you invest in the S&R index for $1000,buy a 950-sfrike put, and sell a loso-stn'ke call. Draw a profit diagram for this position. What is te net option premium? If you wanted to construct a zero-cost cpllar keeping the put stn'ke equal to $950,in what direction would you hay to change th call strike?
3.2. Suppose that you short the S&R index for $1000and sell a looo-strike put. Conseuct a table mimicking Table 3.1 which summarizes the payoffand profit of this position. Verify that your table matches Figure 3.5. For
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3.17. Construct an asymmetric butterlly using the 950-, 1020-, and loso-stn'ke options. How many of each option do you hold? Draw a protit diagram for the position. .(
3.18. Verify tat th.e butterfly spread in Figure 3.14 can be duplicated by following eansactions (usethe option prices in Table 3.4): a. Buy 35 call, sell t'wo 40 calls, buy 45 call. b. Buy 35 put, sell two 40 puts, buy 45 put. c. Buy stock, buy 35 put, sell two 40 calls, buy 45 call.
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INSURANCE,
COLLARS,
AN D OTH ER STRATEGI ES
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One strategy investors are applying to the XYZ options is using thetic stock.'' A synthetic stock is created when an investor simultaneously purchases a call option and sells a put option on the same stock. The end result is that the synthetic stock has the same value, in terms of capital gain potential, as the underlying stock itself. Provided the premiums on the options are the same, they cancel each other out so the ansaction fees are a wash.
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Suppose, to be concrete, that the premium on the call you buy is the same as the premium on the put you sell, and both have the same strikes and times to '
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is $350/oz. If Golddiggers produces no gold, the firm loses its lixed cost, hlls hxed cost of $330/oz. and variable cost of $50/oz., produces gold, the 111111 If $330/oz. Golddiggers lt is better to lose only $30, so Golddiggers will produce and so loses $350- ($330+$50) = variable cost o $50, even when they have negative net income. If te gold price tvere to fall below the l
the gold price suppose
-$30/oz.
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Golddiggerp agrej to t yt a e sold today for $420/oz. jckldyroductonin 1 year. we will assume in atl exiilllj that th toixkard coneact its nanially. fonvaid is the si with phyjical noted earlier, the payoff : .
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seller'' is the sum of te other two lines, @Hedged proht: The line labeled adding them vertically at every gold price. lt is flat at $40/oz.,as we wouldexpect fromrfable 4.2. Aquick way to add the lines togetheris to notice that the cold forward'' rrah has a seller'' graph has a positive slope of 1. and the
seller'' shows o Unhedged profit: Since costis $380/oz.,the line labeled lligher For exnmple, prices. and prot at zeto prost at $380,a loss at lower prices, ground, gold the Golddiggers has a long in at $4209profit is $40/oz. Since it has position in gold. ,* Prtt on the short fomrard position: The sh ort g old fomrd'' tte rpresents the profit from going short the gold forward contact at a fonvard pticeof $420/oz. :
I
.
We profit from locking in the price if prices are lower than $420 and we lose if prices are higher.
.
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$380ii
.
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Hedging with a Fo-ard Contract t; olddiggis can lockin apriceforgoldin 1 yearby enteringinto ashortforFard contract, gtd for delivel'y in year. Suppose tht gpld t b delikred in ageeing tcy to sell its sell fofwfd al1 of and that
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Unhedged seller shot.tgold fonprd Hedged seller
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PERSPECTIVE
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ENT: TH E PRODUCER'S
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W ith
a forward sale of gold.
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ryii Jyyjtytiytyjyjt 2 yliy lktktty) tyqyqyrj(IC#y jt,o; o (yj oisy ry s zu u yqyyy zt-,o ys rosluoi'j l'bkt C/) Ci Hugeu yht mrt! @ trltyyttq tytiy gyiElyljjyyqjijryyygyjyypyjiyjylyylyyjgj#ygyglj
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$40 $40 $40 $40
kytiylqlyylyyqyryyyyryyEiyyryy
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A possible objec:on to hedging with a forward contract is that if go'ld prices do rise, Golddiggers will still rceive only $420/oz',there is no prospect for greater profit. Gold instlrance-iae., a put option-provides a way to have higher profts at high gold prices while still being protected against low prices. Suppose that the market price for a 420strike put is $8.77/0z.2This put provides jloor on the price. Since the ppt premium is paid 1 year plior to the option payoff, we must tale into account interest cost when we compute prost in 1 year. 'I'he future value of the premium
z'rhisuses the Black-scholes formula for the c. 5.5%. J 4.879% and t 1 tyearl. =
=
=
put price with inputs S
=
420, K
=
420, r
=
4.879%,
93
% INTRODUCTION
94
BASIC RISK MANAGEMENT:
RISK MANAGEMENT
To
is $8.77 x 1.05 $9.21. As with the forward contract, we assume financial settlement, although physical settlement would yield the same net income. Table 4.3 shows te result of buying tlzis put. H te price is less than $420, the tlie put. This is put exercised and Golddiggers sells gold for $420/oz.less the cost of Golddiggers sells gold at price is the $420, incom of If net tan greater gives $30.79. market price. the the put-performs better than shorting the forThe insurance strategy-bpying than $429.21. Otherwise the short forward gold of 1 is if price in the more ward year outperforms insurance. Figure 4.2 shows the unhedged position, profit from the put by itself, and the result of hedging with the put. :
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Insuring by Selling a Call
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With tlze sale of a call, Golddiggers receives a prernium, which reduces losses, but the written call limits possible frots. One can debte whether tlzis really constimtes insttrance, but our goal is to see how the sale of a call affects the potential prost and loss
$30.79 $30.79 $60.79 $110.79
$60.79 $10.79 -$9.21
-$9.21
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PERSPECTIVE
willbe in differentregions; thatis, how likely is it thatthe goldprice will exceed$429.21? The plice of the put option implicitly contains information about te likelihood that the gold prie will exceed $420,and by how much. The probability distribution of the gold price is a key factor determining the pricing of the put. We will see in later chapters how the distzibution affects the put price arld how to use information about te probability ' distribution to help us assess lisk. Figure 4.3 compares the profit from the two protectiv sategies we have exnmined: Selling a fonvard contract and buying a put. As you w. uld expect, neither strategy is cleady preferable', rather, there are trade-offs, with each contract outpedbrrning the other for some range of prices. The fact that no hedging sategy always outperforms the other will be true of all fairly priced strategies. J.npractice, considerations such as transaction costs and market views are likely to govern the choice of a sategy.
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PRODUCER'S
What this analysis does not address is the probability tat the gold price in 1 year
=
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THE
for Golddiggers. Suppose that instead of buying a put, Golddiggers sells a 420-strike call and receives an $8.77premium. Golddiggers in this case would be said to have sold a cap.
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Comparison of
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unhedged position, 420-strike put option, and unhedged position plus 420-strike put.
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+
senerhedged sellerhedged
M:.IA forward with put
k
150 100 50 0
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for Golddiggers hedged with a forward contract and hedged with a put option.
$30.79
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Comparison of payoffs
Unhedted seller purchased put Hedged seller
100 50
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450 400 350 Gold Price in l.Year
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($)
1.
550
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. : : : : : : : ::
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: :
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450 350 400 Gold Price in 1 Year ($)
500
%
96
INTRooucTlox
To
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Golddiggers hedging with sale of 420-strike call versus unhedged.
.
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200
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unlwdgedseller written call sellerhedged with
@ Sell some of the gain. Both of these sategies reduce the asymmetry between gains and losses, nd hence lower the cost of insurance. The first strategy, lowring the strike price, permits some additional loss while the second, selling some of tlie gain, puts a cap on the potential
written call
100
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250
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350 400 450 Gold Price in 1 Year
300
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550
The manager's view of the market and willingness to absorb risk will undoubtedly in:uence te choice among tese alternatives. Managers optimistic about the plice of gold will opt for low-strike-plice puts, whereas pessimistic manager; will more likely choose high-strike puts. While corporations per se may not be risk-averse, managers public's may b. Also, some managers may perceive losses to be costly in terms of the of perception boss's them. of f11-11: perception the or the This problem of choosing the appropriate strike price is not unique to corporate risk management. Safe drivers and more careful homeowners often reduce premiums
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.
.
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.
.
.
Figure 4.4 shows the payoff to this strategy. J.f we compute the acmal profit 1 year from today, we see tht if te golcl price in 1 year exceeds $420,Golddiggers will show Prots of
$420 + $9.21
-
$380
=
$49.21
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.
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'rhat is, Golddiggers sells gold for $420 (sincethe written call is exercised by the holder), receives the ftlttlre value of the premium, and has a cost of $380. If t.1+plice t gold is less than $420,Golddiggers will make Pgold+
'
.
Reducing the strike price lowers the amount of insurance; therefore the put option will have a lower premium. Figure 4.5 compares prost diagrams for Golddiggers's hedging using put options with strikes of $400 (premium $2.21), $420(premium 4oo-stlike, low-premium option yields = $8.77),and $440 (premium $21.54). lowest profit if needed the highest proft if insurance is not ttllepriee is high) and theoptipn yields the Oo-strike, high-premium is low). price The insuratlce is needed (the neded. profit is insurance and needed, the if highest lowest protit if insurance is not
$9.21
O -50
g ain
$49.21
50
(
97
* Reduce the insured amount by lowering the strik price of the put option.
jjryyy;jj grjjy;;s yjjty:,,j jgjj-
-
150
'
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PERSPECTIVE
There are at least two ways to reduce the cost of insurance;
lyyjj.ikE't).tf ;'E jjry;f E' ':' ?' C' ')' ('. ,j-jjjjj(,jjjjy'. (' ''j' 1* ryf F' ';' lf EE E' iEC;!fE fj';E E gjjyy,f yjjgyyjjj.f E;' PIi.EqiiqT'. j':'jjjjyj;jjrj.'. f' ,kt' ytjrj-pt.-j;rttf )' 17* Eq'E!'E.(; E j,jjyjjjgy'jjf .jjgyy' .)' jr,'.;jj. 74E* @' t';j'f')'rtf )' (' i''q' tif ((' (lqii:i;llqf jrslf $J' q@ E ()'( ('E' (;7'7:777: ' . CE(.E(((.(E(;'7* r' 1. qip'E!qlF: ((;EE..i E.... qE i;E(;'(@( i E ;!' iliE i'.'@ . !' (l ri sjyf j;'. ;')j' jj'yj yyj'-. y' r.;tf r)'t?!f j')r yyf ('. 4141,* y'. y'y' jtl'tf r' j'y','py-jf 'jq'l).rq . . .'. q' . 'l .. . ' . ... ..E). . ; E. ! ;;; . E ! (. (:E (E;(E. (.E; .E.l:EikE.y E: ( !; j(i...E.)E I E : j (E1. .j.jy. ..!!:j;:::;:;j,. . yy(y ( . .jlji!!!:k ; j . . y (y, ; yi;j; . kj)jy. , fli .. ! i,......a.i-...:..........r... y yjy y.. yj j ; ., lj y;j jyj . yq j j .y.y yy ,( y . .!rj: y . . tyjjy--. y q y g y l-'qi.tq. . y yyyj , jg .yyy . . ..g. . ... . ... . jjj (jj C
)' lll;tf)... tyf t)f ty)y)y'.. t'pjf (j
ENT: TH E PRODUCER'S
BASIC RISK MANAGEM
MANAGEMENT
-
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.
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.
.
On the downside, Golddiggers has exposure to gold but keeps the option premium. By writing the call, Golddiggers keeps the $8.77 call premium and 1 year latef makes $9.21 more than an unhedged gold seller. On the other hand, if the gold price exceeds $420,the call is exercised and te plice Golddiggers receives is thus capped at $420. Thus, for gold plices above $429.21,an unhedged sategy has a higher payoff than that of writing a 420-strike call. Also, for prices below $410.79,being f'ully hedged is preferable to having sold the call.
Adjusting the Amount of lnsurance Consider again Golddiggers's strategy of obtaining insurance against a price decline by purchasinga put option. A common objection to the purchase of insurance is that it is expensive.Insurance has a prernium because it eliminates the risk of a large loss, while allowinga prost if prices increase. The cost of insurance reqects this asymmetry.
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Auric Enterplises is a manufacturer of widgets, a product that uses gold as an input. We will suppose for' simplicity that th jrie of old is the only uncertainty Auric faces. In P articularwe assume that ,
* Aulic sells each widget for a fixed price of $800,a price lnown in advance. * The :xed cost per widget is $340. * The manufacture of each widget requires 1 oz. of gold as an input. @The nongold vadable cost per widget is zero. * The
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by purchasing auto and homeowner's insurance with larger deductibles. This re:ects their proprietary view of the likelihood that the insurance will be used. One important difference between gold insurance and property insurance, however. is that poor drivers would lile smaller deductibles for their auto insurance; this differential demand by the quality of the isured is called adverse selection and is re:ected in te premipms for different deductibles. A driver known to be good would face a lower premium ?orany deductible t11a11 a driver known to be bad. With gold, however, the price of the put is of who is doing te buying. a independent
THE
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BecauseAuric makes a greaterprot if the price of gold falls, Auric's gold position lisk-management is implicitly short. As with Golddiggers, we will examine vadous strategies for Auric. The pro forma net income calculation for Auric is shown in Table 4.4.
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lnsurance: Guaranteeing Price with a Call Optioh 3.You might thirlk that a dealer would charge a higher price for a purchased option if the dealer knew thatan option buyerhad superiorinformation about the market forgold. Howe'ver,in genetal the dealer an ordinary investor will quiclcly hedge tlle risk from the option and therefore has less concern t.11m1 about future movements in the price of gold. '
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Rather than lock in a price unconditionally, Auric might like to pay $420/oz.if the gold price is greater than $420/oz.but pay the market plice if it is less. Auric can accomplish this by buying a call option. As a f'uture buyer, Auric is nattlrally short; hene, a call is
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4.3
V AN AGE
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The Golddiggers andAuric examples illustrate how the two compnnies carl use fozwards, calls, and puts to reduce losses in case of an adverse gold price move, essentially insuring their f'uture cash flows. Why would a flrm use these seategies? to hedge, In Chapter 1 we listed four reasons that firms might use derivaves: pracdce. to speculate, to reduce transaction costs, arld to effect regulatot'y arbirage. In discussed already one of these considerations may be important. We have more t.11a11 that fact market views-for example, opinions about te future price of gold--can te affect the choice of a hedging strategy. Thus, the choice of a hedging strategy can have cite the accounting eeatmnt o a tmnsaction a speculative component. Managers often obviously considertion. transaction and important, costs are as In this section we discuss why ficms might hedge, ignoring speculation, transactions costs, and regulaon (bt we do consider taxes). It seems obvious that managers would want to reduce risk. However, irl a world wit fairly priced derivatives, no trans-
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call has : prmit!p pf $8.77/9z.(recallthat this is the sap gs ipsuran,sppps the wlth . premium te same stnke price). The future value of tli premiumij the on the put $8.77 x 1.05 = $9.21. lfAufic bys the insurance contract, net income on the hedged position wi1lbe as d shown in Table 4.5. If the price is less tha!z $420,the call is worthless at expirtion Auric buys gold at the market price. tfthe price is greater than $420,the call is exercijd and Auric buys gold for $420/oz.,less the cost of tlle call. This gives a profit of $30.49. If the price of gold in 1 yar is less t.11%$410.79,iristtdng the price by buying the locldng in a price of $420. At 1ow prices, the option permits perfonus better t11:11 call of lower gold prices. If the price of gold in 1 year is greater t.11% advantage us to take insuring the plice buying te call perfonns worse than locldng in a price of by $410.79, .
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Whe hedged ivith the fotwrd, Golddiggers will have a proft of $40 whatever price in 1 year. In effect, tlie value of the rduced pro/ts, should the gold price rise, the subsidizes the payment to Golddiggers should the gold price fall. If we use the term ttstat' to deriot apnrt iculagld price in 1 year, we can describe the hdging sategy as shifting dollars from more prostable states (whengold prices are high) to less profitable states (whengold prices are low). 'l'his shifting of dollars from lzigh gold price states to low gold price states will value fbr the s.rm have fthesnn valttes the dollar lntpr in a f/w goldprice state f/ltzn in goldprice state. 'Why might a 11n,nvalue a dollar differently in different states? a /yfg/7 '
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An Example Where Hedging Adds Value Consider a 51%1that produces one unit per year of a good costing $10. Immediately after Production, the 51-mreceives a paypent of either $11.20or $9, with 50% pyobability. has an 'rhus, the 51-mhas eiter a $1.20prost or a $1 loss. On a pre-tax basis, the 51-111
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However, on an after-tax basis, the lil'm could have an expected loss. For exnmple, suppose that when the ftrm reports aprost, 40% of the proft is tued, but when the flrm reports a loss, it pays no taxes and receives no tax reftind. Table 4.6 computes expected after-tax proft under these circumstances. 'I'he taxation of prosts converts an expected $0.10pre-tax gnin into an after-tax $0.14loss.4 Because of tues, the 151-mvalues a dollar of prost at $0.60 ($0.40goeg to the government), but values a dollar of loss at $1. In this simatio, it is desirable for the f11-mto trade pre-tax prots for pre-tax losses. Suppose that there is a forward market for the filnn's output, and that the fomard price is $10.10. If the firm sells forward, prost is computed as in Table 4.7. lnstead of
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an expcted loss of $0.14,we obtain a certain profit of $0.06. Hedging with a fomard transfers netincomefromaless-valued to amorelzighly valued stat, raisingteexpected, value of cash flows. Figure 4.8 depicts how the nondeductibility of losses affects after-tax cash Qows. First, observe that after-tax prost (lineACB) is a concave mction of the output price. (Aconcave ftmction is one shaped like the cross section of an upside-down bowl.) When prohts are concave, the expected value of prosts is increased by reducing uncertainty. We can see this in the graph. lfthe plice is certain to be $10.10,then prot will be given by point C. Howeyt, if plice ap be ither $9 or $11.20, xpected profit is at point D, onthe JineADB at the expected price of $10.10.Because ACB is concavq, pt/f?# D lies below #t?f/ll C, and /2#,j*W incrqases expected proAts-s Some of the hedging rationalej w discuss llinge on there being concav prosts, that value is increased by reducing unertainty. so q
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There are in fact a number of reasons why losses might be more hnrmflll t.1)a1$ profts are beneficial. We now disculs some of those reasons.6
s'rhis is an illustration of Jensen t iltequality whih is discussed in Appendix C, and which we will often in this book. . encounter 6ef'hefollowing are discussed in Smith and Stulz (1985)and Froot et 1. (1994). .'
4.15 asks zlproblem
you to compute proht when losses are deductible.
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MANAGEMENT
WHY
Do
FI RMs MANAG E RISK?
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105
The previous exnmple illuseating the efect of tues was oversimplied in assurning that losses are completely untaxed, btg it is typically the case that governments tax prosts but do not give f'ull credits for losses. Tax systems usually permit a lps; tp be offset against a prost from a different year. However, in present value terms, ttle loss that applied to prots, which still geerates a will have a lower effective tax rate t.11:.11
Increase debt capacity Because of the deductibility of interest expense for tax purposes, irms may find debt to be a tax-advantaged way to raise f'unds.? However, lenders, fearful of banknlptcy, may be unwilling to lend to Erms wit.h lisky cash flows. The amoupt that a firm can borrow is its debt capacity.
income to anotherl, Capital gains taxation (derivativescan be used capital gains incolle as With collarsl, gnd differential taxation defef taxation ross Cafl be tlsed to SIO ifome Colm WCS ldelivatives Som O11e CotllllzlFto aliothefl.
Managerial risk aversion While large, public firms are owned by'well-diversised investors, fil'm managers are typically not well-diversied. Salary, bonus, arld compensation options are a1l tied to the performance pf the fil'm. An individual who is unwilling to take a fair bet (i.., one with an expected payoff equal to the money at stake) is said to be risk-averse. Risk-averse persons are hnrmed by a dollar of loss more tan they are helped by a dllar of gain. 'Fhus, thy benest from reducing uncertainty. The effect is analogous to that shown in Figure 4.8. and have wetll tat is tied to te company, we might lf managers are lisk-averse will they reduce uncertainty. However, matters are not this simple: try tat to expect Managers are often compensated in ways tat encourage them to take more lisk. For example, options given to managers as compensation, which we discuss in Chapter 16, lislier. Thus, a are more valuable, other things equal, when the Erm's stock price is manager's risk aversion may be offset by compesation that is more valuable if the firrli is rislder.
Taxes
motive to hedge. There are other aspects of te tax code that can encourage firms to shif't income using derivatives; such uses may or may not appear to be hedging and may or may not be approved of by tax authorities. TM ruls that may entice firms to use derivatives include the separate taxation of capital and ordinary income (derivatives can be ujed to COIWCXOI1e fol'm
Of
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and distress costs An unuspally large loss can threaten te survival of obvious that a money-losing 51-1':1 fu'm. The is most may be unable to meet reason a hxed obligations, such as debt payments and wages. If a tin'nappears to be in distress, customers may beless willing to purcase its goods. Aouldyoubuy acarorcomputerboth of which come with long-term warranties-from a company that alipears lilely to its warranties'?) would unable and be honor of business ten to out go Act'ual orthreatenedbanknzptcy canbe costly', adollarof loss can costthe company lirms to enter derivatives contracts moretllan a dollar. As with taxes, this is a reason for reducing the probability of loss profit thereby ansfer income states, f'rom states to that Bankruptcy
Afirm that credibly reduces the risliness of its cash flows should be able to borrow more, since for any given level of debt, bankruptcy is less likely. Such a flrm is said to will have raised its debt capacity. To the extent debt has a tax advantage, such a 51711 also be more valuable.
or distress. bnnknlptcy
Firms make lisk-management decisions when they organize and design abusiness. For example, suppose you plan to sell widgets in Europe. You can construct a plant in the United States and exprt t Erope, or you can nstruct the plant in Europe, in which case costs of consuction, labor, interest rates, and other' inputs'will be denominated in the snme cun-ency as the widgets yot! sell. Exchange rate hedging, to take one example, would be unnecessaty Of course, if you uild in a foreign countly you will encounter the cojts of doing business abroad, including dealing with different tax ctis qnd regulatory regimes. Risk can also be aseted by such decisions as lasing versus buyinj equipment, which determines the extent to which costs are Exed. Firmj can choose Qexile producbe reconfgured tion technologies thatmaybe more expensive atte outjet, butFhihcan at low cost. Risk is also affected by the decision to enter a particular line of business in the lirst place. Firms mnking computer mice and keyboards, for example, have to consider the possibility of lawsuits for repetitive stress injulies. Nonfinantial
coslly external financing Even if a loss is not large enough to threaten the survival ofa finn, the fil'mmust pay for the loss, either by using cash reserves or by raising funds externally(for example, by borrowing or issuing new securities). '
Raising funds externally can be costly. There are explicit costs, such s bnnk and undelwriting fees. Thre can also be implicit costs. tf you borrow money, the lender may worry that you need to bon'ow because you are in declze, which increases the Probability that you will not repay the loan. The lender's thinking tlzis way raises the interest rate on the loan. same problem arises even more severely with equity issues. rfhe
At the same time, cash reserves are valuable because they redce
a
srm's need
to
raise f'undsexternally in the f'uttlre. So if the firm uses cash to pay for aloss, thereduction in the in cash increases the probability that the flrm will need costly external snancing future. is costly can even lead the 111-1-r1 The fact that external snancing to forgo investment projects it would have taken had cash been available to use for financing. Thus, however the 51-111 pays for the loss, a dollar of loss may actually cost the firm Hedging dollar. t11a11 can safeguard cash reserves and. reduce the probability of a more costly external snancing.
risk management
106
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The point is that risk management is not a simple matter of hedging or not hedging using:nancial derivatives, but rather a series of decisions tat start when the business is first conceived.
Reasons Not to Hedge There are also reasons why irms might elect not to hedge:
* Transacting in derivatives entails paying and the bid-ask spread. * The flrm must assess expertise.
costs and benests of a given strategy; this can require costly
* The fit'mmustmonitortransactions unauthorized trading. '
costs, such as commissions
'ansaction
andhave managerial coneols inplaceto prevent
* The lirm must be prepared for tax and accounting cmsequences tiO11S. ln Pal-tictllar, Yi S may Colr1 Plicate fekorting.
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Thus, while there are reasons to hedge, there are also costs. When thinking about lisk managecosts and benests, keep in mind that some of what srmsdo could be called derivatives. For example, supposeAuric but obviously involve enters into ment may not agreement with a supplier to buy gold at a fixed price. Will mangement tllink a z-yea1' this of as a derivative? In fact this is a derivative under urrent accounting standards (it is a swap, which we discuss in Chapter 8), but it is exempt from derivatives accountinps Finally, firms can face collateral requirements (the need to post exla cash with their counterparty) if their derivatives position loses money. The box on page 108 illustrates an attempt to manage risk that bacured. '
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Empirical Evidence on Hedging d derivatives use of We know surprisinjly little about the risk-management practic titnns'in real life. It is dicult io tell, from publicly available infofmation, the extent to which fil'ms use derivatives. Beginning in 2000, Statement of Financial Accounting Standards (SFAS) 133 required fil'ms to recognize derikatiVes as assets (jr liabilities on the balance sheet, to measure them at fair value, and to report changes in their market va1ue.9 This reporting does not necessarily reveal finn's hedging psition (forward contracts have zero value, for example). Prior to 2000, fifms had to report notional exposure', hence much existing evidence rlies on dat f'rom the 1990s. Research tries to addrss two questions: How much do stzms use derivatives and ad oter why? Financial firms--commercial barlks, investment banks, brokr-dalers,
Scurrent
delivatves accounting rules contain a purchases and sales'' exemption. Firms need not use derivatives accounting for fonvard coneacts with physical delivery, for quantities likely to be used or sold over a reasonable period in the normal course of business. lenonnal
9See Gmstineauet al. (2001)for a discussion of SFAS l33 and previous accounting rules.
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RISK?
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in derivatives frequently. ltisks are idenloable, and financial institutions-transact risk management. The more open question is the extent to which regulators encourage dedvatives. We can summarize research findings as follows: nfnancial til'msuse
* Roughly half of nonfnancial firms report usin dedvatives, with usage greater among large filnns (Bodnar et a1., 1998', Bartram et al., 2004). * Among firms that do use derivatives, less thn 25% of perceived risk is hedged, wi.thfin'ns likelier to hedge short-term lisks (Bodnar et a1., 1998). * Firms with more investment opporttlnities are likelier to hedge (Gdczy et al., 1997). * Firms that use derivatives have a higher market value (Allayanpis and Weston, 2001; Allaynnnis et a1., 2004., Bal-tram et a1., 2004) and more leverage (Grahnm and Rogers, 2002., Haushalter, 2000).10 Guay and Kothari (2003)verif'y many of these sndings but conclude that for most of minor economic signilicance. derivatives ln their sample of large tifns, firms, use is derivatives t.11%half Among derivatives slightly more report usage. users, the athrs median hedges only of that the about 3% estimate tirm to 6% exposure to interjt rates exchange and rates. Because data are hard to obtain, some studies have focused on particularindustries and even flrms. Tufano (1996),Petersen andThiagarajan (200), andBrown et al. (2003) have examined hedging behavior by gold-mining fmns. Using a uniquely detailed data set, Tufano found that most gold frms use some derivatives, with the median fil'min llis snmple (Nort.h American ftrms) selling folavard about 25% of 3-yearprodction. 'Fifteen' percent of ftrms used no delivatives. BroWn et al. found substantial variation oker time in the nmount hedging by gold finus. Firms tended to increase hedging as the plice rose, and managers reported that they adjusted hedges based on their views about gold prices. Thecurrency-hedging operations of a'U.s.-basedmanufacturingirm areexamined detail by Brown (2001),who finds tat foreign exchange hedging is an integral part in operations, but the company has no clear rationale for hedging. For example, of 51411 Brown reports one manager saying, W do not take speculative positions, but the extent we are hedged depends on our views.'' Faulkender (2005)sndsconsistent evidence for interest-rate hedging in the chemical industry. 'Fhese firms increase exposure to shortbut ocelations term interest rates as the yield curve becomes more upward-sloping,ll between cash Qows and interest rates do not affect behavior. The varied evidence suggests that some use of derivatives is common, especially large flrms, but the evidence is weak that economic t# eories explain hedgipg. at
loGraham and Smith (1999)5nd that after-tax profits are concave for a majority of srms,as in Figure 4.8. However, Graham and Rogers (2002)are unable to find a link between hedging and tax-induced concavity. 1lAn upward-sloping yield curve means that long-tenn bond yields are greater t.11m1 short-tenn bond less expensive., However, we will see in Chapters yields. Tltis appears to make short-term snancing costj, long-term and short-term 7 and 8 that if a company hedges :11 of its future short-term snancing linancing will cost the same.
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We have looked at simple hedging and insurance strategies for buyers and sellers. We now examine some additional strategies that permit tailoring the nmount and cost of insurance. For simplicity we will focus primarily on Golddiggers; however, in every case there are analogous seategies for Aulic. Table 4.8 lists premiums for three calls and puts on gold with 1 year to expiration examples use these values. and tlu'ee different strikes. rl'he
Selling the Gain: Collars
As discussed earlier, we can reduce the cost of insttrance by reducing potentif prot, i.e., by selling our right to profit from high gold plices. We can do tllis by selling a call. If the gold price is above the strike on the call, we are contractually obligated to sell at the strike. This caps our prohts, in exchange for anxirlitial premium payment.
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Compar ison Of unhedged profit for Golddggers versus .. zero-cost collar obtained by buying 400.78-strike put and selling 440.78-strike
-s
200
400.78-440.78
ollar
100
$60.78
stl
!
$20.78
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j 5440
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350 400 450 Gold Plice in 1 Year ($)
unhedgedseller sellerhedged M:.I
150
call.
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500
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5
250
300
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350 400 q 450 Gold Price il 1 Year
I
1
500
550
@)
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The 420 put costs Note tat the 420-440 colla.r still entails paying a of Thus, there premium only is a nd 440 call yields the $2.49. a net expenditure a $8.77, It is probably apparent, thokgh, that we can tinker with tlle strike plices and Of $6 2 pay a still lower net premium, including zero premium, if we wish. The ade-off is that te payoff on the collar becomes less atlractive as we lower the required premium.
.premium.
.
.
collar To construct a zero-cost collar, we could argue as follows: A400and put a uo-sttikecall are equally distant from the forward price of $420. 'This strike that the options should have approximately the snme premium. As suggests equivalence from the table of premiums for different stnke options, the 4oo-strike pui has we can see premium of wlle $2.21, te Oo-strike call has a premium of $2.49. The net premium a receive would buying this collar is thus $0.28.We can constnlct atrue zero-cost from we collar by slightly changing the strike prices, mnking the put more expensive (raisingthe strike) and the call less expensive (alsoraising the strike). With strikes of $400.78for the put and $440.78for the call, we obtain a premium of $2.355for both options. In reality this zero-cost collar of width 40 would be sold at lower strike pricej t.11a11 $400.78 and $440.78..'1'he reason is that there is a bid-ask spread: Dealers are willig to buy a given option at a low price and sell it at a high price. 'I'he ptlrchased put will be bought at the dealer's offer price and the call will be sold at the bid. The dealer can earn this spread in either of two ways: Selling the 400.78-440.78 collar and charging an explicit ansaction fee, or lowering the stlike prices appropriately and charging a zero transaction fee. Either Fay, the dealer enrns the fee. One of the triclk'y aspects of the more complicated derivatives is that it is relatively
A zero-tost
easy for dealers to epbed fees tlpt are invisible to the buyer. Of course a buyer can mitigate tllis problein y alpays sling quotes f'rol dlfferent dealers. t.tl b Wecan eynmlne ff ' epayo s y cons idering seprately the three interesting regions gold prices: of .
Prit
.
g
'
ofgold
<
..
..
'
.
o
$433.78: In this region, Golddiggers can
exercising the put option.
Price ofgold lae/wdepl$400.78 gold at tlm market price.
tw?#
.
jell gold for
$400.78by
$440.78: ln this region, Golddiggers cn sell
i
Price ofgold % $440.78.. In this region, Golddiggers sells gold at $*0.78. It has solda catl, so the owner of e call Svill exercil. This forces Golddiggers to sell gbldto t.1* calt owner for the strike price of $440.78. Figure 4.1 1 grap hs the zero-cost collar against the unhedge position. Notice that between $400.78 and $440.78, the zero-cost collar graph is coincident with the urlhedged prost. Above the 440.78-st11.1(:ethe collarprovldes prost of $60.78,and below the 400.78-stdke, the collar provides prost of $20.78.
The forward contrau Because the put and call with strike as a zero-cost collar prices of $420 have the same premiums, we could also construct a zero-cost collar by buying the $420-st*e put and selling the $420-st1411e call. lf we do this, here is what happens:
112
sEtEc-rlxc
Price ofgold < price of $420.
$420..Golddiggers will exercise the put option, selling gold at the
Price ofgold > $420.. Golddiggers has sold a 420-stIike call will exercise, obligating Golddiggers to sell gold for
call. The owner of that
$420.
ln either case, Golddiggers sells gold at $420. Thus, the collar'' is exactly like a fonvard conact. By buying the put and selling the call at the same strike price, Golddiggers has synthetically created a short position in a forward conact. Since a short forward and 420-420 collar have the same payoff, they must cost the same. This is 1W7ythe #?z??,I/??l- ol the 420-strike options t'lrc the Tllis exnmple is really just -anillustration of equation (3.1). ::420.-420
-rHE HEocE
Rvlo
%:
te call strike ptice below $440.78,in wllich case we would obtain a higher premitlm per call. To offset the higher premium, we could buy less than one call. The ade-off is that we cap the gold price on part of production at a lower level, but F maintain solpe participation at any plice above the stnke. lternatively we could raise the cap level (thesike price on the calll and sell would increase participation in gold price increases up to the than one call. more also effect of generating a riet short position in gold if prices rose the have level, but cap above the cap. rf'his
-twnn
synthetic i orwar d s a t p rices other 'than :420 We can easily extend this example to understand the relationship between option premiums at otlzer strike prices. In the previous example, Golddiggers created a synthetic forward sale at $420. You might think that you could benefit by creating a synthetic fonvard contract at a higher price such as $440. Other things being equal, you would rather sell at $440than $420. To accomplish this you buy the 440 put and sell the zl40 call. However, tere is a catch: 'Ihe zgo-strile put is in-the-money and the Mo-strile call is out-of-the-money. Since we would be buying the expensive option and selling the inexpensive option, we have to pay a premium. How much is it worth to Golddiggers to be able to'lock in a selling price of $440 instead of $42 ? Obviously,it is worjh $20 1 ye ar from toda# or $20+ (105) $19 05 in present value tenus. Since locking in a $410 price is f'ree, it should threfore be the case that we pay $19.05in net premium in order to lock in a $440price. In fct, looling at the plices of the Mo-strike put and call in Table 4.8, we hav premiums of $21.54for te put and $2.49for the call. This gives us ::2
,
Net premium
=
$21.54 $2.49 -
.
.
$19.05
=
Similarly, suppose Golddiggers explored the possibility of locking in a $400 price for gold in 1 year. Obviously, Golddiggers would require compensation to accept a lower price. In fact, they would heed to be paid $19.05-+e present value of $20-today. Again we compute the option premiums >nd we see that th 400-FtIike call sells for $21.26while the 400-stlike put sells for $2.21.Again we have Net premium
=
$2.21 $21.26 -
-$19.05
=
Golddiggers in this case receives the net premium for accepting a lower price.
Other Collar Strategies
Paylater Strategies A disadvantage to buying a put option is tat Golddiggers pays the premium even when the goldplice is high andinsurance was, afterthefact, unnecessary. One strategy to avoid this problem is a paylater sategy, where the premium is paid only when the insurance is needed. Wllile it is possible to constnlct exotic options in which the premium is paid only at expirain and only if the option is in ttze money, the sategy we discuss here is a ratio spread using ordinacy put options. The goal is to find a sategy where if the gold plice is high, tere is no net option prernium. lfthe gold plice is low, there is insurance, but the effecuve prernium is greater than with an ordinary inslzrance sategy. Ifthereis nopremiumwhen thegoldpriceis Mgh, wemusthaveno initialpremium. This means that we must sell at least one option. Consider the following sategy for Golddiggers: Sell a434.6-strikeput andbuy two4zo-strikeputs. Using ourassumptionsk the premium on the 434.6-strike put is $17.55,wllile the premium on the 420-strike put is $8.77. Thus, the net option premium from this strategy is $17.55z- (2 x $8.775)= 0. Figure 4.12 depicts the result of Golddiggers's hedging with a payter sategy. Whentheprice of goldis greaterthan$434.60, neitherputis exercised, and Golddiggers's profit is the same as if it were unhedged. When the price of gold is between $420 and $434.60, because of the written $434.60put, the firl'n loses $2 of profit fpr every $1 decline in the price of gold. Below $420 the purchased 420-.tri1e puts are exercised, and prost becomes constant. The nt result is an insurance policy that is not paid for unless it is needed. Also depicted in Figure 4.12 is the familiar rsult from a conventibnal insurance strategy of hedging by purchasing a single 420-sfrike pt. When the gold price is high, the pay 1a ter strategy wit.h a zero preinium outpebrms the sigte put. When the gold price is low, the paylater sategy does worse because it offets less insurance. Thus, the premium is paid later, if insurance is needed.
4.5 SELECTING
Collar-type strategies are quite :exible. We lave focused on the ease where the ftrm buys one put and sells one call. However, it is also possible to deal with fractional options. For exnmple, consider te 400.78-440.78 collar above. We could buy one put to obtain full downside protection, and we could vary the stn' price of the call by selling fractional calls at strike prices other than $440.78.For example, we could lower 'ke
THE HEDGE
RATIO
ln the Golddiggers and Auric exmuples, we pedbrmed all calculations in terms of one unit of gold, and made two important assumptions. First, we assumed perfectcorrelation between the plice of gold and the price of what each compahy wants to hedge. Second, we assumed that the companies lnew for certain the quantity of gold they would sell and buy. As a result of these assumptions, we effectively assumed that the hedge ratio is one,
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=
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350 400 450 Gold Price in 1 Year
Nlwo'w2 + (H
2 chcdged
I
I
500
550
(5)
'
where te hedge ratio is defined as te ratio of the fonvard position to the underlying E
.
j ( g
assumption-may be valid. We ftrst exnmine the effect of widget ' ln practice, neither price uncertainty on hedging, and discuss cross-hedging. We then exnmine qttantity I/??c'?-lc?lfy
NwPw
=
in te price of widgets
-
(4.1)
N.g Pgold
Ng Pgold+ StPgold
-
-
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'
using an agriculmral exnmple.
gLH N g )20.2 g +
-
-
N
'g
(4.:)
lywpgwg.:
where o'm is the standard deviation of the widget price, o-g is the standard deviaion of the goldprice, and p is the price correlation between widgets and gold. Note that pavo'g is the covariance between widget and gold prices. rfhe variance-minimizing hedge position, S*, is13 H.
aSSC (
Ng
=
-
Nw
po'u)
c'p Equation (4.3)has a saighforward interpretation. 'Fhe flrst tet'm hedges costs-i.e., we second enter into a long position on the nmount of gold we use in production, Ng. term hedges revenue, prov iding the quantity of gold forWards we use to hedge widgets. iS 9om regressing the Widget ljlice ii tli gold p1ice.14 The telql pGwxg the Coecient times the number of widgets, #w, ives us the mber of This regression coecient result ij that the hedge gold conacts to use in hedging widget price risk. tuantity, 'l'he
'l'he
COSS-
gmg
In th Aul'i exampl we assumed that widget prices are sxed.However, since gold is used to produce widgets, widget prices might vary with gold prices. If widget and gold prices vat'y one-for-one, Auric's prots would be independent of the price of gold and Auric would haye tno need to hedge. 12 . . More realisttclly, the price of widgets ould chpnge wit.h the price of gold, but not one-for-one; othr factorq could affect widget plices as well. In this case, Auric might 5nd it helpf'ul to use gold derivaives to hedge the price of the widgts it sells as well as the price of the gold it buys. Using gold to hedge widgets would be an exnmple of crosshedging: the use of a derivative on one asset to hedge anotlzer asset. Cross-hedging adses in many different contexts.
lztrne-for-one''in tlzis context means tllat if the price of by$1 times the amount of gold used to make the widget.
115
The variance of te retllrn on the hedged position is given by
j.
:
%
Suppose that we go long H gold futures contracts, each coveling 1 oz. of gold. I.f F is the gold fonvard plice, the retttrn on the hedged posidon is
j
j
o
where Pw is the widget price.
.
()
RATI
and gold. Conceptually, we can think of hedging widgets and gold separately, and then combining those separate hedges into one net hedget To generalize the Auric example, suppose that buying Ng ounces of gold enables produce Nw widgets. Profit per ounce of gold, without hedging, is to us
.
.... .. . ... .
The hedging problem for Aul'ic is to hedge the dterence
E: 'i'(:.E: EE /73);:E . .
H EDGE
N G TH E
gold rises by $1, the.plice of a widget
lises
l3rrhis can be derived by differentiating equation (4.2)with respect to H*. 14ne term pcrmjo'g measures the comovement of gold and widgetplices, which is typically measured using a linear regression. A common approach' is to regress prce changes on price changes: P..t
-
#u,,J-l
=
+
ppg,l
-
#,g,,-1) +
lf/
'
where the subscript t denotes tlle value at time t. Other specilications, including ttle use of jercentage changes, or regressing levels on levels, are possible. The conect rejression to use dpendj (jn context. In general, regressions using clianges are more likely to give a correct hedge ceflicient since te goal of the hedge is (o haye changes in the price of the asset matched by changes in futures price. In Chapters 5 and 6, we will present examples of hedging stocks and jet fuel, and the appropriate regressions will be returns on returns (stocks)arld chang s on changes et fuell. Regressions of level on level are problematic in mafly contexa. Forexample, in the case of stoclcs, msset pricing models tell us that stock retttnts are related, but we would not expect a stable relauonship between twoprices. The appropriate regression is retul'ns on returns. A comprehensive discussion is Siegel and Siegel (1990, pp. 114-135).
116
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MANAGEMENT
sEt-Ec-rllqc
H*, is denominated in terms of units of gold. The net use of gold futtlres, H*, is the ditference in the number of coneacts needed to hedge gold inputs and widgt outputs. Note that if gold and widgets are uncorrelated-i.e., p = o-the hedge ratio is purchased), while fp > 0, the hedge of gold long fomard per ounce ounce one one (go ratio is less than one because the widget price itself provides an implicit long position in gold that is a partial hedge. When we hedge with H* f'uttlres, ctjgetj is obtained by substimting S* into
equation (4.2):
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Te uncertainty remaining in the hedged position is due to basis risk, which is lisk due to the hedging instzument (gold)and the hedged price (widgets)not moving as predicted. The variance o f pro fits ultimately depends upon te ability to hedge the price of widgets, which, since we are using gold to hedge, depends on the correlation, p, between widgets and gold. The larger that p is, the less is basis lisk. tls section we have shown that the ability to cross-hedge depends upon the 1.11 coaelation between the hedging instrument and the asset being hedged, and that we can determine the hedging amount as a regression coefcient. We will see in Section 5.4 that the same analysis obtains when we use stock index futtlres conacts to cross-hedge a stock portfolio.
QuantityUncertainty
Thequantity afirmprociuces andsells mayvary withtheplices of inputs orottputs. When t'obvious'' hedge ratio (for example, by hedging te expected this happens, usina the quantity of an iput) can increase rather tan decrease lisk. ln this sectin we xamine quantity uncertinty. Agricultural producers commonly face quantity uncertainty because crop size is affected by factors such as weater and disease. Moreover, we expect there to be a correlation between quantity and price, because good weather gives rise to bountiful harvests. What q'uantity of fonvard onacts should a corn producer enter into to minimize the variability of revenue? We will look at three examples of different relationships between price and quantity: The benchmark case where quantity is certain, an example where quandty and price are neyatively correlated, and an example where quantity and price are positively correlated.lD
t
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and quantity for an agricultural producer. Each row is equally Iikely. In scenario A, there is no quantity uncertainty. In scenario B, quantity is negatively c rrelated with prce, and irl scenario C, quantity is positively correlated with price.
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addiion, for each ppssible price of corn there are two equally likely quantities, for a totat o f four possible price-quantity pairs. Table 4.9 illujtrates te three scenados. Note that in scenario B, average quantity is low when price is high (negativecorrelation), correlation). wherea i.nscenario C, average quantity is high whn price is high (positive
First considr ScenarioAwhere quantity iq certain: The producer always produces busels. t-etv and denotetheprice andquantityin 1 year. Revenueis SQ. Without 1m hedging, revenue will be either $3m(if the corn price is $3) or $2m (if the corn price is ..
$2). F
On the other hand, if the producer sells folavard lm bushels at the forward price 2.50 revenue is
=
Revenue
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ls-rhere are futures contracts intended to mitigate the problem of quantity uncertainty in an agriculttlral context. Corn yield futures, for example, traded at tlze Chicago Board of Trade, permit farmers to hedge variations in reoional production quantity, and provide alz aoericulmral eKample of a conact. We discuss quantos further in Chapter 22.
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In all the examples we suppose that the col'n forward plice is $2.50*u and that there is a 50% probabili. ty tat in one year the corn price will be $2/b11or $3/bu. ln
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118
Ilq'rstolluc'rlolq
CHAPTER
MANAGEMENT
Rlslt
To
ln general, if the producer enters into fonvard conacts Hedged revenue
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=-
The variancecminimizing
calculation, we have psc,s have
pso,so'so Gs
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This is the same as the second term in equation (4.3). The formula for the varianceminimizing hedge ratio in equation (4.7)is the negative of the coecient from a regression'of unhedged revenue on price. We can therefore determine the variance-minimizing B and C) in hedge ratios for the negative- and positive-conelation scenarios (scenarios Table 4.9 either by using equation (4.7)directly, or els we can nm a regression of revenue on p S ce. First, consider what happens in Scenario B if we hedge by shordng the expected quantity of production. As a benchmark, column 3 of Table 4.11 shows that unhedged revenue has vatiability of $0.654m.Fromrfable 4.9, expected production in the negative
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hedge can be obtained using equation (4.7). By direct o's = $0.5, and o'sc = $0.654m.16Thus, we
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Column (7) of Table 4.11 shows that variability is reducd to $0.652mwhen hedging optimal hedge quantity is closer to no hedging t.1)a11 hedging. tis amount. to R111 In fact, we gain little by hedging optimally, but we increas: the standard deviation of revenue by 25% if we adopt the plausible but incorrect hedging strategy of shorting 975,040 bushels. Problem 4.21 asks you to venfy' that you obtain the same nswer by On ning a regression of revenue on price. Yo right guess by now that when correladon is positive (Scenario C), the optimal hedge quantity exceeds expected quantity. The fact that quantity goes up when price goes up inakes revenue tat much more varibl than when pric alciil: vaties, and a corrspondigly larger hedge position is reqired. Problem 4.23 askj #o to compute 2 millio bushelseven . j hedge in scenario C. The answer j s to jjmrt mmost th ptlm though production is never tat large.
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119
Ifwe short tltis quantity of corn, column 5 of Table 4.11 shows that there is sti.llvariability irl hedged revenue. Perhaps more surprising, the variability of total revenue acmally itcreases. The reason is that price decreases when quantity increases, so nature already provides a degree of hedging: The increase in qaan:ty pnrfially offsets the decrease in pric. Hedging by shorting the f'UIIexpected quantity leaves us overhedged, with a commensurate increase in variability.
tr2s(s),
the variability of hedged revenue,
%.
cotnelation scenario, B, is
on H units, hedged rev-
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enue, RH),
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SUMMARY
A producer sellig a rislty commodity, such as gold, has ll inherent long position in the commodity Assuming costs are :xed, the firm's profit increases when the price of the commodity increases. Such a f11-mcan hedge profit with a variety of sateocries, including slling forward, buying puts, and buying collars. A 51-r11tat faces price l'isk on inputs hs an inherent short position in te commodity, with prost that decreases when the price of the input increases. Hedocringstrateocriesfor such a 51-minclude buying forward, buying calls, and selling collars. Al1 of the sategies hwolving options can be customized by chanocing the option strike prices. Sateoeies such as a paylater can
l6Because Table 4.11 presents the complete population of outcomes, which are equally likely, it is appropriate to use the populauon estimate of the standard devition. In Excel, this is STDEVP as opposed to STDEV. ne calculation for o'sc is obtained as STDEVPO,1.8, 3, 1.6) 0.6538. =
120
%.I?q-rftoouc'rlolq
To
RlsK
PROB LEMS
MANAGEMENT
provide insurance with no initial premium, but on which the company has greater losses should the insurance be needed. Hedging can be optimal for a company when an extra dollar of income received in times of high prosts is worth less t1::,1:an exa dollar of income received in times of low prosts. Prosts for such a fil'm are concave, irl which case hedging can increase expected cash iow. Concave prosts can arise from taxes, bankrptcy costs, costly external nance, preservation of debt capacity, and managerial risk aversion. Such a 51-mcan increase expected cash flow by hedging. Nevertheless, firms may elect not to hedge for reasons including transaction costs of dealing irl derivatives, the requirement for expertise, the need to monitor and con'ol the hedging process, and complications iom tax and accounting considerations. FURTHER
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l-year continuously compounded 'Fhe l-year forward price of copper is $1/1b. option prices for interest rate is 6%. One-year copper are shown in the table below.l? 'rhe
Strike 0.9500 0.9750 1.0000 1.0250 1.0340 1.0500 In your answers, at a $1.00, $1.10,and $1.20.
,
.
.
minimum
Call
Put
$0.0649
$0.0178 0.0285 0.0376 0.0509 0.0563 0.0665
0.0500 0.0376 0.0274 0.0243
0.0194
consider copper prices in 1 year of
$0.80,$0.90,
4.1. lf XYZ does nothing to manage copper pri risk, what is its prot 1 year from now, per pound of copper? If on the other hand XYZ sells forward its expected copperproduction, what is its estimated prot 1 yearfrom now? Cnstnlct graphs illusating both unhedged ynd hedged profit.
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@ Telco installs telecommunications equipment and uses copper wire from Wirco as an input. For planning purposes, Telco assigns a sxedrevenue of $6.20for each unit of wire it uses.
READING
In this and earlier' chapters we have examined uses of forwards and options, tnking for g'rantedihe pricing of tlpse contracts. Two big unanswered questiops afei How are prices determlned? How does the market for them work? those ' In chapters 5 tlaiough 8, we will explore forwartt and ftures coniacts cticussing Capters 10 througti 13 we will Pricingas well as how market-makers function. 1.11 answer the same questions for optlons. Chapter 14 will discuss how exotic options can be used in risk-management sategies in place of the ordinary puis nd clls discussed in this ch>pter. Wlkarton and irms to assess their hedging. survey noafinanlal cmc regularlyBodnar A recent survey is summ arized in et al (1998) Bnrtram et al (2004)exanune hedgsng bhavior in an internadonal sample of over 7000 firms. Tufano (196, 1998), Petersen andrfhiagarajan (2000),andBrown ei a1. (2003)have smdied hedgingpractices in thegold-mlnlngindustly Otherpapers exnmlnlghedogincludeGczyeti. (1997), Mlayannis and Weston (2001),Maynnnis et a1. (2003), and Allaynnnis et a1. (2004). Guay and Kothari (2003) attempt to quantifyv derivatives usage using information in 51-mnnnual reports from 1997. Brown (2001)provides an interesting and dtailed by one (anonymous)'Iirm and Faulkender (2001) description of the hedging exnmines imerest rate hedging in the chemical industry. Gastineau et a1. (2001)discuss Statement of Financial Accouting Standards 133, wlzichcurrently governs accounting for derivatives. Finally, Reming (1997)relates some of the history of ttheficdtous) Auric Enterprises.
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4.2. Suppose the l-year copper forward price were $0.80instead of $1. If XYZ wer to srll folavard its expected copper production,whpt is its estimated profit on'e year from now? Should XYZ produce copper? What if ie forward copper plice is $0 45?
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4.3. Compute estimated profit in 1 IIIIaI'it'XYZ buys a put opon with a strike of $0.95, $1.00, or $1.05. Draw a graph of profit in each case. 4.4. Compt: estimated prost in 1 IaTiI'XYZ sells acall option Fith a sfrike of $0.95? $1.00, or $1.05. Draw a graph of prost i each case.
.decisions
4.5. Compute estimated profit itl 1 yearif XYZ buys collars with the following s-rikes: a. $0.95for the put and $1.00for tlkecall. b. $0.975for the put and $1.025for the call. c. $1.05for te put and $1.05for the call. Draw a graph of protit in each case.
4.6. Compute estimated profit in 1 year if XYZ buys paylaterpf/l. premium may not be exactly zero):
PROBLEMS For the following problems consider the following three ftrms:
@A'I''Z mines copper, with xed costs of $0.50/1band variable cost of $0.40/lb. * Wirco produces wire. lt buys copper and manufactures wir. One pound of copper can be used to produce one unit of wire, which sells for the price of copper plus $5. Fixed cost per unit is $3 and noncopper variable cost is $1.50.
as follows tthenet
a. Sell one 1.025-strike put and buy t'wo 0.975-strike puts. j
g These
are opion prices from the Black formula assuming that the risk-free rate is 0.06, volatility is 0.l and time to ekpiration is one year. ,
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probability of a $600 loss each year. Firm B has a 50% probability of a $300 proft and a 50% probability of a $100 prot each year.
b. Sell two 1.034-strike puts and buy three l.oo-strike puts. Draw a graph of profit itl each case.
4.7. If Telco does nothing to manage copper plice tisk, what is its profh 1 year from now, per pound of copper tat it buys? If it hedges the price of wire by buying copper forward, what is its estimated prost 1 year from now? Construct graphs illustrating both unhedged and hedged prost. 4.8. Compute estimated profit in 1 year if Telco buys a call option with a strike of $0.95, $1.00,or $1.05.Draw a graph of profit in each case. 4.9. Compute esmatedprofit in 1 year if Telco sells aput option with a strike of $0.95, $1.00, or $1.05. Draw a graph of prost in each case. 4.10. Compute estimated prost in 1 year if Telco slls collars with the following sikes: a $0'.95for the put and $1.00for the call. b. $0.975for the put and $1.025for the call. c. $0.95for the put and $0.95for the call.
a. What is the expected pre-tax prost nekt year for ftrms A and B? b. What is the expected after-tax profit next year for fil'msA and B? 4.17. Suppose thattirms face a40% income tax rate on positive prosts and that netlosses reeive no credit. (Thus, if profits are positi' ve, after-tax income is (1 0.4) x proht, while if there is a loss, after-tu income is the amount lost.) Firms A and B have the snme cash flow distzibution as in the previous problem. Suppose the appropriate effective annual discount rate for both firms is 10%. -
a. What is the expected pre-tax profit for A and B? b. Whgt is the expected after-tax prot forA and B? c. What would Firms A and B pay today to receive next year's expected cash flow for sure, instead of the variable cash Qows described above? For the following problems use the BSCall option pricing function with a stock price of $420 (theforward plice), voladlity of 5.5%, continuously compounded interest rate of 4.879%, dividend yield of 4.879%, and time to expiration of 1 year. The problems require you to vary the stlike prices.
Draw a graph of profit in each case.
4.11, Compute estimated prost in 1 year if Telco buys paylater calls as follows (the net premium may not be exactly zero): '
one 0.975-strike dall and buy t'wo 1.034-st!ike calls.
a. Sell b. Sell t'wo l.oo-sfrike calls and buy three 1.034-sike
calls.
a. Suppose that Auric insures against a price increase by purchasing a +1.0strike call. Vel'ify by drawing aprost diagrnm that simultaneously selling 400-stIike put will generate a collar. What is the cost of yhis collar to a
'
Auric?
Draw a paph of profit in each case.
4.12. Suppose that Wirco does nothing to manage te risk of copper price changes. What is its profit 1 year from now, per pound of copper? Suppose thatWirco buys copper fomard at $1. What is its prost 1 yeay from now? 4.13. What happens to the vatiability of Wirco's prost if Wirco undertakes any strategy (buying ealls, selling puts, collars, etc.) to lock in the price of copper next year? You cn use your answer to the previous question to illusate your response. 4.14. Golddiggers has zero net income if it sells gold for a price of $380. However, by shorting a fonvard contract it is possible to guarantee a profit of $40/oz. Suppose a manager decides not to hedge and the gold price irl 1 year is $390/oz. Did the 511.11 em'n $10 in prost (relativeto accounting break-even) or lose $30 in prot (relative to the prost that could be obtained by hedging)? Would your answer be different if the manager did hedge and te gold price had been $450:7 4.15. Consider te exaple in Table 4.6. Suppose that losses are f'ully tu-deductible. What is the expected after-tu profit in this case?
4.16. Suppose that srmsface a 40% income tax rate on a11prohts. ln particular, receivefull credit. Firm A has a 50% probability of a $1000 proft and
4.18. Consider the exnmple of Auric.
losses a 50%
b. Find the strike prices for a zero-cost collar (buyhigh-sike low-strike put) for which the sikes differ by $30.
call, sell
4.19. Suppose that LA1N lnvestment Bnnk wishes to sell Auric a zero-cost collar of width 30 without explicitpremium (i.e.,there will be no cash payment fromAulic to LMN). Also suppose tat on every option the bid price is $0.25 below the lack-scholes price and the offer price is $0.25 above the Black-scholes price. LMN wishes to enrn their spread ($0.25per option) without any explicit charge to Auzic. What should the stlike plices on the collar be? Note Since the cpllar involves two options, LMN' is looking to make $0.50 on the deal. You ned. to 5nd strike prices tat differ by 30 such tat LNIN makes $0.50.)
4.20. Use the snme assumptions as in thepreceding problem, withoutthe bid-askspread.
Suppose that we want to construct a paylater strategy using a ratio spread. Instead
of buying aco-strike call, Auric will sell one Mo-sfrike call, and use theprerium to buy two higher-sfn'ke calls, such that the net option premium is zero.
a. What higher stn'ke for the purchased calls will generate a zero net option premium? b. Graph the proft for Auric resulting from this strategy. h
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4.21. Using the information in Table 4.11, verify that a regression of revenue on price gives a regression slope coecient of about 100,000. 4.22. Using the information in Table 4.9 about Scenario C: a. Compute ctotal revenue positive.
When
correlation between price and quantity is
b. What is the correlation between price and revenue? 4.23. Using the infonuation in Table 4.9 about Scenario C: a. Using youranswerto theprevious queson, useequatipn (4.7)to compute the variance-minimizing hedge ratio. b. Run aregression of revenue on price to compute the variance-mirlimizing hedge ratio. c. What is the variability of optimally hedged revenue? 424. Using the infonnation in Table 4.9 about Scenazio C: a. What is the expected quantity of production? b. Suppose you short the expected quantity of cor'n. What is tlze standard deviation of hedged revenue? 4.25. Suppose that plice and quantity are positively correlated as in this table: Price
Quantity
Revenpe
$2 $3
0.6m bu 0.934m bu
$1.2m $2.8m
There is a 50% chance of either price. The f'utures price is the effect of hedging if we do the following'.
$2.50. Demonstrate
a. Short the expected quantity. b. Short the minimumquantity. c. Short the maximum quantity. d. What is the hedge position that eliminates variability in revenue? Why?
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k. FINANCIAL
PREPAI D FO RWARD CO NTRACT.S ON STOCK
AN D FUTURES
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From Table 5.1 it is clear that you pay interest when you defer payment. 'I'he interesting question is how deferrinj the physical receipt of the stock affects the ptiee; this d e fen'al occurs with both te folavard and prepaid fonvard conacts. What should cases'p you pay for the stock in those
CONTRACTS
ON STOCK
a foreign Aprepaid forward contract entails paying today to receive something-stocks, permits the prepaid of contracts forward The sale future. a currency, bonds-in the of period possession time. for retaining physical while a owner to sell an asset We will derive the prepaid forward price using three different methods: pricing pricing by present value, and pricing by arbiag. analogy, by
Pricing the Prepaid Forward by Analogy Suppose you buy a prepaid forward contract on XYZ. By delaying physical possession of the stock, you do not receive dividends and have no voting or conol rights. (We ignore here the valpe of voting and conol.)
the possibilility that the arrangements also differ with respect to credit risk. which arises f'rom his or her end of the deal. (And of course the 11511 of will side not transaction other the the person on wonied about yott f'ulfillingyour obligation.) person on the other side of the deal may be
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5.2 PREPAID FORWARD
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In the absence of dividends, whether you receive physical possession today or at time F is irrelevant: In eiter case you own the stock, and at time T it will be exactly as if you had owned the stock the whole time.4 Therefore, w//cn there c?'e no dividends, the #?-fcc ofthe prepaidfonvard ctmrmcl is r/7cstockprice today. Denoting the prepaid fonvard jrice for arl asset bought at time 0 and delivered at time F as FpPw, the prepaid forward price for delivety at time F is
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Pricing the Prepaid Forward by Discounted Present Value We can also derive the price of the prepaid fonvard using present value: We calculate the expected value of the stock at time F and yhendiscount that value at an appropriate stockprice at time F, 5'w,is uncertain. in computing te present rate of rempz. risk-adjusted price, eed appropdate of the stok value to use ap rpte. we If the expected stock price at tlme F based on information we liave at time 0 is EoSv4, then the prepaid forward ljrice is given by 'fhus
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Pricing the Prepaid Forward by Arbitrage Classical arbitrage describes a sitation in which we can generate a positive cash flow either today or in the future by simultaneously buying and selling related assets, with no net investment of funds and wit.h no risk. Arbiage, ill other words, is free money. A.n extremely important pricing principl, which we will use often, is tat the prfcd ofa thqt nt7 arbitrage is possible. be deriyative Hre is an exnmple of arbiiage. Suppose that the prepaid fomard price exceeds P J arbitraoeur w ill uy 1ow anci sell high by buying te stock price-i.e., ' Fo,w > %. 'qlfltf
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130
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To understand the effects of dividends, we will compare prepaid dividends StockA pays no dividend, and othenvise identical stock B pays stocks: forwards on two days dividend from today, just before the expiration of te prepaid forwards. 364 a $5 We know that the prepaid forward price for stock A is the current stock price. What is the prepaid fonvard plice for stock B? Since the $5 dividend is paidjust before the delivery date for the stock 1 year from today, on the delivel'y date stock B will be priced $5 less than ock A. Thus, the price we pay today for stock B should be lower than that for stock A by the present value of Discrete
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the stock for Sz and selling the prepaid forward for &%.This ansaction mkes money arld it is also risk-free: Selling the prepaid forward reqires tat we deliver th stock at time T and buying the stock today nslzres that we have the stock to delivef. Thus, we the buyf of the prepaid earn Fopv s'atoday and at expiration we supply the stock to offset all and futilre fisk. Table 5.2 profts positiv earned today have forwar. We simation. this summadzes
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< &. Now suppose on the other hand that Xpv we can engage in arbiage ' eprning So shorting stock, Oe yar from and the by buying the prepaid forward stock folward arld via tp clse te short we us tat te prepaid now we acquire te stock reversed. simply table abo-ve the afe P osition. Te cash flows in Throughoutthebookwe will assume thatprices are atlevels thatpreelte rbitrage. This raises the question: Ifprices are such that arbitrage is not prohtable, who can alord to become an arbitrageur, watching out for arbiage opporttlnides? We can resolve rbitrage to earn livig, this paradox with the insight that in order for arbitrageurs ttan of degree equilibrium be there from must time; time to opportunities must occur disequilibzium.''s However, you would not expect arbiage to be obvious or easy to undrtAe. The transactions in Table 5.2 are the same as those of a market-maker who is hedging a position. A market-maker would sell a prepaid forward if a cujtomer wished to buy it. The market-maker then has an obligation to deliver the stock t a sxedprice and, in order to offset tis risk, can buy the stock. 'T'he market-maker thus 4ngages in the same ansactions as an arbitragetm except the purpose is risk management, not twptf arbitrage. Thus, the rl't//kmclDn described ?7 Table 5.2-sellilq theprepaidfonvard ??7t7r/l-?nckc?: buying l/kc stock--also describes the acfsn, ofa The no-arbitrage arguments we will make thus srve two funcions: 'Fhey tell us how to take gdvantage of rnispricings, and they desc/be the behavior f market-makers
In general, the price for a prepaid folavard contract will be te stock price less the present value of dividends to be paid over the life of the contract. Suppose there are multiple dividend payments made throughout the life of the forward contract: A stock n. A prepaid is expected to make dividend payments of Dti at times ti i = 1, receiving the receive time but without will entitle the stock F at fomard contract you to prepaid price is the dividends. 'Fhus, forward interim
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The calculation in this example implicitly assumes that the dividends are certnin. Over a short horizon tlzis might be reasonable. Over a long horizon we would expect dividend risk to be greater, and we would need to account for this in computing the present vale of dividends.
132
%.FINANCIAL
FORWARDS
AND FUTURES
FORWARD
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CONTRACTS
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For stock indexes containing many stocks, it is common to contlnuous dividends dividend being paid condnuously at a rate that is proportional to the level the model as of the index', i.e., the dividend yield (the annualized dividend payment divided by the stock price) is constant. This is an approximation, but in a large stock index there can be dividend payments on a large proportion of days.o The dividend yield is rioi likely to be sxedin the short nm: When stock prices tise, the dividend yield falls, at least temporarillt Nevertheless, we will assume a constant proportional dividend fieldfor purposes of this discussion. and the anTo model a continuous dividend, suppose that the index price is the dollar dividend over one day nualized daily compounded dividend yield is is xa
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Tls formula shows thattheforward coneactis apurchase of the stock, with deferred P ayment. The hlterest adjustment compensates for tat deferral.
For the case of continuous dividends, equation (5.7),the annualized forward premium is simply the difference between the risk-free rate and the divideljd yield.
Discrete dividends: To obtain the forward price for a stock that pays discrete dividendj, we tale the f'uture value of equation (5.3). The forward price is the fumre value of the prepaid forward.
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Continuous dividends: future value of equation
A market-maker or arbitragelzr must be able to olset the Iisk of a fonvard contract. It is possible to do this by creating a synthetic fonvard conact to offset a position in the act'ual forward onact. In this discussion we will assume tat dividends are continuous and paid at the rate and hence that equation (5.7)is the appropriate fonvard price. We can then create a , ' synthetic long forward contract by buying the stock and bolwwing to ftmd the position. To see how the synthe:c position works, recall tlzatthe payoff at expiraiion for a long fonvard position on the index is
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Similarly, suppose the market-maker wishes to hedge a long fonvard position. Then it is possible to reverse the positions in Table 5.6. The result is in Tale 5.7. in which you buy the underlying assetand shortthe offsettingforward
Aansaction
Synthetic Forwards in Market-Making and Arbitrage
gjg;i
a forward contract and short a synthetic
forward contract.
hort tailed position in ltock, receiving
We can rearrange this equation to derive other synthetic equivalents.
,j,j,jjjj,
E Eg ( ( E E '' . (E .( . 'j j.q . '.(.'.j...!.((gj(.j(.. .. ...g.j('...y.'.j.).!.j(.Ej.jj.j.g.....(j.'.(...E. j..j(k.j.... g.k.g.. j.jjr...j..jg.E(.......g..(..r j (gj j...g.. ... ..... g.... j...j E(. gg..j..j... g.....j... j g.. j.j.j.. .jg.... jj ... . y.yy.. j..y. . .. ... .y. j .. . .. y.. .. . .... . ... .. .. . . j. j... . . . ... j .. j.'yj... g l'' ' . E....... .. E... E. . Iiliq)ii. .... . j.(.. .g( . j q... . . .yq y-E g ..l('. . q .... q. . .. q... .. ;.. . .. . E.( . y. .... . .. ..E. ... .. g.. .y.. ... . q j..... . . . ! .. y . ..y yr ..iI:j2),k. . y y . y. r . . .y y y ;qE .y .. E g..Ey g .. y . r (y . y. . yy j y... -..j y . yy yy j y i.''.l..:' '';qI1dIqq'!!ii@:#kk'. ' ''' . 1iI;j:!j: . E ' . . . . . .. y y y j . . . ... y . )'.E y ' yy yy. ,....y....y.j... . .. . .. . gj.. j.j. . y. E .... ... .yy.y... y.j y...(.E... y. . yj... . yj. y.y....... y....j... yr . . E ..q . . y. .. . . ( . . .' .' .( '' . . . E.. ....E j..... y. y . .y. y.. E... q y y . ...E... ......E y..E.E...r........E.E.y.y. . y.......;. .... y....y. y.j .. y.. y...-j..y.Ejy......; y.....ryy...y... y.......j..... .y..........yy.. y. .E... ..... y.. .... ...(...... 2. .(.. . (:. y .. ) . . .... .. . .. ........ ..... ....... .... . . . . . ..... . . y .. .. .. . ... ........ . .
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jtyyjjj jgjyjrjtyygj jr,.kjyyjs j;,.jkggrrj;gyjr jr,, jyjjj jj..;j .jy:jtry;jj yjji gjgy kyjjj jr,. jr,gyjjyjkyjyjj jr..;j jrrg;s ::
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Long forward
Stock
.
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Acash-and-canyhas nolisk: Youhave an obligation contractis called acash-and-carry. also the but deliver the asset. The market-maker offsets the short folavard asset own to position with a cash-and-cany An arbitrage that involves buying the underlying asset and selling it folward is called a cash-and-carry arbitrage. As you might guess, a reverse cash-and-carry entails short-selling the index and entering into a long forward position. If te forward coneact is priced according to equation (5.7),then prohts on a cashand-carry are zero. We motivated the cash-and-can'y in Table 5.6 as lisk management by a market-maker. However, an arbitrageur might also engage in a cash-and-cany If the forward price is too high relative to the stock price-i.e., if F().w > Sctr-llr-then an profit. arbitrageur or market-maker can use the strategy in Table 5.6 to make a lisk-free .
138
%.FINANCIAL
FORwARDS
FORWARD
AND FUTURES
A.n arbieageur would make the transactions in Table 5.7 if the fonvard were unW V SCCOFYOVVC to ZC St0Ck--i.C., if ROCSV-JIX > S T As a final point, you may be wondering about the role of borrowing and lending in Tables 5.6 and 5.7. When you explicitly account for bonowing, you account for the opportunity cost of investing funds. For example, if we ornitted borrowing from Table 5.6, we would invest %e-'T today and receive F0,w at time F. In order to know if there is an arbitrage opportunity, we would need to perfonn a present value calculation to compare the time 0 cash flow with the time F cash flow. By explicitly including borrowin g in the calculations, this time-value-of-money comparison is automatic.? Similady, by comparing the implied repo rate with our borrowing rate, we have a simple meastlre of whether there is an arbiage opportunity. For example, if we could borrow at 7%, then there is an arbtrage oppormnity if the implied repo rate exceeds 7% On the other hand, if our bonowing rate exceeds the implied repo rat, tere is no arbitrage oppormity. .
1
'No-Ybitrage
Bollnds with Transaction Costs
Tables 5.6 and 5.7 demonstrate that an arbitrageur can make a costless prot if Fo w + qoetr-llr This analysis ignores eansactioncosts. In pracce an arbieageur will face trading fees, bid-ask spreads, diferent interest rates for borrowing and lending, and the possibility that buying or selling in large quantities will cause prices to change. The effect of such costs will be that, rather than there beilzg'a single no-arbiage price, there will be a no-arbiage boundl a lowerprice F - and an upperprice F + such that arbitrage will not be proftable when the forward price is between thse bonds. Suppose that the stock and forward have bid and ask prices of Sb < S' and Fb < FJ, a ader faces a cost k of transacting in te stock or forward, and the interest rates forborrowing and lending are rb > r. ln this example we suppose that there are no transaction costs at time F, when the forward is either settled by delivery or cash-settled. We will lirst derive F+. An arbitragetlr believing the obselwed forward price, Fo,w, is too high, will undertake the ansactions in Table 5.6: Sell the forward and bnow to buy the stok. Fof simplicity we will assume the stock pays no dividends. 'I'he arbitrageur will pay tlle ansaction cost k to short the fomard and pay (5! + k) to the position is terefore acquire one share of stock. The required borrowing to snance payoff 2k. At time F, is 5'g + the
-(.%a +
lkje
rbT
+ F0,w
Repayment of borrowing
Arbiage
-
Sr +
Value of fonvard
Sv Value of stock
>
Fo
=
a (.% + 2k)c ay.
?ln general, arbitrageurs can bonmv and lend at different rates. Apro forma arbitrage calculation needs to account for te appropriate cost of capital for any particular transaction.
O N STOCK
$
139
Thus, the upper bound re:ects the fact that we pay a high price for the stock ttheask price), pay eapsactioncosts on both the stock and forward, and borrow at a high rate. We can derive F- analogously. Problem 5.14 asks you to vel'ify that the bound below wlzich arbitrage is feasible is F0,w < F -
=
zh
b
rlw
-
zkje
(5.11)
This expression assumes that short-selling the sto'ck does not entail costs other than bid-ask ansaction costs when the short position is initiated. Notice tat in equations (5.10)and (5.11),the costs 811 enter i.n such a way as to make the no-arbiage region as large as possible (for example, te 1ow lending rate enters F- and the high bonowing rate enters F+). This makes economic'sense: Trading costs cannot help an arbiageur make a prost. There are additional costs not reQected in equations (5.10)and (5.11). One is that signicant nmounts of tradlng can move prices, so that what appears to be an arbitrage enters a large ordr. Another challenge may vanish ifprices change when the lbiagetlr risk. tzades do not occur instantaneously, tlie arbiage cpn vaaish tf can be execuuon completed. before the trades are It is likely that the no-arbitrage region will be different for different arbiageurs at a point in time, and different across time for a given arbitragetm For example, consider the ading eansaction cost, k. A large westment bnnk sees stock order llow from a variety of sotlrces and may have wentory of either long or shol't positions irl stocks. 'I'he bnnk may be able to buy or sell shares at low cost by selwing as market-maker for a customer order. It may be inexpensive for a bnnk to short if it already owns the stocks, or it may be inexpensive to buy if the bnnk already has a short position. Borrowing and lending rates can also vary. For a transaction that is explicitly fnanced by boaowing, the relevant interest rates are the arbiageur ds margin al borrowing rate (if that is the sotlrce of f'unds to buy stocks) or lending rate (if stocks are to be shorted). However, at other tlmes, it may be possible to borrow at a lower rate or lend at a higher rate. For example, it pay be possible to sell T-bills being held for some other may effectiyely permit bonowing at a 1ow purpos as a source of short-term funds. Finally, in order to bon'ow money or secures arbitragelzrs must have available rate. Undertnking capital. one arbiage may prevent undertnking another. and I'he overall conclusion is not surprising: Arbiage may be diflicult, lisly, deviations omleieoreticipdcemaybeabiaged, butsmall Largedeviations costly. may or may not represent genuine arbitrage opportunities. E
.
'lhis
Quasi-Mbitrage
is prostable if this expression is positive, or F0,F
CONTRACTS
(5.10)
'l'he previous section focused on explicit arbiage. However, it can also be possible to undertake ilnplicit arbiage by substittlting a 1ow yield position for one with a higher return. We call this quasi-arbitrage. Consider, for example, a corporation tat can borrow at 8.5% and lend at 7.5%. Suppose there is a cash-and-carry ansaction with arl implied repo rate of 8%. There is no pure arbitrage opponunity for the corporation, but it would make sense to divert
140
%.Fl NANCIAL
Fo RWARD
AN D FUTURES
FORWARDS
lending from the 7.5% assets to the 8% cash-and-carry. tf we attempt explicit arbieage by borrowing at 8.5% in order to enrn 8% on the cash-and-cany the transactionbecomes unprostable. We can arbiage only to the extent that we are already lending; this is why it is quasi'tarbitrage.
Does the Forwrd
Price Predict the Future Price?
lt is eommon to think that the forward price re:ects an expetation of the asset's fumre price. However, from the formula for the forward price, equation (5.7),once we lnow the current asset price, risk-free rate, and dividend yield, the fomard price conveys no additional information about the expected f'uture stock price. Moreover, the fomard price systelhatically errs in predictig the future stock price. that has The reason is straightforward. When you buy a stock, yu invest lney interest-earning asset), been in invested othemise could have an oppormnity cost (it an and you are acquiring the risk of t-h stock. On average you expect to earn interest for the time value of money. You also expect an additional retllrn ms as compensation for the risk of the stock-this is the tisk premium. Aljebraially, the expected return on a stock is 'compensation
%=
a
r
-
?-
,
(.12)
Compensation for risk
Compensation for tme
CONTRACTS
$
ON STOCK
141
For example, suppose that a stock index has an expected ret'urn pf 15%, Fhile the l'isk-free rate is 5%. lf te current index price is 100, then on average we expect that te forward plice for dlivery in 1 yea.r will be only 105, index will be 115 in 1 year. of the fomard conact will on average enrn positive This holder however. means that a of bearing the risk of the index.S albeit profits, at the cost This bias does not imply tat a forward contract is a good investment. Rather, /7J. tells it us tat the risk p?'c/?3rfl?ion f/n asset cfln b created at zdm cost ttntf hence value. Though this seems surprising, it is a result from elementaly finnce that a zdm if we buy any asset and bon'ow the RIII nmount of its cost-a tzansaction that requires we earn the risk premium on the asset. Since a forward conact no westment-then has the risk of a fully leveraged investment in the asset, it earns the riskwemium. This proposition is true in general, not just for te exnmple of a folavard on a nondividendPaying stock. rfhe
An Interpretation of the Forward Pricing Formula 'l'he fomard pricing formula for a stock index, equation (5.7),depends on r J, the difference between the risk-gee rate and the dividend yield. This difference is called the cost of carrp Supposeyou buy aunit of the index that costs S andfund theposition by borrowing the risk-free rate. You will pay t'S on the borrowed nmount, but the dividend yield at will provide offsetting income of &S. You will have to pay the difference, (r 84S, on an ongoing basis. This difference is the net cost of carrying a long position in the asset', hence, it is called the 'Tost of carly'' Now suppose you were to short te index and invest the proceeds at the risk-ee would receive S for shorting the asset and enrn t'S on te invested proceeds, You rate. would have to pay S to the index lender. We will call 8 the Iease rate of the but you lease rate of an is what index; it you would have to pay to a lender of the asset. mmualized cash that the make bonower must to the lender. For a payment asset is te nondividend-pgying stock, the lease rate is zero while for a dividend-paying stock, the lease rate is the dividend. Here is an interpretation of the forward pricing formula: -
-
When you enter into a forward conact, there is no invstmnt', hee, you are not compensatd for the time value of mo-ney. However, the fo>ard contract retains the risk lisk. This ???dtwz' that #kdftp/4s/tzlW tontract of the stock, so you must be compensated for ??7lI.l eal'n the ?-'ky?'c/nfl/?/l. If the risk premium is positive, then on average you must expect a positive ret'ul'n from the forwrd contract. The only way this can hppen is if the forward Ijrice predicts too low a stock price. In other words thefonvard coiitract is t/.fl/yd//llf?'c stock price. a biased predictor We ean see this algebraically. Let or be the expected return on a nondividndpaying stock and 1et 1, be the effective annual interest rate. Consider a l-year forward forward price is conact. 'l'he
F() = Sfj(1 + r)
'l'he
Forward price
=
Spot price + Interest to carry the asset
The expected future spot price is
f
)
=
,V(1 +
J)
where Efj denotes as of time 0.3'Thus, the differepce between the forward price and the expected f'uture spot price is -
F0
=
Asset lease rate
Cost of carry OCSI
ttexpectation
.E'0(5'1)
-
S0(1 +
ce)
-
S0(1 + r)
=
Sza
-
r)
1- is the risk p?'c??7!I?n The expression a on the asset i.e., the amount by which the lisk-free asset. This equation veries that thefolnvard asset is expected to outperform the tftr l-lj'i: Tr (7JJ t, t. l-isk (f t)/ /J l-i ()t? is 47 ittlt? b4'l/7 tr :7???t?If?) t tlttr yp?-tr??7ittl71 t7?$ tIltr ttl
The fomard contract, unllke the stock, requires no investment and mkes no payouts and therefore has a zero cost of cany One way to interpret the forward pricing fonnula is that, to the extent the fonvard contract saves ou' having to pay the cost of carry, we is what equation (5.13)says. are willing to pay a higher price. rfhis
-
Accounting fordividends in this example would not change the magnitude of the biks since dividends would lower the expected f'umreprice of the index and te forward price by equal amounts.
142
%.FINANCIAL
FORWARDS
Fu'ruREs
AND FUTURES
%.
cox-rRAc'rs
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Fptures conacts are essentially exchange-eaded fomard contracts. A; with commitmentto buy orsell an underlying assetatsomefumre representa futures contracts they are standardized and have specised exchange-aded, f'utures date. Because are Futures and locations, dates, procedures. may be traded either eleconically or delivel'y orders shouting is alled open sellers to one another ttllis and with buyers pits, in trading clearinghouse The role clearinghouse. of the associated Each exchange has outcry). an is to match the buys and sells that take place during te day, and tp kep track of the obligations and payments required of the members of the clearinghouse, who are called clearing vcp//pua. After matching trades, the clearinghouse typially ecomes the counterparty for each clearing member. Mthough forwards and f'utures are similar in many respects, there gre diferences.
settled * Whereas fclrward coneacts are settled at expiration, futures contracts are calledmarking-torparket. whopis whatto who of owes daily. The determinaon Frequent marling-to-market and settlement of a futures conact can lead to pricing differences between the futures and an otherwise identical fotavard. pjile to offset * As a result of daily setllement, f'utures coneacts are liquid-it ij poidon. ample, For enteling into by opposite date the afnobligation on a given cgncel conact, ftltures 500 you cap your if you are long the September S&P obligation to buy by entering into alz offsetting obligation t selt the jejtember S&P 500 contract. tfyou use $e snme broker to buy ad to dell, yur obliation is cancelled.g of'lic iall y * Over-the-counter forward conacts can be customized to suit the buyer or seller, wherea futures contracts are standardized. For exnmple, available futures conacts may permit delivery of 250 uaits of a particular index in March or June. A fonvard contract could specify Apzil delively of 300 units of te indey. lisk is different with the futures * Because of dily settlemeny, the nature of credit structured ln so as to minl'mize te effects of fact, fmures cpnacts are contract. risk. credit There are typically daily price limits in futtlres markets (andon sme stock ex* changes as well). A price limit is a move in the f'utures price that triggers a temporary halt in eading. Fr exnmple, there is an initial 5% limit on down moves in the S&P 500 fumres coneact. An offer to sell exceeding this limit can trigger effect. I.f that is a temporary trading halt, after wlaich time a 10% price limit is in omplicated, rules be The and 20% 15% limits. subsequent there cn exceeded, are exist. such rules that importnt but it is to be aware E
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We will illuseate futures contracts with the S&P 500 index f'umres contract as a
specific example.
The S&P 500 Futures Contract The S&P 500 futures contract has the S&P 500 stock index as the underlying asset. Futures onindividual stocks have recently begun trading in theunited States. See the box above. Figgre 2.1 shows a newspaper quotation for the S&P 500 index futures contract along with other stock index futures conacts, and Figure 5.1 shows its specifcations. The notional value, or size, of the contract is the dollar value of the assets underlying one contract. In this case it is by desnition $250 x 1300 = $325,000.10 The S&P 500 is an example of a cash-settled conact: Instead of settlin by act'ual delivery of the underlying stocks, the contract calls for a cash payment that equals the proft or loss as (/' the contract were settled by delively of the underlying asset. On the expiration day, the S&P 500 fumres contract is marked-to-market against the actual cash index. This final settlement against the cash index guarantees that the futures price equals the index value at contract expiration.
loBecause the S&P 500 index is a fabricated number a value-weighted average of individual stock prices the S&P 500 index is treated as a pure number rather than a price and the contract is dehned at maturity to have a size of $250x S&P 500 index.
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2000 units of tlle index wlkich we could now sell for only $2000 x 1099. Thus, we lose Suppose tlzat over the flrst week, the futures price (1099 1100) x $2000 drops 72.01 points to 1027.99, a decline of about 6.5%. On a mark-to-market basis, we have lojt
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Specificatons for the S&P 500 index futures
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Months Trading ends
Size
Settlement
.
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lt is easy to see why the S&P 500 is cash-settled. A physical settlement process would call for delivery of 500 shares' (or some large subset thereog in the precise percentage they make up the S&P 500 index. This basket of stocks would be expensive to buy and sell. Cash settlement is an inexpensive alterna:ve.
Margins and Marlcing to Market Let's explore the logistics of holding a fumres position. Suppose the f'uttlres price is 1100 and you wish to acquire a $2.2million position irl the S&P 500 index. The notional value of one contract is $250 x 1100 = $275,000;this represents the amount you are agreeing to pay at expiration perfutures conact. To go long $2.2million of the index, you would million = 8 long futures contracts. The notional value enter into $2.2 m1'11ion/$0.275 of 8 contracts is 8 x $250 x 1100 Q $2 000 x 1100 = $2.2 million. A broker executes your buy order. For every buyer there is a seller, wltich means that one or more westors must be found who simultaneously agre to sell fomard the total number of open positions (buy/sell pairs) same nupber of units of the index. is called the open interest of the conact. Botll buyers and sellers are required to post a performance bond with the broker to j j ensure that they can cover a specified loss on the position. This deposit, which can enrn interest, is called margin and is intended to protect the counterparty against your failure to meet yotlr obligations. The margin is a pelformance bond, not a prmlum. Hence, futures contracts are costless (not counting, of course, commissions qnd the bid-ask 'rhe
spread).
To understand tlze role of margin, suppose that there is 10% margin and weelly
sttlement (in practice, settlement is daily). 'Fhe margin on futures conyacts with a notional value of $2.2 million is $220,000. tf the S&P 500 f'utures price drops by 1, to 1099, we lose $2000 on otr futures position. The reason is that 8 long conacts obligate us to pay $2000 x 1100 to buy
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We have a choice of either paying this loss directly', or allowing it to be taken out of te margin alance. It doesn't matter which we do since we can recover the unused margin balance plus interest at any time by selling our position. lf the loss is subeacted from the margin balance, we have enlmed one week's interest and have lost $144,020.Thus, if the continuously compounded interest rate is 6%, otlr margin balance after one week is
$220 000c0.06x1/52 $144 ()%
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Because we have a 10% margin, a 6.5% decline in the fumres price results in a 65% decline in margin. Were we to close out our posiion by ntering into 8 short index futures contracts, we would receive the remaining margin balance of $76,233.99. The decline in the marginbalance means tlle brokeras signcantly less protection shoull we default. For this reason, pnrficipants are required to maintain the margin at a minimum level, called the mnintenance margin. Tltis is often set at 70% to 80% of te initid margin level. In this example, where the margin balance declines 65%, we would have yopost addiional margin. The broker would malte amargin call, requesting additional margin. If we failed to post additionl mrgin, tlze broker would close the P osition by selling 1000units of the index, and reitlrn to us the remaining margin. In practice, marldng-to-markt and seftling up are performed at least daily. Since margin you post is the broker's protecuon againsi your default, @.major determlnant of margin levels is the volaiility of the undrlying aqset. 'I'he mlnlmum margin on ihe S&P 500 conact has generally been less ttia!l t e 10% F assume in this example. In Aurust 2004. for example. the minimum marcin on tlle X;P 500 futtlres . of the conact. contractwas about 6% of the noonal value 'fo illus eate the egect of priodic setttemet Tale 5.8 reports hypotheticalfutures plice moves and acks the margin posiuon over a period of 10 weeks, assuming weeldy marklng-tp-market and a continuously compounded rlsk-free rate of 6%. As te party agreeing to buy at a dxedprice, we make money when ihe ptice goes up and lose when rfhe opposite would occur for the seller. the price goes down. The lo-weekproft on the position is obtained by subtracting f'romthe final margin theposion balancetllefuturevalue of the originalmargin hwestment. Week-loprofto 5.8 therefore Table is in .
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What if the position had been a fonvard rather than a futures position, but with prices the same? In that case, after 10 weeks our profit would have been
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Conversely, suppose that the interest rate declined as the futures price rose. Then margin balance on a long position grew, the proceeds would be invested at a lower the as Similarly, as the balance declined and required additional hnancing, tls nancing rate. would occur at a higher rate. Here a long futtlres contract would pn average perfonu worse than a long forward contract. Th is comp alison of the folavard and futures payoffs suggests that when the interest is rate pojitively correlated with the futures price, t futures price will exceed the price on an oth.erwise identical folavard contract: The investor who is long futtlresbuys at a higher price to offset the advantage of marking-to-market. Similady, when the interest rate is ngatively correlated with the forward price, the futures price will be less than an otherwise identical fonvard price: The investor who is long f'utures buys at a lower price to offset the disadvantage of marling-to-market. As an empirical matter, forward and futures plices are very similaniz The theortical difference arises frm uncertainty aboutthe intereston mark-to-inarketproceeds. For short-lived conacts, the effect is generlly small. However, for long-lived contracts, the difference can be significant, especially for long-lived interest rate f'umres, for which there is sure to be a c rrelation between the interest rate and the plie of the underlying asset. For the rest of this chapter we will ignore the difference between fomards and
Arbitrage in Practice: S&P 500 Index Arbitrage
Why do the lmres and forward prosts differ? The reason is that with th futures proceeds. Given the prices in Table contrct, interest is eained on the mark-to-market prices on average re below the forwards because 5.8, e loss ij larger for futures t.11% With they ftlnd losses as a fonvrd, by ntrast, losses occur. iriitiz prie and w hv to settlement tugnifies t.11: daily the interest on are not fnded until expiration. Enrning been cosistent gains on there Had contzact. gain or loss compared t!) tat on a fomard exceeded have would the forward prost. prot t e Iiosition in this exanple, te fumres and ultimate futures ontract payoff forward to a lfndix 5.B demonseates that the of number conacts f'utures adjusting the exampleby so s to undo can be equated in this te rriagnlf'y'ing effect of imerest. .
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Comparing Futures and Forward Prices A.nimplication of Ajpendix 5.B is that if the interest rate were not random, t en forward randomly? andfutums prices wouldbe the same. However, whatif theinterestrate varies when the unexpectedly increases interest the rate Suppose, for example, tat on average margin the correlated. balarce positively the t'wo are f'uttlres plice increases; i.e., higher. the interest rate price) f'utures was increased just as would grow (due to an margin balance would shrink as te interest rate was lower. . On average in this case, a long ftlttlres position would outpelorm a long forward coneact.
The S&P500 f'uturescontractprovides a contextfor illustrating practical issues that arise when we try to apply the theoretical pricing formplas to determine the fair price of a futures contract. 1.rlorder to compute the theoretical fonvard price using equatiop (5.7), we need to determine tllree things: (1)the value of the cash index (&), (2) the value of dividends to be paid on the lndex over the life of the contract (J), and (3) the interest We can use readily available information to see whether the observed futtlres price
is close' to that given by equation (5.7). On August 30, 2004, the closing S&P 500 index vale was 1099.15 and th December futures price was 1099.30. The annualized dividend yield on the index was approximately 1.75%, which we will assume is expected to be constant over titne. The December contract expirs on December 17, hence there 0.2986) until expiration. were 109 days (F What interest rate is appropriate? Two interest rates that we can easily observe are which the yield on U.S. Treasury bills and the London Interbnnk Offer Rate ILIBORI, institmions. Ninety-day LDOR can be inferred is a borrowing rate for large snancial from Eurodollar f'utzlres,wlzich we will discuss in Section 5.7. Te yield to maturity on a Treasury bill matuling in December was 1.56%, while implied 90-day LDOR was '
=
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% FI NANCIAL
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Fu-ruREs
AN D FUTURES
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As illuslated by quations (5.10) and * There are transaction costs of arbiage. (5.11), tansaction costs create no-arbitrage regions, ratherthan no-arbitrageprices.
In practice, a representative bid-ask spread on the index f'utures contact rnight be 20 to 30 basis points (abasis point on the S&P futtlres conact is 0.01) and 0.25% to 0.5% on the stocks in the index when traded in signilicant quptities.
*
Different interest rates will ieflect diferences
in issuer credit
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example uses yields as if they were quoted as continuously compounded, which tlley are not. l3-l-his However, forshortperiods and low interestrates, there is almostno dfferencebetween effective annt!al and continuously compounded rates. For example, if the effective annual rate is 2%, te continuously 0.0198, or 1.98%. ln practice, interest rates are quote us n compounded equivalent.is ln(l E vriety of arane conventions for annualizing the rate.
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Listing for the Nikkei 225 futures contract frtlrl til(, bjllII itrt?: t lournal, July21 2004.
With margin
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notie entiresoo-stock * Because of transaction costs, an arbiageurwill usuallybuy it.l4 f'umres contract and the offsetting position index, but instead a subset of in stocks may thus not move exactly together. When buying a large number of
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the Chicago Mercantile Exchangeds Nikkei 225 f'utures contract-see a newspaper quottion in Figure 5.2 and the details summarized in Figtlre 5.3-is a stock index coneact like the S&P 500 conact. However, there is one vel'y important te currency difference: Setllement of the contractis in adifferentcurrency (dollars)t.14:.:1 of denomination for te index (yen).15 To see why this is important, consider a dollar-based investor wishing to invest in the Nilkei 225 cash index. Tis investor must undertak two transactibj: changing dollars to yen and using yen to by the index. When te posidon is dld, the investor reverses these transactions, selling the index and converting yen bak to dollars. There lisk in this transaction: the risk of the index, denoninated in yen, and are two sources of the risk that the yen/dollar exchange rate will change. From Figure 5.3, the Nikkei 225
Future dividends on the s&P 300 stocks are uncefiip. ' Fof plicing a 3-month . . . ftltures contract, one could use equation (5.6),wit.h actal recent ash dividends on underlying stocks for Dti as proxies for fortcbminj dividends.Ttir is still the lisk change over the next 3 ronths. iik is for Th will that dividends jrater a .. . .. longer-dated futures contracts. :
LIFETIME(pEl1 0?Eh1 HIGH L0W SFITLEIH6 HICH t0W ljT
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lzlAnother way to trade te cmsh index is with the use of Standard and Poor's Depository Recepts These are unt investment trusts that are backed by a portfolio intended to mimic the S&P 500. lnvestors can convert units of 50,000 SPDR shares into the actual stock and can convert stock into SPDRS. This keeps SPDRS close to the S&P 500 index, but in practice SPDRS may be mispriced relative to the cash S&P 500just as futures are.
149
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.
There are Jwt? theoretical ptices, depending upon which interest rate we use; te act'ual December futtlres plice, 1099.30, is between these two prices. Does this mean that there is an arbitrage opportunity? There are a number of considerations in interepreting these differences in prices:
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Arbitragetlrs will need to take into account these considerations. Ultimately, the only way to know if arbiage is prostable is to assess specisc prices, ading costs, and bolwwing and lending rates.
priee using LIBOR instead of the T-bill rate is (r-J)T
CONTRACTS
(SPDRS).
lserhere is also ayen-denominated Nikkei 225 futtzres contract that trades at the Osakaexchange. Since it is purely yen-denominated, this contract is priced according to equation (5.7).
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futures contract is denominated in dollars rather than yen. Consequently, the contract insulates investors from currency risk, permitting them to speculate solely on whether the index lises or falls. This lind of contract is called a qttanto. Quantocontracts allow investors in one country to invest in a different countl'y without exchange rate risk.
Nikkei contract provides an interesting variqtion on the The dllar-denominated of construction a futtlres conact. Because of the quanto feature, the pricing fonuulas we have developed do not work for the Nikkei 225 contract. We will discuss quantos and the necessary modilication to plice a quanto f'uttlres coneact in Chapter 22.
5.5 USES OF INDEX FUTURES An index f'utures contract is econorfiically lile borrowing to buy the indey. Why use an index f'utures conact if you can synthesize one? One answer is thai index f'uttlres can actually ading a basket of the permit trading the index at a lower transaction cost t.11a11 stocks that make up the index. lfyou are taking a temporaly position in the index, either for investing or hedging, the transaction cost saving could be signiscant. In this section weprovide t'woexamples of the use of index f'utures: asset allocation and cross-hedging a related portfolio.
Asset Allocation Asset allocation strategies involve switching investments among asset classes, such as stocks, money market instruments, and bonds. Trading the individual securities, such as the stocks in an index, can be expensive. Ourearlierdiscussion of arbitrage demonstfated tatwecan usefomards to create synthetic stocks andbonds. Thepractical implicationis that a portfolio manager can invest in a stock index without holding stocks, commodities without holding physical commodities, and so on.
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General asset allocation We can use forwards and futures to perform even more sophisticatd asset allocation. Suppose we wish to invest ourportfolio inrfreasuly bonds (long-tenu Treasury obligations) instead of stocks. We can accomplish this reallocation with two folward positions: Shorting the fonvard S&P 500 index nd going long te fonvardrltbond. The firstansaction converts ourpor/olio from an indexikestmentto a T-bill investment. The jecond transaction converts the portfolio f'roma'Tlbill investment to a T-bond investmet. This use of futtlres to convert a position from one asset category (stocks) to another (bonds)is called jt futures overlay. Futures overlays can have benehts beyond reducing transction costs. Suppose. invejtment invest managetnet compahy has portfolio managefs Wh sessfully an
they believe t be mispttced. The magers are jttdgd on therelative to the S&P 500 stock index and consistently outperform th indx by 2% per is 2%). Now suppose that new year (in the language of portfolio theoly their clients of the ompany like the performance record, bt want to invest in bonds rather stocks. The invejtpentmanagement company could ;re its stockmanagers andhire t.11a11 bond managers, but )ts..existinl investment manasers are the reason for the compalw's . success. The company can use a futures overlay to continue to invest in stocks, but to provlde a bond ret'urn ipstead of a stock rettll'rl to investors. By investing ip stocks, shorting index futves, and going long bond ftllres, the managers continue to invest in t stocks, but the client receives a bond rettlrn plus 2% rather t.1:a1: a stock return p us 2%. This use of f'uttlres to tfansfonn an outpedbrming portfolio on one asset class into an outperforming portfolio on a different asset class is called alpha-porting.
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swltthing from stocks to T-bills As an example of asset allocation, suppose that we have an investment in the S&P 500 index and we wish to tempornrily invest in Tbills instead of te index. Instead of selling all 500 stocks and investing in T-bills, we can simply keep our stock portfolio and take a short forward position in the S&P 500 index. This converts our cash investment in the index into a cash-and-cany creating a synthetic T-bill. When we wish to revert to investing in stocks, we simply offset the fonvard position. To illustrate this, suppose that the current index price, %, is $100,and the effective l-year lisk-free rate is 10%. The forward price is terefore $110. Supppse tllk4 in 1 yar, th index price could be eithr $80 or $130. 1.fwe sell the index and invest in T-bills, we will have $110in 1 year. Table 5.9 shows that if, instead of selling, we keep the stock and short the forward contract, we enrn a 10% feturn no matter what happens to the vzue of the stock. ln this example 10% is the rate of return implied by the forward premium. tf there is no arbitrage, this return will be equal to the risk-free rate.
Cross-hedging with lndex Futttres Index futures are often used to hedge portfolios that are not exactly the index. As discussed in Section 4.5, this is called cross-hedging. Suppoje that we have a portfplio that is not cross-hedsins with perfeu correlation the S&P 500, and we wish to shift the portfolio into T-bills. Can we use the S&P 500 fumres contract to do this? The answer depends on the correlation of the portfolio with
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%.FINANCIAL
152
AND Fu-ruREs
FORwARDS
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te S&P 500. To the extent the two are not perfectly coaelated, there will be residual risk. Suppose that we own $100million of stocks with a beta relative to the S&P 500 of 1.4. Assume for te moment that the two indexes are perfectly correlated. Pedkct correlation means that there is a perfectly predictable relationship between the two indexes, not necessarily that they move one-for-one. Using the Capital Asset Pricing Model ICAPMI,the return on our portfolio, rp, is related to its beta, pv, by r +
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2. Adjustfor #2c dterence ilt beta. Since the beta of our portfolio exceeds 1, it moves the S&P500 in eitherdirection. Thus we peed to f'urthrincrease our S&P more t.11:z1 500 positin to account for the greater magnitude moves in our porifolio relative to the S&P 500. This gives us x 1.4
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The hedge quantity is denominated in ten'ns of a quantity of fmuies contracts. The second equality follows because Covto, ?'sapl/o'zap the slope oecient when we regress j is te portfoli returri on the S&P 500 rettlrn', t.e., it is the portfolio beta wit mspectto the S&P 500 index. (5.14)is plso the formula we used in copcluding that, with perfect correlation, we should short 509.09 contracts. Notice that the hedge ratio in equation (5.14)depends on the ptio of the market. of alue the portfolio, 1p, to the notional value of the SP 100contract, N. Thus, as the v portfolio chanes value relative to the S&P500 index, it is necessary to change the hedge ratio. This rebalancing is necessary when we calculate hedge ratios using i relationship based on rettlrfls, which at percentage changes. When we add H* futures to the portfolio, the variance of the hedged poitfolio,
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Assume also that the S&P 500 is 1100 with a 0 dividend yield and te effective annual 1166. risk-free rate is 6%. Hence the futures price is 1100 x 1.06 1.fwe wish to alloate from the index into Treasury bills using futures, we need to short some quantity of the S&P 500. There are two steps to calculating the short futures
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Cross-hedging with imperfect correlation The preceding example assumes that the portfolio and the S&P 500 index are perfectly correlated. In practice, con-elations between two portfolios can be substantially less than ne. Using the S&P 500 to hedge such a portfolio would introduce basis risk, creating a hedge with residual lisk.16 Denote te return and invested dollars on the portfolio as rp and Ip. Assume that short S futures contracts, each with a notional. value N. The futures position we earns the risk premium, rstt:p r. Thus, the return on the hedged position is
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q Example 5.5 Suppose we are optimistic about the performance of the NASDAQ index relative to the S&P 500 index. We can go long' the NASDAQ index and short
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iit the s&P 500 futures. We obtain te variance-minimizing position in te s&P 500 by lli regression (fr om June 1999 to June 2004) of the daily till using equation (5.14). A s-year tt( NASDAQ return on the S&P 500 return gives i):r.( /2 0.7188 1l) 1.xAso 0001 + 1 4784 x (?-sap ?-) (g(pto g iE. 'I -
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Listings for varius currency futures contract.s from the bqall Street lournal,July21
11! JIEEE. Ir(-I! 1Iq:.1r!-j,1; jE!:. 117:.. Lot SEFQE. (l1(i Hlll L0W Ih1T
currency Futures Japanese Yen lEl..aFtlrt); $ per svpt Il '?3B 9272 (:39 97,9 vcf Est vclu,c37; gna aia pen int 1,55s, +1M. Canadian Dollartr-gl-:ll ltNzt; $ p:r cr .')2a
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Sept .3I De( 2 p Mf5 26 27552 ,7750 Juo z76 1 25 zEst vc17,817: v:1Mnn 14,87; npenlnt 23,225,+t 229. .7514
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.7152
British Pound ((lgF).-Ji?,J(k $ per L sept 12599 18599 18114 L8437 L2712 16339 De( 12357 123$2 18269 :8295 J- tss 12648 L6859 Est vnl 8,551; vI :0n 14,612:clen Int 29,499, 118. 1$Wis S lf 11 () ((r E)-(HF 1 ; $ per (HF n 153 .81F 50pt g 9e: EWvol9,18; vnIhr:n 2,936:npenipt 58,629, 01 -.9155
-%
($11,878)2
q7,155 19,325 77,452 4,531 733 522 76,979 415
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.8
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the daily. standard deviation of the hedged return is
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.8118
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Australian Dollar ((rE)-Atln ltll; $ pEr Atln :755 sept 032 p1( 2,762: .72):
36,529 258 +588. int 36,937, Est vl 3,219;vnlllnc npen 1/1(llti ()11 63 S () ((??!E)-1XN 5B,0; $ 9er hlhl n e 4 Au p 2 SQ)t 8379 43,123 57 p 0227 Estvnl7,243;vcI r/nn18,659;nponint 45,67z +221 E:u t;l tl61() (3111,r ((1E)-t!5,C; $ p2r f E Sopt 1213 lz 3 LB 95 LZ321 u82 2952 115,611 L2781 LD 5 Det 2I19 12432 1J312 L2311 %z 7R15 71R 1 1:4)5 L145 :152 LN!4 :2729 13$3 Est vn122,053;kol lln 14,973;nszn it 117,:83,
;'
An asset manager who picks stocks is often Rlsk management for stock-pickers maling a bet abot the relative, but not the absolute, performance of stock. For on a riskexample, XYZ might be expected to outpedbrm a broad range of stocks . . . ln value will decline XYZ adjusted basis. lf the economy suffers a recession, however, help used islate in this case to even if it outperforms other stocks. Index f'umres can be the relative performance of XYZ. Suppose the return of XYZ i given by the CAPM: .
=
axsz
+ r +
pxvzt/-,'' -
5.6
CONTRACTS
Currency fumres and forwards are widely used to hedge against changes in exchange rates. The pricing of currency contracts is a straightfomard application of the principles we have already discussed. Newspaper listings for exchange-traded currency contracts are shown in Figum 5.4. Many corporations use currency f'umres and forwards for short-term hedging. An importer of consumer electronics, for example, may have an obligation to pay the manufacmrer 7150 million 90 days in the f'uttlre. The dollar revenues from selling these '
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products are likely kown in the short run, so the importer bears pure exchange risk due to te payable being fixed in yen. By buying 7150 rnillion forward 90 days, tlle importer locks in a dollar price to pay for te yen, which will then be delivered to the manufacturer.
Currency Prepaid Forward Suppose tat 1 yer ior today you
want to have Y1. A prepaid forward allows you to pay dollars today t acquire Y1 in 1 year. What is the prepaid forward price? Suppose th yep-denfninated interest rate is ry and the exchange rate today ($/Y) is ab. We can work backward. tf we want Y1 in 1 year, we must have e-ry in yen today. To obtain tat many yen today, we must exhange xze-ry dollars into yen. Thus, the prepaid forward price for a yen is '
.
.
directly from the time series of returns.
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t?n clf?-/-e/lc,'yis
underlying on stock. By deferring delivery of the fc same as thatfor a #?'c#t7i#/tp?-u?J?W j. . lmmediately, msset,youlose income. In ihecase of currency, if you received the currency you could buy a bond denominated in that currency and enrn interest. The prepaid .
l7You can, of course, also compute the correlation coecient
.7775
(5.16)
r)
The tenn axsz in this context represents the expected abnormal return on XYZ. If we aecording to equation use the S&P 500 as a proxy for the market, then we can select H will position be that on average, we earn ax'xz + ?'. (5.14). The result for the hedged The lisk of the position will be given by equation (5.15). Since the correlation of n individual stock and the index will not be close to 1, there will be considerable remaining lisk. risk. However, the portfolio will not have market
CURRENCY
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FORWARDS
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AN D FUTU RES
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RRENCY CONTRACTS
can pay today for the yen, and borrow in dollars to do so. To have 1 yp in 1 yar, we need to invest
forward price reQects the loss of interest from deferring delivery, just as the prepaid forward price for stock re:ects the loss of dividend income. This is why equation (5.17) is the same as that for a stock paying a continuous dividend, equation (5.4).
-rs.T
Abd
'
in dollars, and we obtain this amount by borrowing. 'Fhe required dollar repayment is '
Suppose that the yen-denominated interest rate is 2% and that the ' .).) current exchange rate is 0.009 dollars per yen. Then in order to have 1 yen in 1 year, we L; would invest today q!)Example
which is the forward exchange rate.
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Suppose that -Y() = 0.009, r), = 2%, and r = 6%. The dollar cost = today is 0.009 x e () ()()ggaz. we def'ey the dollar payment S'i yn j from today of 0.008812: (; ts for 1 (; (jjygj.y sjyu t) by bprrowing at 6%, a cost year )(y. . . . . % J'ktl-ansactionis summarizd in Table 5.11.
5.6 lq Exaeple iq of buying 1
Cttrrency Forwrd
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interestarbit#age.
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arlze a fonkr exchange rate reqects th diffefce i itrst rates denolinated in diffefent currencies. Imagine that you wany to invest $1 fof t yelm You. dollar-denominatd exhapge te dollar into C bopd, r ypu ' an do so by byipg k. ' . . . . ' . .. . MO and buy p b.ond denomtnated in tlaat . ier currecy. vbuca then use c urrency ther rate at which youwill covert the foreign currency fomards t juartee the exchaj I'he principle belzind the plicing of currency forwards is that dollars. back into cun-ency lisk-free risk hedged, pays the same return bonds, Fith the ppsition in cunency forign a risklf'ree dolesti bpds. as TO
Suppose that the yen-denominated interst rate is 2% and the dollarExample 5.5 denominated rate is 6%. The current exchange rate is 0.009 dollars per yen. The l-year fomard rate is =
.
The example shows that boqowing in one currency and lending in another creates If w ffset this bofrowig and lending the same cash flow as a foiward conact. acmat ansactipn lj cllel covered the resulting with fonvald contract, ition an pos
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The forward cuaency rate will exceed the current exchange rate when the domestic lisk-free rate is higher than the foreign risk-free rate.ls
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Notice tat equation (5.18)isjust like equation (5.7),for stock index fumres, with . diderence r 1-4, is the foreign interest rate equal to the dividend yield. The interejtfte the cost of cany for a foreign cuaency (webonow at the domestic rate ?' and ipvest the proceeds in a foreign money-market instrument, enrning the foreign rate ?'y as an offset to our cost). If we wish to bonow foreign currency, ?y is the lease rpte. -
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l8Of course if you thinkaboutit, every currency transaction can be expressed in termsof eithercurrency, for example as yen/dollar or dollar/yen. lf the fonvard price exceeds the curfent exchange rate viewed frofn the perspective of one currency, it must be less from the perspective of the other.
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FORWARDS
CHAPTER
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161
reQects the costs and benets delivery are deferred. The pricing of fonvard conacts delivery. 'I'he seller receives and payment later, so the plice is of this deferred payment owed the seller, and the buyer receives possession later, so the Mgher to re:ect interest prelaid forward conact dividends received by buyer. A reflect the not price ij lower to effects. The hence, it today; price of a prepaid separates these t'wo requires payment forward is
Three-month Eurodollar coneacts have maturities out to 10 years, which means that it is possible to use the contract to lock in a 3-month rate as far as 10 years in the future. The September 2007 futures price in Figure 5.5, for exmnple, is 95.1 1. Apositiop in tlzis coneact can be used to lock in an annualized rate of 4.89% from September 2007 to December 2007. The Eurodollar contract can be used to hedge interest rate risk. For a borrower, for example, a short position in the contract is a hedge since it pays when the interest rate rises and requires payment when the interest rate falls. To see this, suppose that 7 months from today we plan to borrow $1 million for 90 days, and that our borrowing qate is the same as LYOR. The Eurodollar futures price for 7 months from today is 94; this implies a 90-day rate of (100 94) x 90/360 x 1/100 = 1.5%. Now suppose that 7 monts hence, 3-month LIBOR is 8%, which implies a Eurodollar futtlres price of 92. The implied 90-day rate is 2%. Our ex'a borrowing expense over 90 days on $1 million will therefore be (0.02 0.015) x $1m = $5,000. Tlzis extra borrowingexpense is offset by gains on the short Eurodollar contract. The Eurodollar fumres plice has gone down, giving us a gain of $25 per basis point, or $25 x 100 x (94 92) = $5,000. The short position in the futtlres contract compensates cost.20 In the snme way, a long position can be usd us for the increase in our borrowing to lock in a lending rate. The Eurodollar f'utures price is a construct, not the price of an asset. ln this sepse Eurodollar f'utures are different fl'om the futures contracts we have already discussed. Although Eurodollar LDOR is closely related to a numier of other interest rates, there is no one specific identisable asset that underlies the Eurodollar futures coneact. dollars, and comparable rates are quoted LIBOR is quoted in currencies other t.1::.r1 in different locations. In addition to LDOR, there are PDOR (Paris), TDOR (Tolyo), and Euribor ttheEuropean Banking Federation). Finally, you might be wondering why we are discussing LIBOR rather than rates on Treasury bills. Business and bank borrowing rates move more in tandem witlztaliltl)R than with the government ' s bon'ow ing rate. Thus, these borrowers use the futures coneact to hedge. LDOR is also a better measure of the cost of funds ff a markt-maker, so L1BOR.is typially used to price folavard contracts. We will f/ther discuss Eurodollar f'utures in Chapter 7.
The purchase of a stock or other asset entails agreeing to a price, making payment, and taking delivery of the asset. A folavard contract sxesthe price today, but payment and
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FORwARDS
PRo BLEMS
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contracmal differences between forwards and futures can lead to pricing differences, but in most cases forward prices and futures plices are vel'y close. In addition to hedging, forward and f'utures contracts can be used to synthetically switch a portfolio invested in stocks into bonds. A portfolio invested in Asset A can remain invested in Asset A but enrn the returns associated with Asset B, as long as there are fonvard or fumres contracts on A and B. This is called a f'umres overlay. The Eurodollar fumres contract, based on LDOR (London Interbarlk Offer Rate) is widely used for hedging interest rate risk. Because the Eurodollar f'utures conact does not represent the price of an asset (at settlement it is 100 L1BORI,it cnnnot be xpricedusing the formulas in this chapter.
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5.2. A $50 stock pays a $1 dividend every 3 months. with the Erst dividend coming 3 months from today. The ontinuously compounded risk-free rate is 6%. a. What is the price of a prepaid forward contract that expires 1 year from today, immediately after the fourth-quarter dividend? b. What is the price of a forward coneact that expires at the same time? 5 3 A $50 stock pays an 8% continuous dividnd. risk-free rate is 6%. *
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b. What is the price of a fonvard contract that eypirs at the same time?
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Chapter 6 continues our exploration of forward markets by eonsidering commodity forwards, which are diferent from linancial forwards in important ways. Chapter 7 then exnmines interest rate forwards. Whereas forward contracts provide a price for delively at one point in time, swaps, discussed in Chapter 8, provide a price for a series of deliveries over time. Swaps are a natural generatization of forward contracts.
'I'he pricing principles discussed in this chapter will also play important roles when option plicing in Chapters 10, 11, and 12 and financial engineerine in Chap-. discuss we 15. ter . To get a sense of the range of traded coneacts, look at the fumres page of the Ftz/f and also explore te-Web sites of f'umres exchanges: te ClzicagoBoard Street Trade of (wwmcbot-com), the Chicago Mercantile Exchange (wwmcme.com),the York Mercantile Exchange (wwwwnymex.com),and tlle London International New Financial Futures Exchange (wwwoliffe.com),among others. These sites typically provide current prices, along wit information about the contracts: What the underlying asset is, how the conacts are settled, and so forth. The site for One Chicago (www. onechicago-com) provides infonnation about single stock f'uturesin the United States. It is well accepted that forward prices are determined by the models and considerations i tlzis chapter. Siegel an (j Siegel (1990)is a standard reference book on fumres. Early papers that examined fumres pricing include Modest and Sundaresan (1983),Cornell and French (1983),which emphasized tmx effects in futtlres pricing (seeAppendix 5.A), and French (1983),which compares forwards and f'utures when both exist on the same underlying asset. Brennan and Schwartz (1990)explore optnal arbitrage when there are transaction costs and Reinganum (1986) explores the arbitrage pssibilities inherent in time t'ravel. There is a more technical academic literattlfe focusing on the difference between fonvard and futtlres contracts, including Blak (1976a),Cox et al. (198 1). lichard and Sundaresan (1981),and Janrw and Oldlield (1981). W
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PROBLEMS 5.1. Construct Table 5.1 from the perspective of a seller, providing a descriptive name for each of the transactions.
5.4. Suppose the stock price is $35 and the continuously compounded interest rate is 5%. a. What is the Fmonth forward price, assuming dividends are zero? b. If the 6-month forward price is $35.50, what is the armualized forward premium? c. Ifthe forward price is $35.50,what is the annualized yield?
continuous
dividend
5.5. Suppose you are a market-maker in S&R index forward contracts. The S&R index spot price is 1100, the risk-free rate is 5%, and the dividend yield on the index is 0. a. Wh at is th no-arbitrage folavard price for delively in 9 rponths? b. Suppose a customer wishes to enter a short index f'utures position. J.f you take the opposite position, demonstrate how you would hedge your resulting long position using the index and borrowing or lending.
;
c. Suppose a customer wishes to enter a long index f'uttlres position. If you tale theopposite position, demonstrate how you would hedge your resulting short position using the index and borrowing r lending. 5.6. Repeat the previous problem, assuming tat te dividend yield is 1.5%. 5.7. The S&R index spot price is 1100, the risk-free rate is 5%, and the dividend yield pn the index is 0. a. Suppose you observe a 6-month forward price of 1135. What arbitrage would you undertake? b. Suppose you observe a 6-month forward price of 1115. What arbitrage would you undertake? 5.8. The S&R index spot price is 1100, the risk-free rate is 5%, and the continuous dividend yield on the index is 2%.
164
%.FINANCIAI-
FORWARDS
P Ro
AND FUTURES
no-arbitrage
B LEMS
k:
165
bound is given
a. Suppose you observe a 6-month forward price of 1120. What arbitrage would you undertake?
5.14. Vel'ify that when there are transaction costs, the lower by equation (5.11).
b. Suppose you observe a 6-mont.hforward price of 1110. What arbitrage would you undertake?
5.15. Suppose the S&R index is 800, and that the dividend yield is 0. You are an arbitrageur with a continuously compounded brrowing rate of 5.5% and a continuously compounded lending rate of 5%.
5.9. Suppose that 10 years from now it becomes possible for money managers to could travel engage in time travel. ln particular, suppose that a money manager 12.5%. Treasul'y the bill 1981, l-year rate Janual'y when was to a. If time travel were costless, whatriskless arbitrage strategycould amoney managerundertake by avelingback and forth between Janual'y 1981 and January 19822
b. lfmany money managrs undertook this strategy, what would you expect to happen to interest rates in 1981? c. Since interest rates wcrc 12.5% in January 1981, what can you coclude about wheter costless time travel will ever be possible? 5.10. The S&R index spotprice is 1100 and the continuously compounded risk-free is 5%. You observe a g-month forward price of 1129.257.
rate
a. What dividend yield is implied by this forward price? b. Suppose you believe te dividend yield over the next 9 months will be only 0.5%. What arbitrage would you undertake? youbelieveihe dividend yield will be 3% overthenextg c. Suppose would you undertake? arbitrage What
months.
5.11. Suppose the S&P 500 index f'utures price is currently 1200. You wish to purchase four futures contracts on margin. a. What is the notional value of your position? b. Assuming a 10% initial margin, what is the value of the initial margin? 5.12. Suppose the S&P 500 index is cuaently 950 and tlle initial margin is 10%. You wish to enter into 10 S&P 500 futures contracts. a. What is the notional value of your position? What is the margin? b. Suppose you enrn a continuously compounded rate of 6% on your margin balancea your position is markd to market weekly, and the maintenance margin is 80% of te initial margin. What is tlae greatest S&P 500 index f'uturesprice 1 week fwm today at wlzich will you receive a margin call? 5.13. Verify tat going long a forward conact and lending the present forward price creates a payoff of one share of stock when a. The stock pays no dividends. b. The stock pays discrete dividends. c. The stock pays continuous dividends.
value of
the
a. Supposing that there are no transaetion fees, show that a cash-gnd-carry arbitrage is not proftable if the forward plice is less tan 845.23. and that a reverse cash-and-carry arbitrage is not protable if the forward price is greater than 841.02. b. Now suppose that there is a $1 transaction fee, paid at time 0, for going either long or short the forward contract. Show that the upper and lower no-arbitrage bounds now become 846.29 and 839.97. c. Now suppose that in addition to the fee for the fonvard contract, there is also a $2.40fee for buying or selling the index. Suppose the contract is settled by delivery of the index, so that this fee is paid only at time 0.
What are the new upper and lower no-arbitrage bounds? d. Make the same assumptions as in the previous part, except assume that the coneact is cash-settled. This means that it is necessary to pay the stock index transaction fee (butnot the fonvard fee) at both times 0 and 1. What are the new no-arbitrage bounds?
e. Now suppose that transactions in the index have afee of 0.3% of the value of the index (thisis for both purchases and sales). Transactions in the forward conact still have a fixed fee of $1 per unit of th inex at time 0. Suppose the contract is cash-settled so that when you d a cash-andcan'y or reverse cash-and-carry you pay the index ansaction fe both at time 1 and time 0. What are the new upper and lower no-arbitrage boundj? Compare your answer to that in the previous pal't. Hint: To handle the time 1 transaction fee, you may want to consider tailing the stock position.) 5.16. Suppose the S&P500 currently has a level of 875. The continuously compounded return on a l-year T-bill is 4.75%. You wish to hedge an $800,000portfolio that has a beta of 1.1 and a correlation of 1.0 with the S&P 500. a. What is the l-year futures price for the S&P 500 assuming no dividends? b. How many S&P 500 f'utures contracts should you short to hedge your portfolio? What ret'ul'n do you expect on the hedged prtfolio? 5.17. Suppose you are selecting a fumres contract with which to hedge a portfolio. You have a choice of six contracts, each of which has the same variability, but with 0, 0.25, and 0.85. Rankthe f'umres contracts correlations of highest basis risk, from to lowest basis risk. with respect to -0.95,
-0.75,
-0.50,
166
k. FINANCIAL
FORWARDS
5.18. Suppose the current exchange rate between Germany and Japan is 0.026/Y. 'I'he euro-denominated annual continuously compounded risk-free rate is 4% and the lisk-free rate is 1%. What yen-denorninated annual continuously compounded yerl/euro price? forward arld euro/yen 6-month te are 5.19. Suppose the spot $/7 exchangerate is 0.008, the l-yearcontinuopsly compounded dollar-denominated rate is 5% and the l-yeai continuously compounded yendenominated rate is 1%. Suppose te l-year fonvard exchange rate is 0.0084. Explain precisely the eansactionsyou could use (beingcareful about currency of denomination) to make money wit.h zero initial investment and no risk. How much do you make per yen? Repeat for a fonvard exchange rate of 0.0083.
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5.A: TAXES
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5.20. Suppose we wish to borrow $l0.millionfor 91 days beginning next June, and that the quoted Eurodollar fumres plice is 93.23.
APPENDIX
5.B;
APPENDIX
AN D FUTU RES
69,573.25
-86.24
34,813.80
-17.44.
4.35
43,553.99
Appendix available online at www.aw-bc.com/mcdonald.
APPENDIX
FORWARDS
5.B: EQUATING
AND FUTURES
Because the futures price exceeds te prepaid forward price, marling-to-market has the effect of magnifying gains and losses. For example, the futtlres price on a nondividendpaying stock is Fa,w = Sterl'. J.f the stock price increases by $1 at time 0, the gain on crW. Thus, in order to use futures to precisely hedge a the futures contract at time F is F) it is necessm'y to hold fewer fumres time settled being at position (withthe hedge volatility induced by the future the offsetting ffectively extra forward contracts, than 5.13, fewer than eight contraets, to Table long in the example In we can go value factor. marldng-to-market. of effect for male up the Table 5.13 shows the effect of this adjustment to the f'uttlres position and how it is adjusted over time. lnitially, we go long
8 x e -0.06x9/52 y qjess =
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$1979.34, contracts. This number of contracts has a multiplier of $250 y 7.91735 offsets the of number the table. of the conacts Reducipg first multiplier the in the row until increase expiration, time is less there week Each so we of earning interest. effect the number of index units we are long. Prost on this position is 10/52 $43,553.99 $217,727.21c0.06x sj,y6 =
,y()()
-
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If the folavard curve is downward sloping, as with gasoline, we say the market is in
backwardation. Forward curves can have portions in backwardation and portions in contango, as does tat for crude oil. lt would take an entire book to cover commodities in depth. Our goal here is to understand the logic of forward pricing for commodities and where it differs from te logic of Enapcial fonvard plicing. What is the forward curk e iellingus aopt the market ' for the commodity?
.
Sottrce: Futures data from Datastream.
Fluly = 3 14.25d (0.0122-(-0.0919))x(1/6)
FORWARDS
g jjl y5 .
How do we interpret a negative dividend yield? Perhaps even more puzzling, given our discussion of financial futures, is the subsequent drop in the corn futures pricefromluly to September, and the behavior of soybean, gasoline, and crude oi1 prices, which all decline with time to expiration. lt is possible to tell plausible stolies about this behavior. Corfl and soybeans are harvested over the summer, so perhaps te expected increase in supply accounts for the reduction over time in the futures price. In May 2004, the war in traq had driven crtlde oil prices to high levels. We right guess that prod.ucers Would respond by increasing supply and consupers by reducing demand, resulting in lower expected oil prices in subsequent months. Gasoline is distilled from oil, so gasoline prices might behave similady. Finally, in contrast to the behavior of the other commodities, gold prices rise steadily over time at a rate close to the interest rate. Itseems tatwe can tell stolies about the behavior of forward prices overtime. But how do we reconcile these explanations with our understanding of snancial fonvards, in whih forward prices depend on te interestrate and dividends, and explicit expectations of future prices do not enter the fprward price fonnula? The behavior of forward prices can vary over time. Two terms often used by If on a given date the forward commodity traders are contango and backwardation. is upward-sloping-i.e., plices distant in time are higher-then fonvard more curve is in market We obselwe this the contango. pattern with colm in Table 6.1. we say
6.2 EQUILIBRIUM
OF COMMODITY
PRICING
FORWARDS As with forward prices on financial assets, commodity forward prices are the rejult of a present value calculation. To understand tllis, it is helpful to consider synthetic
commodities.
Just as we could create a synthetic stock with a stock fonvard conact and a zerocoupon bond, we can also create a synthetic commodity by combining aforward contract with a zero-coupon bond. Consider the following investment sategy: Enter into a long commodity fonvard contract at the price Fa,w and buy a zero-coupon bond that pays F0,w at time F. Since the forward conact is costless, the cost of this investment strategy at time 0 is just te cost f the bond, or Time 0 cash flow = .-e-rrF
(6 2)
0.T
.
At time F, the strategy pays Sv
-
Fn.w
Fn r
Fonvard contract payoff
Bond payoff
where Sv is the time F price of the commodity. This investment strategy cfeates a in that it hms the same value as a unit of the commodity at time synthetic c/ppn//y, equation F. Note tat, from (6.2),the cost of the synthetic commodity is the prepaid fonvard price, e - r r F() r. Valuing a synthetic commpdity is easy if we can see the forward price. Suppose, however, that we do not krkow the fonyard price. omputing te time 0 vtu of a unit of the commodity received at time F is a standard problem: Ypu discount the expected commodity price to determlne its value today. Let Ezs.j depote te expected time-r price as of time 0, and let a denote the appropriate discount rate for a time-/ cash flow of Sv. Then the present value is E o(Sz')e-C'F
(6 3) .
The important point is tlaat expressions (6.2) tz??l (6.3) repreznf the value. Both would what today receive unit of the commodity re:ect to at time F. pay you oe Equating te two expressions, we have ,tt??lc
e
-rFF
0,r
=
E 0 tky-le-cr
(64) -
;
172
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COMMODITY
FORwARDS
AlqD Fu'ruREs
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RAB l LITY: ELECTRI CITY
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Remanging
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we can write the forward price as )d-t'T Fn w uu:e rTE ()(5'w ICY-XX = X 0 (S T
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(6.5)
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ELECTRICITY
illustratesforwardpricing when storageis notpossible. Theforwardmarketforelectzicity Electricity is produced in different ways: from fuels such as coal and natural gas, or from nuclear power, hydroelectric power, wind power, or solar power. Once it is prduced, electricity is transmitted over the power grid to end-users. Electricity has charteristics
--
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l Historical commodity and futures data necessary to estimate expected cofnmodity returns are relatively hard to obtain. Bodie and Rosansk'y (1980)examine quarterly fumres rettzrns from 1950 to 1976, while Gorton and Rouwenhorst (2004)examine monthly futures remrns frolp 1959 to 2004. plus commodity ftures-and Both smdies construct portfolios of synthetic commodities-eflbills 5nd that these pordblios earn the same average return as stocls, are on average negatively correlated with stocks, and are positively correlated with inllation. These lindings imply tat a portfolio of stocks and synthetic commodities would have the same expected return and less I'iSIC than a diversised stock portfolio alone.
'. . ::
Equation (6.5) demonstrates the EIIIC between the expected commodity price, Efjspl, and the forward price. As with linancial forwards (seeChapter 5), the fonvard price is a biased estimate of the expected spot price, Ezvpj, wit the bias due to the risk premium r.1 on the commodity, a Equation 6.4 deserves emphasis: The #??pc-rft/?aptx/dr?efcc discottnted at fc ?@kofa ttllit t7/c(???2?nt?t'fi/)/ receive at #?nc F. 0 - / -ee rate back to rfl?yd is r/zcpresent vt-lfd This calculation is useful when pedbrrning NPV calculations involving commodities for which fomard prices tre available. Thus, for example, all industrial producer who buys oil can calculate the present value of f'uture oil costs by discounting oil forward prices at the risk-h'ee rate. The present value of future oil costs is not dependent upon whether or not the producer hedges. We will see an example of this calculation later in the chapter. lf a commodity cannot be physically stored, the no-arbitrage plicing principles ' discussed in Section 5.2 cnnnot be used to obttin a fonvard price. Without storage, equation (6.5) detrmines tlze forward price. However, it is dicult to implement this formula, which requires forecasting the expected f'uture spot price and estimating a. Moreover, even when physically possible, storage may be costly. Given the difculties fomard prices and to of pricing commodity fomards, our goal will be to markets. of different economics the understand commodiY In the rest of the chapter, we will f'urther explore similarities and differences betweenforwardprices forcommodities andsnancial assets. Some of the mostimportant differences have to do with storage: whether te commodity can be stored and, i.f so, how costly it is to store. The next section provides an example of folavard prices when a commodity cannot be stored.
6.3 NONSTORABILITY:
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that distinguish it not What is special about First, electricity or else it is wasted.z
only from financial assets, but from oter commodities as well. electricity? is diflicult to store, hence it must be consumed when it is produced Second, at any point in time the muimum supply of electricity is fixed. You can produce less but not more. Third, demand for electricity varies substantially by season, by day of week, and by yime of cy. To illustrate the effects of nonstorability, Table 6.2 displays l-day ahead hourly ptices for 1 megawatt-hour of electricity in New York City. The l-day ahead forward price is $28.37at 3 A.M., and $62.71at 3 P.M. Since you have learned about arbitrage, you are possibly thinking that you would like to buy electricity at the 3 A.M. price and sell it at the 3 P.M. price. However, there is no way to do so. Because electricity cannot be stord, its price is set by demand and supply at a point in time. There is also no way to uy winter electricity and sell it in the summer, so there are seasonal variations as well as inaday variations. Because of peak-load plants that operate only when prices are high, pbwer suppliers are able to temporan'ly incregse te supply of eletricity. However, expectations about supply are already re:ected in the folavard price. Given tese chgractelistics of electricity, what does the electricity forward price represent? The prices in Table 6.2 are best interpreted using equation (6.5). I'he large price swings over the day primarily re:ect changes in the expected spot price, which in t'tlrflreiects changes in demand over the day. Notice two things. First, the swings in Table 6.2 could not occur with financial assets, which are stored. (It is so obvious that snancial assets are stored that we usually don't mention it.) As a coysequence, the 3 A.M. and 3 P.M. forward prices for a stock
z'There are ways to store electricity. For example, it is possible to use excess electricity to pump water uphill and ten, at a later time, release it to generate electricty. Storage is uncommon, expensive, and entai1S losses , however.
J
174
%:CoMuool'ry
FORwARDS
arbitrage, will be almost identical. If they were not, it would be possible to engage in forward price for whereas the P.M. Second, buying low at 3 A.M. and selling high at 3 current infonnation about te it reqects that a stock is largely redndant in the sense 6.2 Table provide prices in fonvard stock price, interest, and the dividend yield, the information we ould not otherwise obtain, revealing infonnation about the future price price discovery, with of the commodity. This illustrates the forward market providing obtainable, about the fmure price of otherwise fonvard prices revealing information, not the commodity.
6.4
FORWARDS PRICING COMMODITY AN EXAMPLE BY ARBITRAGE: Electricity repreyents the exlme of nonstorability. However, many commodities are storable. To see the effects of storage, we now consider te vely simple, hypotetical because they exmple of a forward conact for pencils. We use pencils as an example afotward shuld work. such abouthow preconceptions will have no are fnmiliar and you because it does not exist. The Suppose tat pencils cost $0.20today and for certain will cost $0.20in 1 year. economics of this assumpon are simple. Pencil manufgcturers produce pencils from prduction, more wood and otherinputs. lfthe price of apencil is greaterthan the cost of price the falls, fewer pencils pdce. If the market down driving pencils are produced, market price of pencils thus re:ects te cost of are produced arld te price rises. The pelfectly elastic. production. An economist would say that the supply of pencils is There is notlting inherently inconsistent about assuming that the peneil price is expected to stay the snme. However, before we proceed, note that a constnt price stock. would not be a rasonable assumption about the price of a nondividend-paying wold own A nondividend-paying stock must be expected to appreciate, or else no one commodity and a snancial this difference between obvious there is an it. At the outsek assey.
that, One way to desclibe this ifference between the pencil and the stock is to say ostored. Tllis in equilibrium, stocks and otherfinancial assets mustbeheld by ivestors, for investors to appreciation is necessary appreciates prie stock average', why on the is willingly store the stock. pencils The pencil, by contrast, need not be stored. 'Fhe equilibrium condition for requires that price equals marginal production cost. This distinction between a storage commmodities.3 production equilibrium is a cenal concept in our discussion of
and
COM MODITY
PRICING
AND FUTURES
BY ARBITRAGE:
$
AN EXAM PLE
Now suppose that the continuously compounded interest rate is 10%. What is the forward price for a pencil to be delivered in 1 year? Before reading any further, you should stop and decide what you think the answer is. (Really. Please stop and think
about itl) ne obvious possible answer to this question, drawing on olzr discussion of finanforwards, is that the forward price should be te f'uttlre value of the pencil price: cial c0.l x $0.20 $0.2210.However, common znx suggests r:7l this canat?l be #)d corrccl answer. You know that the pencil price in one year will be $0.20. If you entered into a forward agreement to buy a pencil for $0.221,you would feel foolish in a year when the price was only $0.20. Common sense also rules out the forward price being less than $0.20. Consider the forward seller. No one would agree to sell a pencil for a forward price of less than $0.20, knowing tat te price will be $0.20. 'rhus, it seems as if both the buyer and seller perspective lead us to the conclusion that te forward price must be $0.20. =
An Apparent Arbitrage and Resolution Ifthe fomard price is $0.20,is there an arbieage opportunity? Suppose you believe that the $0.20forward price is too low. Following te logic in Chapter 5, you would want to buy the pencil forward and short-sell a pencil. Table 6.3 depicts te cash flows in this reverse cash-and-carry arbiage. The result seems to show that there is an arbitrage
oppormnity.
We seem to have reached an impasse. Common sense suggests a forward price of $0.20, but the application in Table 6.3 of our fonnulas suggests that any fonvard price less t.1)a11 $0.221leads to an arbitrage oppo%nity, where we would make $0.221 F(),l pencil. per -
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3''ou may be thinking that you have pencils in your desk and therefore you do, in fact, store pencils. inconvenience of going to the store each time you However, you are storing them to save yourself te stock. When be pencils because expect a good financgalinvestment akin to to need a new one, not you for moment, suppose the Thus, time. only a few at a storing pencils for convenience. mu will store convenience in Section 6.6. stofing of for concept the We ret'urn to that no one stores pencils.
FORWARDS
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PRICING
Once again it is time to stop and think before proceeding. Examine Table 6.3 closely; there is a problem. The arbitrage assumes that you can short-sell a pencil by bonowing it today and remrning it in a year. However, recall that pencils cost $0.20today and will cost $0.20 in a year. Borrowing one pencil and returning one pencil irl a year is an interest-free loan of $0.20. No one kvill fdn#l't/ll the pencil w/tlllr charging ytplfan additionalfee. lf you are to short-sell, there must be someone who is both holding the asset and willing to give up physical possession for the period of the short-sale. Unlike stock, nobody holds pencils in a brokerage account. It is straightforward to borrow a financial asset and ret'urn it later, in the interim paying dividends to the owner. However, if you borrow an unused pencil and return an unused pencil at some later date, the owner of the pencil loses interest for the duration of the pencil loan since the pencil price does not change. Thus, the apparent tz?'lpg?w.c in the tltpvc table :tu nothing at all to do w#7 fot-ward ctl//lrcl. on pencils. lf you tind someone willing to lend you pencils for a year, y0u should bonrw as many as you can and invest the proceeds in T-bills. You will earn the interest rate and pay nothing to bonow the money. maldng You might object that pencils do provide a flow of seNices-nnmel having physical of the service llow requires possession tis However, marks on paper. short-seller continues pencil. stock loaned A and it also the to to pencil enrn its a uses up the lender. Consequently, short-seller pencil for the loaned the replrn to enrns no return', the pencil borrower must make a payment to the lendef to compensate the lender for lost time value of money.
Pencils Have a Positive Lease Rate How do we correct the arbitrage analysis in Table 6.32 We have to recognize that the lender of the pencil has invested $0.20 in the pencil. In order to be kept snancially whole, the fc/7#c?-ofa pellcil 1W#reqltire Il. to pay fnrc?'c'r. The pencil therefore has a lease rate of 10%, since that is the interest rate. With this change, the corrected reverse 6.4. cash d-can'y arbitrage is in When we correctly account for the lease payment, this transaction no longer earns prots when the forward price is $0.20or greater. lfwe t'urn the arbitrage around, buying the pencil and shorting the forward, the cash-and-can'y arbitrage is depicted in Table 6.5. These calculations show that any forward plice greater t.1):.1.1 $0.221generates arbitrage
COM MODITY
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Using no-arbieage aruments, we have nlled out arbitrage for fonvard prices less than $0.20(go long the forward and short-sell the pencil) and peater than $0.221(go short the folavard and long the pencil). However, what if the forward price is between $0.20 and $0.221? lf there is an active lending market for pencils, we can narrow the no-arbitrage lease price even further: We can demonstrate that the forward price nlltst be $0.20. lender 10% by buying the and Therefore pencil of pencil 10%. pencil is can earn a rate a lending it. The lease payment for a short seller is a dividend for the lender. lmagine that
BY ARBITRAGE:
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tf instead we use an effective annual interest rate, ?', the effective anntlal lease rate is (1 + ?-) 1 (6.12) t (F0 T/S) 1/W =
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Sometimes it makes sense for a commodity to be stored, at least temporarily. Storage is also called carry, and a commodity that is stored is said to be in a carfy market. One reason for storage is seasonal variation itl either supply or demand, which causes a mismatch between the time at which a commodity is produced and the time at which it is consumed. With some agricultural products, for example, supply is seasonal (there is a harvest season) ut demand is constant over the year. ln this case, storage perrnits consumption to occur throughout te year. With nattlral gas, by contrast, there is high demand in the winter and low demand in the summer, but relatively constant production over the year. This pattern of use and production suggests that tere will be times when natural gas is stored.
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181
In some markets, consistent and reliable quotes for the spot price are not available, or are not comparable to forward prices. ln such cases, the near-term forward price can be used as a proxy for the spot price, S. upward-sloping fonvard curve--occurs when the By desnition, contango-an risk-free downward-sloping forward rate. Backwardation-a lease rate is less than the culwe-occurs when the lease rate exceeds the risk-free rate.
qi r' EL:r?STr:rq?TI)Y 1* jjtf y''yy. yyjf j'l'y' ))-*-* jryf y' ;y' yyf j.'yj)'. (5* ;'jjf (' (t' j'. p' rjjjjyjjjjj,jjjj.d j'jyf )' r:: (. r (;((iq! ttlf 't'jjltyd y' . ;(!(.q( tjt'ytjtj. .gjjjy' . . tjjjyytjjjyjyjj(.'. I g..j.).j.j(j ;.. j j ttr )yy. . ( ryl'. tlqj y k j y y . r y yj j: . . .j casla-and-carry ''ltrx, arbitrage witla a commodity for whcb the Iease . . . . k'ti-:LL...--j,.,.....-,.... t -//1L.' :s Szer-<T restricton is jjtE:
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Storage Costs and Forward Prices Storage is not always feasible (for exnmple, f'resh strawberries are perishable) and when technicily feasible, storage is almost always costly. When storage is feasible, how do storage costs affect forward pricing? Put yourself in the position of a commodity merchant who owns one nit of the commodity and ask whether you would b Willing to store this unit until time T. You face the choice of selling it today, receivin %, or selling it at time F. lf you elect to sell at time F, you can sell forward (toguarantee the price you will receive), and you will receive Fo,w. This is a cash-ud-cany The cash-and-carry logic with storage costs suggests thatyyll vill store (7?11, ifthe present vc/l/c ofselling at l??7d F is at least as great a l/7t# ofselling today. Denote te f'uttlrevalue of storage costs for one unit of the commodity from time 0 yo F as (0, F). lndifference between selling today and at time F requires
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(6.13)
(0 r) ,
the special case where storage costs are paid continuously and are proportional to te 1.11 value of the commodity, storage cost is like a continuous negative dividend of and we can write the forward price as ,
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=
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182
FORwARDS
AND FUTURES
CARRY MARKETS
0), equations (6.13) and (6.14) reduce to our When there are no storage costs ( fnmiliar fomard pricing formula from Chapter 5. When there are storage costs, the folavard plice is higher. Why? The selling price cost of storage (interest) must compensate the commodity merchantforboth the snancial the forward curve can rise faster and the physical cost of storage. With storage costs, dividend negative in that, instead than the interest rate. We can view storage costs as a hold the receiving cash flow for holding the asset, you have to pay to asset. of
-et'ur'n
Example 6.1 supposethattheNovemberprice of cornis $2.50/bushel.the efective 1.'j! l'1mont hl y imerestrateis 1%, and storage costs perbushel are $0.t5/zponth.Assuming that lij lj) is stored Npvepber to Fbwary, the February forward price must compensate
itlf
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=
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;j-@j1 jtEt l ji)
2.50. x.
of storage costs is
($0.05/.01)x
$0.1515
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=
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l Problem 6.9 asks you to verify tat this is a no-arbitrage price. Keep in mind that just because a commodity can be stored should (or will) be stored. Pencils were not storedbecause storage necessary: Aconstant new supply of pencils was available to meet equation (6.13)desclibes the fonvard price vhen storage lccl/r'. cmmodity is stored are peculiar to each commodity.
183
corn, you can sell the excess. However, if you hold too little and run out of corn, you must stop producing, idling workers and machines. Your physical inventory of corn in this case has value-it provides insurance that you an keep producing in case there is a disrnption in the supply of corn. In this simation, corn holdings provide an extra nonmonetal'y return that is sometimes referred to as the convenienceyield-s You will be willing to store corn with alower if you did not enrn the convenince yield. What are the implications t.11:.r1 rate of of the convenience yield for the forward plice? Suppose tat someone approached you to borrow a commodity from which you derived a convenience yield. You would think as follows: I lend the commodity, 1 bealing interest cost, saving storage cost, and losing the value 1 deri've from having a physical inventoly I was willing to bear the interest cost already; thus, 1 will pay a commodity borrower storage cost less the ctwvdnfcncc yield.'' Suppose the continuously compounded convenience yield is c, proportional to the value of the commodity. The commodity lender saves c by notphysically stoling the commodity; hence, tile commodity borrower pays compensating the lender c fgr convenience yield less storage cost. Using an argument identical to that in Table 6.8, we conclude that the forward price must be no less than
=
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k
Storage Costs and the Lease Rate
Fo w
The discussion of commodities to this point has ignored business reasons for holding commodities. For example, suppose you are a food manufacturer for whom corn is an essential input. You will hold an inventory of corn. If you end up holding too much
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Etvirtual
The Convenience Yield
=
,
This is the restlicon imposed by a reverse cash-and-cany in which the arbitageur borrows the commodity and goes long the fomard. Now consider what happens if you pelform a cash-and-cany buying te commodity and selling it folavard. If you are an average investor, you will not enrn the convenience yield (itis enrned only by those with a business reason to hold the commodity). Youcould try to lend the commodity, reasoning that the borrower could be a commercial user to whom you would pay storage cost less te convenience yield. But those who earn the convenience yield lilely already hold the optimal amount of the commodity. There ??l7ybe ?lt? wayforyou to /tz,77 the convenience yield w/3Nlpc/ft/?-/?lfng a caslvand-carly. Those Fho do not earn the convenience yield will not own the commodity. rfhus,-/br an tzvcmgp investor, the cash-and-carry has the cash flow/
does not mean that it was not econornically pencil demand. Thus, Whether and when a
Supjose that there is a c?)rry mrket for a commodity, so that iys forward price is given by equation (6.13).What is the leas rate in this case? Again put yourselfin the shoes of the commodity lender. Ifyou lepd the commodity, you are saved f'rom having to pay storage cost. Thus, the lease rate sould equi the negative of the storage cost. In other words, the lender will pay the borrower! In effect, storage'' for the commodity lender, who the commodity borrower is providing point the the The lender maling a payment to back commodity in future. receives at a dividend. negative borrower the generats a
1: Sze (r-5)r
-
e(r+)F
Szer-bksT '
if there is to be
?'
s'rhe term convenience yield is desned differently by different authors. Convenience yield generally means a return to physical ownership of the commodity. ln practice it is sometimes used to mean te lease rate. ln this book, the lese rate of a commodity can be infen'ed from the fonvard price using equation (6.11). 61n this expresson, we assume we tail the holding of te commodity by buying cF units at time 0, and selling off units of the commodity over time to pay storage costs.
.
k
184
% COMMODITY
FORwARDS
ANo
FUTURES
GOLD !;' ;';' TEjjjjj::j;;k;'E '')4/*2r'1:* #' k,f )' j'(' ;':'',' (('E: ;..' : ! jE )y--,;jjjjj'. ('jjjjy:jyyti q' j')' f' jf jkf ' ' ''(((''''('g('' !'. (' )' yry'r')?rll'lf jjjlf )': j' k jj(g g :j(i.j!(yjkkjjjjkl .:.- i! jg..(.jrq .ji!r .(.(.. ' E.. . . .i. - . p'E . i..- -. - ! E.E. E- .,.!....-..........-.-...............i .
ln summary, from the perspective of an arbitrageur, the price range within which there is no arbitrage is Sze (r+l-c)F
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The conveniene yield produces a no-arbieage region rather tan a no-arbiage price. The observed lease rate will depend upon both storage costs and convenience. Also, as in Section 5.3, bid-ask spreads and trading costs will further expand te no-arbitrage region in equation (6.15). As anoter illustration of convenience yield, consider again the pencil example xof Section 6.4. 1.nreality, everyone stores a few pencils in order to be sure to have one available. You can think of this benestfrom storage as the convenience yield of a pencil. However, because the supply of pencils is perfectly elastic, the ptice of pencils is Exed at $0.20. Convenience yield in this case does not affect the forward price, but it does explain the decision to store pencils. The difficu'lty with the convenience yield in practice is tat convenienc is hard to obselwe. The concept of the convenience yield serves two purposes. First, it explains example, why a commercial user might store a commodity ' P atterns in storage-for when the average investor will not. Second, it provides an additionlll parameter to better explain the forward culwe. You might object that we can invoke the convenience yield to explain alt)' forward curve, and therefore the conceptof the convenience yield is vacuous. While convenience yield can be tautological, it is a meaningful economic concept and it would be just as arbitrary to assume that there is never convenience. Moreover, the ' upper bound in equation (6.15)depends on storage costs but not tlle convenience yild. Thus, the eonvenience yield only explains anomiously low forward pricej, and only when there is storage. We will now examine particular commodities to illusate the concepts from the previous sections.
6.7 GOLD FUTURES Gold is durable, relatively inexpensive to store (comparedto its value), widely held, and actively produced through gold mining. Because of transportation costs and purity concerns, gold often trades in certificate form, as a claim to physici gold at a specisc location. There m'e exchange-traded gold futures, specications for which are in Figure
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AND FUTURES
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is constructed using more expiration dates than are in Figure 6.2.) What is interesting about the gold forward curve is how relatively uninteresting it is, wit the fonvard price steadily increasing with time to mattlrity. From our previous discussion, the forward price implies a lease rate. Shortlsales and loans of gold are common in the gold market, and gold bon-owers in fact have to pay the lease rate. On the lending side, large gold holders (includingsome central bnnks) put gold on deposit with brokers, in order that it may be loaned to short-sellers. 'I'he gold lenders earn the lease rate. The lease rate forgold, silver, and other commodities is computed in practice using reporting services. Table 6.9 shows - equation (6.12)and is reported roudnely by fnancial the 6-month and l-year lease rates for the four gold forward culwes depicted in Figure 6.3, computed using equation (6.12);
Ie 6 Z Here are the details of computing the 6-month lease tate for June 2001. Gold futures plices are in Table 6.9. The June and Septembr Eurbdor ftlttlfes IZCCS On this date Were 96.09 and 96.13. Thus, LIBOR h'om Vne s-mont.h September t and 0.988+, from 91/90 96.09)/400 x September (100 was to 0.978%. The June to December interest December was (100 94.56)/400 x 91/90 1.9.763%, or 1.0197362 annualizd. rate was terefore (1.00988)x (1.00978) 1 Using equation (6.12),the annualized &month lease rate is therefore EXamp
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Gold Investments ((fyou wish to hold gold as part of an investment portfolio, you can do so by holding physical gold or syntetic gold-i.e., holding T-bills apd going long gold futures. Which should you do? lf you hold physical gold without lending it, and if the leas rate is positive, you forgo the lease rate. You also bear storage costs. With synthetic gold, lisk. on the other hand, you have a counterparty who may fail to pay so there is credit lgnorin.j credit risk, however, synthetic gold is generally te preferable way to obtain gold plice exposure. Table 6.9 shows that the 6-month annualized gold lease rate is 1.46% in June 2001. Thus, by holding physical gold instead of synthetic gold, an investor would lose this 1.46% ret-tllql.s Inlune 2003 and 2004, however, the leaserate was about tf storage costs m'e about 0.10%, an investor wouldbe indifferentbetween holdingphysical and synthetic gold. 'I'he f'uttlres market on those dates was compensating investors for storing physical gold. yield from gold. Some nonfinancial holders of gold will obtain a convenience Consider an eleceonics manufacttlrer who uses gold in producing components. Suppose that running out of gold would halt production. lt would be natural in this case to hold a buffer stock of gold in order to avoid a stock-out of gold, i.e., running out of gold. For this manufacturer, there is a retllrn to holding gold-namely, a lower probability of stocking out and halting production. Stocldng out would have a real financial cost, and lease rate-to avoid that cost. the manufacmrer is willing to pay a price-te -0.10%.
Evaluation of Gold Production Suppose we have an operating gold mine and we wish to compute the present value
of future producon. As discussed in Section 6.2, the present value of the commodity rate--of received in the f'umre is simply the present value--computed at te lisk-free the forward price. We.can use the forward curve for gold to compute te value of an --;-' -' -y' 937:1147'744747'/744* ,j'. ..' i'E.q.:' -' -'. jj:::y:,,f --' C' :')' L' k' ....' Etj'lf dl:::::,,f dl:::::,hf 'y''C;' j'k' k'))))yy'.' ).(;'' ?' (E E EE ' ((E -
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B'T'hecost of l ounce of physical gold is %. However, from equation (6.10),the cost of 1 ounce of gold Sf)e-'%T. Synthetic gold is proportionally cheaper by the lease rate, &l.
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188
S EASONALITY:
AND FUTURES
FORWARDS
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As discussed in Section 6.6, storage is an ecopornic decision in which tere is a . trade-offbetwee selling today and selling tomorrow. Ifwe can sell cprn today for $2/17t1 and in 2 months for $2.254m,t storage decision entails comparing the price Wecan get Wecan get in 2 months. ln addition to interest, toda# wit te ' prsilt ' value of ihe plice ''' . . .. . anysis. incllde' need to toragecosts in our we An equtllbrium with olpe current selling and spme stopge tquires thgt col'n tjj pricesb expted to rise t e interest rate plus storage costs, Wllichimpjj es tjyat tjj ere upward trend in the prlce between harvets. While corn is beipg stred, the an willbe . . . fonvrd jrice shoutd behave s in equatipn (6.14),rising at interest plus stopge costs. if supply ad demand Onc th harvest begins, storage is no loger necsary; harvest price will be the fiom remain constant year to year, the same vl'y year. The Iising after the harvest. will that level harvest, only begin again plice fatl to at to corn The market conditions we have described are graphed in Figure 6.5, which depicts a hypothetical forward ctu've as seen from time 0. Between harvests, the fprward price '
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how the possibility of mine closings due to 1ow plices affects valuation.) Note that in equation (6.16),by computing the present value of the forward price, we compute the prepaid forward price. 4*p Example 6.3 t'''j Suppose we hay.e a rnining project that will produce 1 ounce of gold )) jj every year for 6 years. The cost of this project is $1,100today, the marginal cost per compoundd interest rate ounce at the time of extraction is $100, and the continuously y't '1 .hitis 6%. #, We observe the gold forward prices in the second column of Table 6.10, with ')g implied prepaid forward prices in the third column. Using equation (6.16),we can use j . t,! these prices to perform the necessary present value calculations.
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THE MARKET
6.8 SEASONALITY: FORWARD
CORN
Corn in the United States is halwested primadly in the fall, from September through November. The United States is a leading corn producer, generally exporting rather than importing corn. Figure 6.4 shows a newspaper listing for corn futures. Given seasonality in production, whpt should the forward curve for corn look like? Corn is produced at one time of the yem but consumed throughout the year. ln order to be consumed when it is not being produced, corn must be stored. Thus, to understand the forward curve for eorn we need to recall our dijcussion of storage and can'y markets.
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190
AND FUTURES
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of corn rises to reward storage, and it falls at each harvest. Let's see how this graph was constructed. The corn price is $2.50initially, the continuously compounded interest rate is 6%, and storage cost is l.s%/month. The folavard price after n months (where?7 < 12) is ' .
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during this period, with the futures price compensating for storage. A low current ptice suggests a large supply. Thus, when the near-luly price is low, we might also expect storage across the coming halwest. Particularly in th years with the lowest July prices (1999-.-2002),there is a pronounced rise in price from July to December. When the price is unusually high (1996and 2004), there is a drop in price from July to December. Behavior is mixed in the other years. q We can al>o examine the July-December price relationship in the following yetr. In 6 of the 10 years, the distant-December plice (column 8) is below the distant July plice (column6). The exceptions occur in years These patterns are generally consistent with relatively low current prices (1998-2001). with storage of corn between harvests, and storage across harvests only occasionally. Fi n ally compare prices for the near-luly contract (the lipt column) with those for the distant-December contract (the last column). Near-term prices are quite variable, ranging from 216.75 to 435.00 cents per bushel. In December of the following year, however, prices range only from 239 to 286.50. In fact, in 7 of the 10 years, the price is between 251 and 268. The lower variability of distant plices is not surprising: lt is dificult to forecast a hmwest more than a year into the fumre. Thus, the forward prke is re:ecting the market's expectation of a normal harvest 1 year hence. the forward tfwe assume thatstorage costs ar approximately so.o3/month/bushel, price in Table 6.11 never violates the no-arbitrage condition
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GAS
Natural gas is another market in which seasonality and storage costs are important. The namral gas f'utures contract, introdued in 1990, has become one of the most heavily traded f'utures contracts in the United States. The asset underlying one contract is 1 months worth of gas, delivered at a specifc location (differentgas contracts call for delivery at different locations). Figure 6.6 shows a newspaper listing for natural gas f'umres, and Figure 6.7 details the speciscations for the Henry Hub contract. Nattlral gas has several interesting chapcteristics. First, gas is costly to ansport internationally, so prices and fonvard culwes vary regionally. Second',once a given well has begun production, gas is costly to store. Third, demand for gas in the United States is highly seasonal, with peak demand arising from heating in winter months. Thus, there is a relatively steady stream of production with variable demand, which leads to large and predictable price swings. Whereas corn has seasonal production and relatively constant demand, gas has rlatively constant supply and seasonal demand.
'
Sottlce: Futures data from Dataseeam.
gltispossible tohave lowcurrentstorage and alargeexpectedharvest, which wouldcause the December price to be lower than the July price, or high current storage and a poor expected harvest, which would cause the July price to be below the December price.
%.CoMuool'ry
192
FORwARDS
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Natural gas delivered at Sablne Pipe Lines Co. s Henry Hub, Louisiana New York Mercantile Exehang t'ul 10,000 millionBritish thermal units
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Soltrce: Futures data from Datastream.
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Where aded Size Monts Trading ends
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Figure 6.8 displays 3-year (2001)and 6-year (2002-2004)strips of gas futures prices for the srstWednesday in June fz'om 1997 to 2000. Seasonality is evident, with high winter prices and low surnmer prices. The 2003 and 2004 strip shows seasonal cycles combined with a downward eend in prices, suggesting that the market considered prices l that year as anomalously high. For tlze other years, te average pzice for each comitlg year is about the same. Gas storage is ostly and demand for gas is highest in the winter. The steady rise of the forward curve during the fall months suggests that storage occurs just before the heaviest demand. Table 6.12 shows prices for October through December. The monthly increase in gas prices over these months ranges from $0.13to $0.23. Assuming that the interest rate is about 0.15% per month and that you use equation (6.13),storage cost in would satisfy November 2004, ,
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implying an estimted jtorage cst of $0.178 in November 2004. You will find differentimptedmarginal storagecosts ineach year, butthis is to be expectedif marginal stpredr storage costs kary with the tuantity Because of the expense iri nspol-ting gas internationally, te seasonal behavior of the forward curv can kary in differet parts of the world. In tropical areas where gas is usedforcooking and elecicity generatiop, the forward curve is relatively flatbecause demand is reladvely llat. In te Southern hemisphere, where seasons are reversed f'rom the Northrfl hemisphre, the forward curve will pak in June and July rather than =
December and January.
%.COMMODITY
194
FORwARDS
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On the four dates in the figure, near-term oil prices range from $25 to $40, while 7-yeaT frward pfice in each case is btween $22 and $30. The lonprun forward the price is less volatile than the short-run forward ptice, which makes economic sense. ln the short rtm, an increase in demand will cause a price increase since supply is xed. A supply shock (suchas production restrictions by the Organization of PetroleumExporting will cause the price to increase. In the long run, however, both supply Countries EOPEQ) dmand time have to adjust to price changes with the result that price movements and attepuated. The forward curve suggests that market participants in June 2004 did not are price remain the at $40/barrel. to expect
6.11 COMMODITY
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Some commodities are inputs in the creation of other commodities, Which gives rise to commodity spreads. Soybeans, for example, can be crushed to produce soybean meal and soybean oil (anda small amount of waste). A eader with a position in soybeans and an opposite position in equivalent quantities of soybean meal and soybean oil has a the crush.'' crush spread and is said to be Similady, cnlde oi1 is resned to make petroleum products, in particular heating oil and gasoline. The refining process ntails distillation, which separates crude oi1 into different components, including gasoline, kerosene, and heating oil. The split of oil into these different components can be complemented by a process known as :trading
Etcrackinf''
%.COMMODITY
196
FORWARDS
H ED G l N G
AND FUTURES
oil hence, the difference in price between crude oil and equivalent amounts of heating spread. called the crack and gasoline is mixes of outputs. Oi1can be processed in different ways, producing different number of Spread terminology identifes te number Of gallons of oil as input, and the gallons of gasoline and heating oil as outputs. Traders will speak of lt; E5 :E5 1 and :Q-1-1'' crack spreads. The 5-3-2 spread, for example, reflects the prost from tnking of gasoline and 2 gallons of heating oil. 5 gallons of oil as input, and producing 3 gallons oil could use a futtlres crack spread heating and gasoline produeing Apeoleum rener prices. This sategy would entail going long and oil of output to lock in both the cost and heating oil ftltures. Of gasoline of quantities appropriate the noilf'utures and short possible is and it to produce other outputs, prouction other inputs to coursethere are hedge. perfect spread is .not crack the a such as jet fuel, so 'l''he
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STRATEG 1ES
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some cases, One commodity may be used to hedge another. As an example of this W discuss the use of cnlde oil to hedgejet fuel. Finally, weather derivatives provide another example of an instnlment that can be used to cross-hedge. We discuss degree-day index
coneacisas an example
of such derivatives.
Basis ltisk Exchange-traded commodity f'tlttlres coneacts call for delivery of the underlying commpdity at specisc locations and specisc dates. The acmal commodity to be bought or sold may reside at a different location and the desired delivery date may lot match that of the futtlres conact. Additionally, the grade of the delivrable under the fumres conact may not match the grade that is being delivered. This general problem of the t'utures or forward contract not representing exactly what is being hedged is c alted basis ? s k Basis risk is a generic problem wit commodities because of storage and ansportation costs and quality differences. Basis risk can also arise with financial fumres, as for example when a company hedses its oWn boaowing cost with the Eurodollar contract. Section 5.5 demonsated how an individual stock could be hedged with an index futtlres contract. We sw that if we regressed the individpal stock return on the index provided a hedge ratio that mini u ze tj tjje rettlrn, the resulting regression coecient variance of the hedged position. In the same way, suppose we wish to hedge oil delivered on the East Coast with X oil contract, which calls for delivery of oil in Cushing, Ollahoma. the variance-minimizing hedge ratio would be the regression coeflicient obtained by regressing the East Coast price on the Cushing price. Problems with this regression are that the relationsllip may not be stable over time or may be estimated imprecisely. 'i
Suppose we consider buying oil in July and selling gasoline and heatq Eyample 6.4 k.i/j ing oil in August. On June 2, 2004, the July futtlres plice for oil was $39.961-e1, jt( 42 gallons per barrel). The August ftltures prices for unff ti 01- $0 9514/gallon (there are and $1.0171/ga1lon.The 3-2-1 leaded gasoline and heating oil were $1.2427/ga1lon l) :l ')(E) buying 3 gallons of oil and . crack spread tells us the gross margin we can lock in by EE ;(' these prices, the spread is heating of Using oil. 1 and producing 2 gallons of gasoline . tj! tl.y) j i jj = !t'' (2 x $1.2427)+ $1.0171 (3 x $0.9514) $0.6482 i(()j t . . yjy calculation we made no interest adjustment for j or $0.6482/3 = $0.2161/gal1on.In this 11 the different expiration months of the futures contract. % .
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6.12 HEDGING
STRATEGIES
contmodity In this section we discuss some of the complications thatean arise when using 3.3 we discussed f'utures and forwards to hedge commodity price exposure. In Section of quantity uncertainty, where, for example, a one such complication: the problem farmer growing corn does not know the ultimate yield at the time of planting. Other issues can arise. Since commodities are heterogeneous and often costly to transport hedge a risk with a commodity conact that is imperfectly and store, it is common lise lisk to basis rf.k: The price of the being hedged. This gives correlated with the differently tan the price of the commodity underlying the f'utures conact may move cornmodity you are hedging. For example, because of transportation cost and time, the price of namral gas in California may differ from that in Louisiana, which is the .to
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Another example of basis lisk occurs when hedgers decide to hedge distant obligations with near-tenn futtlres. For example, an oil producer might have an obligation to delixer 100,000 barrels per month at a xed price for a year. The natural way to hedge this obligation would be to buy 100,000 barrels per month, locking in the price and supply on a month-by-month basis. This is called a strip hedge. We engage in a stlip hedge when we hedge a strenm of obligations by offsetting each individual obligation with a f'uttlres contract matching the maturity and quantity of the obligation. For the oi1 producer obligated to deliver every month at a fixqd price, th hedge would entail buying the appropriate quantity each month, in effect tnking a long position in the strip. Arl alternative to a strip hedge is a stack hedge. With a stack Vdge,we enter into f'utures contracts with a single maturity, with the number of conacts selected so that changes in the pre-wnt valtte of the future obligations are offset by changes in the value ttstack'' of futures contracts. In the context of the oil producer with a monthly of tlzis delivery obligation, a stack hedge would entail going long 1.2 million barrels using the near-term contract. (Actually, we would want to tail the position and short less than 1.2 million barrels, but we will ignore this.) When the near-term contract mattlres, we
200
k. COMMODITY
PROBLEMS
A&D FUTURES
FORWARDS
There are many other examples of weather risk: ski resorts are hafmed by warm winters, soft drink manufacturers are harmed by a cold spring, summer, or fall, and makers of lawn splinklers are harmed by wet summers. In al1 of these cases, fin'ns could that make payments based upon hedge their risk using weather derivatives--contracts cross-hedge specific l-isk. their characteristics of weather-to realized weather-related delivatives based measurements. The payoffs for weather are on An example of a weather contract is the degree-day index futtlres contract traded at the Chicago Mercan tile Exchange. A heating degree-day is the maximm of zero and the difference between the average daily temperamre and 65 degrees Fnhrenheit. A cooling d.egree-day is the maximum of the difference between the average daily temperature and 65 degrees Fn hrenhe it and zero. Sixty-five degrees is a moderate temperature. Athigher temperatures, air conditioners may be tlsed, whil at lower temperamres, heating may be used. A monthly degree-day index is constructed by adding the daily degree-dys over the month. The futures contract then settles based on the cumulative heating r cooling degree-days (the'twoare separate contracts) over the course of a month. 'I'he size of the contract is $100 times the degree-day index. As of September 2004, degree-day index contracts were available for over 20 cities in the United States, Europe, and Japan. There are also puts and calls on these ftlmres. With city-specisc degree-day index contracts, it is possible to create and hedge payoffs based on average temperatures, or using options, based on rgpges of average temperamres. If Minneapolis is unusually cold but the rest of the contry is nonnal Iflalce a large payment that will the cooling degree-day contract for Minneapolis will consumption of Notice that in this increased the energy. the holder for compensate would provide example) nattlral not price contract (for a sucient hedge, scenario a gas much effect would alone Minneapolis have unusual cold not in on national energy since ,
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For fnancial assets, is the dividend yield. For commodities, is the commodity lease mrc-the return that makes an investor willing to buy and then lend a commodity. Thus, for the commodity owner who lends the commodity, it is like a dividend. From the commodity borrower's perspective, it is the cost of borrowing the commodity. As with snancial fonvards, commodity forward prices are biased predictors of the f'uttlre spot price when the commodity return contains a risk premium. While the dividend yield for a financial asset can typically be observed directly, lease rate for a commodity can typically be estimated only !?),obselwing #?c./?3j)(7?'# the fonvard curve provides important information aboqt the commodity. The price. Commodities are complex because every commodity market differs in the details. Forward eurves for different commodities reiect different properties of storability, stor-
%
201
age costs, production, and demand. Electricity, gold, corn, natural gas, and oil all have distinct forward curves, re:ecting the different characteristics of their physical markets. These idiosyncracies will be reflected in the commodity lease rate. When there are seasonalities in either the demand or supply of a commodity, the commodity will be stored (assumin g this is physically feasible) and the forward curve for the commodity Willre:ect storage costs. Some holders of a commodity receive benests from physical onvenietce yield. The convenience ownership. This benest is called the commodity's ownership different different creates for investors, and may or may not returns to yield price. The yield forward convenience re:ected in the be can lead to no-arbieage regions plice. costly no-arbitrage It also be than to short-sell commodities with a rather can a yield. convenience signifcant ,
FURTHER
READING
We will see in later chapters that the concept of a lease rate-which is a generalization of a dividend yield-helps to unify the pricing of swaps (Chapter 8), options (Chapter 10), and commoditplinked notes (Chapter 15). One particularly interesting application of the lease rate arises in the discussion of real options in Chapter 17. We will see there that if an extractable commodity (suchas oil or gold) has a zero lease rate, it will never be extracted. Thus, the lease rate is linked in an important way with production
decisions.
'
.
.
A useful resource for learning more about cornmodities is the Chicago Board of X and the CBOT) Trade (1998). The Web sites of the valious exchanges (e.g., particular commodities and trading and with information about useful also resources, are hedging strategies. Siegel and Siegel (1990)provide a detailed discussion of many commodity f41t'ures. There are numerous papers on commodities. Bodie and Rosansky (1980)and Gorton and Rouwenhorst (2004)examine the risk apd return of ommodities as an investmept. Brennan (1991),Pindyck (l993b), and Pindyck (1994)examine the behavior of commodity prices. Schwartz (1997)compares the performance of different models of commodity price behavior. Jarrow and Oldfield (1981)discuss the effect of storage costs on plicing, and Routledge et a1. (2000)present a theoretical model of commodity
forward curves.
Finally, Metallgesellschaft engendered a spirited debate. Paperj wlitten about that episode include Culp and Miller (1995), Edwards and Canter (1995j, and Mllo and Parsons (1995).
PROBLEMS 6.1. The spot price of a widget is $70.00per unit. Forward prices for 3, 6, 9, and 12 months are $70.70,$71.41,$72.13, and $72.86. Assuming a 5% continuously compounded annual risk-free rate, what are the annualized lease rates for each maturity? Is this an example of contango or backwardation?
J
202
%.COMMODITY
FORwARDS
P Ro
AND FUTURES
prices for 3, 6, 9, and 12 6.2. 'T'hecurrent price of oil is $32.00per barrel. Forward Assuming a 2% continuously months are $31.37, $30.75, $30.14, and $29.54. annualized rate for each malease the coltapoundedannual risk-free rate, what is backwardation? turity? Is this an example of contango or tisk-free rate of 3% annually, at what lease 6.3. Given a continuously compounded commodity price? tRecall the pencil rate will forward prices equal the current wotlld there be contango or example in Section 6.4.) lf the lease rate were 3.5%,
backwardation? continuously compounded lease 6.4. Suppose that pencils cost $0.20 today and the compounded interest rate is 10%. The continuously rate for pencils is 5%. The costlessly. stored be uncertain.and pencils can pencil price in 1 year is what payment do you have to make 1 a. lf you short-sell a pencil for year, Enancial investor to store to the pencil lender? Would it make sense for a equilibrium? pencils in b. Show that te equilibrium forward plice is $0.2103. prices are ruled out by arbitrage in the c. Explain what ranges of forward and cannot four cases where pencils can and cannotbe short-sold and can be loaned. price is 310.686, and 6.5. Suppose the gold spot price is $300/oz.,the l-yea/fonvard 5%. is compounded-risk-free rate the continuously
a. What is the lease rate? gold is not loaned? b. What is the ret'ul'n on a cash-and-carry in which cash-and-earry in which gold is loaned, earning c. What is the return on a the lease rate? continubusly compounded interest rate is For the next three problems, assume that the (payableat the end of the quarter). quarterly 6% and the storage cost of widgets is $0.03 widgets: Here is the forward price ulwe for 2004 Dec 3.000 6.6.
2005
Mar 3.075
Jun
3.152
Sep 2.750
Dec 2.822
Mar 2.894
2006
Jun
2.968
explanations for the shape of this forward curve? a. What are some possible cash-and-carl'y entered b. What annualized rate of ret'ul'n do you earn on a into in December 2004 and closed in March 20052 Is your answer sensible? of return do you earn on a eash-and-carry ntered c. What annualized rate 20052 Is your answer into in December 2004 and closed in September sensible?
B LE M s
%.
a. Suppose that you want to borrow a widget beginning in December 2004 and ending in March 2005. What payment will be required to make the transaction fair to both parties? b. Suppose that you want to borrow a widget beginning in December 2004 and ending in September 2005. What payment will be required to make the transaction fair to both parties? 6.8.
a. Suppose the March 2005 fonvard price were $3.10. Describe two different transactions you could use to undertake arbitrage.
b. Suppose the September 2005 forward price fell to $2.70and subsequent forward prices fell in such a way that there is no arbitrage from September 2005 and going forward. Is there an arbitrage you could undertake using forward contracts from June 2005 and earlier? Why or why not? 6.9. ConsiderExample 6.1. Suppose theFebruary forwardprice had been $2.80.What would the arbitrage be? Suppose it had been $2.65. What would the arbitrage be? In each case, specify the transactions and resulting cash flows in both November and February. What are you assuming about the convenience yield? 6.10. Using Table 6.10, what is your best guess about the current price of gold per ounce? 6.11. Suppose you lnow nothing about widgets. You are going to approach a widget merchant to borrow one in order to short-sell it. (That is, you will take physical. possession of the widget, sell it, and return a widget at time F.) Before you ring the doorbell, you want to make a judgmentabout what you think is a reasonable lease rate for the widget. Think about the following possible scenatlos. a. Suppose that widgets do not deteriorate over time, are costless to store, and are always produced, although production quantity can be varied. Demand is constant over time. Knowing nothing else, what lease rate might you face? b. Suppose everytlzing is the same as in (a) except that demand for widgets varies seasonally. c. Suppose everything is the same as in (a) except that demand for widgets varies seasonally and the rate of production cannot be adjusted. Consider how seasonality and the horizon of your short-sale interact with the lease rate.
d. Suppose everything is the same as in (a) except that demand is constant over time and production is seasonal. Consider how production seasonality and the horizon of your short-sale interact with the lease rate. e. Suppose thatwidgets cannotbe stored. How does tlzis affectyour answers to the previous questions?
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-u
.z:
-
1TQ
-l3
-a
.yj
-14
-15
za 9.P s ;WRx
eb
-1
-1
P Qa
-2
p
*1
.6
..
-;
-3
-b5l
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mx
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stts
nk
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coupons and are issued at aprice close to their matulity value (i.e.,they are issued at parl. Notes have 10 or fewer years to maturity and bonds have more than 10 years to maturity. The distinctions between bills, notes, and bonds are not important for our purposes; we will refer to all three as bonds. You will also see Treasury inflation protected securities ITIPSIin Figure 7.1. These are bonds for which payments are adjusted for in:ation.
Finally, the most recently issued government bonds ar called on-the-run; other bonds re called off-the-run. These ten's are used ftequently in talking about government bonds since on-the-run bonds generally have lower yields and greater trading volume than ofthe-run bonds. Appendix 7.A discusses some of the conventions used in bond price and yield quotations. In addition to government bond information, there is also a listing for STIUPS. Trading of Registered lnterest and Principal of Securitiej-is A S'rllps-separate a claim to a single interest payment or the principal portion of a govelmment bond. These claims trade separately from the bond. STRIPS are zero-coupon bonds since they make onl# a single payment at mattltity. 'CSTRIPS'' should not be confused with the forward strip, which is the set o? forward prices available at a point in time. We need a way to represent bond prices and interest rates. Interest rate notation is, unfortunately and inevitably, cumbersome, because for any rate we must keep track of three dates: the date on which the rate is quoted, and the period of time (thishas beginning and ending dates) over which the rate prevails. We will 1et ?'? (?I r2) represent the interest rate from time rl to time tz, prevailing on date t. If the interest rate is current-i.e., if t rl-and if there is no lisk of confusion, we will drop the subscript. ,
=
-3
.gj
-yg
*kv
.!j
.
-j,
.yy
r,j
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->
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->
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cll: -
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r=
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Zero-coupon Bonds
.u
-3
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etz'l
..4
..6
-u
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-2p
-
-5
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!2'
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We begin by showing that the zero-coupon bond yield and zero-coupon bond price, columns(1)and (1) in Table 7.1, provide the same infonnation. A zero-coupon bond is a bond that makes only a single payment at its mamrity date. Our notation for zerocoupon bond plices will mimic that for interest rates. The price of a bond quoted at time
5
nci
-7
-z
-27
-22
-7
-b,
-7
-22
-2:
'i
-2?
-
*-b
-22
-z:
-;kv9
-12*24
rb
-%:::
-7
-?'
-;
-9
>22
-:
-!l
-:41
-
-z2
-tu4
*4118
-!
-?
207
208
%.INTEREST
RATE FORWARDS
Boxo
AND FUTURES
maturing at tz, is Ptfj(fl 1:.)(). As with interest lo; with the bond to be purchased at rl and = /1 rates, we will drop the subscript when tfj plice P(0, 1) = 0.943396 means that you would of bond I'he l-year zero-coupon 1 You could also pay P (0, 2) = 0.881659 pay $0.943396today to receive $1 in year. = to receive $1 in 3 years. today to receive $1 in 2 years and P (0, 3) 0.816298 ?'crl//wl on a zero-coupon bond is simply The yield to maturity (orintenzal rate of bond, we end the percentage increase in dollars eacned from the bond. For tlae l-year interest quoting 1 = 0.06 more dollm's per $1 invested. If we are up with 1/0.943396 rates as effective annual rates, this is a 6% yield. 1 = 0.134225 For the zero-coupon z-yearbond. we end up with 1/0.881659 interest effective raye of could call this a z-year more dollars per $1 invested. We lf basis. we want this 13.4225%, but it is conventiontl to quote rates on an annual could assume annual yield to be comparable to tlae 6% yield on the l-year bond, we 2) = 0.065. r(0, that + r(0, 2))2 = 1.134225, wlch implies compounding and ,
.
-
-
In general,
get
(1
P (0, n)
=
1 (1 +
/-(0,
(7.1)
/2)2&
discoltntfactor-. A Note from equation (7.1) that a zero-coupon bolld price is a the future. If you in receive would $1 today to pay zero-coupon bond price is what you of a zero-coupon the ptice multiply it by Ct, you can have a f'uture cash flow at time t, of equation (7.1), Because cash the flcfw. of value l), bond, P(0, to obtain the present l), i.e., r(0, the discounting rate at multiplying by P (0,r) is the same as Ct c, x .P(0, r) r(0, r)) ; (1+
'q' -.'. -' jjf )1* )r)' y(' ,';'q' ).)' j' ;CfjY !t' (' ,':',;' j'jyjf )' )' jrf (' k--' ;,' ;'.''1* t't'jjf ()li r:!(i ,j.-----,#'-' y' t)f )!i E'i)ll l' . ( ((r. '( q! :'IIk-2-;ii,' iiii!E!t,'jy'. (' ryf :i:3i'33'. jjf j't' #' jjylqrr'. 1)* r' @-'. .-' #-.' yj': .. (t'. ptk:'. y' yy'jyjj. jpl-' .'y'yjf
1')' . ( . An nvestor inves yingfor
.'i.'. ..'. '''!?q1' :';; --' -.'r(;'.'r;i..'-.-.' -' T'!'7':TY T!T''I''?Tr'FTY i'jjf -'i II'iEE ' ' : -' 7* :';(' -.'-.. ji)r:', ' :
.' .'.
(qE E'' E' E'' E i EE '(ir'' (';l!E E ' iq (( E('E'E ' E'E'.(E.(iEiE'.E'(('i.qi'i'E!(E!i (!!..i r!! '' C'r' q7!'p'' . .(. .';' . )j:. .! ( E j; 'j qq j ( qyj.. i g '(j . ; y..j.r.qr:.yg... ..'.. .yjryg..jjj)t,,)). .;q yj' g jj .' ' ) )#' yy y. jj ;jg. y yjjg: -''Itt:i;p;Iy..yjy y ..) jlt)?iEg;)7J . y . g (yj . y g jy .jj y r y . .. .g . . ..jjtyykjjj.. yyj . . E . 1IIi:i;Ii::. y.-gy. j j gg . j -r.1IiiF:pp!. . . ; . . iIjI. -j.j E )i. E . i. i . E ::. . . - -).. :-t.-j-. .k..-.r..-.. --..;-y-.... -. .t. .. )- . rk)j:. . . . . ... ..... ...... . . . :
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209
'
:
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%.
BAslcs
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-
2 years has a chptce of buying a z-year zero-coupn bond
gl + Earn r(0, 1)
E1+ ro(0, $12or buying a l-year bond paying 1 + ro(0,1) for 1 year, and reinvesting the proceeds at the implied forward rate, ro(1,2) between years 1 and 2. The implied forward rate makes the investor indiferent
rl 1,
z)j
forward rate, r(1, 2)
Earn implied
Pa/ng
(1 +
r(O, 1)! x
'
()
Tjme
.'I
1
Earn r(0, 2) per year
(1 +
?o,
oj2
between these alternatives.
(1 + /:(0, satisfyy
2)12. Thus, the tlme 0 forward rate from year 1 to year 2,
(1
.+
- (0 1)11(1+
? ()
,
rn(l
2)1 =
,
(1+
?-0(0,
?3(1,
2), should
2)12
=
value factor. The inverse of the zero-coupon bond price, 1/#(0, r), provides a future those in Figure 7.1 such as In coneast to zero-coupon bond prices, interest rates interpretation difcult make their (if you are subjeet to quoting conventions that can doubt this, see Appendix 7.A). Because of their simple interpretation, we can consider 111 of sxedincome. zero-oupon bond prices as the building block for mamrity is A graph of annualiied zero-coupon yields to maturity against time to mamrity yields shows how vary us to called the zero-coupon yield ctlrve. A yield curve yield culwe based on the present is it to In maturity. practice, common time i h to wt bonds, not zero-eoupon bonds.
Or
1+
?3(1,
2)
=
(1+ /-t)(0, 1 + ,-4,(p,
2)12 1)
Figure 7.2 shows graphically how the implied forward rate is related to 1- and z-year yields, If ?-0(1, 2) did not satisf'y equation (7.2),then tere would e an arbitrage opporturlity Problel 7.15 asks you to work ihioug the arbitrage. ln geeral, we have .
()1+
r0(/l
/))Y-''
=
,
coupon
(1+
?-0(0,
I1 +
?3(0,
r2)(lY
rj
)1JI
=
/1)
P,
.P(0, r2)
Corresponding to l-year and z-yearinterest rates, rc(0, 1) and ?3(0, 2), we have prices of l-year and z-yea.r zero-coupon b on d s, P()(0 1) apd /$(0, 2). Just as the interest rates imply a forward l-year interest rate, the bond prices imply a l-yearfonvard zero-coupon bond price. The i.mpliedforward zero-coupon bond price must be consistent with the implied forward interest rate. Rewriting equation (7.3),we have ,
lmpliedForward Rates (1) or We now see how column (3) in Table 7.1 can be computed from either (2). The l-year and z-yearzero-coupon yields are the rates you can enrn from year 0 to is also an implicit rate that can be earned from year 1 and from year 0 to yem' 2. There with the other two rates. This rate is called the year 1 t year 2 that must be eonsistent implied forward rate. . 2. We Suppose we could today guarantee a rate we could earn from year 1 to year for 2 0(0, invested and $1 1)) years earns know that $1 invested for 1 year earns (1+ column
/$(/.1t?.) ,
=
1 (1+ r0lrl
,
/.2)1 , 2
-,,
(1+ ?-0(0, J'1))f' (j. /()((),uljj/a
=
./3(0,
/.a)
=
.j.
The implied forward zero-coupon bond price from zero-coupon bond prices maturing at tg and /1 .
/1
po
,
j.j
)
to h is simply the ratio of the
210
k
INTEREST
Boxo
AND FUTURES
RATE FORwARDS
k
.gs
tt
C
jtt )
=
0.943396 + 0.881659 + 0.816298 6.95485%
E.
j
=
%:
.1
,
=
211
j'yyl Example 7.2 Using the infonnation in Table 7.1, the coupon on a 3-year coupon . (y Etty tj.j bond thatsells at par is jy: j j j 6ggg y('.
Example 7.1 Using information in Table 7.1, we want to compute the implied forward interest rate from year 2 to year 3 and the implied folavard price for a l-yer zero-coupon bond purchased in year 2. The implied forward interest rate, rp(2, 3), can be computed as (1 + 0.07)3 (1+ ?-o(0 3)13 ()84s.74)5 ?-0(0, 2)) a 1 + ?-0(2, 3) (1 + () ()65)2 (1+ =
BAslcs
-
.
or equivalently as ,-0(2,
1+
3)
.Jb(0, 2) =
/$(0, 3)
0.881659 =
0.816298
=
Equation (7.5) computes the bond price by discounting each bond payment at the appropriate for a cash flow with that particular matulity. For exapple, in equation rate (7.5), the coupon occuring at time ti is discounted using the zero-coupon bond price Pt (?,.ti). An alternative way to write the bond price is using the yield to mattllity to discount a11payments. Suppose the bond makes p?l payments per year. Denoting the per-period yield to matufity as y,,,, we have
1.0800705
The implied forward l-year zero-coupgn bond price is 1 .P0(0. 3) 0.925865 ?-0(2, 3) = 1+ P0(0, 2)
k
=
11
Bt (r, F, c,
i= l
Coupon Bonds Given the pdces of zero-coupon bonds--column (1) in Table 7.1-we can price coupon 7.1-the coupon bonds. We can also compute the par coupon--column (4) in Table need to know rate at which a bond will be priced at par. To describe a coupon bond, we of the bond payments, and ed dte the date at wh'ich the bond is being priced, the start principal. Som ptactical of and the amount of the payments, and number amount the discussed complexities associated with coupon bonds, not essential for our purposes, are
in Appendix 7.A.
We will let Bt (h h, c, ?l) denote the time t price of a bond that is issued at h mattllit.y payment, and makes l everlly mamres at tz, pays a coupon of c per dollar of will spaced payments over the life of the bon, beginning at time 11 + h tjj/l. We different 1. is than $ we can maturity payment assume the maturity payment is $1. If the that amount. al1 npltiply by payments just time t Since the plice of abond is the present value of its payments, at issuance the price of a bond maturing at F must satisfy ,
,
-
lt
Bt t, F, c,
??)
/))
cP t,
=
+ Pt (/, F)
With i being the index in the summation. Using where ti t + i CT tj/n, solve for the coupon as (7.5)i we can #, (r, F) Bt t, T, c, n) C n )-!f=lPt t, ) =
-
-
1, so the coupon on a par bond is given by 1 Pf t, T) j7.'l l P J t tt ) -
C=
=
,
.
+
1
(1 +
#,?,)'l
to compute the quoted annualized yield to maturity, y, as y = m x y,?,. bonds, for example, make two coupon payments per year, so the annualized Goverment twtce maturity the semiannual yield to mamrity. is yield to ' ' ' . . . Th diffetence between cjation (7.5)and equation (7.7) that in equation (7.5), at the ppropriate fte for a ash flow occning at each copon payment is discontd tht timek I kaiio (7.7), oe rat is ttsed to dicimnt a1l cash flowj. By definition, kuatin (7.7)can be misleading, thetwo expresssions give the sam price. Ilowkf, invstor the maturity, yield the is not ret'urn since to an earns by buying nd holding y,?,, prvtdes insight into how th cash flowj from a ond. Mfeover, eqtio (7.7) no a bonds. ith be replicate d w zero-coupon bondcan
It is common
ts
'
Zeros from Coupons We have strted with ero-coupon bond prices and dedued the prices of coupon bonds. In practice, the simation is often the reverse: We observe prices of coupon bonds and coupon bond must infer prices of zero-coupon bonds. This procedure in which bond prices of is called bootstrapping. prices are deduced from a set coupon Suppose we observe the par coupons i Table 7.1. We can then infer the first zero-coupon bond ptice f'rom the srstcoupon bond as follows:
equation
1 (1 +
=
=
c
; (1+ j'lt')
'zefo
i= 1
A par bond has Bt
??)
=
0.06).P(0, 1)
This implies that P(0, 1) = 1/1.06 = 0.943396. with a coupon rate of 6.48423% gives us
1 Since we lnow 17(0, 1)
0.0648423P(0,
=
=
Using the second par coupon bond
1) + 1.0648423#(0,
0.943396, we can solve for #(0, 2):
2)
212
k. INTEREST
RATE FORWARDS
#(0, 2)
AN D FUTURES
=
1
BOND
0.0648423 x 0.943396
-
1.0648423 0.881659 =
Finally, knowing P(0, 1) and P(0, 2), we can solve for #(0, 3) using the 3-year par coupon bond with a coupon of 6.95485%:
1
=
(0.0695485x 17(0, 1)) + (0.0695485x #(0, 2)) + (1.0695485x /7(0, 3))
#(0, 3)
=
1
-
(0.0695485x 0.943396)
-
(0.0695485x 0.881659)
1.0695485
= 0.816298 There is nothing about the procedure that requires the bonds to trade at par. In fact, we do not even need the bonds to all have different maturities. ltbr exanple, if we had a l-year bond and tw0 different 3-yellr bonds, we could still solve for the tlu'ee zero-coupon bond prices by solving simultaneous equations.
lnterpreting the Coupon Rate A coupon rate-for example the 6.95485% coupon on the 3-year bod-detennines the cash flows the bondholdaer receives. Howevr, excpt in special cas, it does not , jy earps by hol yjy coaejpond to the rate of retfn that-an ihvestr citlt notjyja jjqyjtj. iinplied ertai; i.e., the i'pr/aidrates for rates interst Suppose ar a momnt tt Imagine and 2. that we buy actually in l that will Table 7.1 the in rates years occur are thef all reinvesting maturity, it bond and hold to the 3-year coupons as are paid. What stoj calulations, let's through the and discuss going eal'n? Before of do we rate rettlrn the intuition. We are going to invest an amount t time 0 and to reinvest al1 coupons by buying more bonds, and we will not withdraw any cash until time 3. f/? eject, Ij?e are ctpn-lrlfcrf?z,ga 3-)'ear zdm-ctplfp/n bolld. Thus, we should ellrn the same return as zero: 7%. This buy-and-hold return is different than the yield to maturity on a The coupon payment is set to make a par bond fairly priced, but it is not 6.95485%. of the acmally return we enr'n on th,bond exept in the special case when the interest rate is constant over time. Consider frst what would happen if interest rates were certain, we bought the 3-year bond with a $100plincipal and a coupon of 6.95485%, and we held it for 1 year. The price at the end of the yepr would be 106.95485 6.95485 Xl = V 1.0700237 (1.0700237)(1+ 0.0800705) ,
%
213
We earn 6%, since that is the l-year interest rate. Problem 7.13 asks you to compute yor z-yearreturn on this investment. By year 3, we have received three coupons, two of which have been reinvested at the implied forward rate. 'I'he total value of reinvested bond holdings at year 3 is
6.95485 x ((1.0700237)(1.0800705) + (1.0800705)+ 1) + 10
=
122.5043
The 3-year yield on the bond is thus
.
wlzichgives us
BAslcs
l/3
122 5043
.j
m
100
gogg
As we expected, this is equal to the 7% yield on the 3-year zero and dlfferent from the coupon rate. This discussion assumed that interest rates are certain. Supppse that Fe buy and hold the bond, reinvesting the coupons, and that interest rates ar not certain. Can we still expect to earn a ret'urn of 7%? The answer is yes if we use interest rate forward contracts to guarantee the rate at which we can reinvest coupon proceeds. Othemise, the answer in general is no. The belief that the implied forward interest rte equals the expeted f'uture spot interest rate is a version of the expectations hypothesis. We saw in Chapters 5 and 6 that forward prices are biased predictors of f'uttlre spot prices when the underlying assethas a risk premium; the same is true for forward interest rates. When you own a coupon bond, th rate at which you will be able to reinvest coupons is uncertain. If the resulting risk. carries a risk premium, then the expected rettlrn to holding the bond will not equal the 7% remrn calculated by assuming interest rates are certaink The expectations hypothesis will generally not hold, and you should not expect implied fomard interest rates to be unbiased predictors of f'uttlreinterest rates. In practice you can guarantee the 7% ret'urn by using forward rate agreements to lock in the interest rate for each of the reinvested coupons. We discuss forward rate agreements in Section 7.2.
'3-year
= 99.04515
Continuously Compounded Yields Any interest rate can be quoted as either an effective annual rate or a continuously compounded rate. (Or in a variety of other ways, spch as a semiannually compounded rate, which is common with bonds. SeeAppendix 7.A.) Column (5)in Table7.1 presents the continuously compounded equivalents of the rates in the yield'' column. ln general, if we have a zero-coupon bond paying $1 at mamrity, we cn write its price in terms of an annualized continpously compounded yield, I'CCCQ, t), as3 'zero
P,
1)
=
e
--rccfn tj; '
The l-period return is thus l-peliod
ret'unl
=
6.95485 + 99.04515 l00
= 0.06
=
1
3In future chapters we will denote continuously compounded interest rates simply as r, without the cc superscript.
%
INTEREST
RATE FORWARDS
FORwARD
AN D FUTU RES
We can comput
(0, t)
1 =
-
l
ln g1(P
(0, /')1
the continuously compounded 3-year zero yield, for example,
1
1n(1/0.816298)
=
as
0.0676586
Alternatively, we can obtain the same answer using the 3-yea.r zero yield of 7%:
ln(l + 0.07)
=
0.0676586
z4??yof the zero yields or implied forward yields in Table 7.l can be computed as effective annual or continuously compounded. The choice hinges on convention and ease of alculation. RATE AGREEMENTS, EURODOLLARS, AND HEDGING
7.2 FORWARD
.5%
211 days 1.5% rquarterly = 2% rquarterly =
Borrow
$100m
+100m
-l01.5m
-102.0m
'
' Depending upon th# interst rate, tbere is a variation of $0.5min the borrowing
cost. How can we hedge this uncertainty?
rate agreement
(FRA) is an over-the-counter
%
215
Firstconsiderwhathappens if theFlkAis settledin Septemln arrears ber, on day 211, the loan repaymentdate. In thatcase, thepaymentto theborrowershould
be
l-IrRAl
(rquarrly -
x notional principal
Thus, if the bonmving rate is 1.5%, the payment under the F'RA should be
(0.015
$100m
0.018) x
-
-$300,000
=
Since the rate is lower than the FRA rate, the borrower pays the FRA counterparty. Similarly, if the borrowing rate turns out to be 2.0%, the paymertt under the FRA should be -
$100m $200,000
0.018) x
=
Settling the FRA in arrears is simple and seems like the obvious way for the contract to work. However, settlement can also occur at the time of borrowing. lf the FRA is settled in June, at the tim FRA seulement at the time of borrowlng the money is borrowed, payments will be less than when settled in arrears because the borrower has time to enrn interest on the FlkAsettlement. In practice, therefore, the FRA settlement is tailed by the reference rate prevailing on the settlement (bprrowing)date. (Tailing in this context means that we reduce the payment to reqect the interest ealmed between June and Septemben) Thus, the payment for a borrower is uarterly (?q1 + rquarterly -
Notional principal x
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If rquarterly = 1.5+, the payment in June is
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Forward Rate Agreements A forward
AND HEDGING
FRA seulement
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We now consider the problem of a borrower who wishes to hedge against increases in the cost of borrowing. We consider a fir'm expecting to borrow $100m for 91 days, beginning 120 days from today, in June. This is the borrowing date. The loan will be repaid in September on the loan repayment date. In the examples we will suppose that or 2%, and yhat the te effective quarterly interest rate at that time can be either l is implied June 91-day forward rate (therate frbm June to September) 1.8%. Here is the risk faced by the borrower, assuming no hedging;
120 days
EURODOLLARS,
rate. The actual borrowing is conducted by the borrower independently of the FlkA. We witl suppose that the reference rate used in the FRA is the same as the acmal borrowing cost of the borrower.
Thus, if we observe the price, we can solve for the yield as rcc
RATE AGREEM ENTS,
-$295,566.50
-$300,000.
contract that guarantees a
borrowing or lending rate on a given notional principl amount. FRAS can be settled either at the initiation or mamrity of the borrowing or lending trapqaction. If settled at mattlrity, we will say the FRA is settled in arrears. ln the example above, the FRA could be settled on day 120, the point at which the borrowing rate becomes known and the borrowing takes place, or settled in arrears on day 21 1, when the loan is repaid. F''ltAs are a fomard contract based on the interest rate, and as such do not entail the
actual lending of money. Rather, the borrower who enters an F.RA is paid if a reference rate is above the FRA rate, and the borrower pays if the referene rate is below the IRRA
In order to fnake this payment, By denition, the future value of this is the borrower can bonow an extra $295,566.50,which results in an extra $300,000loan payment in September. If on te other hand rquanerly = 2.0% the payment is ,
$200,000 $196,078.43 1 + 0.02 = The borrower can invest this amount, which gives $200,000in September, an amount that offsets the extra borrowing cost. If the forward rate agreement covers a borrowing period other than 91 days, we simply use the appropriate rate instead of the 91-day rate in the above calculations.
%.INTEREST
216
Synthetic
RATE FORwARDS
RATE AGREEM ENTS,
FORWARD
AND FUTURES
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Suppose that today is day 0 and we plan to lend money 120 daks hence. By using a forwltrd rate agreement, we can guarantee the lending rate we will receive on day 120. In particular, we will be able to invest $1 on day 120 and be guaranteed a 91-day rettlnl of 1.8%. We can synthetically create the same effect as with an FlkAby trading zero-coupon
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First, let's get a general sense of the transaction. To match the FlkAcash flows, we cash going out on day 120 and coming in on day 211. To accomplish this, on day 0 want will need to borrow with a 120-day maturity (togenerate a cash outflow on day 120) we and lend Fit.h a 211 day mattlrity (to generate a cash inCow on day 211). Moreover, we Want the day 0 value of the bonowing and leqding to be equal so that there is no initial cash flow. This, description tells us what we need to do. In general, suppose that today is day 0, and that at time t we want to lend $1 for the period s, enrning the implied folavard rate ?:(/, t + J) over the interval from t to ?'otr, will denote the lonannttalized t+ t + s. To simplify the notation in this section, percent return from time t to time s. Recall lirst that
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INTEREST
AND FUTURES
RATE FORwARDS
To summarize, we have shown that an FRA is
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(J ! --rt We can synthetically create the payoff to an FRA, r,kj j+o ) (,j 1 l
tl
't...'.t
,u)
) ,
by borrowing to
buy the prepaid forward, i.e., by 1. Buying 1 +
rto (lI
,
.'
FORWARD
tz4 of the zero-coupon bond maturing on day tz, and
2. Shorting 1 zero-coupon bond maturing on day
/1
.
we iszczo,
.
RATE AGREEM ENTS,
EURODOLLARS,
AN D HEDGI NG
$
219
have
-
92) x 100 x $25) x 100
I
I 1 ! t
*.'''
Number of Eurodollar contracts
=
-
A
100 l + 0.0 18
)
j
l
)j
t?
-98.2318
=
98.2318 x (92.8
94) x $2500
-
-$294,695
=
8% (rquanerly2.0%), our total gain on the contracts will be =
.
(92.8
-
(100
-
?'t.lBoRlq
x 100 x
$25
Thus, the payoff on te Eurodollar contract compensates us for differences between the implied rate (1.8%)and actual LIBOR at expiration. To illustrate hedging with this contract we again consider two possible 3-month borrowing rates in June:. 1.5% or 2%. If the interest rate is 1.5%, borrowing ost on $l00m will be $1.5m,payable in September. If the interest rate is 2%, borrowing cost will be $2m. Suppose that we were to short 100 Eurodollarfutures contracts. Ignoling markingto-market prior to June, if the 3-month rate in June is 1.5%, the Eurodollar futures price will be 94. The payment is
((92.8
-
94)
x 100 x
$25) x 100
-$.
=
300,000
We multiply by 100 twice: Once to account for 100 contracts, and the second time to convert the change in tlRe futures price to basis points. Similarly, if the borrowing rate
s'rhiscalculation treats the Eurodollar contract as if it were with daily settlement, discussed in Appendix S.B. associated
98.2318 x (92.8
-
92) x $2500
=
$196,464
Notice that the amounts are different than with the FRA: The reason is that the FRA payment is automatically tailed using the 3-month rate prevailing in June, whereas with the Eurodollar contract we tailed using 1.8%, the LIBO:R rate implied by the initial futures price. We can now invest these proceeds at the prevailing interest rate. Here are the results on day 211, when borrowing must be repaid. lf LIBOR = 6% (lkuarterly = 1.5%), . we save $300,000in bolwwing cost, and the proceeds from the Eurodollar . contract are -$294,695 x (1.015)
-$299,
=
8% (rquarterly 2.0%), we owe an lf LIBOR vested proceeds from the Eurodollar contract are =
=
$196,464 x (1.02)
=
extra
l15
$200,000in interest and the in-
$200,393
Table 7.4 summarizes the result from this hedging position. The borrowing cost is close
to 1.8+.
Convexity bias and tailing ln Table 7.4 the net borrowing cost appears to be little a less than l You might guess that this is due to rounding error. It is not. Let's examine the numbers more closely. .8%.
a forward contract, ignoring the issues 6We
i
! t ! l
be
.
'
(
i
'
'
=
..
1
$200,000
=
This is like the payment on an FRApaid in arrears, except that the futures eontract settles in June, but our interest expense is not paid until September. Thus we have 3 months to arn or pay interest on our Eurodollar gain or loss before we actually have to make the interest payment. Recall that when the FRA settles on the borrowine date, the oavrent is the oreselt vallle of the change in borrowing cost. The FRA is thus tailed automatically as part of the agreement. With the Eurodollar contract, by contrast, we need to tail the position explicitly. We do this by shorting fewer than 100 contracts, using the ihplied 3-month Eurodollar rate of 1.8% as our discount factor. Thtls, we enter into (5
lf LIBOR
q..t
..
..
l
g(92.S
=
Eurodollar futures contracts are similar to FRAS in that they can be used to guarantee FRAS and Etlrodollar contracts, a borrowing rate. There are subtle differences between understand. that however, are important to Let's consider again the example in which we wish to guarantee a borrowing rate for a $100mloan from June to Seplember. Suppose the June Eurodollar ftltures price = j go over 3 luonths. As we saw in is 92.8. lmplied 3-month LDOR is 100-92.8 4 single short Eurodollar the payoff contract at expiration will be5 Chapter 5, on a
:
'
Now consider the gain on the Eurodollar futures position. lf LIBOR = 6% (rquarterly 1.5%), our total gain on the short contracts when we initiate borrowing on day l20 will
Eurodollar Futures
..
assume here that it is possible to short fractional contracts in order to make the example exact.
J
k. INTEREST
220
FORWARD
AN D FUTURES
RATE FORWARDS
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Gain plus interest
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0.200393m
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-101.799m
.
Net
= 1.5%), we pay $1.5m in borrowing cost, pnd we 6%, (rqumerly the Eurodollar contract, for a net borrowing expense of $1.799115m. lose $299,115on ttprofit'' from the Eurodollar hedge, relative to the use of an FRA, of $1.8m Tlzisis a
If LIBOR
=
-
$1.799115m $884. ((fLDOR 8% (rqumerly 2.0%), we pay $2.0min borrowing cost, but male Eurodollar contmct, for a net bonowing expense of $1.799607m.We $200,393 on the FItA, of- $1.8m $1.799607m $393. relative proEt, to an make a =
=
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-
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It appefkrs that we systematically come out ahead by hedging with Eurodollar futures instead of an I7RA. You are probably thinking that something is wrong. IJy the Eurodollar As it mrnsout, what we have just shown is that the ?-t-Ild implied p/-dl/t-I#l, ratefor l/7c san'e ft/t7?s.To FRA (??w/d#./-()?-1?t7?-#.) ctwlpwcfcalnot eqltal the the interest perspective: When rate tul'ns out to be high, consider this, the borrower see proceeds payoff the positive and Eurodollar short has can be reinvested conact the a interest realized the rate t'urns out When the high date rate. the payment loan at until payoff negative and has contract Eurodollar short low, the be we to can fund this loss a until the logn payment date by borrowing at a low rate. Thus the settlement structtlre of the Eurodollar contract works systenatically in favor of the bon-ower. By tvtnling the argument around, we can verify that it systematically works against a lender. The reason this happens with Eurodollars and not FlkAs is that we have to make decision before we lnow the 3-month rate prevailing pn day 120. When we tailing the fixed by tail a amount (1.8%in the above example), the acmal variations in the realized of the borrower and against the lender. The 17RA avoids this by work in favor rate tailing-paying the present value of the change in boaowing cpsts-using automatically date. borrowing the actual interest rate on the the rate In order for the futtlres plice to be fair to bot.h the borrowr and implicit in the Eurodollar futures price must be higher than a comparable FRA rate. This difference between the FRA rate and the Eurodollar rate called convexity bias. For the most part in subsequent discussions we will ignore convexity bias and treat the
tender,
.is
EU RO DO LLARS, AN D HEDGI NG
ENTSZ
$
221
is that in many FRAS as if they are interchangeable. The reason Eurodollar eontract and in a proht of results bias convexity example, small. ln the above cases the effect is For short-tenn contracts, the of several hundred dollars out of a borrowing cost $1.8m. efft can be important.? effect can be small, but for longer-term coneacts te before the final contract settlement. We ln practice, convexity bias also matters markng-to-market a fumres contract can lead to a fumres price saw in Section 5.4 that fumres contract is marked to market that is different from the forward price. When a with the ftitures price, there is a systematic and interest rates are negatively correlated This leads to a f'uturesprice that is greater advantage io being short the f'umres eontract. with the Eurodollar contract in this what exactly happens the forward price. Tltis is t.14a11 receives a payment that can be invested example. When interest rates rise, the bonower interest rates fall, the borrower males a paymentthat at the higher interest rate. When benest. Marldnginterest rate. This works to the boaower's can be f'unded at the lower implied by the why rate the another reason therefore to-market prior to settlement is comparable l7IA. Eurodollar coneact will exceed that on art otherwise
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RATE AG REEM
but The Eurodollarfutures contractis based on LDOR, contract example, the Treasury-bill futures there are other 3-month interest rates. For bill. A borrower could use either the Treasury is based on the pdce of te 3-month hedge their bon-owing rate. Eurodollar contract or the Treasury-bill f'utures contract to Which conact is preferable? potential to default. Thus, LIBOR Banks that offer LJBOR time deposits have the increase in the interest rate that is includes a defaultpremium. (The default premblm an will default.) Pzivate companies compensates the lender for the possibility the borzower will also include adefaultpremium. thatborrow can also default, so theirborrowing rates
I-IBOR versus
T-bills
3-month
then it would be possible to perfectly tail the 7If future interest rates were known for certain in advance, is due to interest on the difference between about the error rates, position. However, with uncertant'y the error is that Given we tail by the forward rate, the realized rate, F, and the fonvard rate, rforward. measured by F ( 7 rorward ) 1 + rfonvard -
The expected error is E
F ( 'F
-
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s
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the standard
(7.2 example can be 2% or 1.5%, so is the vafiance of the interest rate. Rates in our where 0.00252 0.0000062.5. approximate'ly 25 basis points or 0.0025 and the variance is thus deviation is Convexity bias is thus 0.00000625 $603.09 $ l00m x l 18a example was ($884+ $393)/2, or $638.5. The actual average convexity en'or in the =
=
.0
k. INTEREST
222
RATE FORWARDS
DURATIOX
AND FUTURES
The U.S. government, by contrast, is considered unlikely to default, so it can borrow at a lower rate than Enns. In addition, in the United States and other countries, government bonds are more liquid than corporate bonds, and this results bonds.S in higher plices-a liqttidity prc/nflf?n-for government The borrower will want to use the futures contract that has a price that moves in tandem with its own borrowingrate. lt makes sense that aplivate borrower's interestrate will more closely track LDOR than the Treasurpbill rate. ln fact, the spread between corporate borrowing rates and Treasuries moves around a great deal. The problem with hedging borrowing costs based on movements in the T-bill rate is that a private firm's bolwwing costs could increase even as the T-bill rate goes down; this can occur during times of financial distress, when investors bid up the plices of Treasury securities ttllight to quality''l. Thus, LYOR is commonly used relative tp other assets (a so-called private high-quality, interest rate. benchmark, markets in as a Figure 7.3 shows historical 3-month LDOR along with te differnce between LIBOR and th 3-month T-bill yield, illustrating this variability.g It is obvious that the spread varies considerably over time: Although the spread has been as low as a few basis points, twice in the 1990s it exceeded 100 basis points. In September of 1982, when Continental Bnnk failed, the spread exceeded 400 basis points. A private Lmolt-based 'E '( i kjj;'!111::111:)* !1!Ik-k-jII;' y' i't' p' 1* t' jjy'j (' jjk'. jt'tyd (yy'' ( ' E' ql. y!E!!!j,;;f syg-'yf tjjjy,:,,.f 11!1/*. tttd t;yf r' lqc'if j j'.!((j'. ' gq j(.... j:ji j'jjtgtytyj j;;. 41111:::1;.17. .', . .j'.j. j .jjjy ty ! . j. r t!. . . IljlEjz:. .(..q.. .. jjy;.yjjj;y.. yy yg.. .yyt. jjygj-.g.(y.j...yq.,y..gjq y. )!i.r.r: ..2 . . . t-jyty;yyjy, .1lIi2:!!q-.. , . . . . ..
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Date Sottrce: Datastream.
8In the United States, another reason for government bonds to have higher prices than corporate bonds is that government bond interest is exempt from state tmxation. gThe TED spread tt'T-Bills over Eurodollars'') is obtained by gong long T-bill futures and short the Eurodollar futures contract'
coxvExl'ry
k
borrower who had hedged its borrowing rate by shorting T-bill futures in August of 1982 would by September have lost money on the T-bill contract as Treasul'y rates declined, while the act'ual cost of borrowing ILIBORIwould have remained close to unchanged. This example illustrates the value of using a hedging cpntract that reflects the acmal cost of bonowing. The Eurodollar futures contract is far more popular than the T-bill futures contract. Trading volume and open interest on the two contracts were about equal in the early 1980s. H.owever, in August 2004, open interest on the Eurodollar contract exceeded 6 million contracts, while the T-bill contract had zero open interest. This is consistent with LDOR being a better measure of private sector interest rates tan the T-bill yield.
lnterest Rate Strips and Stacks Suppose a borrower plans to borrow $100mby rolling over 3-month debt for a period of 2 years, beginning in 6 monts. Thus, the borrowing will take place in month 6, month 9, month 12, etc. The borrower in this simation faces eight unknown quarterly borrowing rates. We saw in Section 6.12 that an oilhedgercould hedge each commitment individually (a strip) rcould hedge the entire commltmentusing one nakldated contract (a stack). The same alternatives are available wit interest rates. One way to hedge is to enter into separate $100mFRAS for each f'uttlre3-month period. Thus, we would enter into one F'RA for months 6-9, another for months 9-12, etc. This strip hedge should provide a perfect hedge for f'uttlre borrowing costs.
'
Depending on market conditions, using a strip is not always feasible. Forexample, forward prices may not be available with distant mattlres, or liquidity may be poor at distant maturities. Rather than individually hedging the boaowing cost of each quarter, ttstack'' of short-tenn Flus or an alternative in the context of this example is to use a Eurodollar contracts to hedge the present value of future borrowing costs. In the above example, we will be borrowing $100 million per quarter for eight quarters. To effect a stack, we would enter into fprward agreementt on the 3-month rate, maturing in 6 months, for slightly less ihan $800 million (We enter into less than$800 million' of forward rate agreements due to tailing for quarters 2 through 8.) As with the oil example in Section 6.12, the obvious problem with a stacking stratborrowing costs in distant quaders may not move pedctly is egy basis risk: Quarterly with borrowing costs in near quarters. Once we reach the frst quarter of borrowing, a11of the forward agreements mature and we need to renew our hedge. We now face seven quarters with unlnown borrowing costs and therefore we enter into fomard agreements for slightly less than $700million. is exactly The constantrenewal of the hedgingposition necessaly to effect astackandroll like that in the oil example. .
J7
a;q?)
,.
AxD
7.3 DURATION
AND CONVEXITY
An important charactelistic of abond is the sensitivity of its price to interestrate changes, which wemeasureusing duration. Duration tells us approximately how much the bond's price will change for a given change in the bond's yield. Duration is thus a summal'y
i
224
% INTEREST
DURATION
AND FUTURES
RATE FORWARDS
of bonds with different coupons, measure of the risk of a bond, permitting a compmison times to mamrity, and discounts or premiums relative to principal. In this section we also discuss convexit'y, which is another measure related to bond price risk.
tl
#(y)
C/?n (1 + -
=
f
.:/,?7)
izzz1
+
,,
''
price =
-
Upit chgpge in yild
jxzj
=
-
(7/?n
i
(1 +
Unit change in yield 11
1
1
t)'/p?
i -
B(vj 1
-1-
yjln
j.j
??l
l . #(y)
(l
-1.
y/??l)f
+-
Alf'
11 ???
(1
...h
.y//31)/
We obtain another measure of bond price risk-Macaulay duration-by multiThis puts both bond price and yield changes in expression with a clem- interpretation:
i= l
??l
(1 +
c/???.j
i
Macaulay duration
.'.://,1()?1+1
m
-
1 + y/lll
(1 + y/???) '
+
?.1
Arf'
-
z?2 (1 + y/???)''
Example 7.4 Consider the 3-year zero-coupon bond in Table 7.1 with a yield to maturity of 7%. 'Fhe bond price per $100of matulity value is $100/ 1.073 $81.82979. =
$100/1.07013= $81.60691, Atayield of 7.01%, one bsis pointhigher, tebondpriceis per $100of maqlrity value. a change of As an alternative way to derive the price change, we can compute equation (7.9) with C = 0, M = $100,11 = 3, and ??? = 1 to obtain -.-$0.02288
$100
3x 1.073
1 + yjnl Chanee in bond orice x Unit change in yield S(y) -
=
-
(7.9)
q
Equation (7.9)tells us the dollar change in the bond plice for a change of 1.0 in y. It is natural to scale this either to refect a change per percentage point (in which case equation (7.9) by 10,0001. we divide equation (7.9) by 100) or per basis point (divide Equation (7.9) divided by 10,000 is also known as the price value of a basis point (PVBP). To interpret PVBP for a bond, we need to know the par value of the bond.
x
x
-
M
71 =
y/F?l)f+l ?'
1
1 - 1.07
=
=
.y/??1.11
-
?n
Change in bond price
Modifed duration
plying equation (7.10)by 1 + percentage terms and gives us an
The chapge in the bond price for a unit change in the yield, y, islo
hangeiti bond
=
1 #()?)
-
11
i
(7/!??l + ( 1 y/???)f
j..j
??)
/$lr
??
-1-
??2
(1
...1-
yjnln
To interpret this expression, note that (C/??1)/(1 + yjlnli is the present value of the is th bond payment, which occurs in f/??7 years. The quantity C/?n/((1 + 'jlnlBl'lj therefore the fraction of the bond value that is due to the fth payment. Macaulay duration is a bveighted twc/wgc of #7e time (nl/???/Jd?' ofperiods) ulltil #l' boltd ptky/?l'?lll occlfr, with the weights being the percentage of the bond price accounted for by each payment. This interpretiion of Macaulay duration as a time-to-paymnt measure explains why For a zero-coupon these meajures of bond price sensitivity are called bond, eq ation (7.11)implies that Macaulay duration equals time to maturity. Macaulay duration illustrates why maturity alone is not a satisfacto/ risk measure for a coupon bond. A coupon bond makes a series of payments, each with a different maturity. Macaulay duration summarizes bond price risk as a weighted average of these EEduration.''lz
-$228.87
=
a change of 1 basis point, we divide by 10,000 to obtain almost equal to the actual bond plice change. This -$228.87/10,000 k. illustrates the importance of scaling equation (7.9)appropriately.
In order for this to rflect -$0.02289,
=
lo-fhisis obtained
225
-l.
M
(1 + y/,n.)
%
When comparing bonds with different prices and par values, it is helpful to have of price sensitivity expressed ypr/-dollar of 1747/7:'/ btain this by price. k measure a This gives us a dividing equation (7.9)by the bond price, #(y) and multiplying by change in the bond price measure known as modified duratlon, which is lepercelltage for a unit change in the yield:
Dmation Suppose a bond makes ??l coupon payments per year for F years in the amount C(m, and pays M at maturity. Let ),/7?7be the per-peliod yield to mattlrit.y (by convention, y is the armualized yield to mamrity) and 11 = m x T the number of periods until matulity. The price of the bond, S(y), is given by
AN,D CoNvExlTy
by computing fe derivative of te borldprice with respect to the yield, dBlsl/dy.
l lerhis measure of duration is named after Frederick Macaulay, who wrote a classic history of interest rates (Macaulay, 1938). l2The Excel duration functions are Dltration for Macaulay duration and MDttratit for modified duration.
k. INTERES'I-
226
Du RATI o lq
AND FUTURES
RATE FORwARDS
Co
AN D
NvExl'rv
k
227
.0725)3
Returning Ekample 7.5 q'r :. j:gj; )(. . 7% bpnd is ytifor ihe )'.) 'j
to Example 7.4, using equation
' E
1.07
-$228.87
x
-
1)E,
1
(7.11),Macaulay duration
k
-3.000
=
$81.62979
Example 7.6 Consider the 3-year coupon bond in Table 7.1. For a par bond, the yield to mamrit'y is the coupon, 6.95485% in this case. For each payment we have x
(;.4;.54:5
payment 1
1.0695485
%pavment 2 -
=
%payment 3 Thus, with
11
=
3 and
??l
=
(1 x 0.065026) +
=
0.0695485 a
=
0.065026
=
0.060798
(1.0695485) 1.0695485 (1.0695485)a
=
The act'ual new bond price is 0.02% of the bond price.
$100/(1
k.
Although duration is an important concept and is frequently used in practice, it has a concept'ual problem. We emphasized in the previous section that a coupon bond is a collection of zero-coupon bonds, and therefore each ash flow has its own discount rate. Yetboth duration formulas are computed assuming that all cash flows are discounted by a single artificial number, the yield to mattlrity. In Chapter 24 we will examine alternative approaches to masuring bond price lisk.
Dttration Matching Suppose we own a bond with time to matulity 11 price 5.1 and Macaulay duration DI We are considering a short position in a bond with mattlrity tz, price Bz, and Macaulay duration Dz. We can ask the question: How much of the second bopd should we shortsell in order that the resulting portfolio-long the bond wit.h duration Dj and short the bond with duration Dz-is insensitive to interest rate changes? ,
0.874176
1, Macaulay duration is
(2 x 0.060798) + (3 x 0.874176)
=
2.80915
The interpretation of the duration of 2.8 1 is that thebond responds to interestrate changes 2.81 yeazs to maturity. Modihed duration is as if it were a pure discount bond wit.h k. 2.80915/1.0695485 = 2.626482.
Equation
(7.12)giyes
.
us a formula for the change in price of each bond. Let N B l + NB1
and, using equation (7.12),the change in price due to an interest rate change of
(#l(.71+ 6) fl (.:1)1+ = -Dl Bt (.:1)6/(1+ yl ) -
Since duration tells us the sensitivity of the bond price to a change in the interest given change in it rate, can be used to compute the approximate bond pzice change for a changes from y the yield the bond and is bond prie B the on (y) rates. Supppse interest tfle d duration, D, for mpd yield. The formula the small chpng is in where E E, a to y + can b written 1 (S()' + 6) :::::::::: ..i!E((;1p
,
denote the quantity of the second bond. The value of the portfolio is
-
N (S2(y2 +
6)
#D2.P2()'2)c/(1
Eq is
,:212)1
-
+
.y2)
where Dl and D2 are Macaulay durations. If we want the net change to be zero, we choose N to set the right-hand side equal to zero. This gives N=-
.l()/)1
-
.........
$81.060. The prediction enor is about
=
D1 S1 (.,1)/(1 + )?l) D2#2(y2)/(1 + J?2)
.............
..
s(y)
s
Letting Macaulay duration be denoted by Dslac, we have Duac = D(1 + y). We can therefore rewrite this equation to obtain the new bond price in tenns of the o1d bond price and either duration measure:
W'hen a portfolio is duration-matched in this fashion, th net investment in the portfolio will typically not be zero. 'Fhat is, either the value of the short bond is less tn the value of the long bond, in which case additional Enancing is required, or vice versa, in which case there is cash to invest. This residual can be financed or invested in vel'y shortterm bonds, with duration approximately zero, in order to leave the portfolio duration
matched.
lt Example 7.7 Considerthe 3-yearzero-couponbond with aprice of $81.63persloo )!))maturity value. The yield is 7% and the bond s Macaulay duraton is 3.0. f the yield jyt tl were to increase to 7.25%, the predicted price would be ,
y...t
i
/(7.25%)
=
$81.63 (3/1.07)x $81.63 x 0.0025 -
=
$81.058
Suppose we own a 7-year, 6% coupon bond with a yield of 7%, and r1 Example 7.8 )EJ lind duration-matched short position in a lo-year, 8% coupon bond yielding the it want to )).. '.''ti7 5%. Assuming annual coupon payments, the Macaulay duration and price of the two
228
% INTEREST
RATE FORWARDS
AND FUTURES
DURATION
bonds is 5.882 years and $94.611, and 7.297 years and $103.432,respectively. Thus, if we own one of the 7-yea.r bonds, we must hold 5.882 x 94.61 1/(1.07) - 7.297 x 103.432/(1.075)
equation (7.14)with C
0, ??7
=
Convexity
=
1, and M
3 x 4 x
=
-0.7408
=
Using equaton
the lo-year bond. The short position in the lo-year bond is not epough to pay for the 7-yea.r bond; hence, investment in the portfolio is 1 x 94.611 0.74 jy x jts zsr 17.99. If the yield on both bonds increases 25 basis points, the price change of the portfolio is units qf
-
.
-1.289 + (-0.7408)
x
=
Convexity The hedg
ip example 7.8 is not perfect because duration changes as the interest rate to which duration changes as the bond's yield
Convexity
11
1
B ()')
g=l
C/m
i i + 1) ln p,
(1+
.'.:/??7)
j.j.:t
+
/7
(n + 1) ???a
(j
.j.
M yjjtjtt-vg
By +
E:)
=
#(y)
(D x B (y) x
+ 0.5 x Convexity x B)')
x c2
7.@
Consider again example 7.7. We want to predict the new price of a 3-year zero-coupon bond when the interest rate changes from 7% to 7.25+. Using
13At the original yields, we computed a hedge ratio of 0.7408. Problem 7.19 asls you to compute tlle hedge ratio that would have exactly hedged the portfolio had both interest rates increased 25 basis points and decreased 25 bmsis ponts. The two hedge ratios are different, which means that one hedge rato would not have worked perfectly. l4-rhis is obtained by taking the second derivative of the bond price with respect to the yield to maturity, dlBs,jjdl'l, and normalizinR the result bv dividing by the bond price. l5If you recall calculus, you may recognize equation
price. See Appendix 13.%
(7.12)as
a Taylor series expansion of the bond
-
j'jy'yjj;-', '))' 77(* 1* ;'))r' t'(' (' q' 5,* )!..tl:')i (' (q!' ltf E' ?' jyyy', y' j'g.t' jjyd ;'rlf ((' j)' jjjjr:::;;;ky'-' 1* i'lqlft.qiplllqslkllijqrlq)l qjf i IE itryf (;4* yjd j.tt!r))jyy'. g' r?''t. ...'.' )' (' ttf y' j)' r:' t'j't' FE iE 7:7r.rr77)42:77( ' tjyt.yjyjjtdsjtjtyj'jy jjjjyll'tttf )' yjf jyf E l:lliti yjyy'jjj;i:jjj::d 1*.)* y' 1$$* '(tj'' '. .'. '!( E (:' ' E; E ktE ijjjt,,,jjjr'. E ;'('( ''(.!EE''?( t' Er; '. .. !.. ' i ' ... .E.' i. EE (qE..' q' .. (.E.. . (.ti .;:t( ;(.i i('i(:1((:.qi '):ijqq l E E! . .i E.; i ; #!.'- . i'-i lr:ii i!jjj):. (q , . .i E... ...Ei iE. pE :@. .j)! j jtjr ! . @ )..,j.. .r.yyjgtjy.ry(.j.)). E . jq . j( (: C1!i':t)E ; t ytjy;..y. yyy t ..'@t--It)-...y.yq.jy ljjjjj!j, .. j. ..yjjjjq:js jjjjjs j.yy IE .yygyy.q . ):j;;iiqf')., y y yy i.. . (E y. j..j..j.yyg jjy. jjL...,...,...j...-,..,.,.t= .. .yjjyjyy . .. . .' jyy .... .-. (' .
.''
rr-trtr'jjy;f : .
E
where D is modifed duration. Here is an example of computing a bond price at a new yield using both duration and convexity.
Example
=
-jj,'.
We can use convexity in addition to duration to obtain a more accurate prediction of the new bond price. When we include convexity, the price prediction formula, equation 15 (7.12), becomes 6)
(7.15),the price at a yield of 7.25% is
:';'F' -' y'j'ykytjj,'. lji'j,yyr r;--jjyjr')E '!)f I'E :'i':f'
(7.14)
-
10.48 12
=
Figure 7.4 illustrates duration and convexity by comparing three bond positions tathave identicalprices atayieldof 10%. Duration is the slope of thbond price graph at a given yield, and convexity is the curvamre of the graph. The 10% lo-yearbond has the lowest duration and is the shallowest bond price curve. The othertwo bonds have almost
changes.l'Coltvexitymeasures the extent C anges. The formpla for convexity is14 h =
81.63
The predicted price of $81.060is the same as te acttlal price at a yield of 7.25%, to an accuracy of three decimal points. ln example 7.7, the predicted price was slightly lower ($81.058) than the act'ual new price. The difference without a convexity correction occurs because the bond's sensitivity to the interest rate changes as the interest rate changes.l6 Convexity corrects for this effect. %.
Q
-0.004
-1.735
1
x
.07(3+2)
$81.63 (3/1.07)x $81.63 x 0.0025 + 0.5 x 10.48 12 x $81.63 x 0.00252 = $81.060
(7.25%)
.
$100,convexity of the bond is
=
100 l
%.
AN D colqvExl-ry
.
-
.);)....:.-
-
''
i
.
.
' .' : E. E
.--.
.
- . .
.
.
'
- -
'(
'
-)
-
:
: :
'
E' ' '
:
:
.
:
:
.
.
-
-,;j,::jjjj;g-
.
.
-
.
.
.-..........)....,...
,..
'
i.
- , . ,. . .
-....-
.......
. ...,
.
.
;.;((t
,
.
..
-
y
Comparison of the valueof three bond positionsas a function of the yield to maturity: 2.718 1O-year zero-coupon bonds, one lo-year bond . paying a 10% annual coupon, and one 25-year bond paying a The 70%coupon. (D) and duration (c) of each convexity yield of 10% at a bond in the Iegend.
are
.
.
y
y
.
jrjj,skyjj j:gy)j jlyg)j ,(r2: j:yg)j qj.rjjs j:r:jj gjrr;js (yjj,
,-,,,,,,,,,
1.s 1.4 1
'
-
a
-
egj 1.2 ;
1.1
,
.
.......jfj;;,.
j:yjjg
=
=
=
=
=
-w
N.v.
I
*'->.
'j
;:
j:gyjj j;jgj ijyjr.tj:y gjrg;j lgygjts gjj;;j (o n (s c no.91) 1o% coupon lo-yek bond CD 6.14, C 52.79) 10% Coupon 25-year bond CD 9.08, c 139.58) =
1 -<-''-.-
0.9
Nw-.w
() 8
o.7
.
' . ' I ' 0.6 0-06 0.07 0.08 0.09 01 0.11 0.12 0.13 0.14 0.15 Yield to Mattlrity
'
'
'
I
l6You might wonder about this statement since the bond in example 7.7 is a zero-coupon bond, for which Macaulay duration is constant. Notice, however, thatthe bond priceprediction formula, equation (7.12), depends on nlodised duration, which is DMac/(l + y). Modilied duration does change with the yield oft the bond.
k. INTEREST
230
RATE FORwARDS
TREASURY-BOND
AlqD FUTIJRES ' ' E E' : ' IE jyf ;'p' ijf fq'l :r51'l77'r'q7' .'i' l)lk' ': (i::': ' !-kjg'.'' i: :' @ E: : '.'. E.'.' .'.:' .EE'i!E(E.(iq j': (' E'''' E'E'EE:. i' ('E(!: : ((@ ' ':( ('q(ii.!-,(. !E yk'' 4* i';'' 7q:J* 1* )' k' ((' (' jly'.yyji ;q!'q:'q::!!'r'l'g'! jyf j'y' t'q'y' (:':Er ( i. 'i('IE:E;@!!(I('. lqirqE qltlqt' (' ')' r' jyjfjjyfyyjy'yyylyy k. ' .. . . ' '.. yy' ;jy' . . .' ;jj;jjjyjj;;.'. ' (..' ('! (. . I'' .(i- E- E - - - ',-i .y . Ey-. --E -. . E. -. r i!,-r.;- --.. . - E(q. E: ;EEl ;. q . . g .y ( . y y . y. . ; ( y . yy yjk@ ' ..)).. . :.. - :i .E!.: ';j.l); ' jj y yjj. ;yy....yy.;y.jy. y.. y yyj.j. :. ; .. .. .. . ;- iiq:-ji gyg yjy..y. .. . .... .. . .. ..
)' jjjjjjjj-f 'll!:d 7* t'jy'rf yyyf r' rr' )y'j'(' jf 1* tyf j'y )' j'' )t,'' jj-;' y't)yf t'y' ' . . .t' ('t-' (' tjy'tjyjy'. ( '. .j;.-yy . j. r.ttjy )j.jrj.ijjL)i,,i;qjj )jyy. 'rj.j. yy . j.... .,.(jlljjjj::;jy:r.jljjh:;jj::, . g.y... j. jygs ... ''!.'; .y. jjjjj .. ..
t't', tyrf ti'. .'.'
,'-y;'. ,?' ' ''f' -$j' :'y' .-..' !' :':)' '')!,jjj;-jkjjjy'E Eyf
rqual durations at a yield of 10% and their slopes are equlkl in the gure. However, the 25-year bond exhibits greater curvamre: Its price is above the lo-year bond at both lower and higher yields. This greater curvature is what it means for the 25-yeltr bond to have greater convexity.
'
YREASURY-BOND
AND
TREASURY-NOTE
FUTURES
.
-
.
'
.
-
.
-. ... .
.
-.
'.
.
-
.
..
. .
. . . . . .
. -
. . .. . . . .
.. .. ..
.
. .
-
....
-
-. . . . .
-
..
.
-
,
,
.
-
.
.
Specifications for the . Treasurpnote futures
contract.
The Treasurpnote and Treasury-bond f'utures contracts are important instnlments for hedginginterest.rate risk.l? Figure 7.5 shows newspaper listings for these futtlres, and the specihcations for the T-note contract are listed in Figure 7.6. The bond contract is simi'lar except that the deliverable bond has a mattllity of at least 15 years, or if the we will focus bondis callable, has 15 years to 'Iirst call. The two coneacts are similar; tibond'' will discussion and In tis T-note the use the contract. terms we hereon
FUTU RES
$
231
.
-.
yjrjjtrriijk,g . -
.
'
.. . . . . .
. :
AN D TREASURY-NOTE
'
jE
'
.
The idea that using both duration and convexity provides a more accurate model of bond price changes is not particular to bonds, but iy pertains to options as well. This is our first glimpse of a crucial idea in derivatives that will appear again in Chapter 13 when we discuss delta-gamma approximations, as well as throughout the book.
7.4
'
:
.
.
.
..
wheretraded
.
CBOT 6% underlying lo-year Treasury note Size $ 100,000 Treasul'y note Months Mar, Jun, Sep, Dec, out 15 months Trading ends Seventh business day preceding last business day of month. Delivery until last business day of month. Delivery Physical T-note with at least 6.5 years to maturity and not more than 10 years to maturity. Price paid to the short for notesr with other than 6% coupon is determined by multiplying futures price by a cotwersion factor. The conversion factor is the price of the delivered note ($1 par value) to yield 6%. Settlement until last business day of the month.
<Enote''
interchangeably.
The basic idea of the T-note contract is that a long position is an obligation to buy bond with between 6.5 and 10 years to maturity. To a first approximation we can 6% a think of the underlying as being like a stock with a dividend yield of 6%. The futtlres plice would then be computed as w-ith a stock index: The f'uttlre value of the current bond price, less the fumre vtlue of coupons payable over the life of the futtlres contract.
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2004.
Interest
................................. Rate ....Futures
Tfeasufy Bondstnm
af l()2x pts rznds 114-39 121-24 5M,W Sept 19q-:l3 119-14 12-07 l.':l 1I)8..0 107.41 1:7...21 11347 12(1-21 18709 127-28 net (yca Int Dzet wl y:n z'3z?.az Est vd n9,724 ,r2;
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pts rznds cf 1Cl :15-(5515-B 1,333758 l',..ts l5-M 35,374 Dt t -17,W1 Est vl 723,779;vell'cil 3446117npenInt 14368,7524 Qf 5 Yr. Tfeaeury Notes ((BD-$1M1;pts 32* lr 11$41 llthl 1:%.13 :1.1+1.,q -1M 112-15 IA.Z.9 l,lszJM 5ept +3,$$1. Est ynl 408,924;vnIhlon1l,223; cpQR lnt 1,7:5.116, 2 Yr. Treasury Notes ((B.f'b$21%; ;ts 311s ef 11 7 l.tls4l (+1.87 137,212 s'pt 95-235 195-21 (5-162 195-17 vcIhlrt MJM; Q)eR IRt 127,21% Est vcI29,3224 -21.9
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interest rate on the lo-year Treasu!'y l-/-f'he thelo-year note futures are important.
This description masks a complication that may already have occurred to you. The delivery procdure permits the short to deliver any note maturing in 6.5 to 10 years. Hene, the deliverd te can be one of many outstanding not:, Witha rane of oupons Which bond does the futures price represent? andmturities. . atl bonds Of that ttltl be delivered, there will generlly be on: that is tlie most advantageous for the short to deliver. This bond is called the cheapest to deliver. A description of the delivery procedure will demonstrate the importance of the cheapestto-deliver bond. In fulflling the note futures contract, the short delivers the bond in exchange for payment. The payment to te short-the invoice prc for the delivered bond-is the f'utures price times the conversion factor. The conversion factor is the price of te bond if it were priced to yield 6%. Thus, the shol't delivering a bond is paidlB
Example 7.10 Consider two bonds maling semiannual coupon p'ayments. BondA 7% bd Vtll exactly 8 years to maturity, a price of 103.71, and a yield of is opon bond would have a price of 106.28 if its yield were 6%. Thus its conversion 6.4%. factor 1.0628. Bond B has 7 years to mattlrity and a 5% coupon. Its current price and yield are 92.73 and 6.3%. lt would have a conversion factor of 0.9435, since that is its plice at a 6% yield. % .
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AN D FUTURES
I NTEREST RATE FORWARDS
Now suppose that the fumres contract is close to expiration, the observed futures price is 97.583, and the only two deliverable bonds are Bonds A and B. The short can decide which bond to deliver by comparing the market value of the bond to its invoice price if delivered. For Bond A we have
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These calculations are summarized in Table 7.5. Based on the yields for te two bonds, the short breaks even delivering the 8-year 7% bond and would lose moneydelivering the 'F-year 5% coupon bond (theinvoice price is less than the market price). In this example, the 8-year 7% bond is thus the cheapest to deliver. ln general there will be a single cheapest-to-deliverbond. You mightbe wondering why both bonds are not equitlly cheap to deliver. The reason is that the conversion factor is set by a mechanical procedure (the price at which the bond yields 6%), taling no coincidence, two account of the current rlative market prices of bonds. Except by deliver. cheap be equally will to not bonds Also, al1 but one of the bonds must have a negative dlivery value. If tFo bonds ontf no-arbiage had a positive delivery value, then arbitrage would be jossible. The confguration in general has one bond worth zero to deliver (Bond A ip example 7.10) and the rest lose money if deliverd; To avoid arbitiaje, the fumres prie is Plice of cheapest to deliver Fumres price + Conversion factor for cheapest to deliver =
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This discussion glosses over subtleties involving transaction costs (whetheryou already own a bond may affect your delivel'y proft calculation) and uncertainty before the delivel'y period about which bond will be cheapest to deliver. Also the T-note is deliverable at any time during the expiration month, btg trading ceases with 7 business days remaining. Consequently, if there are any remaining open contracts during the last wek of the month, the short has the option to deliver any bond at a price that might be a week out of date. This provides a delivery option fof the short that is also priced into the contract.. There are other complications, but sufice it to say that the T-bond and T-note contracts are complex. The T-bond and T-note futures contracts have been extremely successful. The contracts illustrate some importantdesign considerations for afumres contract. Consider srst how the contract is settled. If the contract designated a particular T-bond as the underlying asset, that T-bond could be in short supply, and in fact it might be possible for someone to corner the available supply. (A market corner occurs when someone buys most or all of the deliverable asset or commoditjc) A short would then be upable to obtain the bond to deliver. ln addition, the deliverable T-bond would change om year to year and the contract would become more complicated, since traders would have to price the f'utures differently to re:ect different underlying bonds for different maturity
dates. An alternative scheme could have had the contract cash-settle
against a T-bond
index, much like the S&P 500. This arrangement, however, introduces basis risk, as the T-bond futures contract might then track the index but fail to track any particular bond. In the end, settlement procedures for the T-bond and T-note contracts permitted a range of bonds and notes to be delivered. Since a high-coupon bond is worth more than an otherwise identical low-coupon bond, there had to be a conversion factor, in order that the short is paid more for delivering te high-coupon bond. 'T'he idea that there is a cheapest to deliver is not exclusive to Treasury bonds. The same issue arises with commodities, where a f'uttlres contract may permit delivery of commodities at diferent locations or of different qualities.
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7.5 REPURCHASE
AGREEMENTS
An extremely important kind of forward contract is a repurchase agreement, or repo.19 A repo entails selling a security with an agreement to buy it back at q tixed plice. It is sale coupled with a long folward contract. Like effectively a reverse cash-and-carry-a it is equivalent cash-and-carry, to borrowing. The particular twist with a reverse any repo is that the underlying security is held as collateral by the counterparty, who has bought the seculity and agreed to sell it at a xed price. Thus, a repo is collateralized bon-owing. Repos are common in bond markets, but in principle a repurchase agreement can be used for any asset.
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l9For a detailed eeatment of repurchase agreements, see Steiner (1997).
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234
RATE FORwARDS
AlqD FUTURES
CHAPTER
!q Example 7.1 1 Suppose you enter into a l-week repurchase agreement for a 9))i)tk. and .. . The cunent price of the T-bill is $956,938, you agree to t month $1m Treasury bill. C repurchase it ip 1 week for $958,042. You have borrowed mony at a t-Week rate of ')q tl. 958.042/956.938 1 0.1 15*. receivine cash todav and oromisine to repay .cash plus (j. )'t interest in a week. The security provides collateral for the loan. k. -
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Called
a term repo. ' . .' . . . The eounterparty is said te have entered into a reverse repurchase agre:ment, or contraci. This is a loan of cash for th'duration reverse repo, and is short the fol-ward of the agreement with a security held as collateral. lt can also be desribd a a cashan
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If the borrower does not repay the loan, the lndef keeps th secutitf. Thus, the counterparty's view of the lisk of the transaction differs according to the quality of the collatei-al.Collateral with a more variable plice and a less litjuid market is lower quality from the perspective of the lender. Because collateral quality varies, evel'y security can have its own market-determined repo rate. government securities, and can Repurchase agreements are most common collateral collateral--called security a spcial as be negotiated to require a specic collateralsecurities of of vadety with any government as a rcpltrc/?tueagreenent--o ?'eptl?'c/?/.d agreement. Generi collateral repos have greater called a general collateral :exibility and hence lower ansaction costs. The repo rate on special collateral repos will generally be below that on general collateral repos. Suppose that there is demand for a specisc bond' as a speculative investment. The owner of such a bond can engage in a repurchase agreement, nd high low repo rate means that the demand for the bond will drive the repo rate down. cash received the for the bond that exceeds the original bondholder can ea.rn interest on . bond. of specialness te repo rate, thereby profting from the In addition to a repo rate that reQects collateral quality, dealers can also char/e a haircut, wlaich is the amount by which the value of the collateral exceeds the arpount of th e l 0an. The haircut reflects the credit lisk of the borrower. A 2% haircgt would mean that the bon-ower receives only 98% of the market value of the security, providing an additional margin of protection for the counterparty. Repurchase agreements are frequently used by dealers to Enance inventory. In the ordinary course of business a dealer buys and sells securitis. 'Fhe puycflase of a jecurity requires funds. A dealer can buy a bond from a customer apd thep repo it ovruight. The money raised with the repo provides the cash needed tl pay te sller.. 'I'he dealer then has a cost of carrying the bond equal to the repo rate.20 The counterparty on this .for
20The repurchase agreement in this exgmple provides financing. The dealer still is the ultimate owner of the bond and thus has plice risk that could be hedged Wt.h futures con%cts.
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eansaction is an investor with cash to invest short-term, such as a corporation. The investor buys the bond, promising to sell it back. This is lending. The snme techltiques can be used to finance speculative positions. Hedgefunds, for example, use repurchase agreements. A hedge f'und speculating on the price difference between two Treasury bonds can fnance the transaction with repos. An example of this is discussed in the Long-rrerm Capital Management box on page 236. How do we engage in a transaction lilie this-long bond A and short bond B-in practice? The answeris thatweundertake thefollowingtwo transactions simultaneously: The longposition: Buy bond A and repo it. Use the cash raised in the repo to pay for the bond (recallthat dealers nance wentory in this fashion). When it is time to reverse the repo, sell the bond and use the cash raised from the sale to buy the bond back and close the repo position tthink of the sale and close of the repo as happening snultaneously). Note that a low repo rate for this bond works to the arbitrageur's advantage, since it means that the reptlrchase price of the bond is low. The arbitrageur also benefts from a price increase on te bond. Borrow bond B by entering into a reverse repurchase agreeThe shortposition: ment. We receive the bond (collateralfor the loan) via the reverse repo, sell it, and use the proceeds to pay the counteparty. At the termination of the agreement, buy the bond back in the open market and ret'ul'n it, being paid the repo rate. Since we receive interest in this transaction, a high repo rate works to our advantage as does a price decrease on the bond.
Since the investor is betting that there will be a reduction in the plice difference between the two bonds, it is necessazy to enter into both legs of the eansaction. The arbitrageur would like a low repo rate on the purchased bond and a high repo rate on the sold bond, as well as a price increase of the purchased bond relative to the short-sold bond. ln practice, haircuts on bot.h bond positions are a ansaction cost. Haircuts yre a capital requirement imposed by the counterparty, which means that an arbitrageur must hkve capital to undertale an otherwise self-hnancing arbitrage transaction. Differences in repo rates on the assets can be alz additional transaction cost. Even if the plice gap between the two bonds does not close, the arbitrage can be prohibitively costly if the difference inrepo rates on the two bonds is suciently great. Cornell and Shapiro (1989) document that in one well-lnown episode of on-the-rurl/off-the-run arbitrage (seethe box on page 236), the repo rate on an on-the-run (short-sold) bond went to zero, making arbitrage costly even though the price gap remained when the on-the-rtm bond became off-the-run.
The price of a zero-coupon bond wit.h F years to maturity tells us the value today of $1 to be received at time F. The set of these bond prices for different maturities is the zero-coupon yield curve and is the basic input for present value calculations. There are
k. INTEREST
236
AND FUTURES
RATE FORwARDS
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the change in the government's borrowing rate may be different from the change in the borrowing rate for a fit'm or individual. Duration is a measure of a bond's l'isk. Modified duration is the percentage change in thbondpricefor aunitchange intheinterestrate. Macaulay duration is the percentage change in the bond price for a percentage change in the discount factor. Duration is not a perfect measure of bond plice risk. Aportfolio is to be duration-matched if itconsists said of short and long bond positions with equal value-weighted durations. Convexity is a measure of the change in duration as the bond's yield to maturity changes. Treasury-note and Treasury-bond futures contracts have Treasury notes and bonds as underlying assets. A complication wtth these contracts is that a range of bonds are deliverable, and there is a cheapest to deliver. The futures price will reqect expectations about which bond is cheapest to deliver. Repurchase agyeements and reverse repurchase agreements are synthetic shorttermborrowng andlending, the equivalent of reverse cash-and-can'y and cash-and-carry transactions.
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Basic interest rate concepts, are ftlndamental in Nnance and ar used throughout this book. Some of the fonpulas in thts chapter will appear again as swap rate calculations in Chapter 8. Chapter 15 shows how to prie bonds that mak payments denominated in foreigncurrencies orcommodities, andhow topricebonds containing options. Whilethe bond price calculations in this chapter are useful in practice, concepts such duration as have conceptual problems. ln Chapter 24, we will Fee how to build coherent, internally a consistent model of interest rates and bond prices. Useful references for bond and money market calculations are Stigum (1990)and Stigum and Robinson (1996).Sundaresan (2002)and Tuckman (1995)are sxed-income int topics in this chapter i more depth. Convexity bias is studied by texts tat Burghardt and Hoskins (1995)and Gupta and Subrahran#am (2000). Grinblatt and Longs'taff (2000)discuss the market for STRIPS and smdy the pricing relationships between Treasury bonds and STRIPS. The repo market is discussed in Fleming and Garbade (2002,2003, 2004). .go
rates, including the par
Forwardrate agreements (FRAs) permitborrowers and lenders to hedge theinterest by locking in the implied fomard rate. lf the interest rate changes, FRAS require a rate payment re:ecting the change in the value of the interest rate as of the loan's laturity day. Eurodollar contracts are an alternative to FlkAs as ahedging mechanism. However, Eurodollar contracts make payment on the initiation date for the loan rather than the maturity date, so there is a timing mismatch between the Eurodollar payment and the interest payment date. This gives rise to convexity bias, which causes the rate implied by the Eurodollar contract to be greater than that for an otherwise.equivalent FlkA.rfreasury bill contracts are yet another possible hedging vehicle, but suffer from basis risk since
PROBLEMS 7.1. Suppose you observe the following zero-coupon bond prices per $1 of maturity payment: 0.96154 ll-yearl, 0.91573 (2-year), 0.87630 0.82270 tzyearl, t3-yearl, 0.77611 ( ' For each matulity year compute the zero-coupon bond yields (effective armual and continuously compounded), the par coupo rate, and the l-year implied forward rate. -year).
7.2. Using the information in the pmvious problem, nd the price of coupon a s-year bond that has a par payment of $1,000.00and annual coupon payments of $60.00.
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238
k. INTEREST
RATE FORwARDS
PROBLEMS
AND FUTURES
7.3. Suppose you observe the following effective annual zero-coupon bond yields: 0.035 (2-year),0.040 (3-year),0.045 l4-yeall, 0.050 (5-year). 0.030 tl-yearl, For each maturity year compute the zero-coupon bond prices, continuously compounded zero-coupon bond yields, the par coupon rate, and the l-yellr implied forward rate. 7.4. Suppose you observe the following l-year implied forward rates: 0.050000 (1year), 0.034061 (2-year),0.036012 (3-year),0.024092 (4-year),0.001470 (5year). For each maturity year compute the zero-coupon bond prices, effective annual and continuously compounded zero-coupon bond yields, and the par coupon rate. 7.5. Suppose you observe the following continuously compounded zero-coupon bond yields: 0.06766 tl-yelrl,0.05827 (2-year),0.04879 (3-year),0.04402 t4.lyearl, For each mamrity year compute the zero-coupon bond plices, 0.03922 effective annual zero-coupon bond yields, the par coupon rate, nd the l-year implied forward rate. '(5-year).
7.6. Suppose you observe the following par coupon bond yields: /.03000(l-year), 0.05174 (5-year).For each 0.03491 (2-year),0.03974 (3-year),0.04629 (4-year), effective annual and conbond plices, mamrity year compute the zero-coupon yields,' bond l-year implied fonvard and the compounded tinuously zero-coupon 7.7. Using the information in Table 7.1, a. Compute the implied forward rate from time 1 to time 3. b. Compute the implied forward price of a par z-yearcoupon bond that will be issued at time 1. 7.8. Suppose that in order to hedge interest rate risk on your borrowing, you enter into an FRA that Willguarantee a 6% effective annual interest rate for 1 year on $500,000.00. On the date you borrow the $500,000.00,the act'ual interest rate is 5%. Determine the dollar settlement of the FRA assuming a. Settlement occurs on the date the loan is initiated. b. Settlement occurs on the date the loan is repaid. 7.9. Using the same information as the previous problem, suppose the interest rate on the borrowing date is 7.5%. Determine the dollar settlement of te F'RA assuming a. Settlement occurs on the date the loml is initiated. b. Settlement occurs on the date the loan is repaid.
%
239
Use the following zero-coupon bond prices to answer the next three questions'.
Days to Maturity 90 180 270 360
Zero-coupon Bond Price
0.99009 0.97943 0.96525 0.95238
7.10. What is the rate on a synthetic FI;A for a 90-day loan commenc'ing on day 90? A 180-day loan commencing on day 90? A 270-day loan commencing on day 90? 7.11k What is th rate on a synthetic FRA for a l80-day loan commencing on day ISO? Suppose you are the counterpar for a bonower who uses the FRA to hedge the interest rate on a $10mloan. What positions in zero-coupn bonds would you use to hedge the risk on te FRA? 7.12. juppose you are the counterparty for a lender who enters into an FlkAto hedge the lending rate on $10mfor a 90-day loan commencing on day 270. What positions in zero-coupon bonds would you use to hedge the risk on the FIA? 7.13. Using the inforiation in Table 7.1, suppose you buy a 3-yea.rp?r coupon boad and hold it for 2 years, after which time you sell it. Assume that rates are c rt aitl not to change and that you reinvest the coupon received in year 1 at the l-year rat prevailing at the time yotl receive the coupo'. Vel'if'y that the z-year ret'unaon this investment is 6.5%.
tnterest
7.14. As in the previous problem, consider holding a 3-year bond for 2 years. Now suppose that intrest rates ca change, but that at time 0,' the rates in Table 7.1 prevail. What transactions could you undertake using forward rate agreements tck ' guarantee that your z-yeaT ret'ul'n is 6.5%:/ 7.15. Uonsiderthe implied forward rate between yer 1 and year j, based on Table 7.1. a. Suppose that 0(1, 2) 6.8%. Show how buying the z-yeaT zero-coupon bond an bocowing at the l-yea.r rate and implied frward rate of 6.8% would enrn you an arbitrage proft. =
7.2%. Show how bonowing the z-yearzerob. Suppose that ?:(1, 2) cou/on bond and lending at the l-year rate arid implid forward rate of 7.2% would earn you an arbitrage prost. =
7.16. Suppose the Septelber Eurodollar f'utures contract has a price of 96.4. You plan to borrow $50mfor 3 months in September at LDOR, and you intend to use the Eurodollar conact to hedge your borrowing rate.
240
%.INTEREST
RATE FORWARDS
AND FUTURES
API:EN D Ix
a. What rate can you secure? b. Will you be long or short the Eurodollar contract? c. How many contracts will you enter into?
7.A:
INTEREST
RATE AN D BON D PRICE
CONVENTIONS
Q
241
7.21. Consider the following two bonds which make semiannual coupon payments: a zo-year bond wit.h a 6% coupon and 20% yield, and a 30-year bond with 6% a coupon and a 20% yield. a. For each bond, compute the price value of a basis point. b. For each bond, compute Macaulay duration.
d. Assuming the tnle 3-month LDOR is 1% in September, what is the settlement in dollars at expiration of the fumres contract? (For pulposes of this question, ignore daily marking-to-market on the fumres contract.)
c. eTor otherwise identical bonds, Macaulay duration is increasing in time to mattllity. ls this statement always true? Discuss. ;'
7.17. A lender plans to invest $100mfor 150 days, 60 days from today. (That is, if today is day 0, the loan will be initiated on day 60 and will mattzre on day 210.) The implied forward rate over 150 days, and hence tlzerate on a 150-day FRA, is actual interest rate over that period could be either 2.2% or 2.8%. 2.5%. 'l''he
a. If the interest rate on day 60 is 2.8%, how much will te lender have to pay if the FRA is settled on day 60? How much if it is settled on day 210? b. If the interest rate on day 60 is 2.2%, how much will the lender have to pay if the I7RA is settled on day 60? How much if it is settled on day 210? 7.18. Consider the same facts as the previous problem, only now consider hedging with the 3-month Eurodollarfutures. Suppose ttle Eurodollar fumres conact that mamres 60 days from today has a price on day 0 of 94.
c. What 3-month LEBOR is implied by the Eurodollar f'utures plice? Approximately what lending rate should you be able to lock in?
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'
7.22. Afl 8-year bond with 6% nnnual coupons and a 5.004% yieid sells for $106.44 with a Maeaulay duration of 6.631864. A g-year bond hMs coupons with a 5.252% yield and sells for $112.29with a Macaulay durption of 7.098302 . ! You wish to duration-hedge the 8-year bond using a g-year bond. How man# g-year bonds must we shol't for every 8-year bond? '7%'armual
7.23. A 6-year bond with a 4% coupon sells for $102.46wit.h a 3.5384% yield. The conversion factor for the bond is 0.90046. An 8-year bond with 5.5% coupons sells for $113.564with a conversion factor of 0.9686. (A11 coupon payments are semiannual.) Which bond is cheaper to deliver given a T-note f'utures price of 113.81? 7.24.
a. Comptlte the convexity of a 3-year bond paying almual coupons of 4.5% and selling at par. b. Compute the convexity of a 3-year 4.5% coupon bond that makes semiannutl coupon payments and that cun-ently sells at par.
a. What issues azise in using the 3-month Eurodollar contract to hedge a 150-day loan? b. lf you wish to hedge a lending position, should you go long or short the contract?
''
c. Is the convexity different in the two cases? Why? 7.25. Suppose a lo-yearzero couponbond with aface value of $100trades at $69k20205. a. What is the yield to mattlrity and modllied duration of the zero-coupon bond? t
d. What position in Eurodollar futures would you use to lock in a lending rate? ln'doing this, what assumptions are you making about the relationsllip between 90-day LEBOR and the 150-day lending rate?
b. Calculate te approximate bond plice change for a50 basis pointincrease in the yield, based on the modised duration yotl calculated in part a). Also calculate the exact new bond price based on the new yield to maturity.
7.19. Consider the bonds in Example 7.8. What hedge ratio would have exactly hedged the portfolio if interest rates had decreased by 25 basis points? lncreased by 25 change. basis points? Repeat assuming a so-basis-point
c. Calculate the convexity of the lo-year zero-coupon bond. d. Now use the formula (equation 7.15) that takes into accotlntboth duration and convexity to approximate the new bond price. Compare your result to that in part b).
7.20. Compute Macaulay and modilied durations for the following bonds: a. A s-yearbond paying annual coupons of 4.432% and selling at par. b. An 8-year bond paying semiannual coupons with a coupon raye of 8% and a yield of 7%. c. A lo-year bond paying annual maturity vklue of $100.
coupons
of 6% with a price of
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APPENDIX PRICE
7.A: INTEREST CONVENTIONS
RATE AND BOND
This appendix will focus on conventions for computing yields to maturity for different linds of bonds, and the conventions for quoting bond prices. When discussing yields to matrity, it is necessary to distinguish on 4heone hand between notes and bonds, which
242
k
INTEREST
AN D FUTURES
RATE FORWARDS
APPENDIX
make coupon payments and are issued with more than 1 year to mamrity, and on the other hand bills, which have no coupons and are issued with 1 year or less to maturity. The quotation conventions are different for notes and bonds than for bills. For a full treatment of bond pricing and quoting conventions, see Stigum and Robinson (1996).
INTEREST
7.A:
RATE AND BON D PRICE
$
CONVENTIONS
price of the bond is $3.5 1 0.032
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The bond-plicing formulas in Exapples 7.12 and 7.13 illustrate that even with a constant yield to maqlrity, thbgnct price will vary with the time until the next coupon Payment. This occuts becase equation (7.5.1)copputes a ond prtce that fully reflects the coming coupon payment Using this formula, the bond prie lises over time as a coupon payment approaches, ten fatls on the coupon payment date, and so forth. The bond plice quoted in this fashion is called the dirt'y price. Intuitively, if you buy a bond three-fourths of the way from one coupon payment to the next, the price you pay should riect tllree-fourths of the coming coupon payment. This prorated Jtmount is acrued interest, which is included in the plice in equation (7.5.1). Accnled interest ls calculated as the prorated portion of te coupon since the last #J d days sipce tle last coupon, accrued interest is C x (#' #)/#'. coupon date. With ln practice, bond prices are quoted nt of accnled interest. The dirty price less accrued interest is the clean pricej which does not exhibit the predictable rise and fall in price due to the coming coupon payment.22
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This can be rewritten as By)
7.13
Consider te same bond as in Example 7.12. Suppose that on the semiannual yield is still 3.2%. There are 96 days until ttze ( February coupon payment and 184 days between the August and February payments. j' . the price for the bond at a 6.4% yield (3.2%seniannual) is Using equation (7.17), 'tj E
ttbonds.'' Bond coupons We frst consider notes and bonds, which we will refer to asjust and yields are annualized. If bond is described as paying a 6% semiannual coupon, this means that the bond pays 6%/2 = 3% every 6 months. Further, if the ond yield 3.5%. Bond is 7%, this means that the bond's 6-month yield to maturity is 7%/2 mulitplying by 2 rather than annualized by by compounding. ad yields are coupons Sulijose abond makes semiannual coupon payments of C/2, and has a semiarmual yield of ,/2. The quoted coupon nd yield are C and ,.21 Let d be the ct'ual number of dys until the next cou'pon, and d' the number of days between the previous and next coupon. We take into account a fractional period until the next coupon by discounting the cash flows for that fractioni period. The price of the bond is
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d', and equation
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This formula assumes t'here is one f'ull period until the next coupon. Consider the bond in Example 7.13. Accrued interest as of Novemq Example 7.14 tl ber 11 would be 3.5 1.674. Thus, the clean price for the bond x (184 96)/184 yj ttf would be i'!@ Clean price Dirty price . accrued inteyest t'.) tl) ' . =
-
Consider a 7% $100 maturity coupon bond that males semiannual February 15 and August 15 and mamres on August 15, 2012. payments on coupon Suppose it is August 15, 2004, and the August coupon has been paid. There are 16 remaining payment. Ifthe semiannual yield, y, is 3.2%, then using equation (7.18),the Example
7.12
21If a bond makes coupon payments nl times a year, the convention is to quote the coupon rate as ??; times the per-period payment. The yield to maturity is computed per payment peliod and multiplied by nl to obtain the annual quoted yield. '
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$105.286 $1.674 $103.612 -
=
zzBecause accrued interestis amortized linearlyratherthan geometrically, ths statementis notprecisely true; see Smitb (2002). '
%
t
% INTEREST
244
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Table 7.6 presents typical Treasury-bill quotations. Suppose today is January l Abond maturing February 13 has 43 days to maturity and one maturing December 18 has 351 days to mattlrity (assuingit is not a leap yel). yields'' (Stigum and Robinson, The yildj'' in this table a.re compafable to Trajury-bond yields. To 1996), intended to malce Treasury-bill #ields obtain the yields, *e frst :nd the market pries of the T-bills. A'ltbill price is quoted on number subtrltcted from 100 t obtain an annualized discount basis. The discount is the the invoice price for the T-bill, #. The fonpula, normalizing the face value of the T-bill
$
x 43 lf you use this fonnula for the 351-day bill, however, you obtain a yield of 4.078 rather than the 4.04 listed in Table 7.6. The bond-equivalent yield calculation for this bill takes into account that a bond with more than 182 days to maturity would male a coupon payment. Hence, to make the bill yield comparable to that for a bond, we need to accdunt for the imaginary coupon. The formula from Stigum and Robinson (1996)is .2
Bills
CONVENTIONS
PRICE
j.0372
=
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Maturlty
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:
:
INTEREST
we see that
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7.A1
APPENDIX
AND FUTURES
RATE FORWARDS
x.ydty a,ys
-
+. g
=
2
( j ( dygs
2 xydtj a5y s
-
:t xtjays 365
-
)
-
j(
j
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l s0 0
j
j .
l 19) -
Applying this to the 351-day bond gives
.
i'bond-equivalent
Vask
to be 100 is %
x days
(jyxouut
X
=
100
-
360
The T-bills in Table 7.6 have invoice prices of
100 100
-
-.r
3.65 x 43/360
=
99.5640
3.87 x 351/369
=
96.2268
Thus, an investor pays 0.995640 per dollar of maturity value for the 43-day bill and tttrue'' 43-day and 351-day 0.962268 for the 351Lday bill. Note that these prices give us yields? what are the discount factors. Given the prices, l00 = 1.004379 or 0.4379% over 43 days. while the 351A 43-day bill yields 99.5640 day bill yields 3.9212% over351 days. Thebond-equivalent yield calculations armualize these yields in a way that makes them more comparable to bond yields. This necessarily involves maldng arbitrary assumptions. Forbills less than 182 days from mamrity, abill is directly comparableto amaturing bond since neithrmakes acoupon paymentover that period. In this case we need only to adjust for the fact that bonds are quoted using the actual number of days (i.e., a 365-day basis) and bills are quoted on a 360-day basis 365 x discount/loo l-i e
where n,estands for
ttbond-equivalent
ZZZ:
360
-
discount/ 100 x dpys yield.'' Applying this formula to the 43-day T-bill,
-
2 x 35 l
a(o
35 1 + L.j x5)
(
2 -
2 x 35 l
( a65
2 x 35 l 365
-
-
(
j.) j.
-
l
tj.xru.
)
xtj.ts(;agy
j
This matches the quoted yield in Table 7.6. ln Excel, the ftlnction TBYLEQprovides the bond-equivalent yield for a T-bill.
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AN EXAM PLE OF A COM MODITY
swAps
and examine interest rate swaps, total retul'n swaps, and more complicated swap examples. '
commodity
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An industrial producer, IP Inc., is going to buy 100,000 barrels of oil 1 year from today and 2 years from today. Suppose that the forward price for delivery in 1 year is $20+-e1 and in 2 years is $21/barrel.We need interest rates in this discussion, sd . suppose that annual interest rates are as in Table 7.1 (see page 206): The 1- and z-year and 6.5%. bond yields 6% are zero-coupon lpcan use forward contracts to guarantee the cost of buying oil for the next 2 years. Speciscally, IP could enter into long forward contracts for 100,000 barrels in each of the next 2 years, committing to pay $20/barrelin 1 year and $21/b-e1 in 2 years. The present value of this cost is $20
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Figure 8.2 depicts cash :ows and eansactionswhen the swap is settled Enancially. Th is settled physically t/?--#?7ancf(7#y. resultsfor #?cbuyer are #?' same w/7cl/pcrthe both cases, the net cost to the oil buyer is $20.483. 1.r1 We have discussed the swap on a per-barrel basis. For a swap on 100,000 barrels, simply multiply all cash llows by 100,000. ln this example, 100,000 is the otional we amount of the swap, meaning that 100,000 barrels is used to detennine the magnitude of th payments when the swap is settled financially. -u/t'/p
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To satisfy this equation, the payments must be $20.483in each year. We then say that the z-yearswap prie is $20.483.Hovever, t7,1.), paylllents that /ptzl'ea pr-td/pr value of
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IP could invest this amount today and ensure that it had the funds to buy oil in 1 and 2 years. Alternativly, J.P could pay an oil supplier $37.383,and the supplier would c'ommit to delivering one barrel in each of the next two years. A single payment today for a single delivery of oi1 in the futtlre is a prepaid forward. A single payment today to obtain mltltiple deliveries in the future is a prepaid swap. Although it is possible to enter into a prepaid swap, buyers might wony about the resulting credit risk: They have fully paid for oil that will not be delivered for up to 2 years. (The prepaid forward has the same problem.) For the same reason, the swap counterparty would won'y about a postpaid swap, where the oi1 is delivered and full P ayment is made after 2 years. A more attractive solution for both parties is to defer payment until the oil is delivered, while still 'Iixingthe total price. Note that there are many feasible ways to have the buyer pay; any payment stream with a present value of $37.383is acceptable. Typically, however, a swap will call for equal payments in each year. The payment per year per barrel, Jr, will then have to be such that -.1t7
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Physical Versus Financial Settlement Thus far we have d.escribed the swap as if the swap counterparty supplied physical oil to the buyer. Figure 8.l shows a swap that calls for physical settlement. Ip this case $20.483 is the per-barrel cost of oil. of the swap. With snancial However, we could also vangefoh'ltclllcialseulellellt settlement, the oil buyer, 1P, pays the swap countelmarty the difference between $20.483 and the spot price (if the difference is negative, the counterparty pays the buyer), and the oil buyer then buys oil at the spot price. For example, if the market price is $25, the
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The swap counterparty is a dealer, who hedges the oi1price risk resulting from the swap. The dealer can hedge in several ways. First, imagine that an oil seller would like to lock in a fixed selling price of oil. ln this case, the dealer locates the oil buyer and seller and serves as a go-between for the swap, receiving payments from one party and passing them on to the other. In practice the sxedprice paid by the buyer exceeds th tixd price received by the seller. This price difference is a bid-ask spread and is the dalef's fee. Figure 8.3 illustrates how this transaction would work with financial settlement. 'he o il seller receives the spot price for oi1 and receives the swap price less the spot price, on net receivipg the swap price. The oil buyer pays the spot price arpd yecives the spot price less the swap price. The situation where the dealer matches the buyer and book'' transaction. The dealer seller is called a back-to-back transaction or bears the credit risk of both parties but is not exposed to price risk. A more interesting situation occurs when the dealer serves as counterparty and hedges the trapsaction using forward markets. Let's see how this would work. After entering the swap with the oil buyer, the dealer has the obligation to pay the price and receive the swap price. lf the spot price rises, the dealer can lose money. spot oil. The dealer has a short position in 1- and z-year The nattlral hedge for the dealer is to enter into long forward or fumres contracts to offset this shol't exposure. Table 8.1 illustrates how this strategy works. As we discussed earlier, there is an implicit loan in the swap and this is apparent in Table 8.1. The net
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The swap price, $20.483,is close to the average of the two oil forward prices, $20.50. However, it is not exactly the same. Why? Suppose that the swap price were $20.50.The oil buyer would then be commiitlng $t.50less than the to pay $0.50more than the forward price the srstyear and would pay /'/?t? relative Thus, price the second r/lc/t/rlszt-lpz bttyer J$)()If/# have forward to cttrve, year. the There implicit lending in the swap. iltterest-h-ee Ioan ct/lfpl/eprclr/y. is to p77t'l#c an Now consider the actual swap price of $20.4831m1e1.Relative to the forWard curve prices of $20 in 1 year and $21 in 2 years, we are overpaying by $0.483 in the first year and we are undepaying by $0.517 in the second year. Therefore, the swap is equivalent to being long the two forward contracts, coupled with an agreement to lend $0.483 to the swap counterparty in 1 year, and receive $0.517in 2 years. This loan has the effect of equalizing the net cash flow on the two dates. 1 7%. Where does 7% come The interest rate on this loan is 0.517/0.483 from? We assumed that 6% is the l-year zero yield and 6.5% is the z-yearyield. Given these interest rates, 7% is the l-year implied forward yield from year l to year 2. (See Table 7.1.) By entering into the swap, we are lending the counterparty money for 1 year beginning in 1 year. ((fthe deal is priced fairly, the interest rate on this loan should be the implied forward interest rate.
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The Market Value of a Swap 'When the buyer hrst enters the swap, its market value is zero, meaning that either party could enter or exit the swap without having to pay anything to the other part.y (apartfrom commissions and bid-ask spreads). From the oil buyef perspective, the swap' consists of two forward contracts plus an agreement to lend money at the implied fol-ward rate of 7%. The forward contracts and forward rate agreement have zero value, so the swap does as well. Ohce the swap is struck, however, its market value will generally no longer be for two reasons. First, the forward prices for oil and interest rates will change over zero, time. New swaps would no longer have a fixed price of $20.483;hencq, one party will tmvind the swap. owe money to the other should one party wish to exit or Second, even if oil and interest rate forward prices do not change, the value of the ltlttil theslist payment is ?pltz#c. Once the first swap swap will remain zero only payment is made, the buyer has overpaid by $0.483relative to the forward curve, and hence, in order to exit the swap, the counterparty would have to pay the oi1buyer $0.483. Thus, even if prices do not change, the market value of swaps can change over time due to the implicit borrowing and lending. A buyer wishing to exit the swap could negotiate terms with the original counterparty to eliminte the swap obligation. An alternative is to leave the original swap in place and enter into an offsetting swap with whoever offers the bejt pri. The priginal swap called for the oil buyer to pay the fixed price and receive the floating tze; the offsetling swap has the buyer receive the fixed price and pay floating. The original. obligation would be cancelled except to the extent that the xed prices are different. However, the difference is known, so oil price lisk is eliminated. (There is stitl credit risk whe the original swap counterparty and the counterparty to the offsetitng swap are different. This could be a reason for the buyer to prefer offsetting the swap with the oliginal counterparty.) To see how a swap can change in value, suppose that immediately after the buyer lises by $2 in years 1 and 2. Thus, the year-l enters the swap, the forward curve for oi1 forward. price becomes $22 and the year-z forward plice becomes $23. 'I'he oliginal swap will no longer have a zero market value. Assuming interest rates are unchanged, the new swap plice is $22.483.(Problem 8.1 asks you to verify this.) The buyer could unwind the swap at this point by agreeing to sell oil at $22.483,while the original swap still calls for buying oil. at $20.483.Thus, the net swap payments in each year are 's
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Cgmpanies use interest rate swaps to modify their interest rate exposure. ln this section we will begin with a simple example of an interest rate swap, similar to the preceding oil swap exapple. We will then present general pricing formulas and discuss Ways in wlzichthe basic swap structtlre can be altered.
However, the interest payment on the loan is due at the end of the yem The interest floating interest payment would therefore occur at the rate determination date for beginning of the period. As with an FRA we can tlzink of the swap payment being made at the end of te period (wheninterest is due). With the finaneiatly settled oil swap, only net swap payments-in this case the difference between LDOR and 6.9548t:,-.-:/e actually made between XYZ and the counterparty. lf on party defaults, tey owe to the other party at most the present value of net swap payments they are obligated to make at current market prices. This means that a swap generally has less credit lisk than a bond: Whereas principal is at risk with bond, only net jwap,payments are at lisk in a swap. The swap in this example is a constnlct, making payments as (f there were an exchange of payments between a sxed-rate and soating-rate bond. ln prtice, a fund might bonds fixed-rate and wish own Eoating-rate manager to have exposure while continuing to own the bonds. A swap in which a fund manager receives a qoating iate in exchange for the payments on bonds the fund continuej to hold is called arl asset swap. 'te
7kSirnple Interest Itate S4vap that every Suppose that XYZ Corp. has $200mof Coating-rate debt at LtBoR-meaning would preferto that Lolkvbut xed-rate debt with year's current have pays could effect mattlrity. There several XYZ change. this 3 years to are ways First XYZ could change their interest rate expospre by retiring th floatipg-rate debt and issuing Exed-rate debt in its place. However, an actual purchase ad sale of debt has transaction costs. Second, they could enter into a strip of forward rate agreements (FRAs) in order to guarantee the borrowing rate for the remaining life of the debt. Since the FRA for each year will lpically carry a different interest rate, the company will lock in a different rate each year and, hence, the company's borrowing cost will vary over time, even though it will be fixed in advance. A third alternative is to obtain interest rate exposure equivalent to that of fixed debt by entering into a swap. XYZ is already paying a Qoating interest rate. They rate therefore want to enter a swap in which they receive a Qoqting rate and pay yhefixed rate, which we will supposr is 6.9548%. This swap is illustrated in Figure 8.4. Notice the similarity to the oi1 swap. ln a year when the fixed 6.9548% swap rate exceeds l-year LIBOR, XYZ pays 6.9548% LIBOR to the swap counterparty. Conversely, when the 6.9548% swap rate is less than LIBOR, the swap counterparty pays LIBOR 6.9548% to XYZ. On net, XYZ pays 6.9548%. Algebraically, the net interest payment made by XYZ is
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There are timing conventions with a swap similar to those for a forward rate agreement. At the beginning of a year, the borrowing rate for that year is known.
pricinjan
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To understnd the pricing of the swap, we will examine it from the perspective of both the counterparty and the firm. Wefirst consider the perspective of the counterparly who we assume is a market-maker. The market-maker is a counterparty to the swap in order to earn fees, not to take on interest rate lisk. Therefore, the market-maker will hedge the transaction. The marketmaker received th fixed rate from the company and pays the ioating rat; the danger for the market-m ke i is that the Qoating rate will rise. The risk in this eansactioncan be hedged by entering into forward rate agreements. We express the time 0 implied forward rate between time ti ad tj as rfjti tj4 nd ihe realized l-yer tate as 15 The currenl l-year rate, 6%, is known. Wit.h the swap rate denoted R, Table 8.2 depicts the lisk-free (but time-varying) cash flows faced by the hedged market-malcer. How is R determined? Obviously a market-maker receiving the fixed rate would like to set a high swap rate, but the swap market is competitive. We expect R to be bid ,
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INTEREST
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Computing the Swap Rate
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The expression 'F71.1 /7(0, tilrt-L ti) is the present value of interest payments implied by the syip of fonvard rates. The expression .P(0, ti) is just the present value of a $1 annuity when interest rates vary over time. S)'=j Thus, the swap rate annuitizes the interest payments on the floatingvratebond. We can rewrite equation (8.2)to make it easier to interpret: ,
11 t'(' jyyy;:jgf j';'jy)f '' i hf j'k,j-'r' k;jyy,rf (,jj;;' ;r' j,y,,.:,jf qE #;'. (r' Tq'ly'l'f .jjyr)' )' k' ,'t'iikiiIiir7i:'' :!jE;;' .jjyyy' 'q' (t' q' ( ')! q'll:!' '' i. (g'E.ik j!gjllgj;f .!jy::' .!jj::' yf )' ( jij:E(((.(';'.('' .. .'(qEi';'.:;'((.. 'r)('q.E'q: (',: '.)( (. i';.(((. ( (E.(E(.Eq''.' '..'' ('('.'E''(iqj.7?'jyyq r' '(Ejjjjjj,' jjf j;:yyy;,,f ijIi(:-'. IE (j.7(! )' t')'((';..* yky'. ljtjtf y' j'j-'-' ' : '..i' ij!!q'j.i ))1 'q'..i.('(i (r(@' , E - - E- E -.; ;)j;' .E kL.qk E :- . . Er:....E :. EE i $. ..E '. . -; E :. .(;; . y ! j ) jy--. i r. ( y . . . q . . y q j q yj.', . . q ( : y ( 71.1j ;-7tir . g . ( y ( y . . :.yyy jg . - j(-y..y .yyy..jy (j-,j-k. . . . . .q ;. . . .. rr .lkjj;y yt,yg.k(. (.r-..- . ; j.,.--(.(i. .. r . ...r. . . . ..
.
,
R
'=''*',,. '?q' ,,j-,,,-jj,,,,'
General
We now examine more carefully the general calculations for determining the swap rate. We will use the interest rate and bond pfice notation introduced in Chapter 7. Suppose there are ?? swap settlements, occurring on dates ti, i /?. The implied forward 1, interest rate from date ti-k to date ti, known at date 0, is rztil ti). (We will treat /-()(4-1 ti) as l'qt having been annualized; i.e., it is the return earned from ti-L to ti. The price of a zero-coupon bond mamring on date ti is #(0, ti). The market-maker can hedge the ioating-rate payments using forward rate agrements. The requiremeny that the hedged swap have zero net present value is
uwt?p
.'!'.........#''>'-.. -' -t'( ,,j,,;,,kk,,,,' ;'. -' ;''''7j'F' j,:yyr:j,f ,j:yyyr,' jj::rr;,,f :'1** j,:ry::,,f '-f
257
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Since the terms in square brackets sum to one, this form of equation (8.2)emphasizes that the lixed swap rate is a weighted average of the impliedfomardrates, wherezero-coupon bond prices are used to determine the weights. There is another, equivalent way to express the swap rate. Recall from Chapter 7, equation (7.4),that te implied folavard rate between times and tz, ?'()trl r2), is given rl by the ratio of zero-coupon bond prices, i.e., ,
?-()trl?2) ,
=
#(0,
/.1)/#(0,
?2)
-
1
.
%. SWAPS
INTEREST
.'
%. 259
RATE swAps
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Therefore equation ?1
(8.1)can -
.''.
be rewritten
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=
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This is the valuation equation for a bond priced at par wit.h a coupon rate of R. The conclusion is that the rate is l cottpon rate t??l a pJ?- coupon !747,?4@. This' result is intuitive since a firm that swaps frortl floating-rate to exposure sxed-rate bond. ends up wit the economic equivalent of a sxett-rate 'wcw
The Swap Cttrve As discussed in Chapter 5, the Eurodollar f'umres coneact provides a set of 3-month forward LYOR rates extending out t0 years. It is possibl: to use thij set of forward interest rates to compute equation (8.2)or (8.3).As discussed in Chapter7j zero-coupon bond prices can be constructed from implied fonvard rates. The set of swap rgtes at different maturities implied by LIBOR is called the swap clf?qzd. There is an over-the-counter market in interest rate swaps, wltich is widely quoted. The swap curve should be consistent with the interest rate curve implied by the Eurodollar fumres conact, which is used to hedge swaps.l . Here is how we construct the swap cul've using the set of Eurodollr prices.z Column 2 of Table 8.4 lists 2 years of Eurodollar f'uttlres prices from June 2004. The next column shows the implied 91-day interest rate, beooinning in the month in column 1. For example, using equation (5.19), the June price of 98.5558 implies a June to
l'T'heEurodollarcontractis a futures contract, while a swap is asetof fonvard rate agreements. Because of convexity bias, discussed in Chapter7, the swap curve constructed frop Eurodollaffutures contracts following the procedure described in this section will be somewhat greater t.11%th obsen'ed swap Bu'rghardt and Hoskins (1995)and Gupta and Subramayam (2000). curve. This is discussed by zcollin-Dufresne and Solnik (2001)point out that the credit risk implicit in the LIBOR rate underlying the Eurodollar fumres contract is different than the credit risk of an interest rate swap. EEBOR is computed as an average 3-month borrowing rate for international banks wit.h good credit. Banks tat experience credit problems are dropped from the sample. Thus, by construction, the pool of banks represented in the Eurodollar contract never experience a credit downgrade- A firm with a swap, by contrast, could be downgraded.
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P (0,ti ) + P (0,??,)
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You may recognize this as the formula for the coupon on a par coupon bond, equation (7-.6),from Chapter 7. This in t'ul'n can be rewritten as R
.
! ::
i .
:
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Setting this equation equal to zero and solving for R gives us l #0(0, r,,) R
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1.461 1%
0.9964
1.7359%
0.0063
0.9914
2.0000% 2.2495%
Ma1=05 Jun-05 Sep-05
97.495 97.025 96.600 96.235
0.0075 0.0086
0.9851 0.9778
0.0095
0.9695
Dec-05 Mar-06
2.6997%
95.910 95.650
0.0103 0.01 10
0.9603 0.9505
2.8995%
2.4836%
3.0808%
Source: Eurodollars futures prices from Datastream.
Septembr quarterly interest
rate of
(100
-
98.555)
91 1
4 o0 --59
=
0.0037%
Column 4 reports the corresponding implied zro-coupon bond price. In the second row, the prie is the cost in June of $1 paid in September. The third row is the June cost of $1 paid in Decembervand so forth. The furth row is 1
1
1
0.9851 1.0037 x 1.0050 x 1.0063 which is the June cost of $1 paid in March. The December swal rate, expressed as a quarterly rate. is the 'Iixe d q uarterly intrest rate from Vne through March, with swap payments in June, September, and Decerber (the months in wlzich the quarterly rate prevailing over the next 3 months is lnown). This is computed using equation (8.3): =
0.9851 = 0;54)% + 0.9914 + 0.9851 Multiplying this by 4 to annualize the rate gives the 2.00% in the swap rate column of Table 8.4. 1
-
0.9964
In Figure 8.5 we graph the entire swap curve againstquarterly forwardrates implied by the Eurodollar curve. The swap spread is the difference between swap rates and
%.swAps
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INTEREST
RATE SWAPS
%
261
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receives floating. This investor is paying a high rate in the early years of the swap, and, hence, is lending money. About halfway through the life of the swap, te Eurodollar forward rate exceeds the swap rate and the loan balance declines, falling to zero by te end of tlle swap. The sxedrate recipient has a positive loan balance over the life of the swap because the Eurodollar f'umres rate is below the swap initially-so the fixed-rate rcipient is receiving payments-and crosses the swap price once. The credit risk in this swap is therefore borne, at least initially, by the Eked-rate payer, who is lending to the 'Iixed-rat recipient. The implicit loan balance in the swap lsk illustrated in Figure 8.6.
.. E. q - y E-. -
-
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-
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-
Forwrd (3-mntli
interest rate cul-ve mplied by the
Eurodollar strip, swap
rates, and constant maturity Treasury yields for june 2, 2004.
Deferred Swaps We can construct a swap that begins at some date in the future, but for which the swap rate is agreed upon today. This type of swap is called a deferred swap. To demonstrate this type of swap, we can use the information in Table 7.1 to compute the value of a z-period swap that begins 1 year from today. The reasoning is exactly as before: The swap rate will be set by the market-m er so that the present value of the :xed and
Maturity Soltrce: Datastream.
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The Swap's lmplicit Loan Balance An interest rate swap behaves much like the oil swap in Section 8.1. At inception, the swap has zero valu to both parties. If itlterest rate! change, the prsent value of the 'Iixedpayments and, hene, the swp rate will change. The market kalue f the swap is the difference in the present vale o? payrents between the old dhkp rate and the new swap tat. For example, conjider th 3-year swap in Table 8.3 (seepage 256). If interest rates lise after the swap is entered into, the value of the existing 6.9548% swap will fall for the party receiving the sxedpayment. Even in the absence of interejt rate changes, however, the swap in Table 8.3 changes value over time. Onc the first swap paynzent is npade, 4he swap acquires negative value for the market-paker (relativeto the use of forwaos) becauje in the second year the market-maker will make a net cash payment. Similarly, the swap will have positive value for the borrower (againrelative to the use of forwards) after the 'lirst much'' (relative payment is made. ln oyder to smooth payments, the borrower pays curve) the forward in the first and receives refund in the second to year a year. The l-telltillji ?-tf cl g/'t? $)t7/7 ij trt)l t JJ':/ I?ll /t? t!?l rt??-/? (7,7 t lll /4/ p7t??? l6; 2 ? -1$?:7 Ifll l :7 il tlt)?l ttr t? tltli ti (7, 7:7 l tl-cl r 421
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Treasury-bond yields for comparable mamrities. Thus, Fijure 8.5 also displays yields on government bonds.
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Soltrce: Dataseam.
262
% swAps
INTEREST
floating payments is the same. This gives us R
-
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=
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Solving for #, the defen'ed swap rate is 7.4854%. ln general, the fxed rate on a deferred svttp beginning in k periods is cmputed as '
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i=k
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(0 ti )
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RATE SwAras
%.
263
Credit risk does not vanish; it is still mostly held by the long-term bondholders. The swap counterpacty faces credit risk since the 'lirm could go bnnkrnlptwhen the value of the swap is positive to te counterparty (thiswould occur if interest rates had risen). The notiohal principal of te loan is not at lisk for the swap counterparty, however. so the credit risk in te swap is less than for a short-term loan. Thus, by swapping its interest rate exposure, tle 'Iirm pays the short-tel'm interest rate, but lhe long-tenn !7tp??t7t7J#d?'J continue to bear ?3/47.1 oftlle credit risk. If it seems odd to you that the 111711 can use a swap to convert a high fixed rate into floating rate, recognize tat any time there is an upward-sloping yield low a curve, te short-term interest rate is below the long-term interest rate. lf you reduce the period for which your borrowing rate is fixed (whichhappens when you swap 54ed for floating), you bon'ow at the lower short-term interest rate instead of te higher long-tenu interest '
rate
Swaps thus penuit separation of two aspects of borrowing: credit risk and interest ratezisk. To the extentthese lisks are acquired by those most willing to hold them, sWaps increase eficiency.
Amortizing and Accreting Swaps y S:vap lnterest Itates? Managers sometimes say that they would like to borrow sort-term because short-term inier s t rates are on average lower than long-term interest rates. Leavinj aside the question of whether this view makes sense theoretically, let's tale for granted the desire to bormw at short-term interest rates. The problem facing the manager id that th fi.rl'f may be unable to borrow signiscant amounts by issuing short-term debt. When a ;rm borrows by issuing long-term debt, bondholders bear both rate risk and the credit risk of the firm. lf the firm borrows short-term (forexample, by issuing commercial paper), lenders primarily bear credit lisk. In practice, short-term lenders appear unwilling to absorb large issues from a single bbrrower because of redit lisk. For example, money-market mutual funds that hold commercial paper will not hold large amounts of any one irm's commercial paper, prefelning instead to diversify across frms. This diversiscation minimizes the chance that a single banlruptcy will signiicantly reduce the fund's rate of return. Because short-term lenders are sensitive in this way to credit risk, a f'm cannot bonow a large amount of moey short-tenn without lenders demanding a higher interest example, pension funds and rate. By contrast, long-term lenders to corporations-for insurance companies-Fillingly assume both interest rate and credit risk. Thus there lre borrowers who wish to issue short-term debt and lenders who Itre unwilling to buy it. Swaps provide a way around this problem, permitting the firm to separate credit risk and interest rate risk. Suppose, for example, that a fnnborrows long-term and then swaps into short-rate exposure. The Erm receives the lixed rate, pays the fxed rate to bondholders, and pays the :oating rate on the swap. The net payment is the short-term rate, which is the rate the ;rm wanted to pay.
tnterest
We have assumed that the notional value of the swap remains tixedover the life of the swap. However, it is also possible to engage in a swap where the notional value is changing over time. For example, consider a floating-rate mortgage, for witich ever# payment contains an interest and principal component. Since the outstanding principal is declining over time, a swap involving a mortgage would need to acount for tis. Such a swap is called an amortizing swap because the notional value is declining over time. lt is also possible for te principal in a swap to grow over time. This is called an accrethzg swap. Let Qt be the relative notional amount at time t. Then the basic idea in pricing a wit.h a time-varying notional amount is the same as with a fxed notional amount: swap The present value of the fxed payments should equal the present value of the floating payments: 11
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rate is still a weighted average of implied forward rates, only now the
weights also involve changing notional principal. Many other structures are possible for swaps based on interest rates or otherprices. One infamous swap structure is described in the box on page 264, which recounts the 1993 swap between Procter & Gamble and Bankers Trust.
'
% swAps
264
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To understand these alternatives, let's consider the example of a dollar-based firm that has euro-denominated 3-year xed-rate debt. The annual coupon rate is p. The finzl is obligated to make a series of payments that are fixed in euro terms but variable in dollar tens. Since the payments are known, eliminating euro exposure is a straightforward hedging problem using currency forwards. We have cash flows of each year, and -(1 + p) in the mamrity year. lf currency forward Ilices are Fp,;, we can enter into long euro forWard contracts to acquire at a lnown exchange rate the euros we need to pay to the lenders. Hedged cash Ilows in year t are -pFt. As we have seen in other examples, the fonvard ansactions eliminqte lisk but leave the firm with a variable (but rislless) streltm of cash llows. Th'e variability of hedged cash flows is illustrated in the following example.
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debt payments in a which entails making debt payments in one currency different cunency. There is typically an exchange of principal at both the start and end of the swap. Compared with hedging the cash :ows individually, the cul-rency swap generates a different cash flow stream, but with equivalent value. We can examine a currency swap by supposing that the 51411in Example 8.1 uses a swap rather than forward contracts to hedge its euro exposure. and receiving
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Currency Swap Formulas Currency swap calculations are the same as those for 4heother swaps we have discussed. To see this, consider a swap in which a dollar annuity, R, is exchanged for an anpuity in another currency, R*. Given the foreign annuity, R*, what is A? W e star twith the observation that the presept value of the two annuities mustbe the same. There are 11 payments and the time-o forward price for a unit of foreign currency delivered at time ti is Ftt This gives .
y Example 8.2 Make the same assumptions as in Example 8.1. The dollar-based 'lirm !,(1 enprs into a swap where it pays dollars (6% on a $90 bond) and lece ives euros (3.5% i'l.t is eliminated. The market-maker receives tl) on a e100bond). The firm s euro exposure !i1dollars and pays euros. The position of the market-maker is sumtnarized in Table 8.6.
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There are other kinds of currency swaps. The preceding examples assumed that al1 borrowing was fixed rate. Suppose the dollar-based borrower issues euro-denominated a
268
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8.4
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At the beginning of this chapter we looked at a simple two-date commodity swap. Now we will look at commodity swaps more generally, present the general formula for a commodity swap-showing that the formula is exactly the same as for an interest rate swap-and look at some ways the swap structtlre can be modihed.
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Quantityand
Price
It might make sense for a gas buyer with seasonally varying demand (for example, someope buying gas for heating) to enter into a swap in which quantities vary over time. For example, a buyer might wapt tlee times the quantity in ihe wlnter months as in the summer months. A buyer atso might be willing to flx different prices in different seasos-for exnmple, if there is seasonal variation in th price of the output produced using gas as an in p ut How do we determine the swap price wit.h seasonally varying .
quantities?
The Commodity Swap Price The idea of a cornmodity swap, as discussed in Section 8.1, is that we use infonnation in the commodity forward curve to fix a commodity price over a period of time. We can derive the swap price following the same logic as before.
269
Think about the position of the market-mAer, who we suppose receives the hxed payment, X, makes the floating payment, and hedges the risk of the :oating payment. If there are ?) swap payments, the resulting hedged cajh flow is
Swaps with Variable COMMODITY
swAps
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Let Qt;denote the quantity of gas purchased at time ti Once again, we can think about this from the perspective of the competitive market-maler. The market-maker who hedges the swap will enter into varying quantities of forward contracts in different months to match the variable quantity called for in the swap. The zero-prot condition is still that the fixed and :oating payments have zero present viue, only in this case they .
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An option to enter into a swap is called a swaption. We can see how a swaption works by retunling to the two-date oil swap example in Section 8.l The z-yearoil swap price was $20.483.Suppose we are willing to buy oil at $2.483+=e1. but we would like to speculate on the swap price being even lower over the next 3 months. Consider the following contract: If in 3 months the hxed price for a swap commencing in 9 months (1 year from today) is $20.483or above, we enter into the swap, agreeing'to pay $20.483and receive the floating price for 2 years. 1f, the other hand, on the market swap price is below $20.483,we have no obligation. If the swap price in 3 months is $ 19.50, for example, we could enter into a swap at that time at the $19.50 price, or we could elect not to enter any swap. With this contract we are entering into the swap with $20.483as the swap price only when the market swap price is greater; hence, this contract will have a premium. In this example, we would have purchased a payer swaption, since we have the Iight, but not the obligation, to pay a fixed price of $20.483for 2 years of oil. The counterparty has sold this swaption. When exercised, the swaption commits us to transact at multiple times in the future. It is possible to exercise the option and then offset the swap with another swap, converting the stream of swap payments into a certain seam with a fixed present value. Thus, the swaption is analogous to an ordinary option, with the present value of the swap obligations (theprice of the prepaid swap) as the underlying asset. The strike pdce in this example is $20.483,so we have an at-the-money swaption. We could make the strike plice different from $20.483. For example, we could reduce the swaption premium by setting the strike above $20.483. Swaptions can be American or European, and the terms of the underlying swapfixed price, qoating index, settlement frequency. and tenor-will be precisely specified.
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This equation makes perfect sense'. If we ar going to buy more gas when the forward 111. price is high, we have to weight more heavily the forward price in those onths. When the quantity is not varying. Q 1, the fonuula is the same as equation (8.11), when let the summer swap price be We can also permit prices to be time-varying. If we denotd by # an d the winter price by #., tlien the summer and winter swap prices can be any prices that stisfy the market-maker's zero present value condition'. =
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ll After3 months, the xed price on the underlying swap is $21.50.We exercise tlie option, '' obligating us to pay $21/barrelfor 2 years. If we wish to offset the swap, we can enter lti)into a swap to receive the $21.50fixed price. In year l and year 2 we will then receive ry.J) . '' y $21.50and pay $21,fora certain net cash flow each yearof $0.50.The :oating payments tl cancel. :
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A receiver swaption gives you the right to pay the floating price and receive the would exercise when the fixed
fixed strike price. Thus, the holder of a receiver swaption swap price is below the strike.
Although we have used a commodity swaption in this example, an interest rate or currency swaption would be analogous, with payer and receiver swaptions giving the right to pay or receive the fixed interest rate.
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k Example 8.4 ABC Asset Management has a $2 billion investment in the S&P stock !.) index. However, fund managers have become pessimistic about the market and would );)y ))) like One way to do to reduce their exposure to stocks from $2 billion to $1 billin. y ).(q retain stock ftlnd position but the of stocks. However, the billion is sell can this to 1) $1 l',t total engaging stocks of the in transfer by ret'urn the )t) hnancially swap, obligating a return swapped stocks, capital gains) the the fund to pay the total remrn (dividendsplus on li i.j; . ) while receivinc a tlotttina-rateret'urn such as LDOR on the swapped $1 billion notional i' avoids the transaction costs of qtsale of physical stock. i amount. This ') j. Table 8.7 illustrates the payments on such a swap. ln year 1, ABC earns 6.5% ( . . the 6.5% in )t)on the s&P index. However, on .the portion it has swapped, it must pay exchange for the 7.2% floating rate. The net payment of 0.7% leaves ABC as Welloff as .)y.j.j floating rate. In yeltr 2, ABC ljj if it had sold the index and invested in an asset paying the lt' receives l 8%, compensating it for the difference between the 7.5% floating rettlrn and S'ilt tl the 10.5% loss on the S&P index. Finally, in year 3 the S&P index does well, and ABC zllt ') pays 16.5% to the conterparty. .
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Atotal rern swap is a swap in which one party pays the realized total retul'n (dividends plus capital gains) on a referene asset, and the oter party pays a floating return such as LIBOR. The two parties exchange only the difference between these rates. The party paying the retul'n on the reference asset is the total ?'erIf?-?l payer. As with other swaps, tere are multiple settlement dates over the life of the swap. The umulative efect for the yotalreturn payer is of being short the refer nc psset and long an asset payig the floating rate. The cash :ows o a total return swpp are illustrated
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This obseryayion is correct, butnotice that the fund is paying the difference between the index return atd a short-term interest rate-this difference is the t'isk premiurp the on index. In Section 5.3, we had a similar result tor a forward contract: On akerage short a position in a forward contract on a stock index loses money because the Iisk premium has zero value. The average loss associated with swapping a stock index for LIBOR is the jame as the average loss associated with selling the stock and buying a :oating-rate note paying LIBOR. It is just that the swap makes the loss obvious since it requires a payment. Some investors have used total return swaps to avoid taxes on foreign stocks. In impose Withholding many CaSeS, Cotlntfies ttues On foreign investors, meaping thqt if a firm in Countl'y A Pays a divided, fOr example, country A withholds ftiiop of that a dividend from investors bed in Country B. A total reii'n swap enables a country-B investor to own country-A stocks without plysically holding then, and ths in many cases without hving to pay withholding taxes. For exainjle, a U.S. investor could first swap out of a U.S. stock index and then swap into a Eurpan stok inde, effectively becomillg the ounterparty for a Eropean invstof wnti to swap ot of European stock expostlre. Becase net swp jayments are not always recognized by withholding ftl les tls asaction can be more taxl-'fltciet than holding the foreign stocks directly. not er uje o f total remrn swaps is the managnieht of creditrisk. Afund lanager holding corporate debt can swap the return on a particular bond for a iogtig-rate rettlna. If the company that issued the bond goes banlrupt, the debt holder receives a payment on the swap compensating for the fact that the bond is worth a fraction of its face value. lf you tlink about this use of tptal return swaps, it is a erude tool for managing The prolem is that bond prices also change due to interest credit isk speclcally. rate changes. A corporate bond holder might wish to retain interest rate risk but not balzkruptcylisk. Thus, there are products called default swaps. These are ssentially default options, in which the buyer pays a premium, usually amortized over a series of payments. If the reference asset experiences a event'' (for example, a failure to make a scheduled payment on a particular bond or class of bonds), then the sellr fnakes a payment to the buyer. Frequently these contlucts split the return on the b6nd into the portion due to interest rate changes (withTreasury securities used as a reference) and ,
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You rnight wonder about te economics of a swap lik this. The stock index on average earns p higher return than LIBOR. So if the fund swaps the stock index in exchange for LYOR, it will on average make payments to the counterparty.
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the portion due to credit quality changes, with the swap maldng payments based only on te latter. We discuss these swaps in Chapter 26.
UMMARY
same formulas used to price swaps will appear again in the context of structured notes, which we will encounter in Chapter 15. We will discuss default swaps in Chapter 25 .
Litzenberger (1992)provides an overview of the sFap market. Turnbpll (1987) discusses argpments purporting to show that the use of swaps can have.a positive net present value. Default swaps are discussed by Tavakol.i (1998).Because of convexity bias (Chapter 7), the market intrest rate swap curve is not eactly the same as the swap cttrve constructd from Eurodolll f'utures. This is discussed irt Burgardt and Hoskins (1995) and Gupta and Subrahmanyam (2000).The SEC complaint agninst J. P. Morgan Chase is at hup://-.sec-gov/litigation/complaints/comp I 8252.12m.
PROBLEMS Some of the problems thgt follow use Table 8.9. Assume that th cun-ent exchange rate
is $0.90/6.
payment is R
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8.1. Consider the oil swap example in Section 8.1 with the 1- and z-yearfonvardprices of $22/bre1 and $23/b-e1. The 1- and z-yeaT interest rates are 6% and 6.5%. Verify that the new z-yea.r swap plice is $22.483.
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Table 8.8 summarizes the substitutions to make in equation (8.13)to get the vatious swap formulas shown in the chapter. This formula can be generalized to pennit time vriation in the notional amount and the swap price, and the swap can start on a deferred basis. of net An importapt charactelistic of swaps is that they require only $heexchaj if plincipal. So a firm enters an interest rate swap, for payments, and not the payment of example, it is required only to make payments based on the difference inj interest rates, bonds. noton the underlying principal. As a result, swaps have less credit risk t anfloitng exchang ?or the rate retul'n on an asst Total ret'ul'n swajs involve a ing suchas LYOR. The term sbvap is also used to describe agreements likethe fcter & :' t':' ' '' IELLIE j'-';j)jj)-';q;,;.jj-'. E-lll; E'E'' : 1@ :: ' : :5 ' ' C'E : :' EE'iE'' F:EE i ' :' ' E:'' E''':. :. i' E Cq lrf 1;* j''(' :. ' : E: E E iE ;'y.'y' j;' j-'zg.g 177* 7!).11177571 (' 96* i ' .q' i' PE@E 1*)ElE '(' ''i''i'77:7'* (jlf @E.! ;)i ' . q7''E'qF7'q':''q'T:('r(57' (!i'((-:'.-Etyr!.t.-...jjjjjj;,) .(i.! EEEE.! .!'' '( ;Ef?E . .':! .(. @
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A swap is a contract calling for an exchange of payments, on one or more dates, determined by the difference in two prices. A single-payment swap is the same thing as a cash-settled forward contract. ln the simplest swaps, a sxedpayment is periodically exchanged for a :oating payment. A finzl can use a swap to lock in a long-term conunodity price, a fixed interest rate, or a fixed exchange rate. Considering only the present value of cash flows, the same result is obtined using a strip of fomard contracts and swaps. The difference is that hedging wit.h a strip of forward contracts results in net payments that are time-varying. In contrast, hedging with a swap results in net payments that are constant over time. The value of a swap is zero at inception, though as swap payments 'are made over time, the value of the swap can change in a predictable way. The fixed price in a swap is a weighted average of the corresponding forward prices. The swap formulas in different cases a1l take th same general folzm.Let htij denote the forward plice for the floating payment in the swap. Then the fixed swap
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275
Gamble swap (page 264), which requ'ed payments based on the difference in interest rates and bond prices, as well as default swaps. FURTHER
HAPTER
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8 2 Suppose that oil forward prices for 1 year, 2 years, and 3 years are $2 $21 and $22. 'Fhe l-year effective annual interest rate is 6.0%, the z-yearinterest rqte is 6.5%, and the 3-year interest rate is 7.0%. *
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a. What is the 3-year swap plice? b. What is the price of a z-yearswap beginning in one year? (That is, the first swap settlement will be in 2 years and the second in 3 years.) 8.3. Consider the same 3-year oi1 swap. Suppose a dealer is payipg the tiyedprice and receiving :oating. What position in oil forward contracts will hedge oil price risk
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in this position? Velify that the present value of the locked-in net cash qows is Zero.
8.4. Consider the 3-year swap in the previous example. Suppose you are the fixed-rate payer in the swap. How much have you overpaid relative to the forwardprice after the first swap settlement? What is the cumulative overpayment after the second swap settlement? Verify that the cumulative overpayment is zero after the third payment. (Be sure to account for interest.) 8.5. Consider the same 3-year swap. Suppose you are a dealer who is paying the sxed oi1price and receiving the floating price. Suppose that you enter into the swap and immediately thereafter a1l interest rates rise 50 basis poits (oilforsvrd prices are unchanged). What happens to the value of your swap position? What if interest rates fall 50 basis points? What hedging instrument wold have protected you against intrest rate lisk in this position? 8.6. Supposing the effective quarterly interest rate is 1.5%, what are the per-barrel and 8-quarter oi1 swaps? (Use oil forwld prices in swap prices for zquarter Table 8.9.) What is the total cost of prepaid 4- and B-quarter swaps? i
j
8.7. Using the information about zero-coupon bond prices and oil forWa.f prices in Table 8.9, construct
the set of swap prices for oi1 for 1 through 8 quarters.
8.8. Using the information in Table 8.9, what is the swap price of a l-quarter oil swap with the hrst settlement occurrng in the third quarter? 8.9. Given an 8-quarter oil swap price of $20.43,construct the implicit loan balance for each quarter over the life of the swap. 8.10. Using the zero-coupon bond prices and oil forward prices in Table 8.9, what is the price of an 8-peliod swap for which two barrels of oil are delivered in evennumbered quarters and one barrel of oil in odd-numbered quarters? 8.11. Using the zero-coupon bond prices and natural gas swap prices in Table 8.9, what are gas forward prices for each of the 8 quarters? 8.12. Using the zero-coupon bond prices and natural gas swap prices in Table 8.9, what is the implicit loan amount in each quarter in an B-quarter namral gas swap?
8.13. What is the sxedrate quarter2:7
in a
interest s-quarter
rate swap with the first settlement in
8.14. Using the zero-couponbond yields in Table 8.9, what is the lixed rate in azquarter interest rate swap? What is the sxedrate in an 8-quarter interest rate swap? 8.15. What B-quarter dollar annuity is equivalent to an B-quarter annuity of 61? 8.16. Using the
assumptions
in Tables 8.5 and 8.6, velify that equation
(8.9)equals 6%.
8.17. Using the information in Table 8.9, 4- and B-quarter swaps?
what are the ettln-denonlillated
B I-EMS
k.
277
hxed rates for
8.18. Using the information in Table 8.9, verify that it is possible to derive the 8quarter dollar interest swap rate from the 8-quarter euro interest swap rate by using equation (8.9).
Pariy
Oytipn Rcktipnskip
this chapter we bgin to study option pricing. Up to this point we have primari1y studied contracts entailing/rpl comniitments, such as forwards, futures, and swaps. These coneacts do not permit either pacty to back away from the agreement. Optionality transactions. The prinoccurs when it is possible to avoid engaging in unprotable value ?/?c right to back t7lvt7y cipal question in option priing is: How do hnm a ct?p????r??7c??r? .w11
Beforewedelveintopcingmodels, we devote this chaptertoresning ourcommon about options. example, Table For contains call andput prices for IBM forfour 9.1 sense different strikes and two different expiration dates. These are American-style options. Her'e are some observations and questions about these prices:
* What determines the difference between put and call prices at a given strike? @How would te premiums change if these options were European rather thanAmerican? * It appears that, for a given strike, the Janual'y options are more expensive than the November options. Is this necssarily
true?
@Do call premiums always decrease as the stlike price increases? Do put premiums always increae as the strike price increases? * Both cll and putpremiums change by less than the change in the strike price. Does this alwayj happn? These questions, and oters, will be answered in this chapter, but you should tale a minute and tI'IiIZkE about the answers now, drawing on what you have lenrned in previous chapters. While doing so, pay attntion to how you are trying to cbme up with the answers. What constimtes a persuasive argument? Along with sndingthe answers, we want to understap how to think abopt questions like these.
9.1 PUT-CALL
PAJVTY
Put-call pality is perhaps the single most important relationship among option prices. ln Chapter 2 we argued that synthetic forwards (createdby buying the call and selling the put) must be priced consistently with actual forwards. The basic parity relationship for
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k. PARITY
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=
P(#,
F) + Sje -s.
-
pv 0.w LK)
(9.2)since
$2.78
=
$1.99+ $4o $zoe-0.0sx0.25
4
-
Why does the price of an at-the-money call exceed the price of an at-the-money by put $0.79?We can answer this question by recognizing that buying a call and selling cash flows than an a Put is a synthetic alternative to buying the stock, with different outright purchase. Figure 9.l represents the cash Cows for a synthetic and outright purchase. Note that the synthetic purchase of the stock entails a cash outflow of $0.79today and $40 at expiration, compared with an outlight purchase that entails spending $40 today. Also, both positions result in the ownership of the stock 3 months from today. With the outright purchase of stock, we still oWn the stock in 3 months. With the synthetic will exercise the P urchase,' we will own the stock if the plice is above $40 because we call. We will also own the stock if the price is below $40, because we sold a put that
K)
call with sike price K and time to expiration T) is the plice of a European put, Ftl w is the forward price for the underlying asset, K is the strike price, F is the time to expiration of the options, and PV(),wdenotes the present value over the life of the options. Note that e-rl'h p is the prepaid forward price for the asset and e-rTK is the prepaid fonvard pfice for the strike, so we can also thirlk of parity in tenus of prepaid forward prices. The intuition for equation (9.1) is that buying a call and selling a put with the strike equal to the fomard price (F(),w = K) creates a synthetic forward contract and hence must have a zero price. lf we create a synthetic long forward position at a price K4 since this is the benest lower than the forward price, we have to pay PV0,r(Fo,w of buying the asset at the strike price rather than the fonvard price. Parity generally fails forAmerican-style options, which may be exercised prior to maturity. Appendix 9.A discusses a version of parity for American options. We now consider the parity relationship in more detail for different underlying
=
iE1
K)
-
F)
1) This is consistent with eqution
strike price)
PVo,r(Fo,r
-
,
EEE : .(
(3.1)from Chapter 3 expresses this more precisely: CK,
PV0 w(Div))
-
1* p Example 9.-1 Suppose that the price of a nondividend-paying stock is $40, the l() @1 and options have 3 months to expiration. 'l))continuously compounded interest rate is 8%, zo-strike and European call sells for $2.78 a 4o-sttike European put sells for $1.99. 'ii A
European options with the same stlik: price and time to expiration is put
(&
q'
' jk
-
T) +
=
Sollrce: Chicago Board Options Exchange (wwwocboeacom).
Call
,
where Sz is the current stock price and PV0,w(Div) is the present value of dividends Sz PV0,w(biv) P ayable oer the life of the stock. For index pptions, we knowthat %e-bl' Hence, we can write
0.25
80
PK
=
y' @' )' j'' tlf :'(' ;''j!')' qf (i' p' r' ttf ;#'r ()' ))')' k,--,'. ..1111.*. t'#' $'t,' ll'tf !t#' r' )' )(' (jp/; ' ! ( j(( r(j: (.'' l).: j )ri@j'(j' t'. .q@ .((.(' ( ..' .k ( . ji'( ' .;(j.(IEI)I . .. ... . ).!1Ii'iiji. .. p;-.. l11E4:tIi:: lillt!d:Cijjjjjjjji
.'. .'.'.... . -' ---'' :'s' S' $(' '-;(tjk),'.' F' ,':q' :(i' ;-' :'-'I.III:::ICii-'.X -#' ,'' yff y' tj-' (' ;'j'k' i' ' ' :gi;: jl(E ' i q :
,,' -' ..' -'. ' -jf
.'
! (E : 7:7q4' i ! E 'E' : (rli!' JII'!'!))!!'T:'!Iq!S:!I ' q7)7:. ' i( i ' ': : j'.( i'E'E((j! ; (. . (..'.' '('q(.E'Eq'(: ' .;E E . ('! E'. g. . (.'. . qj('.(;jiE. (('q . . .. E . (t. . .- . . . -gk-,.. - . 4(';.!)i.E.(.j.yjjj)jtE;(E!j . . .- .. . -.- ..-.. . . -q j ..yy(jj -ry . . q- .E y .-..; -. . . . - i .. (..( .j(j.y((yjk..:jjj;jjE jyjjt i..(:jkj..(y,gjjjjj E.((jy(.yy;,yjjy4yyjjj:; ; (kgtjjjjj (-qyy .ygyy. y.jjgjsjyassyjsjsjsyyytyysjygsa,yjyjyyy .y g s. ( (... y . . j... .jgjtt . j . y ..;j. jy j . .. .jjy .... tjjgyj.sj ' . . ;;; ! - E7 -.. . ! -. ).. -. - - '. -.- -. -- . -. -.;. ; . - . - ---.... .. '
:
:
:
. . .
''
-
''
'
.
. .
:.
;
' ..
:
.'
..
y j:...y .. r)..y )).jj,))j,).j.)) ;;. : kk. k(. rr j . ; .)!' .. . . .. . .. .
.
-.
.
' ..
,.
-
. .,
- .
.
.
.
-
:
:,-
.
.
... . '.
.-
.....
.
.
. -. . . .
.
.
Cash flows for outright purchase of stock and for synthetic stock created by buying a 4o-strike call and selling a 4o-strike put.
. tjy.yja tjjjj (jyygy,
yjjys yjgy yj.;j,ytggyyyjyy yygjjg;j jkjygjtgy;j jjs yj.yyja gjgy tjyy ;jg. yjjjjyjjjyjj .jgy
.
gjy.yj,
oay o
.jjy
yyygyj yggy
.
,.
. .
oay 91 Own 1 share
-$40
'
Buy Call, Sell Put
'
k. PARITY
284
AND OTHER
OPTION
RELATIONSHIPS
PUT-CALL
will be exercised', as the put-writer we have to buy the stock. In either case, in 3 months we pay $40 and acquire the stock. Finally, the dollar risk of the positions is the same. In both cases, a $1 change in stock price at 3 months will lead to a $1 change in the value of the position. ln other the both positions entail economic ownership of the stock. You can verify that the words,
Rewriting equation
is the same by drawing a prost and loss diapam for the two positions. Thus, by buying te call and selling the put we own the stock, but we have deferred the payment of $40 until expiration. To obtain this deferral we must pay 3 ronths of interest on the $40, the present value of which is $0.79. F/?c optil p?w??II???. dter !?y interest t??)the #'./?'?w/ ofpaynlentfor the stock. lnterest is the reason that at-the-money European calls on nondividend-paying stock always sell for more than at-the-money Europeap puts with the same expiration. Note that if we reverse the posiion by selling the call and buying the put, then we synthetically short-selling the stock. In 3 months, the options will be exerciqed and are will receiv $40. ln this case, the $0.79compensates us for deferring rceipt of the we stock price. There are differences between the outright and synthetic positions. First, the stock dividends and the synthetic does not. This example assumed that the stock paid no pays dividends. lfit did, the cost of the actual stock would exceed that of the synthetic by the present value of dividends paid over the life of the options. Second, the actual stock has voting rights, unlike the synthetic position. '
'
.''..
.
.
'
.
))';q Example )' l.j't
''
285
Parity provides a cookbook for the synthetic creation of options, synthetii stock stocks, and T-bills. The xample above shows that buying a call and selling a put is like buying the stock except that the timing of the payment for the stock differs in the two eases.
Iisk
.
k
PARITY
(9.2)gives
S0
Ct#
=
us
F)
.,
-
PK
,
F) + PVnw(Div) + e-''TK ,
To match the cash flows for an outright purchase of the stock, in additin to buying the call and selling the put, we have to lend the present value of the strike and dividends to be paid over the life of the option. We then receive the stock in 91 days. ......,j,..,,,.'
r Example 9.3 ln Example 9.1, PV(),n.25(ff) $40/-0.08x0.25 $39.21. Hence, by . . the put for $1.99,and lending . .invest . . . total . We i)) buying the call for $2.75,selling $39.21, a of today. In 91 opiions have the $40 days, worih and two we lk. a T-ill $40. We acquire ) the stock via one of the exercised ptions, using the $40 T-bill to pay the strike price. .y.jt =
).')
.
.
jg
.
=
.
.
. .
.
E
'
,).
.
.
.
.
.
%.
synthetic T-bills If we buy the stock, sell the all, and buy the put, we have purchased the stock and short-sold the synthetic stock. Tliis transaction gives us a hedged ppsition that has no risk but requires investment. Pmity shows us that
&+
9.2
Make the same-assumptions as in Example 9.1, except suppose that )1)the stock The price of the European call dividend just before expiration. $5 pays a yjt .('@.1 is $0.74 and the price of the European put is $4.85. These prices satisfy parity with i)t).l . () dividends, equation (9.2):
PK,
T)
-
CK,
T4
PV0,r(#)
=
+ PVn,w(Div)
We have thus created a position that costs PV(#) + Pvrmiv) FW,r(Div) at expiration. This is a synthetic Treasury bill.
and thi pays K +
i'('. :
ty ittj 11*1
-0.08x0.25
$0.74 $4.85 -
=
($40 $5d -
-0.08x0.25
)
-
$40/
)
j' .'
Example 9.4 ln Example 9.1, PVn,(t25(A-) $39.21. Hence, by buying the stock, )g buying put, and selling the call, we ca create a T-bill that costs $39.21and pays $40 1(-. ktin 91 days. k. IE! '
The call price is higher than the put price by interest on the strike ($0.79)and lower by the present value of the dividend ($4.90),for a net difference of $4.11. )t)@ %
.
In this example, the at-the-money call sells for less than an at-the-mony put since dividends on the stock exceed the value of interest on the strike price. lt is worth mentioning a common but erroneous explanation for the higher premium of an at-the-money call compared to an at-the-money put. The prot on a call is potentially unlimited since the stock price can go to infnity, while the profit on a put can be no greater than the strike price. This explanation seems to suggest that the call should be more expensive than the put.l However, parity shows tat the true reason for the call being more expensive (as in Example 9.1) is time value of money.
E '
.j
=
'''''
.
.
''
Since T-bills are taxed diferently than stocks, the ability to create synthetic Treaa sury bill with the stock and options creates problems for tax and accoting authorities. How should the return on this transaction be taxed as a stock transaction or as interest income? Tax rules call for this position to be taxed as interest, but you can imagine taxpayers trying to skirt these rules. The creation of a synthetic T-bill by buying the stock, buying a put, and selling a call is called a coniersion. If we short the stock, buy a call, and sell a put, we have created a synthetic short T-bill position and this is called a rverse conversion.
synthetic 1ln fact, the m'gument also seems to suggest that every stock is worth more than its pn'ce!
options
CK,
Parity tells us that F)
=
s'o -
PVp.w(Div)
-
PW.w(S)
+ PCK, F)
286
% PARITY
AND OTHER
RELATIONSHIPS
Op-rlolq
GENERALIZED
pndthat PCK, F)
=
CK,
T4
Note that for a pure-discount nondividend-paying stock.
+ PV(),w(#) + PW,w(Div)
us'a
-
The hrst relation says that a call is equivalent to a leveraged position on the underlying asset (u% PVn.w(Div) PV(A-)), which is insured by the purchase of a put. The second relation says that a put is equivalent to a short position on the stock, insured by the purchase of a call. '
Options on Cttrrencies 'Suppose we have options tobuy euros by paying dollars. From ourdiscussion of currency forward contracts in Chapter 5, we know that the dollar forward price for a euro is xjer-retT
'
OPTIONS
$
287
bond, the parity relationship is exactly like that for a
-
-
F(): w =
PARITY AND EXCHANGE
where ab is the curient exchange rate denominated as
$/6, ?x is the
euro-denominated interest rate, and r is the dollar-denominated interest rate. is then relationship for options to buy one euro by paying
.
Yheparity
-p
CK,
T)
-
P (K T) ,
=
xze-rT
-
PARITY OPTIONS AND EXCHANGE
The preceding section showed how the pality relationship works for different underlying assets. Now we will generalize pality to apply to the case where the strike asset is not necessarily cash but could be mzy other asset. Tllis version of parity includes all previous versions as special cases. Suppose we have an option to exchange one asset for another. Let the underlying asset, asset A, have price $, and the strile asset (the asset Which, at our discretion, we surrender in exchange for the underly.ipg asset), asset B, have the price Qt Let F)vS) denote the t'ime t price of a prepaid forward on the underlying asset, paying denote the time t price of a prepaid forward on asset Sp at time F, and 1et F,Ps(:) B, paying Q.r at time F. We use the notation Ct Qt, T t) to denote the time t price of an option with F t periods to expiration, which gives us the right to give up asset B in exchange for asset A. PCS; Qt F t) is defined similarly as the right to give up asset A in exchange for asset B. Now suppose that the call payoff at time F .
Ke-rT
Buying a euro call and selling a euro put is equivalent to lending euros and borrowing dollars. Equation (9.4)tells us that the difference in the call and put premiums simply re:ects the difference in the amount borrowed and loaned, in the currency of the country in wlzich the options are denominated.
Suppose the current $/6 exchange rate is 0.9, the dollar-denominated Example 9.5 interest rate is 6%, and the euro-denominated interest rate is 4%. By buying a dollardenominated euro call wit a strike of $0.92and selling a dollar-denominated euro put with the same strike, we construct a position where in 1 year we will buy 61 by paying $0.92. We can accomplish the same thing by lending the present value of C1 today (wit.h 04 $0.864 ) and pay ing j or this by borrowing ihepiejert value a dollar cost of $0.9, -0 ($0.92c-0'06 of $0:92 $0.8664).The proceeds from borrowing exceed the nmount we need to lend by $0.0017..Equation (9.4)pelforms exactly this calculation, giving us a difference between the call premium and put premium of 'j
'
9.2 GENERALIZED
=
-
,
-
-
,
,
is
C(&,
Qv,0)
Psr,
Qr, 0)'
=
maxto, Sp
=
max(0,
Q.)
-
and the put payoff is
Qv
5'w)
-
Then for European options we have this fonn of the parity equation:
=
ne-rq-
Ke-rT
-
=
0.9$/e x
ec-0.04xl $0.92 x -
c-0.o6xl
= $0.8647 $0.8664 = -
-$0.0017
*
%
Cst
,
Qt ,
T
-
t)
-
Pzt
,
Qt ,
F
-
/)
=
Fkppsl
'''''
.
.
.
.
Finally, we can construct the parity relationship for options on bonds. The prepaid forward for a bond differs from the bond price due to coupon payments (whichare like dividends). Thus if the bond price is Bz we have CK,
T)
=
PK,
F) +
E#() -
Pvwtcouponsl)
.f-
PV(),w(#)
(9.5)
Ft%Q)
The use of prepaid fonvard prices in the parity relationship completely takes into account the dividend and time value of money considerations. This version of parity tells us tat there is nothing special about an option having the strike amount designated as cash. In eneral, options can be desined to exchange any asset for any other sset, and put and call premiums are determined by prices of prepaid forwards on the the relative . . undyig ad stzik assets. To proveeqation (9.6) we can use a payoff table in which we bu# a call, sell a sell put, a prepaid fonvard on A, and buy a prepaid fonvard on B. This transaction is illuseated in Table 9.2. . . j' lf the strategy in Table 9.2 does not pay zero at expiratlon, ther is an arbitrage opporttlnity. 'Thus, we expect equation (9.6) to hold. A11European options satisfy this formula, whatever the underlying asset. '''k
Options on Bonds
-
.
%.PARITY
288
GEN ERALIZED
RELATIONSHIPS
OPTION
OTHER
AxD
-'. -' S))' 5* ';'4,* y' -' -j'
dj:,,f (,::2r:1j' l;.f :'r'!' t'?' (' 'r;f ;';q't))' ;(' k;,,,jf (.()i' jqf r' k' ()' '* k' 'Iq..':l.f llr...'.kf ::iE;;' llr...r.kf j'pit)f t$' 7q''lf qtl''r' 'i'rf E':)' (jjr'tf (' t'((411111112::,* )' ;';))' k' jjy:::,,f rkr'gjjjjjjgrji,k (yy'r.yr t't!llpf iiif .,It::)' r' jqf ';j,' .'i''i'jjljjjjyly'ly' rtq'pqf y' j))t't')':iIiq.ii1k)'. (' yl'iilf ryy)f y;f ?.(E! E!'' (' EEE'E( j EiE'(IIE i'E' !'
.' .'.,j:jjjjjjjj'. ....'..
-1,* .,jj::.jjy::' ''
t'r' '..iE'
jj
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.
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. -
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-
. -.
,
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'!:q'qr!r'
5*
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y ty.jjjjjkk-,,. j.tttitj . . ; ty ((.' . ;.k.. . E;.!#. t,--
.
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.
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.
-
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-... -
'
.
.y
-
.
-
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. . . . . ..
.
opportunity unless .-CUt,Qt, T Jy(Q) 0. T t) + #tpr(.)
.
' E.. . - . . - . E...( . .... .
..
..
.
.
-
iiil.. llr'..
t) + Pk, Qt,
E -. EE
E
-
E. . .. EE.
-
..
.
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..
.
E .. .
:
.
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Eq . . ..
-
.
.
-
.E. . - E. .. y. .. .q . . . .
-
.q.. . y. .q.y
........).
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-
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Ey
r:iikik:ij ; p (( .. j tii:ltjl, (jIIyj:i,i. ; (.( . . .. .... .. . )t. .j. .....y..,,,. ..( 3jL;,::,?.. r......y.... j. .. .. .
.
.
l l' ' ' . .. ...'.'. . ..... '. . .. . . . ....'.. .. . .'.' .. .'.. ' . . .. . ......... ...
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-CSt
Buy call
.
.
.
. .. ...' ... ,
,
,
T
,
F
Qt Qt
FhS)
on A
.
.
.
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.
. .. . . .
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...
.
.
-
.
.
...
....
-
.. . . . .
-
-......yr-.-..
..
..
P&
Sell put sell prepaid Frward
.
-
.....
-.
.
-..
-
..'-. ;.. E.-E E . . . . E. y y
.
maxlo,
:j gj( jry j : j - ! j ( ! ( ( r r i j E Lili:z. g E-. . E j . . . - .. -. j j j ::iE!i1. (iii. . @ Eiilikk:. E. . E i E. .. E. . . . . . E. !.. .. -.. . ..i . .. . . E. .. .. .-... .r . E. - . . q ....E. .. - . ; ...-. . . . . . .. . ... . . . q. . . .y ... q . . . .. ... .... 3,,..:3' j...j).')jj.... . y j. ... ......y . .. ..t yy .....,-) . .......... . yyry.......y y y:j-..-.;. (..............,.. . . . ,..... . . .-rtE ..-... .:j. :.... y... .q . .j$;2f.. g;j,-..,jj . yt,.-.t .L..... .... ( y......... y ,,:.,t . . : .. . j.;.. .. y. .. .-.. jk.kii,. ......k-.,. j.. :(.. ,k2:,,,. .....t y. . jE.....L.L. .... -. .j 411,! q...y ... .(1!:1t751: jf.:. r. .(2::!ji::. ..j-.. y ..y..j.. . y !!;;:trry.. y... -: .. .y( . y.. ..(.....t.. y ..-. y.... )Lt;.. ... .j..... .y L3... y.. . . . ?jf.:.. . . . . . . . . . . . . t
(
Sp
t4
0
t4
s'w Qv -
-
Qp
0
Qv
->7w(p)
Forwafd on B '
-
C LS r) T :J,
+P&, Total
A-bje.rs)
-
Qv
t)
?) F Fkevq) -
-
O
j'('.(
Suppose that nondividend-paying stock A has a price of $20, and r Example 9.6 )# of $25. Because neither stoc k p a y s dividends, t( nondividend-paying stock B has a price equal lf A is the underlying assei and B is the prices. their 1i their prepaid forward prices ti) k tl strike asset, then put-call parity implies that '
:
( . )(.
.j
(.
' ::j
Call
)j i -
.
-
put
=
$20 $25 -
-$5
=
i
ilitThe put is $5 more jj,)) ).(,!)
expensive than t he call for any time to expiration of the options.
'
s'oooo sksy'rl -
and Puts?
The preceding discussion suggests that labeling an option as a call or put is always a al1 the time in matter of convention. lt is an important convention because we use it being calls interpret general Nevertheless, puts, and in options. as about can we talking analogy. using why by an vice versa. We can see When you go to the grocery stofe to obtain banaas, you typically say that you involves handing cash t the grocer and are buyiltg bananas. The act'ual transaction receiving a banana. This is an exchpnge of one asset (cash)for another (a banana). We could also describe the transaction by saying that we r sllink ctu/? (in exchange for bananas). The point is that art exchapge occurs, and we can describe it either as buying the thing we receive, or selling the thing we surrender. Any transaction is an exchange of one thing for another. Whether we say we are buying or selling is a matter of convention. Tlzis insight may not impress your grocer, but it is important for optins since it suggests that the labeling we commonly use to distinguish calls and puts is a matter of convention. To see how a call could be considered a put, consider a call option on a stock. This worth is the right to exchange a given number of dollars, the strike price K, for stock > K, we S, if the stock is worth more than the dollars. For example, suppose that if S S K. We can view this as either of t'wo ansactions:
earn
-
exercise when S * Buying one share of stock by paying K. In this case w This is a call option on stock.
Options to Exchange Stock Executive stock options ar sometimes constructed so that the strike price of the option option is the price of an index, rathf than a lixed cash amount. The idea is to have an competitfs, fgther yhat thq outperforms pays e the wh company that pays offonly of this, krfijl hyjthtil As al1 stock have plices a gone up. off simply because . off compenation that given is optlons of Microsoft, pay chairman suppose Bill Gates, options and if if only exercise these will He outperforms Google. only if Microsoft i.e., of share Googl, the exceeds price of Mierosoft, skslrr, the share price s'hlsp'T > From Gates's perspective, this is a call option, with the payoff ksooc,
qcooc.
maxto,skslzr s'coocl -
289
%.
This is a call from Schmidt's perspective. Here is the interesting twist: Schmidt's Google call looks to Gates like a Microsoft put! And Gates's Microsoft call looks to Schmidt like a Google put. Either option can be viewed. as a put or call; it is simply a matter of perspective. The distilctioll /pdrks/dcn t'l??# d-vwap/d depends I1p-l what I#?c label #?' underlying asset a pltt and a call 11 thl's what Jvc label as the strike asset.
at Are Calls
'
Buy prepaid
OPTIONS
Now consider the compensation option for Eric Schmidt, CEO of Google. He will receive a compensation option that pays off only if Google outperforms Microsoft, i.e.,
=
-
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PARITY AN D EXCHANGE
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of stock. Again we exercise when @ Selling K dollars in exchange for one share worth less than the stock. This is a put sell when dollars the we S > A'-i.e., are sike the of stock asset. share as option on dollars, with a Under either intepretation, if S < K we do not exercise the option. If the dollars
sell them for the stock. are worth more tha the stock, we would not exchange Now consider a put option on a stock. The put option confers the right to < K; we earn K S. We dollars. Suppose S one share of stock for a given number of can view this in either of two ways: @ Selling one share of stock at the price K. -
290
%.PARITV
AXD OTHER
Op7'loN
RELATIONSHIPS
GEN ERALIZED
@ Buying K dollars by paying one share of stock. This is a call right to give up stock to obtain dollars.
where we have the
PARITY AN D EXCHANGE
$
OPTIONS
291
Though they will be exercised under the same circumstances, the dollar-denorni-
nated euro call and the euro-denominated dollar put differ in two respects:
If S > K we do not exercise under either interpretation. If the dollars are worth than th stock, we would not pay the stock to obtain the dollars. less
* The scale of the two options is different. The dollat-denominated euro call is based and the euro-denominated on one euro (whichhas a current dollar value of $0.90) dollar put is based on one dollar. * The currency of denomination is different.
Currency Options
W can equate the scale of the two options by holding more of the smler option or less of the larger option: We can either scale up the dollar-denominated eur cltlls, i of them, or we can scale down the euro-denomiated holding 92 dollqr puts, holding .0 0.92 of them. To see the equivalence of the euro call and the dollar put, consider the following two transactions:
The idea that calls can be relabeled as puts is notjust academic; it is used frequently by ansaction involves the exchange of one kind of currency currency aders. A currency for another. In this context, it is obvious to market participants that referring to a particular currency as having been.bought or sold is a matter of convention. Labeling a particular option a call or a put depends upon which currency you regard as your home
1 Buy I l-year dollar-denonzinated euro call options with a strike of $0.92. lf we exercise, we will give up $1 for 'Ctl1ju The cost is 1 x $0.0337 $0.0366. 9.92 . 2. Buy one l-year euro-denominated put option on dollars with a strike of 61.0870. 't'he cost of ihis in dollars is 0.90$/6 x 60.04t7 $0.0366. When the option expires, convert the proceeds back from euros tp dollarj.
Ctlrfency.
In the ollowing example we will show that a dollar-denominated call option on which gives you the right to pay dollars to receive euros, is equivlent to a euros, . euro-denominated put option on dollars, which gives the ri g ht to sell a dpltar for euros. Obviously, the strike prices and option quantities must be chosen appro)ri j tely yor tjjere to be an equivalence. We will say that an option is if the strike prie ;nd Ifmium option is denominated in dollafs. An if the strike priceand are premium are in euros. Suppose the current exchange rate is -Y0 = 0.90$/E, and consider ihe following two options: 2
=
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E'dollar-denominated''
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%.PARITY
AND OTHER
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RELATIONSHIPS
OpTloN
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is P.f&.(;p rate is ..p7 (;k exchange the prices are related by .:
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calls in one currency are the same as pts in the other-is commonplace among currency traders. While this observation is interesting in and of itself, its generalization to all options provides a fresh perspective for thinking about what calls and puts actuatly are.
This insight-that
TO STYLE,
OPTIONS MATURITY,
WITH RESPECT AND STRIKE
We now exnmine how option prices change when thre are changes in option charactelistcs, such as exercise style (Amelican or Ettrupean), the strik jiice, ynd time to iand without exp i ra tiplj Reparkably, we can say a great deal without a pricing rodl making any assumptions about the disibution of the underlying asset.ao Thus, vhatever the particular option model or stock price distribution used for valuing a given option, we can still expect option prices to behave in certain ways. Here is an example of the lind of questions we'will address in this section. Spppose have three call options, with strikes of $40,$45, and $50. Hpw dp the premiums on you these options differ? Common sense suggests that, with a call option on any underlying asset, the premium will go down as you raise the stlike plice; it is less valuable to be able to buy at a higher plice/ Moreover, the decline in the premium cannot be greater tan $5. (The zight to buy for a $5 cheaper price cannot be worth more than $5.) Following tltis logic, the premium will drop as we increase te strike from $40 to $45, and drop again when we increase the strike further from $45 to $50. Here is a more subtle question: ln which case will the premium drop more? lt t'urns out that the decline in the premium from $40 to $45 mltst be greater than the decline from $45 t $50, or else there is an arbitfage opportunity. In tis section we will explore the following issues for stock options (someof the properties may be different for options on other underlying assets): .
* How prices of otherwise identical American and European options compare.
3The so-called of rational option prcing,'' on which this section is based, wtts lirst presented in 1973 by Robei't Merton in an astonishing paper (Merton (l973b)). Ths material is also superbly exposited in Cox and Rubinsten (1985). 'theory
4If
RESPECT
TO STYLE, MATU RITY, AN D WRI KE
* How option prices change as the time to expiration
* How option prices change as the strike price
Cstab, K, F)
9.3 COMPARING
M PARI N G OPTI ONS WITH
you are being fastidious, you will say the option premium ctllot acrct7-p as the strike goes up. Saying that the option premium will decrease as the stlike increases does not account forthe possibility that a1l the premiums are zero, and hence the premium will not go down, but will remain unchanged, as the stzike price increases.
$
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changes.
changes.
A word of warning before we begin this discussion'. f.f you exnmine option price listings in the newspaper, you can often find option prices that seemingly give rise to atbitrage oppormnities. There are several reasons for this. One is that some reported option price quots are stale, meaning that the comparison is among option prices recorded at diffent times of the day. Moreover, an apparent arbitrage oppo>nity only becomes genuine when bid-ask spreads (seeTable 9.1), commissions, costs of short-selling, and market impact are taken into account. Caveat ar itl'ag eur!
Ettropean Versus American Options Since an Amelican option can be exercised t any time, valuable as an otherwise identical European option. (By that the two options have the spme underlying asset, strike piion pprplipte to a Eurojtan Any eyeycise jtotegy ption cannot be lls an Americ ofitioi Th Ameria
it must always be at least as othelavise identici'' w mean price, and time to expiration.) a alwys be duplicated with valuable. Thus ave
CAmer(S,K, F)
kCEurt5',
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F)
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We will see that ther >r times when the right to eady-xercise Anrican and uropean opttons have the same vdue.
(9.8a) (9.8b) is wprthless, and, hence,
Maximum and Minimum Option Prices hw expensive or inexpensive an option can be. Here
It is often usef'ul to understandjust are some basic limits. Calls
The price of a European call option
e Cannot be negative, because the call need not be exercised. * Cannot exceed the jtock price, because the best that can happen with a call is that you end up owning the stock. value. * Must be at least as great as the price implied by parity with a zero put Combiningthese statements, togetherwith theresult
aboutAmerican
optons never
being worth less than European options, gives us S k CAmer(S,K, F) k
where present
CEurt5',
values are taken
K, F) k maxgo, PV0,w(F0,w)
over the life of the option.
=
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294
% PARITY
AN D OTH ER OPTION
RELATIONSHI
Co
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RES PECT TO STYLE, MATU RITY, AN D STRI KE
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A third effect is the possible loss from deferring receipt of the stock. However, when there are no dividends, we lose nothing by waiting to tale physical possession of the stock. We have demonstrated that if a stock pays no dividends, you should never see an optionselling for less than St K. In fact, equation (9.11),like equation (9.9),actually that you should never see a call on a nondividend-paying impliesthe stronger result Ke-rT-ts What happens if you do observe lln option selling than S stocksell for less S'J < and the option is American, you can buy the option, price? lf C K for too low a and K exercise it, (Qmertu?ff, r 1) > 0. Howevr, what if the option enrn & therefore and cannot be exercised early? In this case the arbitrage is: Buy is European stock, and lend the present value of the strike price. Table 9.4 option, short the the arbitrage this case. The sources of proft from the arbitrage are the in the demonstrates equation in those identifed (9.11). same as It is important to realize that this proposition does not say that you must hold the option until expiration. It says that if you no longer wish to hold the call, you should sell it rather than early-exercising it.5
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ssome options. such as compensation options, cannotbe sold- ln practice itisommon to seeexecutives exercise options prior to expiration and then sell the stock. The dscussion in this section demonstrates that such exercise would be irrational if the option could be sold, or if the stock could be sold short.
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OPTIOI.I
That is, if interest on the strike price (whichinduces us to delay exercise) exceeds the present value of dividends (whichinduces us to exercise), then we will for certain never early-exercise at that time. If inequality (9.12)is violated, this does not tell us that we kvillexercise, only that we cannot rule it out. lf dividends are suciently great, however, early exercise can be optimal. To take for $100, an exeme example, consider a go-strike American call on a stock selling $90 to acquire the which is about to pay a dividend of $99.99. If we exercise-paying $100 stock-we have a net position wol'th $10. If we delay past the ex-dividend date, the option is worthless. 1.fdividends do make early exercise rational, it will be optimal to exercise at the last moment before the ex-dividend date. By exercising earlier than that, we pay the strike price prematurely and thus at a minimum lose interest on the strike price.
for puts When the underlying stock pays no dividnd, a call will not rnight be. To see that early exercise for a put can make eady-exercised, but put a e economic sense, suppose a company is bankrupt and the stock price falls to zero. Then until expiration will be worth P%,pK). lf we could a put that would not be exercised early-exercise. we would receive K. If the interest rate is positive, K > PV(#). Therefore, early exercise would be optimal in order to receive the strike price earlier. We can also use a parity argument to understand this. The put will never be exercised as long as P > K S. Supposing that the stock pays no dividends, parit.y for the put is Early exercise
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vzueless (asin the aboveexample of abanknlptcompany), parity cannot rule out eady exercise. This does not mean that we vill early-exercise', it simply means that we callnot rule it out. We can summarize tllis discussion of early exercise. When we exercise an option, necessat'y we receive something (the stock with a call, the strike plice with a put). condition for early exercise is that we prefer to receive this something sooner rather than later. For calls, dividends on the stock are a reason to want to receive the stock earlier. For puts, interest on the strike is a reason to want to receive the sike price erlier. Thus, dividends and interest play similar roles in the two analyses of early exercise. In fact, if we view interest as the dividend on cash, then dividends (broadlydesned) become the sole reason to early-exercise an option. ln Similarly, dividends on the stlike asset become a reason npt to eady-exercise. of strike and dividend the in the case puts, asset, on the case of calls, interest is the dividends on the stock are the dividend on the strike asset.
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when make some general statements about The point of this section has been to conditions it might occur. Early exercise is a under what underlying early exercise will not occur, or the strile price, dividends on the of trade-off involving time value money on outwhen liguring to exercise the position. ln genefal, 10 asset, andthe value of insurance on in Chapters further exercise will discuss early requires an option pricing model. We and 11.
Time t Expiration
expiration? lf the options are change as we increase time to expiration. 1.f How does an option price decline with an increase in time to American, the option price can never increase time to the price can go either up or down as we the options are European, expiration. expirationis atleastas valuable AnAmerican callwithmoretimeto American options expiration. An American call with 2 years identical call with less time to as an otherwise option with 1 year to example, can Etlways be t'urned into an American to expization, for the after 1 year. Therefore, z-yearcall is at least expiration by voluntarily exercising it l-year call. as valuable as the longer-lived Amedcan put is always worth at least as The same is true for puts: A much as an otherwise equivalent European put. nondividend-paying stock will be at least as A European call on a European options call with a shorter time to expiration. This oecrs' valuable as an otherwise identical otherwise identical call has the same price as an because. with no dividends, a European options may be less longer-lived European American call. With dividends, however, it optimal make that options. Economic forces valuable than shorter-lived European short-lived than a worth option mor European early can make a to exercise an option long-lived European option. weeks stock that will pay a liquidating dividend 2 To see this for calls, imagine a since it is value will have call with l week to expiration will from today6 A Euppean have expiration call with 3 weeks to exercisable prior to the dikidend. A Europeanexpiration. This is example of a longeran value at the stock will have no no value since that if the options were Note option. shorter-lived lived option being less valua le than a diyidepd. exercise the 3-week option prior to the Amezican, we wouldsimply shorter-lived Euyopean valuable than Longer-lived European puts can also be less the present worth will be is banknapt company. The put expiration. Logerputs. A good example of tltis a value calculated until time to value of the strike price, with present American, they options were lf shorter-lived puts. the lived puts will be wort.h less than strike price. the worth and hence would be would al1 be exercised immediately
shareholders. A tirm is worthless Iirm pays its entire value to 6A liquidating dividend occurs when a after paying a Iiquidating dividend.
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present value of the strike price therefore decreases with time to mamrity. Suppose, Ker however, that we keep the present value of the strike constant by setting Kt When the strik grows at the interest rate, the premiums on European calls and puts on maturity.? We will demonstrate this a nondividend-paying stock increase with time to calls. identical for is demonstration the for puts; To keep the notation simple, let Pt) denote the time 0 price of a European put Ker We want to show that #(F) > Pt) if mattlring at time r, with strike price Kt will demonstrate > 1. show an arbitrage if #(F) S P (r). -F To this, we if P (F) :; Pt/l-buy the put expensive-i.e., longer-lived not is put more lf the expiration. At tim: with the t the sell and expiration t put to years F with years to shorter-lived the and is ignore value > Kt its St expire. If will can we zero written put option from thij point on. If & < Kt the put holder will exercise the short-lived option and our payoff is St Kt Suppose that we keep the stock we receive and bonow to second option expires at time nance the strike price, holding this position until the F Here is the important step: Notice that the time-r value of this time-r payoff is erT-tj s r K? sT Kp. Table 9.5 summarizes the resulting payoffs. By buying te long-lived put and selling the short-lived put, we are guaranteed not to lose money at time F. Therefore, if P (r) : P (F) there is an arbitrage opportunity. A practical application of this result is discussed in the box on page 299.
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TO STYLE, MATU RITY: AN D STRI RE
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We discussed at the beginning of this section some statements we can make abnut how option prices vary with the strike price. Here is a more formal statement of these propositions. Suppose we have three strike prices, #1 < Kz < Ks, with corresponding call option prices CLKL ), C(ff2), and C(A%) and put option prices /'(/('1), PKqj, and Plj. Here are the propositions we discuss in this section:
1. A call with a low strike price is at least as valuable as an otherwise identical call with a higher strike plice:
(9.13)
C(fC1) R: C(fQ)
A put with a high strike price is at least as valuable as an otherwise identical put with a low strike price:
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These statements are all true for both European andAmerican options.8 Algebraic demonstrations are in Appendix 9.B. It t'urns out, however, that these three proposidons enterinto an op:on spread, are equivalent to saying tat there are no f'ree lunches: Ifyou there must be stock prices at which you would lose money on the spread net of yotlr original investment. Otherwise the spread represents an arbiage opportunity. These three propositions say that you cannot have a bull spreatt a bear spread, or'' butterfly spread for which you can never lose money. Specifically:
1. If equation (9.13)were not true, buy the low-strike call and sell the high-sfrike call (this is a call bull spread). If equation (9.14)were not true, buy the high-strike put and sell the low-strike put (aput bear spread). 2. lf equation (9.15)were not true, sell the low-strike call and buy the lgh-strile call (a call bear spread). Ifequation (9.16)were not trtle, buy the low-stn'ke put and sell the high-sike put (a put bull spread). 3. If either of equations (9.17)or (9.18)were not true, there is an asymmetric buttey spread wit positive profits at a11prices. We will illustrate these propositions with numerical examples.
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There is a different proposition, strike price c-lvdA/y. 11 and Ks,' between is equation Kz Since (9.17). way to wlite the convexity inequality, is that strikes, of other the two we can write it as a weighted average
Now consider the
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where
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With this expression for
Example 9.7 Suppose we observe the call premiums in PanelA of Table 9.6. These values violate the second property for calls, since the difference in strikes is 5 and the difference in the premiums is 6. lf we observed these values, we could engage in
The difference in
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SIn fact, if the options are European. the second statement can be strengthened: option premiums must be less than tlle preselt lz(Jlt? of the difference in strikes.
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change in the option premium ($6) exceeds the change in the strike price ($5).Panel B shows how a bear spread can be used to arbitrage these prices. By Iending the bear spread proceeds, we have a zero cash flow at time 0; the c.ash outflow at time F is always
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'l@Example 9.s (( l..
.
If
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=
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302
PARITY AND OTHER
OPTION
RESPECT
wITH
To
STYLE, MATU RITY, AN D WRI KE
Q
303
The formula for may look imposing, but there is an easy way to hgure out what is in any situation. ln this example, we had the prices 50, 59, and 65. lt is possible to express 59 as a weighted average of 50 and 65. The total distance between 50 and 65 is 0.6 of the total distance. Thus, 15, and the distance from 50 to 59 is 9. which is 9/15 59 write as can we 59 (1 0.6) x 50 + 0.6 x 65
'iy Call prices must then satisfy l tj C(59) :G 0.4 x C(50) + 0.6 x C(65) t)' .y.yytj observe the call premiums in Table 9.7. The change in the option premium l'it' jyysupp ose we 0.567, and the change from 59 )) per dollar of strike price change from 50 to 59 is 5.1/9 ')t 65 is 3.9/6 0.65. Thus, prices violate the proposition that the premium decreases to E .. :
.
Op-rlolqs
CoM PARING
RELATIONSHIPS
.
=
=
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ll l'ilat a decreasing rate as the strike price increases. ) ) To arbitrage this mispricing, we engage in an . )!'r1
-
This is the interpretation of in expression (9.20). Here is an example of convexity with puts.
asymmetric butterfly spread: Buy jy, j) calls, buy six 65-strike calls, and sell ten 59-stfike calls. By engaging in t) fqur so-strike tt $) a butterfly spread, Panel B shows that a prot of at least $3 is earned. Q. ''i
)q Example 9.9 )) and Ks 70. tr) t'l since t't E =
55, 50, Kz See the prices in Panel A of Table 9.8. We have #1 violated 0.75 and 55 0.75 x 50 + (1 0.75) x 70. Convexity is =
=
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calls.
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k: PARITY
304
AND OTHER
In this case, we always make at least 4. Figure 9.2 illustrates the necessary shape pdce. of culwes for both calls and puts relating the option premium to the stlike
Put-call pality is one of the most important relatioris in option pricing. Pmity is the observation that buying a European call and selling a European put with the same strike price and time to expiration is equivalent to maling a leveraged investment in the underlying asset, less the value of cash payments to the underlying asset over the life of the option. Different versions of parity for different underlying assets appea. in Table 9.9. ln very case the value on the left-hand side of the parity equation is the plice of the underlying asset less its cash llows over the life of the option. The parity relationship can be algebraically rean-anged so that options and the underlying asset create a synthetic bond, options and a bond create a synthetic stock, and one lind of opti3n together with the stock and bond synthetically create the other kind of option. 'l'he idea of an option can be generalized to permit an asset other than cash to be strike asset. This insight blurs the distinction between a put and a call. The idea the and calls are different ways of looking at the same contract is commonplace in that puts
Exercise and Moneyness If it is optimal to exercise an option, it is also optimal to exercise an othelavise identical this option that is more in-the-moneykConsider what would have to happen in order for not to be true. Suppose a call option on a dividend-paying stock has a strike price of $50, and the stock price is $70. Also suppose that it is optimal to exercise the option. This means that the option must sell for $70 $50 = $20. Now what can we say about the premium of a 4o-strike option? We lntw from the the discussion above that the change in the premium is no more than tlae change in that oppormnity. This arbitrage means is there else sfrike price, or an -
C
(40)S $20 + ($50 $40) $30
currency markets. Option prices must obey certain restrictions when we vary the strike price, time to maturity, or option exercise style. American options are at least as valuable as European options. American calls and puts become more expensive as time to expiration increases, but European options need not. European options on a nondividend-paying stock do become more expensive with increasing time to maturity if the strike price grows at the interest rate. Dividends are the reason to exercise an American call early, while interesy is the reason to exercise an American put early. A call option on a nondividend-paying stock will always have a price greater than its value if exercised', hence, it should never be exercised early. There are a number of pricing relationships related to ehanging strike plices. In particular, as the strike price increases, calls become less expensive with thir price
=
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SUMMARY
CHAPTER
RELATIONSHIPS
OPTION
45 40
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306
k. PARITY
AND OTHER
OPTION
deereasing at a decreasing rate. The absolute value of the change in the call price is less than the change in the strike price. As the sttike plice decreases, puts become less expensive with their price decreasing at a decreasing rate. The change in the ppt price is less than the change in the strike price. FURTHER
PROBLEMS
RELATIONSHIPS
READING
cltapters. Two of the ideas in this chapter will prove particularly important in later calls we also understand tatif we ideais put-call parity, which tells us The srstkey option since the understand pricing easier to understand puts. Tllis equivalence makes it applicable dlrectly option of to the are pricing techniques and intuition about one kind another-also for exchange options--options asset one to other. The idea of exchange will show up again in later chapters. We will see how to plice such options in Chapter 14. A secpnd key idea that will prove important iq th determination pf fgtprs influecing early exercise. As a practical matter, it is more work tp price p.lzAmerican than a European option, so it is usef'ul to lcnow when this exa work is not necessaty Less obviously, the determinants of eady exercise will play a key role in Chapter 17, where we discuss real qptions. We Fill see that certain kinds of inveqtmentprojects are analogous to options, and the investment decision is like exercising an option. Thus, the early-exercise decision can have important consequences b.eyond the realm of hnancial options. Much of the material in this chpter ean be traced to Merton (1973b),which contains an exhaustive eeatmentof optipn prqperties that must hold if there is to be no arbithis mgterial. trage. Cox and Rubinstein (1985)also provides an excellent eatlent of .
%
307
c. What difference between the call and put prices would eliminate arbitrage? d. What difference between the call and put prices eliminates arbitrage for strike prices of $780,$800,$820,and $840? 9.4. Suppose the exchange rate is 0.95 $/6, the euro-denominated continuously compounded interest rate is 4%, the dollar-denominated continuously compounded interest rate is 6%, and the price of a l-year 0.93-strike European call on the euro is $0.057l What is the price of a 0.93-strike European put? .
9.5. The premium of a loo-strike yen-denominated put on the euro is 78.763. The current exchange rate is 95 Y/C. What is the strike of the corresponding eurodenominated yen call, and what is its premium? 9.6. The plice of a 6-month dollar-denominated call option on the euro with a $0.90 strike is $0.0404.The price of an otherwise equivalent put option is $0.0141.The annual continuously compounded dollar intrest rate is 5%. a. What is the 6-month dollar-euro forward price? b. If the euro-denominated annual continuously compounded interest rate is 3.5%, what is the spot exchange rate? 9.7. Suppose the dollar-denominated interest rate is 5%, the yen-denominated interest rate is 1% (both rates are continuously compounded), the spot exchange rate is 0.009 $/Y, and the price of a dollar-denominated European call to buy one yen with 1 year to expiration and a stlike price of $0.009is $0.0006. a. What is the dollar-denominated European yen put price such that there is no arbitrage opportunity? b. Suppose that a dollar-denominated Eropean yen put with a strike of $0.009 has a premium of $0.0004.Demonstrate the arbitrage. .
PROBLEMS 9.1. A stock eurrently sells for $32.00. A 6-month call option with a strike of $35.00 has a premium of $2.27.Assuming a 4% continuously compounded risk-ee rate and a 6% continuoujdividend yield, whatis t e priee of the associatedput option? 9.2. A stock currently sells f6r $32.00.A Gmonth call option wit.h a strike of $30.00 has a premium of $4.29,and a 6-month put wit.h the same sttike has a premium of $2.64. Assume a4% continuously compounded risk-free rate. Whatis the present value of dividends payable over the next 6 months? 9.3. Suppose the S&R index is 800, the continuously compounded risk-free rate is 5%, and the dividend yield is 0%. A l-year 815-strike European call costj $75 and 8l5-stIike European put costs $45. Consider the sategy of buying the a l-year call, and buying the 815-seike put. the selling 815-stdke stock, position held until the expiration of the a. What is the rate of rettlrn on this options?
b. What is the arbitrage implied by your answer to (a)?
c. Now suppose that you are in Tokyo, trading options that are denominated in yen rather than dollars. lf the price of a dollt-denominated at-themoney yen call in the United States is $0.0006,what is the plice of a yen-denominated at-the-money dollar call-an option giving the right to buy one dolllm denominated in yen-in Tokyo? What is the relationship of tlzis answer to youranswerto (a)? Whatis the price of the at-the-money dollar put? 9.8. Suppose
call and put priees are given
by
Strike Call premium Put premium
50
9
55 10
7
6
What no-arbitrage property is violated? What spread position would you use to effect arbitrage? Demonstrate that the spread position is an arbitrage.
;
308
% PARITY
OPTION
AND OTHER
PROBLEMS
RELATIONSHIPS
50 16 7
55 10 14
What no-arbitrage property is violated? What spread position would you use to effect arbitrage? Demonstrate that the spread position is an arbitrage. 9.10. Suppose call and put prices are given by Strike Call premium Put premium
50 18
7
55 14 10.75
100 9
4
21
Demonstrate an arbitrage.
14.45
105
5 24.80
Find the convexity violations. What spread would you use to effeet arbitrage? Demonstrate that the spread position is an arbitra/e. '
.
.
9.15. Thepriceof anondividend-paying stockis $100andthecontinuously compounded sk-freerate is 5%. A l-year European call option with a strike price of $100 x l of $11.924.A 112 year Etlropean call option e0.05x $105.127has a premium ,0.05x1.5 of price strike $107.,788has a premi.um of $11.50. $100 x witha =
9.50
9.11. Suppose call nd put prices are given by 80 22
c. Now suppose that Apple is expected to pay a dividend. Which of the above answers will change? Why?
=
60
Find the convexity violations. What spread would you use to effect arbitrage? Demonstrate that the spread position is an arbitrage.
Strike Call premium Put premium
309
b. supposeyou have a put option that permits you to give up one share of Apple, receiving one share of AOL. In what circumstane might you th is pu t? Would there be a loss from not eay-exercising early-exercise Apple had a zero stock plice? if
9.9. Suppose call and put prices are given by Strike Call premium Put premium
k
'
. . '
9.12. ln each case identify the arbitrage and demonstrate how you would make money by creating a table showing your payoff. stock with the same time to a. Consider two European options on the same the and 95-strike call costs $4. call The $10 costs go-strike expiration. b. Now suppose these options have 2 years to expiration and the continuously compounded interest rate is 10%. The go-stlike call costs $10 and the 95-st1ikecall costs $5.25. Show again that there is gn arbitrage opportunity. Hil'tt--It is important in this cs that the optins are European.) call sells fors l5, a loo-strike call sells c. Suppose that a go-strike Europeap sells call for $6. Show how you could use an los-strike Of and f $10 a asymmetric butterlly to proht from this arbitrage opportunity. 9.13. Suppose the interest rate is 0 and the stock of XYZ haj l positive dividend yield. ls there any circumstance in which you would early-e:rcise an Amerian XYZ call? Is there any circumstance in which you would early-exercise an American ,
XYZ put? Explain.
9.14. ln the following, suppose that neither stoek pays a dividend. permits you to receive one share of a. Suppose you have a call option that Apple by giving up one share of AOL. In what.circumstance early-exercise this call?
might you
9.16. Suppose that to buy either a call or a put option you pay the quoted ask plice, denoted Ca K, T) and Pa (A', F), and to sell an option you receive thebid, CbK, F) and PbK, F). Similady, the ask andbid prices forthe stock are Sa and %.Finatly, suppose you can bon'ow at the rate ru and lend at the rte L. The stock jys no dividnd. Find the bounds between which you cannot proEtably perform a parity arbitrage. 9.17. J.nthis problem we consider whether parity is violated by any of the option prices in Table 9.1. Suppose that you buy at the ask and sell at the bid, and that yotlr continuously compounded lending rate is 2% and your borrowing rate is 4%. lgnore transaction costs on the stock, for which the price is $106.79. Assume that (IBM is expected to pay a $0.14 dividend in early May. For each sfrike and expiration, what is the cost if you a. Buy the call, sell the puk short the stock, and lend the present value of the strike price? b. Sell the call, buy the put, buy the stock, and bon'ow the present the strike ptice?
9.18.
.value
of
onsider the April 95, 100, and 105 call option prices in Table 9.1.
a. Does convexity hold if you buy a butter:y spread, buying at the ask plice and selling at the bid? b. Does convexity hold if you and selling at the bid?
sell a butterlly spread, buying at the ask price
c. Does convexity hold if you are a market-maler either buying or selling a butter:y, paying the bid and receiving the ask? d. Suppose the ask plice for the loo-strike call had been $9.20 instead of $9.00. ls there a eonvexity violation? For whom?
.
PROOFS
310
%
PARITY AN D O'rH ER
Ola-rlorq
REl-A-rlohlsiqlfzs
APPENDIX 9.A: PARITY BOUNDS OPTIONS FOR AMERICAN
P R: C + PV(#)
S
-
would be worth K S. For However, suppose that the put were exercised early. Then it buying the call qnd selling the example. if we thitk of synthetically creating the stock by eket the stock price in the expiration, put, there is a chance that we will pay K before the present value replace plummets and the put is early-exercised. Consequently, if we Of the strike with the undiscounted strike, we have a valid upper bound for the value of argument) that the put. lt will be true tand you can vel'if'y with a no-arbitrage -
-
S
When there are no dividends, we have C + K S as an upper bound on the put, and European parity as a lower bound (sincean American put is always worth at least as the much as a European put). The parity relationship can be written as a restriction on put price or on the call price S C + K S t P 2 C + PV(#) P + S PV(fC) R: C R: P + S K Thus, when there are no dividends, European palit.y can be violated to the extent of interes't on the sike price: Sinc this will be small for options that are not long-lived, European parit.ycan remain a good approximation for American options. Dividends add the complication that the call as well as the put may be exercised early. There exists the possibility of a large parity violation because of the following tlwhipsaw'' scenario: The eall is exercised eady to capture a large dividend payment, the stock price drops, and the put is then exercised early to capture interest on the strike price. 'Ihe possibility that this can happen leads to a wider no-arbitrage band. With dividends, the parity rlationship beeomes (Cox and Rubinstein, 1985, p. 152) -
-
-
-
-
S S k: P k: C + PV(#) C + K + PV(D) S PV(D) P + S PMLK) k: C A: P + -
-
-
-
=
=
options. 'I'he exact pariyy relationship discussed in Chapter 9 only holds for European especially when parity, put-call However, Amelidan options oftn come close to obeying options have shol-t times to expiration. With a nondividend-paying stock, the call will not be exercised early, but the put receipt of the strike might be. The effect of early exercise for the put is to aceelerate the times to maturity, parity will price. Since interest on the strike price is small for short nondividend-paying stocks. American options on come close to holding for short-lived American options. The put can We now 1et P and C refer to prices of Ameriean have and the European valuable than we put, be more
P S C+ K
dividends. The lower bound bound as in the European case with no and it may be optimal to optimal to exercise the call to avoid dividends, be not may early-exercise the put. $100 and S $100, and the K Consider the worst case for the call. Suppose D $100). We will not exercise the stock is bout to pay a liquidating dividend (i.e., exercised after the dividend is paid, will be call, since doing so gives us nothing. The put relationship then states The worthless. So P = $100. once the stock is 100 0 100 100 + l00 C R: P + S D K
-
K
The upper bound for the call is the same as in European parity, except without dividends. The intuition for the upper bound on the call option (the left-hand side) is that we can avoid the loss of dividends by early-exercising the dall; hence, it is the same
=
-
-
-
=
in this case. And indeed, the call will be worthless
PROOFS OF APPENDIX 9.B: ALGEBRAIC RELATIONS STRIKE-PRICE wwmaw-bc.com/mcdonald. Appendix available online at
-
=
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which early and showed that the method provides a tractable way to price options for exercise may be optimal. The binomial model is often referred to as the Rubinstein pricing model.'' We begin with a simple example. Consider a European call option on the stock does not pay dividends and of XYZ, with a $40 strike and 1 year to expiration. XYZ Iisk-free interest rate is 8%. We its current price is $41. The continuously compounded wish to determine the option price. could expire Since the stock's ret'urn over the next year is uncertain, the option price is stock either in-the-money or out-of-the-money, depending upon whether the valuation for the option should take into account more or less than $40. Intuitively, the each in assign values and possibilities case. lf the option expires out-of-the-money, both its value is zero. lf the option expires in-the-money, its value will depend upon hw far charcterize the unceftainty in-the-money it is. To price the opticm, then, we need to about the stock plice at expiration. Figure 10.1 represents the evolution of the stock price: Today the price is $41.and is in 1 year the price can be either $60 or $30. This depiction of possible stock prices the option. price called a binomial tree. For the moment we take the tree as given and ttcoyukoss-
Later we will learn how to construct such a tree.
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yh payoff for Portfolio B, we assurpe that we sell tvheshares at the ynd p rice market rliatwe repay the borrowed amount plus interest ($18.462x c'08 $20). Thus we hye In computipg
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PortfolioA. Buy one call option. The cost of this is the call premium, Whichis what we are trying to determine. lisk-free rate.l Portfolio B. Buy 2/3 of a shlre of XYZ and bon'ow $18.462at the This position costs
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Note that Portfltos A and B have the same payoff: Zero if the stock price goes down, in which case the option is out-of-the-money, and $20 if the stock price goes up. Therefore, both portfolios should have the same cost. Since Portfolio B costs $8.871, then given our assumptions, the price ofone t?/#(??2 mltst be $8.871. Portfolio B is a synthetic call, mimicking the payoff to a call by buying shares and borrowinj. The idea that positions tat have the same payoff should have the same cost is called the Iaw of one price. This example uses the 1aw of one price io detennine the option price. We will see shortly that there is an arbitrage opport-unity if the laF of one Price is violated. The call option in the example is replicated by holding 2/3 shqres, which implies tha t one option has the rijk of 2/15 shares. The value 2/3 is the delta (A) of tlte option: the option payoff. Delta ' is a key concept, and we that replicates the number pf shares '''' '''' ''' ' . . ''' . ... . . . .. . . . will say much more about it later. Finally, we can say sofnething about the expected return on the option. Suppose XYZ has a positiye risk premium (i.e.,the expectd ret'urlz on XYZ is greater than the rist-free rate). jince we creie the synthetic call by borrowing to buy the stock, the call is equivalent to a leveraged position in the stock, and therefore the call will have an expected ret'ut'n greater thap that on the stock. The option elasticity, which we will discuss in Chapter.lz, measures the amount of leverage implicit in the option.
'T'he Binomial Solution l Of course, it is not possible to buy fractonal shares of stock. As an exercise, you can redo this call options (Portfolio A) to examplee multiplying al1 quantities by 3. You would then compare three = (Portfolio B). buying two shares and borrowing $ I 8.462 x 3 $55.387
In the preceding example, how did we lnow that buying 2/3 of a share of stock and replicate a call option?
borrowing $18.462would
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BINOM
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Note that whep there are dividends, the fonuula adjusts the number of shares in the replicating portfolio, A, to offset the dividend income. Given the expressions for A and B, we can derive a simple formula for the value of the option. The cost of ereating the option is the net cash required to buy the shares the cost of the option is LS + B. Using equations (10.1)and (10.2), and bnds. ,
rfhus,
have . we
LS + B the stock if it goes up, In this tree 11 is interpreted as one plus the rate of capital gain on dividends, the total there down. capital if it are loss of (1f goes and # ij one plus the rate dividend.) return the capital gin or loss, plus the the option whe the stock goes up or down, of value represent the and Cd Let Cu the respeetively. The tree for the stock implies a corresponding tree for the value of 0ption:
ts
-
=
e
-rh
c
e
(r-)
,,
fl
The assme 81stoc tt f rice movements opportunities. In particular, we require that 11 >
e
d
-
#
-
,
If
cd
+
jj
e(r-)/l
-
( jtl 3) .
1/ r-
d
and d shoutd not giv rise to arbitrage q .
(r-J)/l
>
( jtj y;
d
.
To se why this condition pust hold, suppose 8 = 0. lf the condition wre viplated, we 1/), erh or we would borrow to buy th tock (if k: would shozt the stock to hld bonds (if *ay we wpuldearn an arbitrage prost. Therfore the assi'ed process d A:.erh). 10.21 asks #ou to vel'if'y equilibrium. Prblei could not be consistet Vthany possible also hold when &> 0. ihat th coditipr lst A is the number of shares in th replicating prtfolio, it can also Not iat bcse in th ' '( ttjck prie, If tlie stock interpreted a the jenjitivity of the option to a chane . . . .' . . ' . erice by A. This inteiretation changes by $1, ihenthe option plice, LS + B, hafi/ P will be quite importt later. 'tiihr
,
E.
erl' The problem is to solve lf the'length of a period is h, the interest factor per period is the option for A and B such that our portfolio of 2 shares and B in lending duplicates payoff. The value of the replicating portfolio at time h, with stock price% is .
.
.
,
Z S/1 V Z B
At the pliees %
=
ds and
SIt
=
portfolio will satisfyz a., a successf'ul replicating
(A x ds x e h) + (2 x US x e h) +
( x
erh)
=
(s
erhj
=
.p
x
cs c ,f
Example 10.1 Here is the solution for A, B, and the option price using the stock 1.4634, # $60/$41 price tree depicted in Figure 10.1. There we have 11 of $40 and strike price had call addition, the a opon $30/$41 0.7317, and 8 0. In and Cd 0. Using /? 1. Thus C,f $60 $40 $20, 1 year to expiration-hence, equations (10.1)and (10.2),we have $20 0 :/3 h $41 x (1.4634 0.7317) =
=
=
=
=
=
-
'
j
-
=
.
-
proportional 2The term eSh arises in the following equations because the owner of the stock receives a reinvested shares. in that is dividend we assume
=
=
=
1.4634 x $0 0.7317 x $20 1.4634 0.7317 -
s
=
,-0.08
-
-$18.462
=
,
% BI NOMIAL
318 '
A Oh1E-PERI o D BI N o M IAL
I
PRICING:
OPTION
.
). Hence, the option price is given by gj
j.
,.)
j
%
.
hs + B
t).
=
$41 $18.462 $8.871
2/3 x
=
-
it is
to
't/h?e./b?-
A
Note that t/-l/care illterested t???fyill #?' optiol #/#cc, lnow only the option price, we alld B; that is just an intermediate step. If we want to can use equation (10.3)directly: :0.08 1.4634 :0.08 0.7317 :-0.08 $0 x 1.4634 $2n x 1.4634 0.7317 + ts + B 0.7317 ?2t)/. llecessaty
k.
319
Tlle option is underpriced Now suppose that the market price of the option is $8.25. We wish to buy the underpriced option. Of course, if we are unhedged and the stock price falls at expiration, we lose our investment. We can hedge by selling a synthetic option. We accomplish this by reversing the position for a synthetic purcased call: We sho rt 2/3 shares and invest $18.462of the proceeds in Treasul'y bills. The cost of this is
-$8.25
2/3 x
Option premium
$41
$18.462
-
$0.621
=
lnvest in T-bills
proceeds short-sale
At expiration weahave
-
-
=
TREE
=
-
= $8.871 interested Throughoutthis chapterwe will continue to report A and B, since we are replicating portfolio. not only in the price but also in the
Purchased call 2/3 short-sold shares Sell T-bill Tbtl payoff
Arbitraging a Mispriced Option
price? Because we have What if the observed option price differs from the theoretical usin the stock, it is pojsible to take advantage of the a way to replicate the option to buy low and sell high. mispricing and fulfill the dream of every trader-nalely, pdce option is anythig thr than the The following examples illustrate that ifthe theoretieal price, arbitrage is possible. Suppose that the marketprice forthe option is $9.00,instead lisk thpt the stock plice at but tis leaves us with the option, of $8.871. We can sell the the''stock. expiration will be $60 and we will be required io deliver ( at the same time we sell the option ynthet zisk buying bf c a we can address this to create the synthetic optin by buying 2/3 actualoption. wehave already seen how sell th emal option and buy the simultaneously shares and borzowing $18:462.If we synthetic, the initial cash flow is
The option
is overprked
2/3 x
$9 00
$41 + $18.462 $0.129 =
Borrowing
costor shares mispriced. We earn $0.129,the amount by which the option is expifation. We have Now we verify that there is no risk at Receive option premum
Stock Price in 1 Year (&) Wzitten call
$30
:
2/3 Purchased shares Repay loan of $18.462 Total payoff
$0 $20 j
=.$
0
$0
By hedging the written option, we eliminate risk.
..-$20
-$2
$60
$40 j $0
Stock Price in 1 Year (&) $30 $60 -$20'
$0
'
$20
-$40
$20 $0
$20 $0
We have earned the amount by which the option was mispriced and hedged the
1
risk associated with buying the option.
A Graphical Interpretation of the Binomial Formula The bipprnial solution for A and B, eqpations (10.1)and (10.2),is obtained by solving two equations in two unknowns. Letting Clt and % be the option and stock valge after one binomial period, and supposing J = 0, the equations for the portfolio describe a line with the formula Clt '
:=i
X
. .
Sh
'
rlt e B '
'
.
.
.
. E
.
; !' .
'
'
Thiq is graphd Msline AE D in Figure 10.2, which shows the optiop payoffas a fupction of the stock price at expiration. We choose A and B to yie td a p ortfolio that pays Cd whe % = ds and C,, when = % uv. Hence, by conitructiop this line rlws through points E and D. We can control its height by the slope of a payoff diagmp by yarying th number of shares, L, varying the number of bonds, B It is apparent that a line that runs through both E and dS). Also, the point A is the value of the D must have slope A = cu Cdjlus when % = 0, which is the time- value of the bond position, cr B. Hence, portfolio /' er B is tlw y-axis iptercept of the line. You can see by looking at Figure 10.2 that ally line replicating a call will have a positive slope (A > 0) and a negative intercept CB < 0). As an exercise, you can verify graphically that a portfolio replicating a put would have negative slope (A < 0) and positive intercept LB > 0). 'and
.
-
-
A ONE-PERIOD
%
320
OPTION
BI NOMIAL
(' '(' 4(* jjjjg:qqy'. t'. t'(kj(g;jjjjgj'..t' tj . t)!r'.. @li;.;.;t'.
j'jkr': jjd jj': p' l'!ti y' (' f' '.' tlf .r' ?' )(' EE E E' iEE qE E ' EE' E ! iEi ! ;;( yt(jjj'. q' )?' L;L,I?'L'''I;)L;;';;'L' jjjjt,,kjjjjf !!1jrj::;:;;r.' jlf ftj' k i:t'5E t'l)r (' .y' y(' y' tl' q)(;(i);'$r' @' i:)'' ii'('E)( ' ( ' r@' jtd rl;f t'1* j'y' E( 7E ' qIF;IE j)('. y' jjd yjd yjyjy'.' yjd jtg.yd '7'lt')'7j!)7* r' )i' (' ('. (! ;p' :i E 'i'((!E' i.E E.iE.(..9( . .. ; (E .E'. q'.('. i'( .(' . .( ' ' j):)!: EE g(' l !( i ( E E ! 'j; j yj g(y ';ry j y Eq' l:yyi ).1(! .yE.. li 'E:'i(...(' E'EE EE.Ei ! Ejjtj(:(i''i;ij ri.' Ei(j E (:;j:42 y)tg j ': j)jji j .j).. j j yy(ty. j y jr j . . yj . j j j . j . . . j . . r,yyl j yygj y. jyys : jjjjjj!jrj y;y . jjjrq:tjjr:y .jr;. . .y yyjy ....................q.E..tE..... (- ,k-,-y.k(y .. . g j .yy jjjjrs l2rq(. j(g.jjyj..jyyyy . , ... j. jy.gy.g . . )ir)).-
rgt-kjj-k-------it-kjk-yyiytitt'yf !' y' E(' ;'y' j'jyf y,' td j'1* ';'Eil)f jy)d ;'t'':'''t' :qll ' ' Ei E
.'.'
'' .
:
.
'g((
:
' iE
E
E' ' '' . :
:
.. .
i
E
:
:'
.
.
.
..
'
: 'i
'
:
: : .j,jjjjjg. .
'
' ' '' : . ..: .
'
'
y
.
. : :
.
.
r
.
.
:
.
.
.
------.
tmdiscounted What happens if we use p'e to compute the expected Doing this, we obtain p*)dS = er-&bh.s = Ftgl-lt /? *11S + (1
'( '
'
.
'. 'i Ej!jjjjggjjjjk.( :. E: :
.....).-..-..--:
,.........:..;...-............
.......
'
'
:
.
.
..
.
-
! :. . : i: ..
: 'y
y
j.
'g : :
g
:. .
r
y.
, .
,
-
.
,
... .
The payoff to an expring cail option s the dark heavy line. The Payoffto the option at the points d. and us ar Cdand (at point D). The portolio consisting of A shares and B bonds has intercept erDB and slope h, and by construction goes through both points E an d D The slope of the Iine is calculated as Kttaun between points f and D, which gives the formula for A.
c,,
=
stock price?
ltS -K
o
----------------=----
1 1 I I I
k
=
.
1
.:
(Q
=
0
16
K
M
x tltrlrjept t
.
' ,
:'=
Run
its d -
.
ltise
=
%
Cd
-
.>,
(stock
lce naer tme petiod)
t
in Chapter 5, equation 5.7.) Thus, wc calt ct//l7./pl/ld about p* is that it is the probability for which the expected tree. In f. ct, one way to think stock price is the forward price. stock price. We will call p* the risk-neutral probability of an increase in the risk-neutrgl pricing will more discuss and we Equation (10.6)will prove very important in Chapter 11.
Constructing a Binomial Tree t<ee.3 Recall that the goal of the tree We nOw explain thr construction f the binomial stock the price in an economically reasonable about is to charactelize future uncertainty way. uncertainty about the future As a starting point, we can ask: What if there were no period must equal the forward plice stock ncertainty, the next stock price? Without is the price for forward the formula pdce. Reeall from Chapter 5 that F;
going up or down. J.n fact, So far we have not specifed the probabilities of the stock calculations. Since the strategy ptice probabitities were not sed anywhere ilt the option whichever way the stock moves, of holding A shares and B bonds replicates the option irrelevant for pricing the stok is the probability of an up or down mbvement in the
option.
Although probabilities are not needed forpricing the option, there is a probabilistic er-sjh (10.3). Notice tat irl equation (10.3)the terms in terjretation of(Ifequation from follotvs positive and both ctr-xl/f #)/(If are d) sum to 1 tthis #) and prbabilities. Let inequality10.4). Thus, we can intemret tlese terms as -
-
-
er-&tlt
#
,j
=
d
-
. .
j/
..
(10.5)
d
Equation (10.3)cat then be written as
c=
(10 7)
the expeeted stock price equals the
Risk-Neutral Pricing
-
$
when w use p* as the probability of an up move, for the fomard priee of the stock fotwardprice, er-ths. wederived this expressionthefonval-dprice ll-l,g the lpfl/t/?nft'I/
slope
'
TREE
-
Opdoxl
rayoa
.
BINOMfAL
l
PRICING;
-r
e
(,
'.c
,f
+
(1
-
pslca)
(10.6)
Itis peculiar, though, This expressionhas the appearance of adiscountd expectedvalpe. risk of the option is at the though risk-free rate, discounting even at the because we are position in the stock leveraged option is call stock the of a risk (a least as great as the probability that since B < 0). In addition, there is no reason to think that p* is the true ' the stock will go up; in general it is not.
,+?,
=
s'fetr-ll/' P
( l 0.8)
this, under certainty, Thus, without uneertainty we musthave St-v = Ft.twl,. To interpret plice must rise stock the risk-free Thus, the rate. the rat of return on the stock must be at the risk-free rate less the dividepf-tyttld,-r-c-wt.what we man by Now we incomorate unciiainty but we Erst need to desne is the annltalized about return stock the unertainty of uncertainty. A natural measpre ?zp/?w, which stock comptnded '/tz?/#t'I?'# dviation we wil'l denote ofthe ct???/'#?4Itx/,Wy the that return will be stock how deviation sure we are measures by t7'. T. e standar of a rettln chance will greater have with larger a o' a close to te expected return. Stocks far from the expected retutn. and down We incorporate uncertainty ipto the binomial tree by modeling the up difference from the with forward the the Tlice? plice to relative moves of the stock 11.3 that Section will in We see related to i'Vvi-kidfd forwardprlce being length h is of deviation period standard is over a the if the annual standard deviation c, the o'.I. In other words, the standard deviation of the stock return is proportional to aqltare root of time. We now mod.el the stock price evolution as
zeviation.
,+,,,+0.47f
ltz z-,tsie-c.'ii dz F, =
=
n-depth discussion. 3Thl-sdiscussion is intended as a quick overview. Section 11.3 contains a more
.
322
%.BI N o M IAL
OpTl
o h1 PRIcI
Two o R M o RE B l N O M lAL P ER IO DS
I
Nc:
yjtyf I!IEEEEE,X .')(y';' llliq:ill,:.. t . rr. . .. lrfp .
.'....' '.t . --;t!i#'. ..--' :' -'-' -.t'
Using equation (10.8),we can rewrite this as 11 #= =
-
c(r-d)/l+(rx/#
-.
(10.10)
e(r-J)/J-o.x/'
.
binmial tree in Wq began this section by assuming that the stock plice followed the selected and to make the of prices stock $60 were $30 Figure 10.1. The up and down is the same everyting where example example easy to follow. Now we present an and down the moves. up except that we use equation (10.10)to construct 1, so that Suppose volatility is 30%. Since the period is 1 year, we hake equation ( Using and 0. 1' 0.08, (10.10), o'x$41, 0.30. We also have Sj =
=
=
we get = =
..
:. :.
E
: :
-
. :
1+().3x4
$zjjelgs-ox 1-0.3x4T $41, (oms-ojx
,$.j9
= =
954.
.
$ap,jlts
(10.11)
.
10.1, the option price will be different
Because thebinomial tree is'different than iFigure
= $59.954/$41 = Using the stoek prices give in equation (10.11),we have lt = $40, we have Ctt = $59.954 1.4623 and d = $32.903/$41 = 0.8025. With K $40 = $19.954,and Cd = 0. Using eqtlations (10.1)and (10.2),we obtain
.
. . .- .. .. .
i
.
.. :
E
.
:
-'
. .
''
.
::' :.' ...
. .
E
:
-
. ..
. ,L
:
' .- . -.
:.: . . ...
.
-
,
. .
, ... . .
-
.
.
: ! .
:.
:
-
'
-
.
-
-
!
.
..
-.
.
:.
..
'-71111.
'
.
-. . . . .
.
.- . .
.
-
.
.
.
-
Binomial tree fr pricing a European call option; assumes .S' $41 #= $40.00, o' 0.30, r 0.08, F= 1 years, J 0.00, and h 1 000. At each . node the stock price, opton price, A, and B are given. Option prices in bold ltwlfc signifythat exercise is optimal at that node.
659.954
.00,
=
sp-ps4
.00
$41.00 $7.839
=
=
=
=
Example
k(S dz
.
.
=
Ross-lkubinstein tree.
=
.
:
'''
.
=
' ''
i
.
volatility if This is the formula we will use to construct binomial trees. Note that we set volatility, = = with zero equal to zero (i.e.,c = 0), we will have 11St dSt Ft,t-gt Thus, volatility does not bill. Zero with Treasury rise time, still a will just as over the price that prices are klovn fn advance. mean that prices areflzt- it means torward tree.'' In We will refer to a tree constructed using equation (10.10)as a Coxthe Section 11.3 we will discuss alternative ways to costruct a tree, including
Another Oneeriod
%. 323
')'liE1q@' il'tl (' ?' ;f 'fEjjjjy(jy'yjjjy'. j'qf ;'ftljt'tjyjjf tff yt' E' .111:::11,.*----.* :.'(' jtjf )j' )' jlf pjpi.f ql'l.li '#' q' lsrllil-'klylllki:iljij. !)' Tpf rkf qE!' s' (('.)i E i E(()))15 (1;1r714:':r55551((;*6): #' ' j . jj!tr r' )t'. q' (' k' t'4111k,::111'*. j'yjf )' yjkqrf ;'tyf (:E(('Eiq'.!';;' E k ..' jjtj'(yjjjj.(. ''. rg'..jf j.jjj)' jjg'yj)f ,'q' q5''@)@ ''E.i'i'E'. .TE!EE.E ' E(-( (... Ei E.(E'E( . E..:( (E'E( '.; !((( .(T!' liqqi t':f''E')1* ;:k#))-'' i'r p)'ljl.'. j'ijfkjr'. ;'1111.*. y' p!' (jj)'i:. t@ ..;. E:'!.. g((j. ,;'!EiE!!1.r-'---------;:.'' )' )yyy(t;yyjy'. y'. yijtf tj'tjy' .r. r.. ... . t(. ..E.(q..y.EE(.E( . k(Ekk.'..iir . . . '''E . ttt. . ;:. (i ..E . ;i (E; TEjtE fjjjy (!jtrt. !: jyEq t.E! .tjE . (.. ( E.. t.q . . y-; j . r ,;..j j; )( (ytj . ' jyj.:;jyj . . ' i .@1I1EqFi).y., . i -E !1111:::::;1--. E . -E. l.hi . i ;:.)(.. .E;.. :i.; .. ; .ipi ....t(-: . -r.!! ..: 8k)i ' .'-'
F
s
0.738
x-yzz..,jts
$a,'z.9()a sO.0O0
* In order to price an option, we need to lnow the stock price, the strike price, the siandard deviationof retunas on the stock (in order to compute lt and #), the
dividendyield,
and the
lisk-fTee
rate.
* Using the risk-free rate, dividend yield, and (r, we can approximate the futtzre distrlbutionf ihe stock by creating a binomial tree usig equation (10.10). it is possible to plice the option using equation . Once we have the binomial (10.3). 'I'he solution alsp provides the recipe for synthetically creating the option: Buy A shates of stock ( quation 10.1) and borrow B (qptttion10.2). 'ee,
* The formula for ihe option price, equation (l.3), can be written so that it has the appearance of a discounted expected value.
-
$19.954 0 -
. 2.
=
$41 x
=
(1.4623 0.8025)
0.7376
-
1.4623 x $0 0.8025 x $19.954 1.4623 0.8025
There are still many issues we have to deal with. The simple binomial eee seems simple to provide an accurate option price. Unanswered questions include how to too handle more than one binpmigl period, how to price ptg options, how to price Amelican hand, wecan now t'ul'nto those questions. options, etc. Withthebsicbinmialfonnulain
-
B
=
e -0.os
-$22.405
=
-
Hence, the option prke is given by L.S + B
=
0.7376 x $41
OR MORE
BINOMIXL
ERIODS
We now see how to extend the binornial tree to more than one period. -
$22.405 $7.839 =
This example is summarized in Figure 10.3.
Summary .
10.2 TwO
We have covered a great deal of ground in this section, so we pause for a moment to review the main points:
A Twoeriod
Ettropean Call
We begin flrst by adding a single period to the tree in Figure 10.3*,the result is displayed in 'Figtlre 10.4. We can use that tree to price a z-yearoption with a $40 strike when the current stock price is $41, assuming a1l inputs are the same as before. Since we are increasing the time to matulity for a call option on a nondividendpaying stock, then based on the discussion in Section 9.3 we expect the option premium
:
Q BINOMIAL
324
Two o R
l
PRICING)
OPTION
F' :' (iF'iqlf qElrf lrttf q' (' g,' 6* jjf j'g'. ' E i'' E( ' i'E tf 'q' ')r1*,'7!7( yyjf j'y' 14* d!rli;kf k'. )' J' t'1* ti' q' F' ' ltr *47:7(':)* (77q* (' 1* g' jyt.yjjf E' ' ' EEE;E' E FlE; E ('.;'('E.E.;.E'yE.(.( ' gjy'. r' .y.jyjjrjjjyjy,'..' !!g(;.,' t')')' y' 1* )' j)td j'?'11*. llllt---illrf :;E!y).g '' . . E('''' r'! E E y' jsyiyr'' 11:E1;;* (' .-' '1iI:1:1i,k' ((Ej tjf r' j'j)'. t)jf L'-'jik'nk,Ljl-(1111* '(. . ';;.lli . 'pq (JE yjj(jj'. y' ')t'7* '(('. E'. ((.( . .(f;'.' E..(.jtE. .jj.E.i:g;.'. ...(@ ' ijii!ii!tf . g y' r;' ( E. (y y ;k''j E'q. (. E ; i g. EE. i( y y yy jy.jy. yj Ey.y'. E EjE ( .. y.jj .jjjq)(gy.y ;fi : EE l 1 . ( . i. jEE!'E @ ( . k':1t1l-. g j ' . (' yyj, j ( '' y (jy (Ejj .IIII:j:;I;IE. j jjyj. : EE E(.1 . . ..jjgEjy.y.jjjjj.gj;g.jj.j.j..E.j.Ey j.jj llryE. E. .q( ....E' !'. E-(. . l'. 11II::pliIi( E; p ' . .;. ((ri..; .. ) . ; g. r !!.. ... . . . ('l. .. . ( . t ... . ... .. .. .. ..... ... . . . .... . .... ..... r y .. . . . .
,'' :'-2,kj4;2* :;:' -k-f'-'.j' ;':'j,rf r' -' -,r' -,y' -).y)' :7* 7* (.':E C' ;':' :'i'7)* ''('
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Binomial tree for prcing
a European call option; = $41 assumes 5% # = $40.00, tr = 0.30, .00,
-
.
-
-.
-
.
. -.
,,.
-
. . .
(r:)I
--:!ij;:-
j,r
=
,$23.029
B
0.08, T = 2.00 r 0.00, and years, J h = 1.OOO.At each node the stock price, and B option price. are given. Option prices in bold ztwllc signify that exercise is optimalat that node.
/.
1.000
s
-$36.925
=
=
sud
$41.:0: (s.) 510.737
=
B
.%
A
=
=
usu,
11If
=
$48.114 sttd Sdtt) $8.114
=
4/2
A
0.374
=
=
To see how to eonstruct the tree suppose that we move up in ,
$59.154. lf we reach this price, t en we can move further up or down
(10.9).We get
s
sttd
Year 2, Stock 'rfc ma.xlo, S K) =
=
=
$41
=
$8t.669 x
$41
$48.114
=
$26.405
=
*
''''
.
=
=
..
are at expiration, the option value is
$48.114 Again we are at expiration, so the option value is
$26.405Since the option is out $59.954At this node we use
of the money, the value is 0.
equytipn (10.3) to compute Year 1, Stock Price the' option value. (Note that once we are ar this node, the stock price, IIS, is stock price, dS, is $48-114.) $87.669, and the
$59 954,0.08-0.3 $48 114
'tup''
tldown''
.
.
=
x
$87.669 Since we
$47.669.
Year 2, Stock 'zce $8.114. Year 2, Stock Prce
$59.954c0.08+0.3$87.669
The subscript tttt means that the stock has gone up twice in a row and the subscript @.pdthen down. means that the stock has gone up once Similarly if the price in one year i s Sa = $32 93 we have
Sdu
e(0.08-0.3)
''''
=
=.
=
How do we price the option when we have two binomial the call option . periods? The key insight is tht we work back-wardthrough the binorial iee. In order up and down to use equation (10.3),we need to know the option prices resulting il moves in the subsequent period. At the outset, the only period where we lnow the option plice is at expiration. Inowing the price at expiration, we can determine the price in period 1. Having determlned that price, we can work back to period 0. Figure 10.4 exhibits the option plice at each node as well a's the details of the replicatinj portfoli at each nde. Remember, however, when we use equatin (10.3), it is not necessary to cofpute A and B in order to derive the option price.s Here are details of the solution:
-
=
c2x(0.08-0.3)
.
.
-
1111
=
x
$41
.
tis pfice of $10.737, compared to increase. J.nthis exanple the two-period tree will give a 10.j. t $7.839iri Figure
year 1, to Su according to equatiri
$41
=
e(0.08+0.3)
Prking
-$9.111
B
tree
x
x
=
=
$26.405 fku) 50.000
=
$41
d x $41
x
If
x
c2X(0.08+0.3)
-$19.337
yaz-qt)a sdj $3.187
construuins th.
325
.
II2
=
5'w/
=
0.734
$
We lso could have used equation (10.10)directly to compute the year-z stock c0.08+0.3 = 1.462 and d = /0.08-0.3 = () 803. we have Prices. kecallthat 11
$59.954 (.6,1) =
BI N O M IAL PERI O DS
followed by an up move, we would have a nonrecombining tree/ A recombining tree has fewer nodes, which means less computation is required to compute an option price. We will see examples of nonrcombining trees in Sections 11.5 and 24.4.
:
..
,
.
. . ......
:'
.
. g
.
:
'
.
,.
:
:
: .
MORE
l(d e
-0.08
$47 669 x .
e0.08
j () 803 + $8.114x 1.462 1.462 0.803 .462
-
-
'
-
-
e0.08
0.803
=
$23.029
,
$32.903c0.08+0.3 $y8 jj4. =
.
and Sdd = $32 903c0.08-0.3 .
=
$26 405 .
generates the same stock Note that an up move followed by a down move (u%,#) recombining price as a down move followed by an up move sdttj.This is called a than different price a down move tree. lf an up move followed by a down move led to a
4In cases where the tree recombines, the representation of stock prce movements is also (and,some argue, more properly) called a Iattice. The term tree would then be reserved for nonreombining stock movemenB.
5As an exercise, you can verify the A and B at each node.
326
% Bl hlo
M IAL
O p'rl olq
Year1, Stock Prce value:
PRI
Two o R M O RE BI N O M IAL P ER I O DS
cl Nc: I
$32.903Again we use
=
(10.3)to compute the option
equation
t't1*. k' ;'tyyyyjjj;f ' ttyd tjd r)','. ttyyyyf t'. rr'. ii'.:iill:f tty)', t!r,r1@iiiq'. .
.'' .'..' . --r'.. -'.. -..t' -'(E'IX tqf
'
::
,0.98
-
=
.
-
-
=
-
-
=
-
-
'
'
'
'
:.
jr
;'
E' ' . ..
'
-.
- E
.. .
'
'
'.
yyj yl: E' E
. -
-
:
:'
-
'
:
E :''
'
E
:
.
E'
.
: '
:
-;
.
yyy j. ty . (g y jyy .. yg tiiid!i!t . . E:
.
- .. -.
- -,,. , ---2..
.
'-tlilt.r
..
. -
EE-
-
-
-
.
-);-.
.
.
. - -. . .. .
-.-
-
.
-
-.
. --
.. . -
. .. .k Binomial tree for priclng a Europepn call option; assumes .S' $41 # $40.00, tr 0.30, r= 0.08, T= 1 years, J 0.00, and h 0.333. At each node the stock price, option prie, A, and B are given. Option prices in bold ltolk signifythat exercise is Op timal at that node. . ... .
. .
.
.-
. ...
- .. .
. .. .
.
---..-.
.--.
.
-
.
.
..
::;;;'-. LL;;;''.
$34.678 $61.149 $22.202
.00,
::2:
=
-
A
=
B
.00
=
* The option price is greater for th
than for z-yea1'
e l-year option, as we would
t
expect.
* We priced te
option by worling
backward through the tree, starting at the end and
working back to the first pedod.
A
B
'
=
=
=
'
y
=
$52.814 $12.814
0.922
-$33.264
A
B
=
$43.246 :5.700
0.706
A
-$21.885
B
=
=
0.829
-630.139
=
$35.411 $2.535
A
B
=
537.351 $0.000
0.450
-$13.405
=
$30.585 $0.000
A
-.
The generalizaton to many binomial periods is saightforwardk We can represent only spreadsheet or computr programcan a small number of binomial periods here, but a handle a vel'y large nurtlber of binomial nodes. A.n obvious objection to the binornial calculations thus far is that the stock can option only have two or three different values at expiration. lt seems unlikely tat the time problem is to to divide the price calculation will be accurate. The solution to this expiration into more periods, generating a more realistic tree. To illustrate how to do this. at tle same time illustrating a tree with more than two periods, we will re-exnmine the l-year European call option in Figure 10.3, which has a $40 sfn'ke and initial stock price of $41. Let there be three binomial periods. Since it is will assume that other of a period ij /7 1. c We a l-year call, this means that the length 0.3. inputs stay the same, so ?' 0.08 and tz Figure 10.5 depicts the stock price and option price ee for this option. The option price is $7.074, as opposed to $7.839in Figure 10.3. The difference occurs because
1.000
=
s4)..(j(m $7.074
nodes. ln particular, at a given point * The option's A and B are different at different into f'urther j the money. in time, A increases to 1 we go eady exercise was pot permittd. However, * We priced a European option, so made no ditf erence. At evet'y np de p rior would have permitting eprly exercise S K; hence we woutd not have to expiration, (he option price is peater t.11a.11 exercised even if the option hal been American. option it is straightforward to value an option @ Once we understand the two-peliod The importantplinciple is to workbackward periods. binomial two using more t.111111 through the tree.
Many Binomial Periods
= -538.94
=
$50.071 $12.889
=
Notice the following:
327
,--8211-.* .-.' .'
-1:,!1;* -' --' F' jy'yyyf L.,jf )' j'jy'y 7* 1* 7)* il'. jfd j;d (.(.j)' (' yyf ''!' q' )(' jqf (ftf :pii!!;;f :,ii;;' j'#)' ';'fiif (' ;'t'@' t'())* k(; ttf E E' E )' )j(' q' ,.,:'' jj'. jjyf g' ! ' '' !E'' E' E 7* ;'.j'k' I.II::jII.i'. qqlikqljt'. 'IIIk::iIIi'-' !''I:i::;1ki'j'(j'. ;;' !( ( 'E'TE i.(''iiCij'(''' .'E'EE'i'('.E''E'('!.(E''.('; CJSTI7rI'C/IITIIE'')'';E;7JTI'IT7T'F)'')' E r,f -'1I!I:4;-. y'. i'tj j.4' y' tyf (p''!( (i.'g.'.(i;''' ' .i'' '(E''E.(E'(E' '(' i'E(E.i q ;(..'E ': tiqlpgpllpiilqlqg .y.' ' . ' '. #i''' j5 (( E.. ;'((;;...( gjq.kk, 2E...j;.j..;., .jj.( k. r' qi ' E1 (..( ' i ! ! (Ei j;lj yqqj.yg (y;. ()): .jgyyjEtt.,. jyy .-t y . jyy)tt(.' y --)i.. . . yy . y; ( )-.j -11111.... i. yj. -.t, . ,t((--.. - ...y-.yyyy - j-..j. r-r --. ;- q - .:.) ;).;@ ! (:..-( (.. - . ... ttk.-. . .. . .. . . .
1.462 e0.08 0.803 + $3.187 $0 x 1.462 e $8 114 x 0.803 1.462 0.803 $41 Again using equation (10.3): Year0, Stock f'rcc ,0.08 1.462 c0.08 0.803 $10.737 + $3.187 x -0.08 $23.029x 0.803 1.462 1.462 0.803 -0'0S
$
B
=
=
0.000
$0.000
.
$26.416 $0.000
the nurperical approximation is different; it is quite common to see large changes in a binonzialprice when the number of periods, ?l, is changed, particularly when n is small. Sice the lgth of the binomial period is shorter, tt and d are smaller than before 1). Just t be clear about (1.2212 and 0.8637as ppposed to 1.462 and 0.803 with h second-peliod the nodes computed: how procedure, here is the are =
s tt ' Sd
.j
=
yyjstsx
jyaoao.p
2277
()0sx 1/3-0.3../-/3'
$41,
-
=
y5(jwj .
$35.411
The remaining nodes are computed similarly. Th: optiop ptice is computed by working backward. The risk-neutral probability of the stock price going up in a period is 1/3 e().08x
1.2212
-
-
() 863.7 0.8637
=
0.4568
AMERICAN
%.Bllq o M lAL O p-rl o lq
328
PRI
cl lq c: I
The option price at the node where S
e-0
'
08xl/3
=
(($12.814
x 0.4568) +
-
$5
=
%:
329
node where the stock To illuseate the calculations, consider the option price at the computed is node as ptice is $35.411.The option price at that e0.08x1/3 j (pjp, 8637 0.08x1/3 d () = $5 046 08x 1/3 363 x $8 0 + $1.4()1 x e1.2212 0.8637 0.8637
$43.246,for example, is then given by
($0x (1 0.4568)j)
Op-rlolqs
,74)o
-
.
-
'
'
.
'
.
1.2212
Option prices at the remaining nodes are priced similarly.
10.3 PUT OPTIONS Thus far we have priced only call options. The binomial method easily accommodates option prices using put options also, as well as other derivatives. We compute put option call plices; the only the same stock price tree and in lmost the same way as of computing the Instead expiration: difference with a European put option occurs at price as mtxto, S K4, we use maxto, K S4. Figure 10.6 shows the binornial tree for a European put option with 1 year to expiration and a.strike of $40 when the stock price is $41. This is the same stock pzice tree as in Figure 10.5.
-
-
previously had to consider. Figure 10.6 does raise one issue that we have not the price is stock $30.585, option pdce is $8.363. lf Notice that at the node where the make would sense to exercise at that node. The option this option were American, it = expiration, but it would be worth $40 $30.585 when held until is worth option should be more $9.415 if exercised at that node. Thus, in this case the American will option. We now see how to use valuable than the otherwise equivient European options. the binomial approach to value American -
*$8.363
-
-
OPTIONS
10.4 AMERICAN
optimal, the binomial Since it is easy to check at each node wheter early exercise is of the option if it is left metod is wemsuited to valuin Ameriean options. The value it for another period, equation (i.e.,unexercised) is given by the value of holdingby maxto, S K4 if it is a call (10.3). The value of the opion if it is exercised is given and max(0, K S4 i.fit is a put. Thus, for an American put, the value of the option at a node is given by I'alive''
l'.E ;'(y' g' (' ,'(' 4* @:q' j'q(')' )' ''''' 1/4(1* 1* ')'. )-' j')'(p'-ptk-f 4* y' r' ((' (11611* )' ).)-)'. y' ''' qrqd t:d E E ti ! ' ! ' !q! ' .(i()(. '
.'-
.-.' .''. --.' '(;lr'q7!qrq'' -.!15!1.* -' -':'-'-' II;'F!!'?X -' -'-'.' ,',' ;':';'
pi-' -'i1:I:-:;i1r'' .'!.iIr::1;:;;'. kL'. q' .j;j' liliti!!:f lllli:ril!!f j'j'; .kjjjgg;jjjj''jlf 1* y' jE'' (' ('''' !
' '
:
'i i j'(;
('q q'(' j((' ';'q'('Tq'7' (
'
'
EE' ' ' ' ' ' ' : E' ' ' qE( Jj.E E EEE ' ' ' ' ' ''EE'''E'' E : ' E: : : E (i E(. E. E1) ! ;.:. :(. E:EE. . (.Ei (E. E!!!E. ' .( (.EE.(: (g E. :(E. . ... . .. . !.;: t);;)i(J))'i'qiEEl.)E(i . : -. . - y t.E))!j1(:')()./1E' (. ... E.. yEq (: q. yy...;. y y: y.. yyy qy..E- yq yy j- y-y 2 Ejllryjt.:. . . y y -. y- y yy. y .. .. y g . .y y . y y . ;1.()E)E :-' y -(@ . -r 'lE'..;.q(E . - -i-y.E . ; -. -..-- -..--. -. - . . -.. . . . . : . . . .. . . . . . :'
(
(rl '(:.. . !.' j..( . . .. .. . . . y ; y .j jyy j ( y y.jyjy jy. yy(y-yjj)jjq)j. jtyyy.y.jyjj.yj.yjjy..y..j..j)yjjyyyj.jyj.yyyjjs . l:. .!11!. y.. y .. ..ytj y jyj.i . !; yy . 7 :;i .t-. )... .k. . .i.; t,F-.; tr. .'. . . .. '
'
.
.
-.
''
'
'
:
'
'
.
((
'
'.
. ,;gjjjj..
..
-
:
.
.
.
.
.
. -
-
:
:
-
.
-
.
-:!ijq;:
.'
.
.........
.
....;.
. .. . .
...........
..
......
...
. ................
.
-
... . . -.
.. ..
....
Binomial tree for pricing a, European put option; asjumes .S' 541 # = $40.00 , c = 0.30 ! r= 0.08, T= 1.O0 0.0b, and Years, = At each 0.333. h node the stock price, option price, A, and B Option are given.boid llwllc in Prices signifythat xercise is optimal at thai node.
::;;;;-. ::;;;-'' -.,:E11;. .-q:.,-.
y(j.(;(s $61.149 50.000
.00,
=
A
Ps,
K, t4
=
maxLK
0'000
= .$0.(jx
.n
-
-
u,
e-rh
r, t + ltlp. g#(,/5',
where, as in equation (10.5), $so.oy1 $0.741
=
$41
.
A
B
=
$52.814
-0.078
A
=
B
=
,$0.000
.$4.659
s4a.z46 $1.401
cc()
$2.999
,
*
-0.294
=
$15.039
A
= =
$8.809 .$37.351
$35.411 $5.046
$2.649
-0..550
B.
= =
$24.517
$30.585 $8.363
A
-1.000
cs
B;=
(r --J)/, tt
---
--
K, t + /,)(1
-
p*)j)
(10.12)
tj
tf
of the put option valued Figtlre 10.7 presents the binomial tree for the American version where the stock price node the at in Figum 10.6. The only difference in the eees occurs value. Amelican optio at that point is worth $9.415,its early-exercise is $30.585. unexercised is the option if We have just seen in the previous section that the value of 'rhe
-0.171
B
#=
e
+ pds,
t38947 526.416 $13..584
$8.363. through the eek When The greater value of the option at that node ripples back is $35.411,the value is price stock the where ode omputed at the the option price is the reason is that the price is greater at the greater in Figure 10.7 than in Figure 10.6., subsequent node Sdd due to early exercise. the European The initial option pdce is $3.293,greater than the value of $2.999for node. Sdd exerise early the at entirely is to value due option. This increase in ekample. At each In general the valuation of American options proceeds as in tlzis when exercised, we node we check for early exercisek H the value of the option is greater of option unexercised. assign that value to the node. Otherwise, we assign the value the We work backward through the tree as usual.
'
%.Bl N o M lAL
330
O PTl o N
PRI Cl NG:
op-rloxs olq OTHER
l
' E: E ' ' E E' 5*'' E' ' E :' ' E E' ' )' (' ;'t' j':'' :'q' @' ,'y' ''i' 7)/* jif ' : ''IE'''q7. qrqq' EE: : 'I(E'E'Eq( EE k' q'. ,'t)f ,-.t'.j' yy'' )' ;'p' (' F:77'7q4. l'qljf @' :' E E E'(.( ;':! i(qq' (' :,ji!,k' jyy' y'.'j?' t';f )' kj'')i p' ll'rjElqq':pljll ' CE ll.: 'p (Ei i(ii ( 'E. . .(:(E.'E'E.('i.' (j.'-i(EEi.E.E!;i!i.E(EE!' FE'' . '( . . .i ..(i .. :. .:('': i:1: . . . - .-.- .- .E . --. ..- -. .... -. .. . E.-.. i. -..E.E. !':. . .. -. E :.rEE. i ;E . q q . (..,j.. j k. . . . - r! ty r . , q . ! ;E . y y ( y .....y q-y j . . -. ( y-. yy y . E . -. )(' . E!E ; El'i1::::;;,.. -. . .. ;. ...... .- -. -. ; - -. --..;;.. r. jyjt--.E.E! . .)(E r. . .. !..E . . .. . . ..... ! ... .
.'.-!.(11-.* ,'')' 77* ''--;k' ;;,r' F' ;''Ekjjjrjrjjjj'i -' :''t'E'
djrE!;kf ,;' )' )('. ))yy'. .,).!y' j'. )' jyj'gj t'rr!p!l;'f iff r' )'. tyjf y' yy't tj'rj. ';'(' C-'. '-' @-' 11111* r:;;;;,.f t''1I1k:--jI1p' li'tf )'
.'' ..' tpf t.'lf t't.'1-!
'
'
.
'
.
'
'
'
'
:
'.' .
.
.
.
'.
'.
'
-
-
.
..
:
:
. .
..
.
.
. .
-
..
.. .
.
.. .
.
. . .
. .
.
.
')y .
.
.y
-
.-
. . . . . . . ..... .
. . . . . .. .. . . . .
.
-
-
.
...
.
.
-
'
.
-
-
-
.
.
.
-.
.
-
.
,
.
-
.
..
.
-:,114;-
.
.
-
,-.
-
$61.149 $0.000
=
0.000
=
.00,
=
.n $().cc() =
.$40.00,
=
=
::;;;--.
s().(o()
.
cr0.05xl/3 :.45.7x $87.74-7+ (1
.00
5 sc 0,/1
=
A
=
B
=
331
0.457) x
-
=
$32.779j $56.942 =
56.942, we exercise the option at that node.
>
352.814
-0.078
.
.
Since 57.101
.
50.741
%.
(10.2), and the option price by equation (10.3).The risk-neutral probability is given by equation (10.5).6 Figure 10.8 displays a binomial tree for an American call option on a stock index. Note tlptbecause of dividends, early exercise is optimal at the node where the stockprice 0.457.,hence. when S $157.101, is $157.101.Given these parameters, we have p' unexercised is of the option the value
:
'. ( (.yjy-j .7.. 1111F44!1. r .q r llllkipllti. .. jjLiii,btq... .. yj. j' j,;j.r.j(y.ry.jjjy.j'. kk#it1I). iiiikdijk. (...; ..l(. .. ;.(ti r-. . . . . . . .. . .. . .. Binomial tree for pricing an American put option; assumes 41 .K 0.30, o. 1 T= 0.08, r years, J = 0.00, and h = 0 333 At each node the stock price, option price, A, and B are given. Option prices in bold ltwlic signify that exercise is - () Ptimal at that node. .
AssE'rs
0.0O0
$4.659
;'r' 1111:* :!iiF;k' (j'. pll'y)j ttf k' yyf .;' j)'yy!. y)y'-)'jijjkj,-jjjr' y'' lllli:pttf r. r'. tlf t'1* kgjjjjjjjji-jf r' '. (. jy)y)yj@ . (.iill:;llit. (..... r. t-... ( yjjjj ky Iliiij!!:. 7 #1 t.(j..(... ...g;r......-yy!r.k( . .; . 1(. ... .... .
tf '!
:':.' ;;' :,' ;.' -,yr' -j:'.':' -.' j';';:.' -' ;';' ''#' C' Cif ' )' f' p.yf 71* :''(SY 1' E )::: :.(E ''EpE : : ;
.'.
'r' t.@' r.yyf ,')' lrr' ;'(7* r)f /'i't't'-)'i! (74E itf j'j))).#j;yt-(y'kjtr();)-. ')y!' kp-!,' ;'@' (' p' )!7*:1* (7117r1q324)*1(!(t* :1i5r;' y' )' ' l('' i!'((E'E (:i1). i(((E' t'. (E(:'.li'.i ;'';qE'E'(;'E'.q.EEEE(EiFE(EE:.q''qE'.'.iE'E'' .(bb-'E''( lE'!'7':C'S'F'T7'III''!C'I?'CCII')'!TC ' qj .' .l' . E IE E. - - - E ; E --. : --- i ; jjjjjr:jjjk; - ---- -i 5. . - .y E- r. ' '- . t::. ;r-!t'1::::;;;kE. rrr.. :.)----)---:,-,y -; g.(.r -g r; j@-. . E :r.. qE, .:k;jjjjg. ( (,:;; ;!. . . y i ..... i yy j r))i . ..-. j. : j)).-. j. .-i-..;j-., Ei:. -; t..,. i!. q .. E ... ..!. . .. . . .. .
-' ..' .-.' .--:j11--* ......-)' ''
'
$4.a.:,46
5 41 (0tj .
s A
-0.332
=
B
$16-891
=
$8.809 ,37.351 $2.649
.
$5.603 -0.633
A
.2
=
1
:
'
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''
:' ..
'
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. ..
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i
'qqq'lq ' '
' ''
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:
-
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-
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:
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.
-y
.
-
:
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.
:. .
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.
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.
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-
-
.
.
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.
.
Binomial tree for pricing an American call option on a stock index; assumes . $1 10.00 # $100.00, c 0.30, r 0.05, 'F 1 years, t 0.035, and h 0.333. At each node the stock price, option price, A, and B are given. Option pricej in bold ltwllc sighifythat xerise is optimal at that node. =
=
'
.
.
'
-.
-
.
'
:
-0.171
.$35.411
:
.
-
=
=
B
'
1.401
$3.293
'
'
-
$157.101 $.57.101
A
,
=
::;;;),' ::;;;;-. q!l).. ::;;;.-.
$87.747
B
=
=
0.988
-698.347
=
.00
=
530.585 $9.415 A
B
=
ON OTHER
$1a1.4s8 $33.520
=
-1.000
A
=
=
$38.947 $26.416 $13..584
10.5 OPTIONS
=
ASSETS
tpderlying 'l'h model developed thus far an be modifed easily to price pptions op examples of this seyion we present assetp other than nondiyidend-paying stpcks. In curppcies, and futures cpptracts. how to do so. We yamine optiops on stock indexes, In every case the general procedure is the same: We compute the option plice using equaion (10.6). Th diffrelce for different underlying assets will be the construction of the bipomial tree apd tlw risk-neutral probability. The valuation of arl option on a stock thatpays discrete dividends is mor involved and is covered in Chapter 11.
B
= -146.185
$132.779
0.911
$32.779
=
s1 1() ()co
s).11.1()6
.
$18.593
A
B zz
= -:$57
$14.726
0.691 4O8
A
B
.
=
0.833
-$77.871
=
$qz j).g() $6.616
$,a,,4)4 $0.000
.
A
B
=
0.447
-534.984
= ;
$78.576 50.000 A 0.000 B $0.000 =
=
$66.411 $0.000
Option qn a Stock lndex Suppose a stock index pays ontinuous dividends at the rate &. This type of option has in fact already been covered by our derivation in Section 10.1. The up and down index replicating portfolio by equations (10.1) and moves are given by equation (10.10),the
Inmitively, dividends can be taken into account either by (1) appropriately loqkeringthe nodes on the lisk-neutral probabilities unchanged, or (2) by reducing the risk-neutral jrobability tree and leaving and leaving the ee unchanged. The forward tree adopt the srstapproach.
332
%.Bl rqoM
IAL
Op-rlox
PRIcI NG:
op-rloxs otq OTHER
I (l':)i j).1@1 q' f' ..(jyyt'yjjyjy'' (' .j' jyyyf jty'kj. (1* 'J'' 1)* ;'lf 11* p' j'J' t'. r' y' E lrli':r'jf (jf ')f 1111k---1110*.* t'y'tf (' y' /')'!r?qprr7;qrsf 1!* (Cq.q i (' ('(('; f ' '('lrrlr'lq:l'rr?!ll!r'l q (qlik );f.it: @ yyf yyj'yyjj ?' j'( #' ;'t' i' ( q@ q( . ;. (' ! (' jyy'.' t'rrf y' jj': j'#' g.)'y' qtjt'i)ki (241L..* )))' lll'icf .j' 'lf ?p)i'. .q'qr r'Jlplqiiillrq-.. tq'..' . it).. tt.((t;'. q q I r (qi . ;' y ( .. ; .. ;. . j y. y I yyjj. y ry . rjy yj. ry.l1ii:itiIiil. .. yyyj .jg ti. ' (11:1 jyy.yygjjy..j.j.yjjy jj.; j.yytj. jyyj...yyg . j yjg g ( . 1IiiiE!!i-. . E ' i E ),(l1IIiFq5F!. (k.d!1I::::;;ki... 'Ei ..:! '. !tj,; :jtjkk--. . )jEj).. i.tl!'. . t(. . .. . ::..;E'.r.!i(!.;i;.jr .. . . yrjyyj'jjjtyf 'r' ---' -.' E' L' ?' j',':'(' ''(' '4'j,yjjjj', 11:::::,.* j'@)' 'C' rjjfzj'fE. :'(js'jgsf (#' tyf )' ;';'j'jjj;gggjjj;'r' tgf ''. y' T)IT;EE)E;E '7 E E E '' ' E : EE (iE'. '':..E E''' :i jE E E trrTE !E i;. .t (''i'i EE '!: EE: . E :(Ei '.!'' i . i .' i' EE EE . ' y-y.' . j E(-lyjky yyyg jq y q.g. g..q: E : yE.q. !, .gq.r. . y.(. : j;. i. ; !q y . . y j j E ; E
.'.'' .'.
'.'-' .,.'
Options on Cmrencies xzer-rfjh spot plie. Ab, the forward price is F() /, With a cqrrency with .. (. the oreign interest rate. Thus, we construct the binornial tree using =
,
Ifx
where r.f is
=
-
.
(r-ry)/l If
-
-
-
(E: E qE :
'
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.
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;
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Binomial tree for prcing an American put opton on a curreny; assumes
-:ji5;;,
.
.Q5Iq, . # $1 0, c 0.10, r 0.055, T= 0.5 years, & 0.031, and h 0.1 67. At each node the stock price, OPtiOri Price 2. and B are given. Option prices in bold ltollc signifythat exercise is optimal at that node.
'
#
$1.148 A
s
=
=
=
.j
(j ().j )
#
B
=
$1.107 $0.000
q
A
=
$0.867
$1.(sa $o.oaz
B
-0.915
= =
$1.009
$1.012
$1: 02O yatuo
$0.088
.
-0.995
Z
=
X
=
1 O90
A
B
jtl.ngj so zas -0.995
=
=
$1.090 $0.940 $0.160
. (.
..
' .
. .
.
..
.
Solving gives?
Options on Futures Contracts
c
---
We now considt- options on futures conacts. We assume the forward price is te same based on the forward price, we simply add as the f'tlturej pric. Since we build the tree Thus, the nodes are constructed as up and down movements around the current price. lI
= ezr
d
=
-a4' e
Note that this solution for u and d is exactly what we would get for an option on a stock index if J, the dividend yield. were equal to te risk-free rate. In constructing the replcating portfolio, recall that in each peliod afutures contract requ'ed to enter a f'utures pays he change in the futtzresplice, and there is no investment and the lending, #, contract. rheproblem is to find the number f fumres contracts, that replicates the option. We have ,
-
-
F4 + er /, x B F) + er x B
=
Cd
=
C tt
fI
AH
k
A x dF A )M: (IfF
.s().()(o
$0.525
-0.774
=
O.0O0 0.000
-0.459
A
$().oss B
=
y1.(s8 $0.021 =
sjmsj
,
Notice that if we think of ?'y as the dividend yield on the foreign currency, these two equations look exactly like those for an index option. In fact the solution is te same as for an option on an index: Set the dividend yield equal to the foreign risk-free rate and the current value of the index equal to the spot exchange.rate. Figure 10.9 prices a dollar-denominated American put option on the euro. The cuaent exchange rate is assumed td be $1.Q5lq and the strike is $1.10/C. The eurbdenorninated interest rate is 3.1%, and the dollar-denominated rate is 5.5%. Because volatility is low and te option is a-the-money, early exefcie i t/timal at three nodes prior to expiration.
'
(:!1t.. ::;1!,:.
$0.000
=
,
333
!- . gq ! y
:. :
.
.1
A x dxerfh + erh x B = Cd A x uxerlh + erh x B = ClJ The risk-neutral probabilityof an up move in this case is given by d
.
=
There is one subtlety in creating the replicating portfolio: Investing in a income obligation denominated in means investing in a money-market fund or sxed in Chapter 5.) Talinj into account previously that cuaency. (We encountered this idea obligation, the two equations are interest on the foreign-currency-denominated
=
'
..
.
=
*
'
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:
.
. .
....' . .. '
. .
. -
..11111...
.$1
ttcurrency''
p
.
.
xe r-rytl-b. xer-rfth-vs
=
dx
3
'
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AssE'rs
F(II
s
-r
= e
c
1 ,,
-
11 -
-
d
yf
#) u
d d oc If
-
-
1 d
. W'hile A tells us how many f'utures conacts to hold to hedge the option, the value of the option in this case is simply B. The reason is that the f'umres contract requires no
Rerhe
interpretation of A here is the pumber of futures contracts in the replicating portfolio. Another interpretation of A is the price sensitivity of the option when the pfice otthe underlying asset changes. These two interpretations usually coincide, but not in the case of options on futures. The reason is that thefumrespliceattime/ reflects apricedenominated infumredollars. Theeffectontheoption priceof a l f'umrs pricechange today is given by e-r ; .To see this, consideran opton that is one bnomial period contillkled)
334
k
BI Nou
IAL OPTION
op-rlolqsolq OTHER
1
PRICING:
.'p . .
.y'k .y'; )' t't' .,11:t::;;;1..*.* 4.::111;* :'tjyj;f @' )' ;',rt' t-,qi'iljq',i ''irf !' j:yf Flrqi.lif F' 67* ' ii ' E iC( j)'i )' 'y' k' ),r$' ql;f )4(* i'qr'.)f i'tl t23'1;* ' )t)i)t. 11* p!' E i ' '' i ; @ E ll()lSi ' 'irl:li f!E i@! jjr,;))y'. jjf gj'ykf rlf .'('#'(qf (q:1ti h' qE(79. llrpl7lllpp7sqiipli;E I' (r(E'(:7.'i;(!' fi: )' .1iI:::Il,s'-' ytf ,:55E!1;* 1!IIE:!q' ))' )t' f'jj;j;;-. (?' .(C!' t'(')'.'l't'. 1111r* )'j)'(.)ypjjjk,kkjjj;'. l'q (' y),tf jlll#itill:)f r' tjf )jj.'-t-. t','?p.,pf rjity.jf .ri. '. . ..... . . . . ' '. ' . .. ' t.. ))t)(. Fi (i,i (..y. ..!jy i (.. .r'ktl g . r ( . t-.j.j : t-; yyrrj. ).t !j(j; yy . j 'y)).-------()).-j j y Ilil!EE!k. t y y . t))y. i. . :--1111!:. t, . r!. iy.; . )-.;.t. '.pl';y)... ) ....E y. . . ... .;.....
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ll?l'ti7')).. .'.'.
bond. We can again price the investment, so the only investment is that made in the oP tion using equation (10.3). probability of an up move is given by The lisk-neutral
p*
l
=
d #
-
If
-
.
(10.14)
-
E' ' : .E ! :
-
-
. .
:
E
E
.
.
.--- -.. ..'. -- E .. .
.
..
Options o Cmmodities
' . ..
it also possible to have Many options exist on commodity futures conacts. However, is and bonowing the lending market for is a options On the physical commodity. lf tere saightforward. is option priclng such an commodity, then, in teory, is concepmally similar Recall from Chapter 6 that the lease rf7/d for a commodity commodity, the lease rate. lf you buy the yleld. you pay If you bon'ow to a dividend the perspective of from Thus, lease the receive rate. the cornmodity and lend it, you stock index, with the commodity like is the option, a someone sythetically creating the lease rate equal to the dividend yield. 10.8 (imagin Because this is concepmally the same as the pricing exercise Figure 30%), we volatility of and of 3.5%1 lease a rate ine a commodity with a price of $110,a do not present a pricing example. the physical commodity ean ln practice, pricing and hedging an option based on could use it to market-maker exists, contract a be problematic. If an appropriate futures commodities often have physical in transactions hedge a commodity option. Otherwise, commodity Short-selling may not assets. a fnancial greater transaction costs than for dicult. then is Market-maldng discussed 6. in Chapter be possible, for reasons
E
-
. -
'
..
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E
:
E(E' E
'
E
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f : .!
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':
E: . :
-
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-
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E'
E -. .
--
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rrrlllltg--lllliilpp,-, tree fr . pritihj Binomial
- qE(E)))-. ,r,.rsi ....L...Ety.-.........:....-.,.................,......-......j.E-...
.
..
. .
.
........
.
-- -. .
.
.
.
--- -
-
. ....
-
..
.
.
-. -.
.
an American call optton on a futures contract; $300.0 0 assumes # $300.00, tr 0.1 0, r 0.05, F= 1 years, 8 0.05, and h 0.333. At each node the stock prie, option price, A, and B are given. Option prices in bold ltwllc 5i9 nify that exercise is optimal at that node. -
option on a gold futures contract. Figure 10.10 shows a tree for pricing an American call for early exercis is Early exercise is optimal when the plice is $336.720.The inmition holder option pays nothing, is that when an option on a f'uttlrescontract is exercised, the differnce of nyered into a f'utures contract, and receives mark-to-market proceeds the bility to the exercise is between the strike price and the futures price. The motive for mark-toumarket pruceeds. earn interest on the .
iqilsE
=
-
-
--.
:,:111;. ;,!i5;;. ::!11,,::;;;'-' ::E11!,-
. -
...
$.5.6.7.53 $336.720 $36.720
A
,
=
B
=
=
1.000
$36.720
=
.00.
$317.830 521.843
=
=
A
=
B
=
actl.tlto
$aco.(oo $12.488
A
B
=
=
$317.830 $17.830
0.768
$21.843
=
0.513
A
612.488
B
$8.51s =
0.514
$8.515
=
yp,8a.17c A
B
sz8a.17o
$4.066
=
=
$0.000
0.260
$4.066
$267.284 $0.000
A
B
=
=
0.000
50.000
$252.290 $0.000
Options on Bonds Finally, we will brie:y discuss options on bonds. We devote a separate chapter later to discussing fxed-income d rtva t.ives, but it is useful to understand at this point some of the issues in pricing options on bonds. As a srstapproximation we could just say that bonds are like stocks that pay a discrete dividend (a coupon), and price bond options using the binomial model. However, bonds differ f'rom te assets we have been discussing in two important respects.
from expiration and for which
lf
F
>
#F A
K- Then
>
lIF
=
K (#F F(lt #) -
-
-
K) =
1. The volatility of a bond decreases over time as the bond approaches maturity. The prices of 30-day Treasury bills, for exnmple, are much less volatile t.11%the prices of 30-yer Treasury bonds. The reason is that a given chmzge in the interest rate, other things eq'ual, changes the price of a shorter-lived bond by less.
1
-
But bve also have B
=
e-rl'
F
(,I
.z?-r/, (s
-
-
K)
l lt
-
-
d + dF d
-
K)
If JJ
-
-
l
d
gcl
the futures price changes by $1, the option price From the second expression, you can see that if e-rh changes by
2. We have been assuming in all our calculations tat interest rates are the same for a1l maturities, do not change over time and are not random. While these assumptions may be good enough for pricing options on stoks, tey are logically inconsistent for pricing options on bonds: If interest rates do not change unexpectedly, neither do bond prices.
k
336
%.BI NoM
lAL OPTION
CHAPTER
l
PRICING:
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'.' 'sii-r.: .;'. ,''' ;,'
much. bond, these issues On the other hand, if we have a 3-year option to buy a s-year bond payments are discrete, so coupon might be quite important. Another issue is that approximation. the assumption of a continuous dividend is art different nugh that ln general, the concepmal amd practical issues with bonds are 24. Chapter bonds wgrrnt a separate treayment. We will remrn to bonds in
Summary discussed in this section. Here is the general proeedure covering the other assets the price of the underlying asset using * Constnct the binomial tree for Fk.t+l, e tt Ft.t-vlte+(rJ# If St or S: (10.15) F',t+ll --.(ws4- Or # e dSt Ft,t-he St price formulas, the Since different underlying assets will have different forward underlying assets. 'ee will be different for different option is unexercised, can then be computed * The option price at each node, if the as follows: Ft,t-h 1St d ff # tt (10.16) ..yo.x/k7
=
=
-c.v'#
=
=
-
--
r-&lll
=
d
-
11 d -
and, as before, 2*
c
=
e-rh
(/5/
sc
11
+
(1
-
pslcdj
(10.17)
the current node. For where Ctt and Cd are the up and down nodes relative to the value if American option, at each node take the peater of this value and
an exercised.
risk-neutral Pricing options with different underlying assets requires adjusting the Mechanically, underlying asset. of the probability for the borrowing cost or lease rate stock index with an this means that we'can use the formula for pricing an option on a summalizes 10.1 the substitutions. Table yield. appropliate substimtion for the dividend .
.
:
:
: .:
.
-
..
.
:
.
:
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.
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j
stock index.
k (t E.ii:t''' ;.tiii1:-:11Ili:. 'i1It::. ..111(E:..t11..ii1i::''.Illlii!li;' i '. . .' .. .. . . . . . . . .. . . . Domestic Stock index
( :) g ( ; : E- ,.- .. ..- . . . .
;'.'...'
'' :. :.. . .
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:
using the simplebinomial ln some cases, itmay be reasonable to plice bond options option on a 29-year bond. The model in this chapter. For example, consider a 6-month the volatility underlying asset in this case is a 29.5-ye% bond. As a practical mattey, sftptl. because Al$o, difference betwqen a 29.5- and a 29-year bond is likely to be very inierest short-term sensitive the to it is short-lived, this option will not be particulady will maqer interest not rate and the 6-mont.h rate, so the correlation of the bond pzice
%.
SUMMARV
-.
.
.
-
..
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-.
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.. .
.
-
-
. . ..
.
-
umres contract Commodity Coupon bond
Domestic
tisk-free
-
.... . .
rate
Domestic risk-free rate Domestic risk-free rate
Currency
:
g
-.
. .
rate
Domestic risk-free rate
.
-. .
( L ( j j j jljl( (( ; ( !,IIE'iIIL.. i1.Ei!,,-. iiii.. -'I::i!iII... ' k''ti!!i-. !lll '. ..I:tE'IIC.. il i'. . '. .. ' . .. .. . . .. .. Dividend yield '''iii;.''iii.-
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-.
.
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rate
Domestic risk-free rate
Commodity lease rate Yield on bond '
In order to plice options, we must make an assumption about the probability distribution binomial distribution provides a particularly simple stock of the underlying sset. point distribution: At in time, the stock price can go from S up to If S or down price any If and d are given by equation (10.10). the dS, where factors movement to binomial stock price Given movements, the option can be replicated by holding A shares of stock and B bonds. The option price is the cost of this rplicating portfolio, A,C + B. For a call option, A > 0 and B < 0, so the option is replicated by borrowing to buy shares. For a jut, A < 0 and B > 0. If the option price does npt equal this theoretical plice, arbiage is possible. 'I'he replicating portfolio is dynmic, changing as the stock price moves up or down. Thus it is unlike the replicating portolip for a fonvard contract, which is fixed. The binomii option pricing fonnulahas an intemretation as a discounted expected value, with the risk-neutral probability (equation10.5) used to compute the expected payoff to the opiion and the risk-free rate used to discount the expected payoff. Tlzis is lcnopn as risk-neulal pricing. Thebinomial modelcanbe usedto priceAmericap andEtlropean calls andputs on a variety of underlying assets, including stocks, indexes, f'utures, currencies, commodities, and bonds. rfhe
FURTHER
REAUING
This chapter has focused on the mechanics of binomial option pricing. Some of the undrlyingconcepts will be discussed in more detail in Chapter 11. Thre W Will have more to say aboutrisk-neutralpricing, the Enkbetween thebinomiltree and the assumed stock price distribution, how to estimate volatility, and how to llice options when the stock pays a discrete dividend. The binomial model provides a foundation for much of what we will do in later chapters. We will see in Chapter 12, for example, that the binomial option pricing
'
'
338
k. BINOMIAL
OPTION
PRICING:
PROBLEMS
l
%.
339
0. Let tt 30%, r 1.3, 8%, F 1, and ts 10.8. Let S $100, K $95, tr option. /3 the binomial for American At 0.8, and 2. Construct tree put d an premium, and node provide A, B. each te
small. formula gives results equivalent to the Black-scholes formula when 11becomes the understand also pricing, you Consequently, if you thoroughly understand binomial generalize binomial will trees to see how to Black-scholes formula. ln Chapter 22, we tmcertainty. handle two sources of and Bartter ln addition to the original papers by Cox et a1. (1979)ad Rendleman binomial the exposition of excellent (1979), Cox and Rubinstein (1985)provides an model.
=
=
=
=
=
=
=
=
=
10.9. Suppose s'g $100,K 1. and r =
=
$50,r
=
7.696%
(cntinuously compounded),
=
0,
=
1.05. What is the 1, we have II' 1.2 and # a. Suppose that for h binonaial option price for a call option that lives one period? ls there any problem with having d > 1? =
PROBLEMS
=
=
0.6. Before comptlting the option l and d b. Supjose now that tt pri, what is your guess aboui how it will change from your previous answer? Does it change? How do you account for the result? lnterpret your answer using put-call parity. 1.4 and # 0.4. How do you tIIiII:Z the call option price c. Now let 11 will change from (a)? Hciw does it hange? How do you gccount for thiSb Use Ijt-ill parity to explain your aswer. .4
=
a11rates are In these problems, 11 refers to tlze number of binomial periods. Assume otherwise. xplicitly states continuods ly compoun ded unless the #roblem
10.1. Let and 11 u
$100, K
=
$105,1,
=
8%, F
=
0.5, alzd J
=
=
0. Let
u
=
1.3, d
0.8,
=
.1.
=
$100, K
=
$95, r
=
8%. F
=
0.5, and
=
=
t
r
0. Let If
=
1.3, d
0.8,
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
0. Let 11 1.3, 1, and $95, o' 30%, r 8%, F 10.6. Let S $100, K 2. Construct the binomial tree fpr a European put ption. At d 0.8, and /) each node provide the premium, A, and B. of 10.7. Repeat the option price calculation in the previous question for stock prices else lixed. What happens $80, $90, $110,$120, and $130,keeping everything increases? price stock the A to the inital put as =
=
=
10.10. Let S $95, ?$100, K 3. and n = 0 F 1 year,
=
=
=
=
=
=
=
'' .'
=
8%
(continuouslycompounded),
o'
=
30%,
=
=,
7
1.
=
call is.$16.196. a. Velify tlat the price of a European arbiage? b. Suppose you observe a call price of $17. What is the price of $15.50.What is the arbitrage? c. Suppose you observe a call 10.3. Let S $100, K $95,r 8%, F 0.5, and $ 0. Let tt 1.3, d 0.8, and ?1 1. is a. Venfy' that the price of a European put $7.471. b. Suppose you obselwe a put price of $8. What is the arbitrage? of $6. What is the arbitxage? c. Suppose you obsetwe a put price 1.3, 1, and 8 0. Let If 30% r 8%, T 10.4. Let S $100, K $95, tr node each option. At call for binomial tree a 2. Construct the d 0.8, and l provide the premium, A, and B. of 10.5. Repeat the option price calculation in the previous question for stock plices happeps else evelything What keeping and sxed. $130, $80, $90,$110,$120, to the initial option A as the stock price increases? =
=
=
and B for a European call? a. What are the premium, A, b. What are the premium, L, and B for a European put?
10.2. Let S and ?;
=
a. Vel'ify that the binomial option price for an American call option is $18.283. Verify that there is never eady exercise', hence, a European call Wouldhave the irie plice.
b. Show that the binomial option price for a European put option is $5.979. Venf'y' that put-call patity is satised.
c. Verif'y that the price ' of an American put is $6.678. (
E
10.11. Repeat the previous problem assuming tllat the stock pays a continuous dividend compounded). Calculate theprices of theAmelican of 8% peryear (continuously and European puts and clls. Which opons are eady-exercised?
10.12. Let S .
F
'
=
=
$40, K
=
0.5 year, and
$40, ?12
=
=
8% (continuouslycompounded),
o-
=
30%,
=
0,
2.
a. Consttuct the binomial tree for the stock. What are b. Show tat the call price is $4.110. c. Compute the prices of Amrican
tt
and #?
apd Europea!l puts.
10.13. Use the snme data as in the previous problem, only suppose that the $5 instead of $4.110.
catl price is
a. At time 0, assume you write the option and fonu te replicating portfolio to offset the written option. What is the replicating portfolio and what are the net cash flows from selling the overpriced call and buying the synthetic equivalent?
;
340
% BI rqo M IAL O p'rl olq
APPEN D IX
P RI cl lq G:
b. What are the cash flows in the next binomiz period (3 months later) if the call at that time is fairly pliced and you liquidate the position? What would you do if the option continues to be overpliced the next period? c. What would you do if the option is underpriced the next period? 3%, If 10.14. Suppose that the exchange rate is $0.92/6. Let ?'s 4%, and rq and K $0.85. d 0.9, F 0.75, n 3, =
=
=
=
=
=
1.2,
=
$1.00.
dollar interest 10.16. Suppose that the exchange rate is 1 dollar for 120 yen. is 1% compounded) and rate (continuously te yen rate is 5% (cpntinuously compounded). Consider an at-the-money American dollar call tat is yendenominated (i.e.,the call permits you to buy 1 dollar for 120 yen). The option has 1 year to expiration and the exchange rate volatility is 10%. Let n 3. =
a. What is the price of a European call? An American call? b. What is the pzice of a European put? An American put?
=
prices for American options instead of
10.20. Repeat
the previous problem ciculating European. What happens?
AND OPTION
10.A: TAXES
PRICES
It is possible to solve for a binornial price when there are taxes. Suppose that each form of income is taxed at a different rate: interest at the rate rf capital gains on a stock at the rate Tg, capital gains on options at the rate To, and dividends at the rate Td. We assume that taxes on a11forms of income are paid on an accrual basis, and that there is no limit on the ability to deduct losses or to offset losses on one form of income against gains on another form of income. We then choose Lt and Bt by requiling that the afer-tx ret'ul'n on the stock/bond portfolio equ the after-tx rettzl-rlon the option in both the up and down states. 'Fhus we require that
g&-mTgvt-v
10.17. An option has a gold f'utures conact as the underlying asset. The current 1year gold ftltures price is $300/oz.,the strike plice is $290,the risk-free rate is 1. What 6%, volatility is 10%, and time to expiration is 1 year. Suppose ?1 replicating the for the gold? What portfolio is price call option of is the on a entails call always option VtReplicating option? Evaluate the statement: call a asset.'' underlying buy.the borrowing to =
5%, J
=
5%, F
=
1,
The solutions for
=
=
=
a. What is the price of a European call option with a stlike of $95? b. What is the price of a European put option with a strike of $95?
L
and B are then
1
-
=
vo
E1+
+
?-/,(1
z.)1 Bt
-
.
(10.18)
(.%+/,) /,(x%)1 14,+/,
To
-
) 4 , sib) / j (5.-1 -
S*l 1 -L r # S-1 1/ 1 (S ) A # $ I (S ) $ To 1 d u -
B
=
1 1 Ti JL ?-/, 1 ro -
-1--
/
-1
-
Tg
us'p
-
1
-
-
T
-
Tg
-
o + J(1
-
Tdl
-
This gives an option price of
$t
=
=
AJ
z $t+h(.%+/,)
A
=
$1000?Why do a. What are the prices of European calls and puts for K equal? plices be the find to you $10002 b. What are the prices of American calls and puts for K c. What are the time-o replicating portfolios for the European call and put? 5%, 8 3. Let 3%, and T 30%, ?' 10.19. For a stock index, S $100, c ?7 3.
z'd)1
-
-
=
30%, r
+ &&(1
,%) -
-
c. How do you account for the pattern of early exercise across the two options?
=
=
,
'rhe
=
=
=
=
APPENDIX
a. What is the price of a g-month European put? b. What is the price of a g-month American put?
=
341
.
10.15. Use the same inputs as in the previous problem, except tlwt K
=
$
10.21. Suppose tat tt < er-&th showtlaat there is an arbitrage opportunity. Now er-nsll showagain tlaat there is an arbitrage opportunity. supposethat d >
a. What is the price of a g-month European call? b. What is the plice of a g-month American call?
c
TAXES AN D OPTI ON PRI CES
$100, o' 30%, ?- 3%, and & 5%. (You $95, K c. Now 1et S have exchanged values for the stock price and strike price and for the interest rate and dividend yield.) Value both options again. What do you notice?
=
10.18. Suppose the S&P 500 futures plice is 1000, and /1 3.
1O.A:
=
1 1 Ti 1 + rlt 1 ro -
('%+/,)+ Lp*$t+lt +
(1
# jstwi, (S,+/,)1 +
-
-
(10.19)
-
where 1+ +
#=
?-/,
1 1
-
Ti -
-
Tg If
-
8 d
1
-
Td -
1
-
To
# (10.20)
In practice, dealers are marked-to-market for tax purposes and face the same tax rate on a11fonns of income. ln this case taxes drop out of all the option-pricing expressions. When dealers are the effective price-setters in a market, taxes should not affect prices.
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and tr 0, any in-the-money option should be aexercised In the special case when r = when the stock price is twice the exercise xercise 0.5r, tlten we l'mmediaiely. If J =
=
price.
Thedecisionto exerciseis morecomplicatedwhenvolatility is positive. lnthis case the implieit insurance has value that varies with time to expiration. Figure 11.1 displays $100, the price above which early exercise is optimal for a s-yearoption with K 500 usirig computed binomil volatilies, different 5%, and J 5%, for thzee ?' stock price, stepsk Recall from Chapter 9 that if it is optimal to exercise a eall at a given shows 11.1 the lowest prices. thus stock Figure all then it is optimal to exercise at higher which is price, this lowest oscillation in stock price at which exercise is optimal. The approximate that binomial and down movements evident in the figure, is due to the up the behavior of the stock', with an ifnite number of binomial steps the early-exercise schedule would be smooth and continuously decreasing. Compacing the three lines, we obsel've a signilicant volatility effect. A s-yearoption with a volatility of 50% shold only be exercised ifthebtockplice exceeds about $360. Ifvolatilit is 10%, theboundary =
=
=
' :
.
.-.-
-
''' -
'' .
-
.
. .r.
!. . . . ;: . : . . .. -..:.111-,211. :
:.' .
. -
!
-
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.
-'
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i
-
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. . ,.-1j11).
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.
. .
-
: ..:
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:.
.
-
.r...,..;
;.
'
. -
-
. ..
.
-
. -. : -)jj,....
-
: : E-. .. -
.
.
.
-
,-j..
.
-. .
-
i
.
-.
E'
. . . .
-
.
'
: -' ' -'. .
.
100 90
''
.
o-lo
=
.
'
'
...,. . ..
au .
.
.
.
.
.
#-. q
.
.
.....z.
..
C'
.
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y'
.
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,)
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.
.
=
=
J
=
..
5 ogo.
z .
.
'$ ,q&,,
-
vo '
;
'.
.
.
80
...
.
.
boundaries for of 10 o/o, volatllltles . 300/o, an d 50% f'or a 5-year Amerlcan p t O tlO n t In a II Cases P /( $ 100, r 50/0,an d
'
' '
.
:
.
.-;::;;(;;ty.
.:-.y
.
' .
''
'
:'E .' .' . :
E'
-
;
. ' .' . Early-exercise .
.
<
G&.
.
60
c.
0.30
=
Z*j
p.!a),-(a k jt/, %z; ,
tjjjjur'z..ry?t,,.z,jyg'
jpjt/, g-kx.' V'ff ; yjjygyzmooksmmygsj: r ..#
so
Lgtyy,
yjs, yy. f,jrsttjtk ty, )jq .. .
vsj ...
..j-yj
.
yyyu ys
j..
.
40
,j4
, ,, y
e
=
.
;
,
0.50
'
30 20
0
0.5
1.0
1.5
2.0 2.5 3.0 3.5 Time in Years
4.0
4.5
5.0
D IN G
UlNl D ERSTAN
346
k. Bl lqo
M lAL OPTI
olq
PRl
cl lq G: 11
equal, eady-exercise criteria Figures 11.1 and l 1 also show that, other things occurs because the value of insurance become less stringent closer to expiration. expiration. diminishes as the options approach where J = r, the overall While these picmres are constructed for the special case rfhis
conclusion holds generally.
+
;j'
PRICING
Solving for p*
written 10, we saw that the binomial option pricing formula can be ln qhapter e-rhk sc = /7 tt + (1 pslcd
c
d
p
*
(r-)
-
=
-
-
probability? p*, showing We will bgin our exploration of risk-neutral pricing by iterpreting that gives probability butratherthe thatit is notthe tl'ue probability that the stockgoes up, that it show then will rate. We risk-free the stock an expected rate of ret'urn equal to the calculations cash flow discounted is possible to compute an option price using stndard cumbersome. using tlle true probability that the stock goe up, but that doing so is
.
.
'
Probabilities Pricing an Option Using Real ''
' .
.
.
d
d
,
,
' '
-
tt p' the equation in (11.2).This is Fhy we rfer to the asstoek p' of This is exactly the deEnition I@. It is te probabi ty ihat '-isk-lktttl-alpl-obabilin'//?&/ the stockprice will go risk-neutral world. liskprice would increaje in a equauon (t1.2),be used in a probaity, risk-neutral &otonly would the risk-free rate. Thus te discounting would take place at the also a11 world. but investors are neutral be said to price options as t oplion pricing formula, equaton (11.1),can assuming that investors are l'isk of being repetitious, we are not risk-neual. At the expected to assets are actually rislty that assuming of the acttlally tisk-neutral, and we are not interpretation p?-fcfl/.gis tw3 of rettlna. Rather, riskmeutral rate risk-free the that portfolio earn arise from finding the cost of the fonnulas above. Those formulas in turn practical replicates the option payoff. lterpretation of the oprion-pricing procedure has great Interesngly, this methods otherpricing are t'isk-neual pdcing can sometimes be uked where for Monte importance; risk-neual plicing is the basis *i1l see irl Chapter 19 that too dicult. We sinulatd uder the surrtion that assets Wllich sset jtices are Carlo valuation, in u:d to vale the option. t'isk-fre rate, and these simulated prics ' r: .. j. the earn
t'isk-llelttl'alpl'obabilin' that the stock will go up. Equation (11.1)has We laeled p* the the expected value calculation the appearance of a discounted expected value, where lisk-free rate. the uses #* and discounting is done at reassuling, of result the plice is a present value calculation is that option idea The an cash flow discountd standard A puzzling. but at the same time equation (11.1)is that pmbaility uslng the value true expected calculation would require compting an expected using the ret'um done would then be the stock price would go up. Discpnting lijk-free lisk, V oreov V what is p*? Is it really a rate. the equivalent of not on an asset
.
Iislt
what to emphas ize that investors are risk averse. To se lt is common in snance probability with qffered either (b)$2000 $1000, or (a) aversion means, suppose you are since alternative 0.5, and $0 with probabilit'y 0.5. A risk-averse invstor prefers (a), will require a of investor value ldnd This expected as (a). (b) iS ri Sky and has the same equal. values are premium to bear risk when ekpected thing and a risky bet with an A risk-neutral inkestor is indifferent between a sure Arisk-neutral investor for example thing. of expected payoffe qual to the value the sure (jr alternative (a) (bj. will be equally happy with assttming l/?7r Before proceeding, we need to emphasize that at nt? point zrc wc rik-nplfrrcI/. Now 47/2# throttghout J'/7c book the pricing ct'lfcl//cl/tln- are investorsare ctpn-ilcnl with 'nve-/pr. being rik-t-lver-c. risk-neutral Having said this, let's eonsider what an imaginary world populated by investors are only about expected returns investorswould be like. In such a world, ,
(r-J)/l
D
d
.
e
*
#
'l'he Risk-Neutral Probability
(
ocrives
#=
-
where
347
$
=
-
11.2 UNDERSTANDING
PR l CI N G
be would have no risk premium since investprj wuld lisk-free and not about ziskiness. Assets rate. expected rettu'n equal to the willing to hold assets with an probability of the stock dsk-neutral world, we can solve for the ln this hypothetical risk-free rate. In the binomii is expected to earn tlae going up, p*, such that the stock lf ttS or down to dS. the stock is to eat'n model we assume tat the stock can go up to stock will go up, p*, must then the probability that the the tisk-free rettlrn on average, satisfy rIt $h &l' zy e S p If c + (1 p ld'e
.2
RISK-NEUTRAL
RISK-NEUTRAL
,
'
stgndard discounted cash option pricing cpnsistent Fith We are left With the quesion: ts itistribution for the fumre Ttie answer is yes. We can use the tr llow calculations? ttkeoption. This expected payoff can computng the expected payoff to in price s bsed on the stock's required return. then be discounted with a rate price Uptionsbecuse there is no Discounted cash flow is not used in pratice to in order to compute te to colpute the option price valuing is neessry It so: do to reason of an option using However, we present two xamples rate. disnt ct ad algo to understand coni using real probabilities, real probabilities tb.see the difliculW tisk of an option. stock is a and how to determine the continuously comppunded xpected rettu'n on the Suppose that th probability of the stock going dividends. Then if p is the tnze that the stock does not pay consistent with u, z/. and a: up, p must be (11.3) c/'us pld. plt S (1
tock
-t-
-
=
348
%.B1No
M IAL
Op-rlo rq PRIcl
Nc:
U N D ERSTAN
II
@))'t'
p
'
'
-
-
(f
(11.4)
d Forprobabilities to be between 0 and 1, we must have tt expected payoff to the option one peliod hence is euh d Cu + /C,, + (1 p)Cd u d -
-
de'
>
>
-
.
-
tt
eah
-
tt
-
d
(11.5)
Cd
. .
''
E
.. (11.6) e e + SA + B S'A + B We can now compute the option price as the expected optipn payof, equation (11.5), discounted at the appropriate discount rate, givep by equaion (11.6).This gives euh jj da d c-F /' Cd C'1 + (11.7)
-.--! .
-:
.
.
. . .
.
.
.
. . ... .
. .
.
.
--. .
-
--
.00,
.
.
-
tt - d
tt
d
-
Ju#d?,/?7?l#lg' calcuthe ?-g/c-?effr?-tW lt tllcns out that tLisgives If- the tJl?ld option price equivalent equation to (11.7)is ft-l/sn. Appendix 11.A demonsates algebraically that the risk-neu'al calculation, equation (11.1). The calculations leading to equadon (t1.7) started with tlie assumption ihatthe expected return on the stock is a. We ten derived a consistent probability, #, and discount rate for the option, y.. You may be wopdeting ifit matters whether we have the tcorrect v alue of a' to start with. 'T'he answer is tat it does not matter: /?zyconsistent pair of a and y will give the same opion price. ltisk-neual pricing is valuabl because serling a r results i.n the silnplest pricing procedure. ''
=
To see how to value an option usipg tnl Prc?yabilities, we will example compute two examples. First, consider the one-period binomial example in Figure 11.3. = 15%. Then Suppose tat the continuously compounded expected return on XYZ is a equation i s from going of stock probability the (11.4), up, the true '
A one-period
-
= 1.4623
-
:.80:2,5 0 8025 ,
=
0.5446
Brealey and Myers
(2003,ch. 9).
349
L.g
-
-.
.
.
$59.954
$19.954 $41.000
-
g.g.8aq
=
A
=
=
0.738.
-yzz.4(s
s
=
.
$32.903 $0.000
payoff to the option in one peliod, from equation ( ' '
The exected
''''''
'
.. .
.
0.5.446x $19.954+ (1 0.5446) -
x
$0
(11.5) is
$10.867
=
The replicating portfolio, A and B, does not depend on p or a. In this example, A =$22.405. discount rate, y, from equation (11.6)is given by 0.738 and B
=
'he
=
e
yh
0.738 x $41 () j.s e 0.738 x $41 $22.405 .
=
-$22.405
..y
0.738 x
-
e
(l.(s
$41 $22.405 -
= 1.386 Thus, y
ln(1.386)
=
32.64%. The option price is then givep by equation
=
e -0.324 X $j() 86.y .
=
(11.7):
$,7.839
This is exactly th plice we pbtained befpre. A and Notice tht in order t orpute the ctiscount rate we firjt had to comput B. But once w h ave compp ted A and B we can simply compute the option price as need fof further lppiatins. is lt n b hlpful to liow the acmal LLS+ B (jfk pottless. fetrn vlation it is but piion, fof expected ,
,
rrhere
*
.
.
To demonstrate that this methd of valuation works over multiple periods, Figure 11.4 presents the same binomial as Figure 10.5, wit.h the addition that the true discount rate for the option, y, is reported at each node. Given the 15% contintlpusly compounded discount rate, the true probability df an up move in > E Fi g ure 11 4 is . example
A multi-period
'ee
.
'
.
jyjs
x j /3
1.
.(;
*
g6gg
=
0.5247
.
To comput te p t ttae nod Whre the ftk ljrice ij optifl yricete aext period ai 26.9%. This gives
expected
e-0.269xl/? x $34.678+ (1 0.5247) (0.524.-/ -
x
$6t.149,W
discount the
$12.814) $22.202 =
$43.246,the discount rate is 49.5%, and the option price is ,-0.495x1/3 x $12.814+ (1 0.5247) x $0j $5.700
W'hen the stock prie is
lsee,for example,
$
:
.
:g y . yy - -- - - - . ..
.. -
PRl C1 N G
'
j( g
:
-
-
,0.15
y
y
fhr
=
'
. .
. : :
,
---).
'
E
.
:
:
:
=
iS
#
.
''
.
.00
ihe
e
.
''
EE E ' '
'
:
=
tisk-neual probabilides: At Now we face the proble> with using real as opposed to discount te opon whai rate do we discount this expected payoff? It is not conect to equivalent to a leveraged is the opdon stock, because expected the ret'urn the a, at on the stock. investment in te stock and, hence, is riskir t1):.11 Denote the appropliate per-period diseount rate for the option as y. To compute y, weighted average of the we calz use the fact that the requiredremrn on any poAolio is te porlblio.l In Chapter 10, w saw that an option is etpivalent ret'tlr'nson the assets in the exfected remrn to holding portfolio consisting of A shres f stock and B bods.
lz/:
'
.
:
-
=
=
-
#. Using p, the actual
'
.
.
.
. .. . .' Binomial treefor pricing a European call option; assumes :=F 41 # $40.00, o' 0.30, r 0.08, F= 1 0.00, and years, h 1.OOO.This is the same as Figure 10.3.
.
=
If
MCFYYiO On tYS
.'. .'..' .''
E'!E : E' E ' E ; . EE .. . . E 'y' ggy . (j;Eq yj. gyy y. y .y . y y --.'
: '
L.L.. :.
.
ah
RISK-NEUTRAL
:'#' tj'yyj. y' 'j'q' (' j7* 5)* ff ;q(;t;'. .j' ')' i'q!' ?' r' 1* if :774r:y' yq''!qlrrr'qqlq:qqll (/k t'j'('(1* qt' p;f (.;((('@ (j(;.j. ; ..(qq (tq.'')rqiqpp))i y(.'j' (::1 k( ' (' ( ij q'qE((ill i E.('( (( !'.'i( rq:' ?' yjjj'( ' i'!t:' ifi@( $* l'r' plilt-:-.dl!f L-?iLL-'. j'y' ;';'(kl))'. )' j'. jjj'. jjf (' .i5. i'p' i'. 1111(*. . t'r'7IiiI.(' .'.( '.. ' . . . .. . jj. . ..j t()j)jj).' )-',' t'. jyj' j.j . jy..y'.j.j.j. .y(.'j..yy'. .li. .' tqil.E...( yyy..(..jyy. j. 'jy tyyg' L;L . .). j r..ygjgj yjjjg' ;: . jjjj.jj j .. j j j g j g . g j . .. . . j ! . t . 1IIiiii.. (E .jj .. j j : . . . illlk:tliii. i '. !!!!iqqit-. t .. ' (.:. .....-.'. : p- r.. )-... .;...t......)-.., (.. .''yijj'y '.' '''r:' '' -.-.'ki.. --.-----' -' ;''F' ' i 'I.II::::Cii..X --' ---j'-' -' y' $),* y',,;jyjjjy j','gjjjgjjjjy. lff .4* (' @E 'E : E :E E ; :E E E : . . : . . . . . . . . (E !'E).t.ijy.EL;E'CiI. iEq)@-f/i)@
Solving for p gives us
D l NG
y.5:2,,(j,
-
=
%.BlxoMlAL
350
TH E
11
Pnlcllqc:
OpTlox
)' ;'!' (rr.'!ri ;j'* j'jjf y' q' 1* $' srl' '11:* @' )' 1.1* l't: E' qr!f )5;* q' y' j('. )'. q' iff t.;Ff j(' )' tyf t'tif i'1* 77!717'r'''q!' qq!rf 1(* r'jjjjtgjgjjjj'i tf ttlf (' jgjjjjy;d y' t:jd j'jg;'.yyyyy. ;yj'. y' r' jf yyd j'q j:' jjgfyjyjjyjjjjgtjjy'yi'y j'. )'. (jj' ryjjj;;d yj'tr (jj'r(yjy'. yj'. j'?:'jjd y'. E !E .(' (' y' ' ( !';)i2 ' ( !.E!'(.'((..'('' i.!'i' 'i prl !E(: . ; .' i' E!i.!!.i'(.pi(:: ( :; t'y'. . . '.;' . . . '. .. (.( ( y;(. yyy (j(' p:i;kl. ;yq:jty jy;y. jjytjjrrjjjjjjyy . . j .j j.f) .!-sl.; . . ; : : j j( ( ;j,jjjjj, j yjjy;j.jjjy gyjk jjjj.yy jyl j j.yyy j.yjgy y . . y . jj jy. j. . g (jy-jyjyi tti . yj.y jjjjjj;jj;j,. yjy . .. )i.-'t'E .jyygjy.yygg. E!).--?.t).F-.-......I...-..:...d.-..-'-''.'.'.'.'i:.. ;y .gjj.y.y . tj; .y ygjj.gjj.j.j..y.y . . .. ;',.y' jjrgjjjjd :'('ilfllhLflli'Elprlilrl''lilrl;E)f ky'. ,,,4444,,* ..yjj' ,g;;;;j,'
.' '
E jE : .
.
EE tE
E:' ' :' '' .
;
E: . .
:
E'E 7 iE E E!E E .E. i!
:: E: EEE' E
:
:
'' : .. .
. ...
,,;jjjjj.
.
.
.
y
.$jjy,.
'..)!,-.
...-.;:2ii))...
....
.
. y.
y
yj
yjy y yy y y kik---t y .g y g gyy . . . -
...
...................
Fi'
:
E' EE: iE. E.'i.. .y .. . .
:
.
'
' 5;
j( ,,..-.-.'.L-..i-....,..,..-.'.L.L-..----11.LX.i
. .. , g....E..g jjjjjgjgj,E E E E EE . -
'
. .
,
,
:. . :
-
.y
:
y
,-,
-
.
.-.
$34.678
.
Binomial tree for pricing an American call opton; assumes .S $41 # $40.00, c 0.30, r 0.0% T= 1 0.00, and years, h 0.3334 The continuously compounded true expected return on te stock, a, is 1594. At each node the stock 'price option price, and continuouslj compoun d e d true discount rate for the OPtion v, are given.
y
x/x
=
661.149
$22.202 v 0.269
.00,
=
11.3 THE BINOMIAL
TREE
M lAL TREE AN D LOGN ORMALITY
Q
AND LOGNORMALITY
The usefulness of the binomial pricing model hinges on the binomial tree providing a reasonable representation of the stock price distrlbution. In this sectio we discuss the motivation for and plausibility of the binomial tee. We will define a lognonnal distribution and see that the binomial tree approximates this distltiution.
=
=
=
gg;;;j,,
:.
Bl NO
.00
$52.814 $.12.814
sso.()71 $12.889
=
=
y= 0.323
=
y
.
o.a.s7
y
0.495
=
$37.351 30.000
,$35.411
62.535
,
NlA
=
$43.246 $,5x()
$41.000 $7.074 =
y
v
=
O.4qs
y
wx/x
The Random Walk Model It is often said that stock plices follow a random walk. In this section we will explain what a random wall is. In the next section we will apply the random walk model to stock prices. To understand a random wall, imagine tat we flip a coin repeatedly. Let the random variable J' denote the outcome of the qip. If the coin lands displaying a head, F = 1. If te coin lands displayig a tail, F = If the probability of a head is 50*, the coin flips, with is After th flip denoted Yi, the cumulative total, the fair. n we say Zn is -1.
.
$30 s8s $0.000 .
')'
=
(11.8)
X/X
,
$26.416 50.000 y NIA =
;''
lisk-neual These are both the same option plices as in Figure 10.5, where we used Plicing. though the e. To compute th price at the node We coptinue by . wt Z. ng back . . . . option jrice 4henext period xpected where the stock prie is $50.071,we dlsctwnt the at 32.3%. Thus, -0.323X1/3
e
+ (1 g() sg4'jy $:2,p,.:2,g:). .
-
0.5247) x $5.7004F:r $12.889
Again, this is the same price at this node as in Figure 10.5. The act'ual discount rate for the option changes as we move down the ee at a point i.ntime and also over time. The required return on the option is less when the stock when it is $43.246(49.5%).The discount rate increases price is $61.149(26.9%)t.11:.r1 equivalent to a leveraged position in as the stock price decreases because the opton is option moves out of the money. the the stock, and the degree of lvrage increases as possible obtain opton prices using standard to These exapples illustrate tat it is iehnlqus. denerk owekef, ihereij noraqn to dp so. Moredisconted-cash-qw risk-neuttat pricinj Worl mns that it is not necessa to estimate over, the fact that option. Sinee expected remrns are a, the expected ret'urn on the stock, when pricingan option pricing a great deal.easier. hard to estimate preciqely, this makes Appndix 11.B goes into more detail about risk-neu'al p/cing.
It t'urnsout that the more times we :ip, on average, the farther we witl move from where we start. We can understand inmitively why with more iips the average distance fl'oll the stnrting point increses. Think about the first flip and imagine you get a head. You move to +1, and as far as the remaining :ips are concerned, this is yt?lfr ncw starting point. Afier te second flip, you will either be at 0 or +2. If you are at zero, ti is as 1.f you started over,' however if you are at +2 you are starting at +2. Continuing in tllts way your average distance from the starting point increases with the number otdips.z Mother way to represent the process followed by Z,, is in terms of the change in Z?,
-
Z?,-l
=
F,,
explicitlyas
We an rewlite this more
Heads:
Zn
Tails :
Z,,
-
-
Z,;-I
=
Z?;- I
=
+1 -
(1 1.9) (11.10)
1
zzkfter?? llips, the average squared distance from the starting point will be ll. Conditional on the rst flip being a head, your average squared distance is 0.5 x 0 + 0.5 x 22 2. Ifyour lirst flip had been a tail, your average squared distance after two moves would also be 2. nus, the unconditional average squared distance is 2 after 2 Qips. lf Dtlt represents your squared distance from the starting point, then =
Dn2 = 0 5 x (D ?:.
l
+ 1)2 + 0 5 x (D lt .
-1
-
1)2
=
2)2
a-)
+ j
Since D()2 0, this implies that Dcl n. 'Fhis idea that with a random walk you drift increasingly farther from the starting point is an important concept later in the book. =
=
%.Bl N o M IAL
352
O p'rl olq
PRI
cl N G: 11
TH E Bl NOM IAL TREE AN D LOGNORMALITY
With heads, the change in Z is 1, and with tails, the change in Z is 1. This random walk ls illusated in Figure 11.5. 'I'he idea that asset prices should follow a rgndom walk was articulated in Samuelson (1965) In ecient markets, an asset price should reflect all available infrmation. by dehnition, new infonnation is a surprise. In response to new information the price is equally likely to move up or down, as with the coin flip. The price fter a period of time is the initial price plus the cumulative up and down movements due to informational .
3. The stock on average should have a positive return. The random walk model taken literally does not permit this. It ttlrns out that the binomial model is a variat of the random walk model that solves a11 these problems at once. The binomial model assumes that continuously rehtrns are a m??#t7?nwalk. Thus, before proceeding, we 'Iirst review some ct/n7pt7lf?7#c# compounded rettlrns. of continuously properties
surprises.
.of
Modeling Stock Prices as a Random Walk rf'he idea that stock prices move up or down randomly makes sense', howe'ver, the description of a random wnlk in the previous section is not a satisfactory description of stockprice movements. Suppose we take the random walk model in Figure 11.5 literally. Assume the beginning stock jrice is $100,and the stock price will move up or down $1 each time we flip the oin. There are at least three problms with this mbdel:
Continuously Compounded Returns Here is a summary of the important properties of
?-/,/.1-/,j.nSt+kIS:) =
((f we know the conThe exponential function computes prices from returns exponentiating both sides of obtain St-vltby compounded tinuously reprn, we can This gives equation (11.11).
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Illustraton of a random walk, where the counter, Z, increases by one when a fair coin flip comes up heads, and decreases by one with
tails.
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.
.
compounded returns.
Let St and St-o be The logarithxnic function computes returns from prices stock prices at times t and t + /7. The continuously compounded ret'urfl between t and t + /7, l't,twl, is then
2. The magnitude of the move ($1) should depend upon how quiclcly the coin Qips occur and the level of the stock price. If we flip oins nce a second, $1 moves li E ' :IEi 77:77777:77')777:. !'' LE E Eq'' i iql ti!E.';i. i';(''(.'.(EE !'.l'FiE:r.E' E' ' EE i( E /1 pjl 711)!r1i.EEEE q! jj)l l '''f-k (j:1)9: . 'E . ; (Ei'E.i ;! '!(E ';EEE.!'Ei' EE 'q; . . ig E E. !. E ltjjE (i E.@ ; i ! E-E ;. . E E ;-,jjjj.! t...E i!.; .E.Ej.);j (: :E..yi.p. i.j jjLILL. E! . E. .( ( ; y . . g j)). . y;q; IE j( jg .. jj.;(qj;j )g)(.. .j .;,-(!-j;;.. yy y;..yj.yEg;j.! -.E i;,. lq yjy. (qy yj l'lEj . ;j. ijEq.. .))Eh1p)....(; : .. .. . llllq:piliil ; ., . .(. !(.:;E. y( ()))y)k)).. ....y. j q y-i ... ... ..... r. . . ... .. .r.. , .r..
continuously
(See also Appendix B at the end of this book.)
1. If by chance we get enough cumulative down movements, the stock price will become negative. Because stockholders have limited liability (theycan walk away from a bnnknlpt srm),a stock price will never be negative.
.
353
are excessive; in real life, a $100 stock will not typically have 60 $1 up or down movements in 1 minute. Also, if a $1 move is appropriate for a $100 stock, it likely isp't appropriate for a $5 stock.
-
ijqd td t(;';' (:'I1I1k2L:IIir'' r' j;'. )' j'yyd '.ytyy'yy'tr;iyyyq' t'.:': (' (;' i'trd lllt!i!jkd y' -.1.-J111r*);,lkiiiiq)'.' rjjtd tjj')jy .t)' td 1il':
Q
(11 1g )
yy
Continuously compounded returns are additive compounded retul'ns over a number of periods-for
.
Suppose we have continupusly
example, h,twlt, r?+/,,r+2/,, etc. The continuously compounded ret'urfl ovet long period is the sum of continuously compounded returns over the shorter periods, i.e.,
o
11
-5
.13)
( 11
.
?-l
l-t--
==
l lltgt-ih
i= l
-10
A ontinuously Continuously compounded returns can be less than compounded rettlrn that is a large negative number still gives a positive stock price. The reason is that er is positive for any r. Thus, if the log of the stock price follows a random wnlk, the stock price cannot become negative.
-100%
-15 -20
Here are some examples
-25 -3O
-nh
0
50
100
150 200 250 300 350 Number of Coin Flipsyn
400
450
500
illustrating these statements.
Supposethe stockprice onfourconsecutive
Example 11.1
days is $100,$103,$97,
and$98. The daily continuously compounded retul'ns are ln(103/100)
=
0.02956; ln(97/103)
-0.06002;
=
111(98/97)
=
0.01026
THE BINOMAL
k. Bl NoM
354
lAL
11
op-rlorqPRlcllqc:
ret'urn from day 1 to day 4 is ln(98/100) continuously compounded returns: daily of ))jThis is also the sum the = tt + ?'2,3 + r3,4 = 0.02956 + (-0.06002)+ 0.01026 compounded
t) The continuously
)E)
-0.0202
k..
r1,z
TREE AND
3d5
$
LOGNORMALITY
with the square root of time. This is why The standard deviation therefore scales pricing model. appears in the binomial
-0.0202.
=
o'.ll i
k The Binomial Model
Suppose that the stockpdce today is $100and tat 1 yearfwm today qj)) Example 1 However, the 100)/100 't,(it is $10. 'Fhe percentage rettzlaa is (10 compounded continuously a (1:continuously compounded return is 1(10/100) .jj.retunsof k. .2
-1
=
=
-
understand the biqmial model, whih is We are now in a position to better
-90%.
-0.9
SL
.
yjt
s
cauu xzy
g
(p-jjjysg
ujy
-2.30,
=
),'
-230%.
Taking logs, we obtain 1n(&.g,/& )
=
(?' -
jh
(T''UV
:1:
(11.16)
.
compounded ret'urn from t to t + rt,t-vlt, the Since Lnst-vlt/'t) is the continuously continuouslycompoundedremm. the binomial modelis simply aparticularway to model certain of is which g(?' ()J,), and the other of which is That returp has two parts, one pdce moves (+cVI1. uncel-tainand generates the up and down stock tvee problems in the random wnlk solves the equation (11.16) Let's see how discussed eadier: down the binomial tree stock price cannot become negative. Even if we move 1. compounded retul'n continuously negative, resulting large, many times in a row, the will give us a positiv Iric. smaller, therefore up and down 12 2. As stock price moves occur iore frequently, gts volatility is the sarr n rtter how nl cqjtnlti, moves get smaller. By thpeicentage ,
Example 11.3 Suppose tlaatthe stock price today is $100 and that over 1 year the Using equation (11.12),the end-of-year continuously compounded ret'urn is = percentageret'urrt is price will be small but positive: ,I = 100=5'00 $0.6738. -500%.
'rhe
0.6738/100 1 -
%.
-99.326%.
=
'l'he
The Standard Deviation of Returns equauon Suppose the continuously compounded return over month i is rmonthly,f From nnnual the return is (11.13), we can sum eontinuously compounded returns. .
'rhus;
- 12
t-nRon jjj y, L
p.annua j z. i==1
The variance of the
nnnual
return is terefore rm'ontluy'f
(11.14)
f= 1
Now suppose that remrns are uncorrelgted over time; that is, the realizaion of the return expected retut'ns in subsequent periods. With this in one peliod does not affect each assumption, tlle variance of a sum is the sum of the(7.2variances. Also suppose tat nnnual variance, denote lf let ten te we month has the same vaziance of remrns. from equation (11.14)we have 'the
2
=
(7.2
12 x
mOn
gjjy
Taldngthe squarerootof both sides andrearranging, deviation in terms of the annual standard deviation: tr Jmonfllly =
thrare. sincretisftlowafndomwalk, marlybinomiplperiods the stock pric is $t00 r $5. whether the same Plicechange ts probability of an up move, so we 3. There is a (r J) term, and we cart choose the the stock price is positive. -
12
Vartrannual) = Var
'
-
we can express the monthly standard
./--12
If we split the year into /7 periods of length 12(so that 12= over the jeriod of length h, (r2,, is
1/77),the standard deviation
expected change in ean guarantee that the thought of #s a random FnlkqFigtqe 11.6 Tclillumate that th binprnial 7ee cm be when the continuously compounded rt'urn folloWs illuspts the stock pri that results particular path through a 500-step binomial ee, with is a ranom walk. The hgure one of coin flips as in Fi'gtlre 11.5. the par' ticular path generated by the same sequence
Lognormality and the Bpmial
lognormal distribution, which is commonly used to The binomial tree approximates a model stock prices. lognormal distribution is te probaFirst, what is the lognormal distribution? The assumption that continttously ct/l/lpt/lln#c# rcrffm, bility distribution that arises from 4he tree, we are nonnally ditributed. When we averse the binomial ::lu on the stock t7?z l of h 8 X-hIn the o' ) components ' implicitly adding up binomial random remrn variables random of binomial the 0), sum (x) or, the same thing, h limit (as ?1 compounded rettlrns in a binomial tree are continuously is normally distributed. Thus, -
-.>
-->
.-1,
o-
=
c
Model
(11.15)
.
%.BINOM
356
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IAL OpTlox
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Illustration of a
11:2. ;tjisi;-
e
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11O
particular path through a 500-step binomial tree, w here the up and down moves are the same is in Figure 11 Assumes S 5100, r 6%, o' 30%,
100 .
.5.
90
=
=
=
80
N
T = 5 years, and h 0.01 =
Bl NOM
$
IAL TREE AN D LOGNORMALITY
357
neutral probability of reaching each nal period node. There is only one path-sequence tlze top or bottom node (IfIIIf or ddd), but there are of up and down moves-reaching intermediate reaching node. For example, the ftrst node below the top each three paths huld) can be reached by the sequences IIIf#, ltdlt, or dutt. Thus, there are more pats that reach the intermediate nodes than the extreme nodes. We can take the probabilities and outcomes from the binomial tree and plot them against a lognormal distribution with the same parameters. Figure 11.8 compares a three-period binomiz approximation with a lognormal distribution assuming that the initial stock price is $100, volatility is 30%, the expected ret'urn on the stock is 10%, and the time horizon is 1 year. Because we need different scales for the discrete and continuous distributions, lognormal probabilities are graphed on the left vertical axis and binornial probabilities on te light vertical axis. Suppose that a binomial tree has 11 periods and the risk-neutral probability of an up move is p*. To reach the top node, we must go up n times in a row, which occurs with a probability of (p*)'l The price at the top node is Sttn There is only one path through the tree by which we can reach the top node. To reach the flrst node below the top node, we must go up 12 1 times and down once, for a probability of (p*)'i-1 x (1 p*). The price at that node is Since the single down move can occur in any of the this there periods, can happen. The probability of reaching the t.h node are n ways /7 p*)f The price at this node is Sun-id The number of is the below top (p*)'?- x (1 reach node this is ways to
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1.0
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2.5 Time
3.0
3.5
4.5
4.0
5.0
-
(years)
.
distributed? w'ich means that the stock is lognormally discmplete iscussion of this to Chapters 18 and 20, but we can defer tlted. We a more jj j with yy an examp e ow oorks see The binomil rodel implicitly assigns probabilities to the various nodes. Figure 11.7 depicts the constnlction of a tree for three binomial periods, ong with the risk-
normally (approximately)
'
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Number of ways to reach fth node
where ?1 !
.
x
?7
=
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.
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=
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=
i
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.
.
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() (jyz .
5
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-
p*)
() zs 0.20
o ls '
0.10
0.002 O S0 dltt
Ad
K
0.004
svdu 3//(1
-
p*)2
0 50
100
150
200
Lognormal
250
300
350
0.05 0 400
N Binomial
Ad.z (1 pmp -
t
.
3The expression
can be computed ('') f
0.35 0.30
0.008 0.006
Au 'tj
': : :
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.
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in Excel usl-ng the combinaton-al function, Cb??I&zII,
f).
k. BINOMIAL
358
OpTloN
We can construct the implied probability distribution in the binomial tree by plotting the stockprice at each final period node, Sun-id against the probability of reaching that node. The probability of reaching any given node is the probability of one path reaching that node times the number of paths reaching that node: Probability of reaching fth node
=
p*'' -i
(1
-
p
zy
77 !
i
) (n
-1.
(11.17)
i )! i !
Figure 11.9 compares the probability distribution for a 25-period binomial tree with the orresponding lognormal distribution. The two distributions appear ctose; as a practical matter, a 25-peliod approximation works fairly well for an option expiring ip a few
months.
Figures l 1.8 and 11.9 show you what tlle lognormal distribution fpr th stockprice looks like. The stock price is positive, and'the distribution is skewed to the light; that is, there is a chance that exeemely high stock prices will occur.
Mternative Binomial Trees :
'
Thre are other ways besides equation (11.16)to constnlct a binomlal tree that gpproximates a lognormal distribution. An acceptable tre must match the standard deviation 'pf the continuously compounded returfl on the asset and must generate a.n appropriate distribution as the length of the binomial period, /?, goes to 0. Different methods of constacting the binomial lzree will result in different 11 and # stock movements. No matter how we construct the tree, however, we use equation (10.5)to determine the lisk-neutral ' probability and equation (10.6)to determlne the option value.
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The best-known way to constnlct abinomial binomial tree tree is that in Cox et al. (1979),in which the tree is constructed as
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approach is often used in practice. A problem with tltis ap'I'he Cox-loss-lkubinstein if that 11 is is large or o' is small, it is possible that erh > CE'Win which however, h restriction in equation (10.4). In real applications case the binomial tree violates the problem does this small, not occur. In any event, the tree based on te would be 12 so equation price violates (10.4). fomard never
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Another alternative is to construct te tree using 11
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'I'llis procedure for generating a tree was proposed by Jarrow and Rudd (1983) and is sometimes called the Jarrow-lkudd binornial model. lt has a vely patural motivation that you will understand after we discuss lognonnality in Chapter 18. You will 5nd in computing equation (10.5)that the risk-neu'al probability of an up-move is generally close to 0.5. l l three me t.ho ds of constructing a binomial tree yield different option prices for finite n, but approach the same price as ?) x. Also, while the diferent binomial trees all have different up and down movements, a11have the snme rat'ib of u to d: ---)*
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Is the Binomll Model Realistic? Any option plicing model relies on an assumption about the behavior of stock prices. As *e seen in this section, the binomial model is a form of the random wnlk stck prices. The lognormal random wnlk mdel in this adapted model, to lodeling othek stck plice nmng things, that volatility is constant, that section assumesj A1I of thse independent time. tht and do returns are over not occur, movements the data. violated in assumytions appear to be We will discuss the behavior of volatility in Chapters 18 and 23. However, there is evidence that volatility changes over time (seeBollerslev et aI., 1994). It also appears binomial model has the property that on occasion stocks move by a large amount. period length, h, becomes smaller. smaller the that stock price movements become as terefore a feature of the data inconOccasional large price movements-ljumps''-are sistet wtth the biomial model. We will also discuss such moves in Chapters 19 and 21. Finally, there is some evidence that stock rettlrns are correlated across time, with positive .have
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360
k. BI No
M IAL
017-r10rq PRI cl Nc:
STOCKS
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J.n pratice we need to figure out what parameters to use in the binornial model. 'I'he directly. most important decision is the value we assign to t7', which we cnnnot objen?e t One possibity is to measure o' by computing the standard deviation o condnuously compounded historical returns. Volatility computed from historical stock retxlrns is historical volatility. Table 11.1 lists 13 weeks of Wednesday closing prices for te S&P 500 composite index and for M, along with the standard deviation of the continuously compounded returns, computed using the StDev fucdon in Excel.4 Over the l3-week period in the table, the weelly standard deviadon was 0.0309 and 0.0365 for the S&P 500 index and IBM, respecdvely. These are weekly standard dviations since they are computed from weeldy returns; they therefo/e measure the variability in welly returns. We obtain nnnualized standaid devi/ons by tuliplying the weeldy standard deviations by W-i giv hlg mmual standai d deviauons of 22.32% for the S&P 500 index and 26.32% for IBM. ' We can now use these armualized standard deviations to construct binomial eees with the binomial period, h, set to whateker is appropliate. Don't be misled by the fact that the standard deviauops were estimated with weelly data. Once we annualize the esthnated standard deviations by multiplying by VV Fe can then muluply again by .Wto adapt the annual standard deviation to any size binomial step. 'I'he procedure outlined above is a reasonable way to esdpate volatility when conand identicalty distribtd as in thelogatinuously compoundedreml'ns areindependt arenotindepepdent-as randomwalkmodel in S.ectipn11.3. However, ifretllrns lithmic with som commodiies, forexample-volatility estimation beomes mom omplicated. If a high price of oil today leads to decreased demand and increased qupply, we would expect prices in the futum to come down. In this case, the volatility over F years will be t7'uf, reQecting the tendency of prices to revert from exeme yalues. Extra less than care is required with volaety i.f te random walk model is not a plapsible economic model of the asset's price behavior. ,
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correlations at the short to medium tenn and negative correlation at long horizons (see Cnmpbell et al., 1997, ch. 2). 'I'he random wnlk model is @useful stnrling point for thinking about stock price behavior, and it is widely used because of its elegant simplicity. However, it is not sacrosarlct.
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Al t ough it may be reasonable to assume that a stotk index ljys dividndj contiuously, stocks pay dividends in discrete lumps, quarterly or annually. In addition, individt!al How ovef shrt horizons it is freqntly possible to predict the amount of te dikidd. should we plice an option when the stock will pay a known dllr dividpd dudng the life of the option? The procedure we have alrady developed for creating > binomial 'ee can accommodate this case. However, we will also discuss a preferable alternative due to Schroder (1988).
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4We use weelly rather (11m1daily data because compudng daily statistics is complicated by weekends and holidays. ln theory the starldard deviatlon overthe 3 days fromFriday to Monday shouldbe orter t111111 over the l day from Monday to Tuesday. Using weekly data avoids this kind of omplicauon. Further, using Wednesdays avois most holidays.
Modeling Discrete Dividends When no dividend will be paid between time t and t + //, we create the binomial tree as in Chapter 10. Suppose that a dividend will be paid between times t and t + and that
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lAL OpTl
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$32.216 $26.380
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Problems with the Discrete Dividend Tree 'Fhe practical problem with this procedure is that the tree does not completely recombine after a discrete dividend. ln all previous cases we have examined, we reached the snme price after a given number of up and down movemepts, regardless of the order of the movements.
Figute 11.10, in which a,dividend wit a period-z value of $5 is paid between periods 1 and 2, demonstrates that with a discrete diyidend, the order of up ap tl down povements afects the price. In the third binomial period there re six rather than four possible stock prices. To see how tlle tree is constructed, period-l prices are l/3+0.3x4'f7'z
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The peliod-z prices from the $50.071node are -
ud and du nodes do not recombine. There are six distinct prices in the final ljefiod as each set of ex-dividend pricej generates a disunct tree (tltreepricej arise frl iv top from the bottom two prices i pefiod 2). twoprices in period 2 and three prices arike Each liscre te dividend auses the to bit-urcate. There is also a concpt'ual problem wit.h equation (11.20).Since the amount of the dividend is fixed, the stock price could in principle become negative if ther have been lprge downward mokes in the siock prior to the dividend. . This exmnple demonstrates that handling fixed dividends requirs care. Wenow constructing a tree using eqgation tttrn to a method tat is comptationally easier t.1)a14 (11.20) and that iill not generate negative stock prices.
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Repeating this procedure for the node S = $35.411gives prices of $3t.300 and $26.380. You can see tat there are now four prices instead of three after two binomial steps: The
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A Binomia l Ree Uslg the Prepaid Forward Schroder (1988)presents an elegnt method of constructing a tree for a dividend-paying stock that solves both problems encountered with the method in Figure 11.10. The key insight for this method is that if we lcnow for certain that a stock will pay a sxed dividend, then we can view the stock price as being the sum of two components: the
% Bl NoM
364
Op-rlo?q
IAL
CHAPA'ER
11
PR1cl Nc:
dividend, which is like a zero-coupon bond with zero volatility, and the present value of the ex-dividend value of the stock-in other words, te prepaid fonvard price. Since te dividend is ltpwn? all vqlatility is atibuted to tlle prepaid folward component of te stock price. g where F is Suppose we know that a stock will pay a dividend D at time To < the prepaid stock price base of option. Then expiration movements date the o the we The one-period fonvard pdce for the prepaid fomard pricek Fff'w St De-rlb-tt Fherh. this gives us up and down movements of before, As is Ft,t-v fonvard
have a greater volatility. We use the approximate cF
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correction
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365
.
De-rTn-ts However, the actual stock price at each node is given by st = + Figure 11.11 shows the construction of the binomial tree for this case. Bot the observed stock price and the stock price less the present value of dividends tte prepatd forward price) ftre included irl the sgure.Note tat te volatility is 0.3392 rather t11:.11 0.3 as in Figure 10.$. The reason for this difference is that the random wnlk is assumed to apply to the prepaid forward price. If the acmal stock price is observed to have a volatility of j0%, then the prepaid forward price, which is less than the stock price, must
:'j'r' gl':k yy'. )' 7* !' 7q7q(1* !)7q!!r7777'!q!!(q' (7q7:)1774'7r* )t' ,.'. q'. d))' 11* F' 'iitd i)' (' )' f' :piii!l;' ).'-ki-...'. yyd j'y';'('(1* !ltd y' q' :' IEyy.Lj'JjX (qd qd f!d r' t'.-' r.' -1'E151;* p:'. p' ''!!rlI:i::;k:. iI1Ii-:-1idI.' pf jj'y rid :!i1E;;' il::E::,'f )' 7'q!!!q:-:-'. ))' 1ii1Ei5!:)'. j'. y' yjjd t?' C' (' IIiII'.. IlIi!iq'. yy'y j'yyd )' '?' yjd t((;'. y;j'y' r'
k.
1.2492. Look at the node where the stock price is $61.584. Tlzis is a flrst that If cum-dividend price, just before the dividend is paid. The nodes in the last period are constzucted based on the ex-dividend price, for example
=
=
x
You pay be wondering exactly how te
-
tt
ty s
$41 = 0.3 x $36.26
,
=
=
SUMMARY
($61.584
-
$5) x 1.2492
=
$70.686
As a final point, we obtain risk-neutral probabilities for the tree in the snme way as the absence of dividends. It is important to realize that we cons%ct the binomial tree in probability of Leprepaidfolward for wllichpays no dividends. Thus, the lisk-neutfal equation prepaid price is given by the moke forward in (10.5)jjust as in the case n p paying stock. nondividend of a
'-' ,'''.' '' ?'
-'-' .-' ''7* -' -'. ,'y' jttz'yrd :y' j'!' f'
)--'..-')r' ): .q'!@.( (' rf.$r(171.E1/1j1171141(:(58): (.
: i ' i( E EiE EEE '' E IEEE qE 'i'E'EEi ! qr. EE .Ct! f:( !' i' 'i.'.'(.':..'.'E'E!E'q'q!(' ' @'! ((Ej.ji(E '(i(.'( li '''. ( 1* 'r;'i E . y-. (. g ;.j.()j.).L.;Lj;;LL);L;LqLL;) jyyjyy. y y. .. ; , ...y . ( .j)y.yy yy,. j yj. yrj . ;.yy y. j .y. yjj gjyyj; q.;ljEE y,.k jj;.j((..y.k,jj(; ;.y . yjyyyjy jy .. yy . jj jj. jg . g y .y:j,;jyjj, . .yy yyy. j . y j;t: . Iliit:tli:ll . - . . -- j.- .- .-j- -.jyE-:j.jj-)( /:/: ..q LkbbiqLkLk. . . . ' i;.,.! ... r.!. .. . E' .
:
.
.
.
:
.
-
'
'
- . -.: . E.
.
.
-
-..
'
.
.
t .
-
y
'
'
'
:
'r '
:
.
..
::
yy
'
:.
-;(jj-yj,.
.
.
.
-t(rrq!!!!)-------.q.
-..
-;t
.
.
.j
-
-
-
-
' :
i
.
.
.
,,
.
y
7:;;;)-. -1:E511;.
,
,
.
-
-
..-
.. . .. .. . . . Biomial tree for pricing an American call option on a stoc k paying a distte dividend of $5 in 8 months; assumes
-.. -.. .. .
.
: ': ' : : i
--.
,;
'
. ..,. .
-.
.
. . ... . . . . .
-. ..
. .. ..
.. . ..
5. $41 #= $40.00, tr 0 r3392 r
. -
-. .
-
.
.-.
.
FP
70.686
=
$.50.686 FP
$61.584 =
56.584
$21..584
.00,
.$50.164
=
=
,
FP
1 years, = 0.00,' and h 0:333. At each nde .the stock price, prepaid forward price, and opton price are 9iven. ption prices in bold ltwllc signfy that exercise is optimal at that node.
$47.777
45.296
=
pp
$11.308
0.08,
=
.4:.7y,:
,$'
y yyy .
.00
T
.
=
'
FP
$41.000 =
FP
36.260
$43.246 =
38.246 4ly
E
sa
$5.770
-
FP
$35.485 =
$32.293
Ff'
30.616
$1.501
32.293
:z
$0.000
$30.851
F j,
=
:,s
.
a,s)
stlc(o
FP
$21.827 21.827 50.000 =
Both call and put options may be rationally eyercised prior to expirauon. The, eadyexeicise decisionweighs threeconsiderations: dividends onthe underlying asset, lnterest on the sfrikeprice, andtheinsurancevalue of keeping tlzeoption alive. Calls will beearlyexercised in order to capture dividends on the underlying stock; interekt and insurance weigh agninst early ek ercise. Puts will be early-exercised in order to captttfe interest on the strike price; dividends and insurance weigh agninst eady exercise. For both calls and puts, the early-exercise criterion becomes less stringent as the option has less time to matllrity. -isk-neual optton valuation is consistent with valuation using more tradition plicing it is not lecessazy to estimate discounted cash flow methods. With l'isk-neueal tlzeexpected refllrn on the qtock in order to price an option. With traditional discounted cash flow methods, the coiect discount rate for the option varies along the binomial pricing. with liskrneulal tree; thus, valuation is considerably more complicated t.11a11 'l'he binomial model, which approximates the lognormal distri u tin is a random wnllcmodel adapted to modeling stock prices. The model assumes tat the continuously compounded refllrn on the stock follows a random wnlk. The volatility needed for the binomial model can be estimated by computing the standard deviation of continuously compounded returns qnd almualizing the result. 'lil e btnomial model can be adapted to price options on a stock that pays discrete (i divi en s. bi scre te dividends can lead to a nonrecombining binomial tree Ifwe assume that the prepaid forward price follows a binomial process instead of the stock price, te tree becomes recombining. ,
t
.
366
k. BI NoM
FURTHER
IAL
Op-rlorq
PRlcllqc:
k
PRo BLEMS
11
11.4. Consider a one-period binomial model wit.h 12 1, where S $100,130%, and 0. Compute American put option prices for K o' $110, $120,and $130. =
READING
model, which we discuss The binomial model can be used to delive the Blaekcholes lisk-neutral pricing will become evident in of in Chapter 12. The practical importance valuation hinges upon risk-neutral pricing. Chapter 19, when we see that Monte Carlo and show how the option price may 11.4 ln that chapter we will also reexamine Figure
0.08,
=
$100,
=
a. At which stdkets) does early exercise occur? b. Use put-call parity to explain why early exercise does not occur at the other strikes. .
be computed as an expected value using only stock prices in the linal period.
The issue of how te stock price is distributed will also arise f'requently in later chapters. Chapter 18 discusses lognormality in more detail and presentq evidence that ha pter 20 will examine in more stock prices are not exactly lognormally distributed. what happens when gets particular in detkil the question of how the stock plice moves. vel'y small in the binomial model. We will ret'ut'n to the detenninants of eady exercise in Chapter 17, when we discuss real options. 'T'he literature on risk-neutral plicing is fairly technical. Cox and Ross (1976)was pricing and Harrison and Krepj (179) stdied the the ftrst paper to use lisk-neutral good treatments of this topic are Huang and Litzeberger economic underpinnings. Two inspired Appendix 11.B-and Buter and Rennie (1996). (1988, ch. 8l-their treatment and Campbell et al. (1997) Cochrane (2001)summarize evidence on the distribution of stock prices. The original Samuelson work on asset prices following a random walk (Snmuelson, 1965) remains a classic, modern empirical evidence notwithstanding. Broadie and Detemple (1996)discuss the computation of Amrican option plices, and also discuss alternative binomial approaches and their relative numerical eciency.
=
=
=
367
c. Use put-call pality to explain why early exercise is sure to occur for all strikes greater than tat in your answer to (a). 11.5. Repeat Problem 11.4, only set J 0.08. What is the lowest stlike prie at which early exercise will occur? What condition related to put-call parity is satisfed at this strike price? =
11.6. Repeat Problem 11.4, only set ?' 0 and J 0.08. What is the lowest strike price (if there is one) at which eady exercise will occur? If early exercise never occurs, explain why not. =
=
For te following problems, note that the Binomcall and Bilompttt functions are array functions tat rettzrn the option delta (A) as well as the price. If you know A, you can comppte B as C SL. -
11.7. Let S $100, K $100, o' 30%, 1, 0.08, t stock has Suppose te an expected ret'ul'n of 15%. =
=
=
=
=
1, and J
=
0. Let l
=
10.
y
a. What is the expected ret'urn on a Europemz call option? A European put option? b. What happens to the expected ret'urn if you increase the volatility to 50%:/
PROBLEMS Many (butnot a11)of these questions can be answered with the help of the Binomcall and Binomlhtt functions available on the spreadsheets accompanying this book.
$'1t0 1, where S 11.1. Consider a one-period binomial model with h prices call option ?orK G' 30% and J 0.08. Compute American $80, $90, and $100. =
=
=
,
=
.
,
?' =
=
0
,
$70 ,
0.08. What is the greatest strike 11.2. Repeat Problem 11.1, only assume that 1' occur? will exercise What con dition relaye t pt-call parity price at which early price? is satisEed at this strike =
=
0.08 and
t5
=.
0. Will
=
=
=
=
all c. Use put-call parity to explain why early exercis: is sure to occur for lower strikes tan that in your answer to (a).
?'
=
=
a. At which strikets) does early exercise occur? b. Use put-call parity to explain why early exercise does not occur at the higher strikes.
11.3. Repeat Problem l 1.1, only assume that ever occur? Why?
11.8. Let S $100,o' 30%, ?' 0.08, t 1, and & 0. Suppose the true expectd /1 stock is the 15%. 10. Compute Eufopean call prics, A, and B Set return on sikes of and ff $70,$80,$90,$100,$110,$120, $130. For each jtrike, cmpute ihe exyected ret'urn on the option. What effect does the sike hav on the option s
early exercise
y
expected
return?
11.9. Repeat the previous problem, except that for each sttike price, compute the expected retfn' on the option for times to expiraion of 3 months, 6 months, 1 year, and 2 years. What effect does time to mamrit.y have on the optiqn's expected feuarn? '''
.
. .
. .
.
.
.
11.10. Let $100,o' 30%, r 0.08, t 1, and & 0. Suppose the true expected ret'urn on te stock i 15%. Set ?1 10. Compute Europea put plices, A, and B for stlikes of $70,$80,$90,$100,$110,$120,and $130. For each stlike, compute the expected ret'urn on te option. What effect does the strike have on the option's epected return? .%
=
=i
=
=
=
=
11.11. Repeat the previous problem, except that for each strike price, compute the expected remrn on the option for times to expiration of 3 months, 6 months, 1 year,
368
k
Bl lqoM lAL OpTloN
11
PRIcI Nc:
APPEN DIX
and 2 years. What effect does time to maturity have on the option's return?
expected
0. Using equation (11.17) 1, and 11.12. Let S $100, c' 0.30, 1. 0.08, t tenminal node and Sudn- to compute to compute the probability of reaching a lisk-neutral distribution of year-l stock prices as in the price at that node, plot te ?1 10. 3 and n Figures 11.8 and 11.9 for
11.13.Repeat S3
the previous problem for n $80? 5'1 > $1202
50. What is the risk-neual
=
ure 11.4.
0.08, and & a. Using S $100,r prices? l-year forward and
expected stock plice using the binomial ee in Fig-
tisk-neual
=
0, what are the 4-month, 8-month,
=
lisk-neual
expected sttwk b. Velify your answers in (a)by computing the equation second, in the and binomial Use period. third price flrst, (11.17) to determine the probability of reaching each node. plice using the
11.15. Compute the l-year fomard lem 11.13.
so-stepbinomial
tree in Prob-
=
$
PRICI NG WORK?
369
What do you observe about the pattern of historical volatility over time? Does historical volatility move in tandem for different assets? stock 1. 0.08, apd t 0.30, r 11.20. Suppose that S $45, g $50, K will pay a $4 dividend in exactly 3 months. Compqte the price of European and A me lican call opions using a four-step binomial tree. 'l'he
=
=
=
=
=
11.A: PRICING OPTIONS WITH TRUE PROBABILITIES In tlus appendix we demonseate algebraically that computing the option price in a way using a as the expected rettlrn on the stock gives the correct option price. consistent dennition of )z, equation (11.6),we can rewrite equation (11.7)as the using
1
(A5' + #)
=
=
=
=
d
(,W
d
e-o'./f
=
rlt
-
d
C,, +
tt
rh
=
e e Cd + d
(z
-
e
r(t
Cu
-
Cd4
!1 11 d ctzA5' + e s,s rj d Since LS + B is the call price, we need only show that the expression in large parentheses is equal to one. From the desnitions of A and B we have -
erh d C,,+ jj JI tt d -
We can rewrite
j
(hs
erh
-
-
d
-
-
Cd=e rIt (A5' + B4
(11.4)as
=
11
e
-
1, compounded), t 8% (continuously 11.16. Suppose $95, r $100, K using the Explicitly 30%, arld & construct an B-periodbinomial tree 5%. cr Cox-lkoss-Rubinstein expressions for 11 and #: =
RISK-NEUTRAL
probability that
'
'
DOES
APPENDIX
11.14.We saw in Section 10.1 that the undiscounted equalsthe fonvard price. We will verify tis =
WHY
=
=
<
=
=
=
=
=
11.B:
+ B4
co
e
hs + erha
gcrtl,+
B) +
tcz
-
erhlhsj
=hs
+ B
This follows since the expresdion in large parentheses equals one.
Compute the prices of European and Amelican calls and puts. 1, 8% (continuouslycompounded), t 11.17. Suppose S $100, K $95, ?' binorniat using Explicitly and J 5%. 8-peliod the tz'ee construct 30%, an tr lognormal expressions for 11 and #: =
=
=
=
=
=
tt
=
e
r-n-.so'ljlt+Jl;
d
=
er-t--so.zjll-cx-
DOES RISK-NEUTRAL
There is a large and ltighly technical literattlre on risk-neutral pricing. The underlying economic idea is fairly easy to understand, however,
Compute the prices of Etlropean and Amelican calls and puts.
11.18. Obtain at least 5 years' worth of daily or weelly stock price data for a stock of your choice. a. Compute annual volatility using all tlle data. b. Compute nnnual volatil for each calendar year in your data. How does volatility val'y over tne? c. Compute annual volatility for the flrst and second half of each year in your data. How much variation is tere in your estimate? of daily data for at least three stocks' and if you can, one 11.19 Obtain at least s'years currency. Estimate armual volatility for each year for each asset in your data. .
APPENDIX 11.B: FHY PRICING WORX?
,
Utilityased
Valuatio
The starting point is that the wembeing of investors is not measured in dollars, but in titility Utility is a. measure f satisfaction. Economists say th t jny tors exhl'bit declining ?ncl/'*p7! tttilityl Starting from a given level of wealth, the uiility gaied from adding $1 to wealth is less tan the utility lost from takipg $1 away from wealth. we expect tat more dollars will male an investor happier, but that if we keep adding theprevious dollars. dollars, each additional dollarwill make theinvestorless happy t11a14 Declining marginal utility implies that investors are risk-averse, which means tat an investor will prefer a safer investment to a rislder investment that has the same rfhus,
370
k
BI NoM IAL OPTION
PRIcI NG:
11.B:
APPEN D IX
11
expected return. Since losses are more costly than gains are benehcial, a risk-averse investor will avoid a fair bet, which by desnition has equal expected gains and losses.s To illustate risk-neutral pricing, we imagine a world where there are tFo aisets, stock and a lisk-free risky bond. Investors are risk-averse. Suppose the ecnomy in a of will period be in one two states, a high state and a low state. flow do we value one We such world? need to know three things: a assets in 1. What utility value, expressed in terms of dollars today, does an investor attach to te marginal dollarreceived in each state in the f'umre? Denote the values of $1received in the high and low states as Uu and UL, respectively.6 Because the investor is risk- averse, $1 received in the high state is worth less than $1 received irl the low state, hence, UH < Uz,
SZCC
the risk-fl'ee bond Pays
WHY
DO ES
Price of bond
Qu x
=
1+
r
Qu QL
=i
=
p x Uu (1 pj -
Since there are only two possible states, we can value any f'uttlre cash flow using these state prices. The price of the lisky stock, Sftis
1+
a
Price of stock
=
QH
x
CH
+
x
(Q
QL
(11.23)
x 1
cash flows by the price.
Price of bond
(11.24)
1 QH
QL
+
Cu + (1 PICL Price of stock
px
=
-
CH
p x
=
X CH
QH
(1
+ E
+
-
(11.25)
plCz
QL X CL
Standard Discounted Cash Flow 'l'he standard discounted cash flow calculation entails cpmputing te security price by I the cas: of the stock, discounting the expected cash flow at te expected rate of gives this us 'eiurn.
#
X CH
+
(1
1+
a
a rewriting of equation bond price is
'his is simply the
1 .y
QL
371
The expected retunz on te stock is
(11.21)
X UL
$
1
=
=
3. What is the probability of each state oceurring? Denote the probability of the high btate as p.
'
1+
We can calculate rates of return by dividing expeted Thus, the iisk-free rate is
lislty
We begin by defning a state price as the price of a seeurity tat pays $1 only when particular state occurs. Let Qu be the price of a security that pays $1 when the l'tigh a state occurs, and QL the price of a security paying $1 when the low state occurs.? Since Uu and Ug are the value today of $1 in each state, the price we would pay is just the value times the probability that state is reached:
WO R K?
$1 in each state, we have
.
2. How many dollars will an asset pay in each state? Denpte the payoffs to the stock in each state Cu and CL.
RlS K-NEUTRAL P R1Cl N G
j
(11.22)
?.
-
#)W
= Price of stock it is obviously correct. Similarly,
hence, (11.25)', = Plice
Of
bond
Risk-Neutral Pricing s'rhisis arl example of Jensel : Ineqltality (seeAppendix C at the end investorhas a concave utility functon, which implies that e-(&(-r))< ULE(a-))
of this book). A risk-averse
the utility from receiving the The expected utility associated with a gamble, E'(&(-'r)), is less t.118.11 value of the gamble for &('@)J. sure, expected G-lkchnically Uu and &&are ratios of marginal utilities, discounted by the rate of time preference. However, you can think of them as simply converting fumre dollars in a particular state into dollars
today. 7erheseare
often called 'tArrow-Debreu'' Kenneth Arrow and Gerard bebreu.
securities,
named after Nobel-prize-winning economists
'l'he point of risk-neutral pricing is to sidestep the utility calculations above. We are looking fpr probabilities sch that when we use those probabilitieq to ompute expected expliit adjustments, and discount that ekpecttib at the riskcsh flsk ulity correci the will get free rate, then we answer. rl'hetrick is the following: lnstead of utility-weighting the cash :ows and compututilitjwkveight fc probabilities, creaing probaing xpetaiins, new we Use riskuneutral i this perform the Nw will how context. to bilities. jriing we see lzigh risk-neual probability of the define state, prices equation the in (11.21)to state # * as 'fltfr
t'lisk-neutlal''
,
,
p=
p
p x Uu Js + (1 p) x x -
=
UL
Qu pzz+ QL
k. BI NoM
372
PRI
Op-rl o?q
IAL
APPEN D IX
cl ?qG: 11
Now we compute the stock price by using the risk-neutral probabilities to compute expected cash llow, and then discounting at the risk-free rate. We have
# +Cu + (1 p 1 ?-
+
QH
4CL
cH
ps-l-t?s
=
QL
1
-l-
QHCH =
QH QHCH
=
RISK-NEUTRAL
PRICI NG WORK?
$
373
=
$180+ 0.48 x $30
0.52 x
=
$108
The expected ret'urn on the stock is therefore
L
a
QLCL
(1 + r)
By de:nition,
$95.544.
QLCL
+
E (.j)
$108 $95.544
?-
QL4
DOES
The expected cash flow on the stock in one period is
-l-
+
+
c
+ ps-l-ls
WHY
11.B:
shows that we can construct which is the price of stock, from equation (11.22). risky price them assets. to risk-neutral probabilities and use
if
=
-
13.037%
1=
13.037%, we will get the price
we discount E (S1) at the rat
'
rfhis
Risk-neutral
#
Example Table 11.2 contains assumptions for a numedcal example. Using equatipn 11.21, the state prices are Q;-= 0.48 x $0.98 = $0.4704.
state prips $0.4524, and r'
Qu
0.52 x $0.87
=
'
.
-
Bz
,
nus, using
Qu +
=
(11.26)
$0.4704 $0.9228
$0.'4524+
=
The risk-free rate is
?. rislty s'oclt
valuing the the stock is
=
1 0.9228
We can also verify that a call option on the stock can be valued using risk-neutral the value computed using tl'ue 0 52 x 0.87 x maxto,
Using risk-neual
$180 $130)+ 0.48 -
x
0.98
x maxto,
-2.
$30 $130)
(11.27)
$95.544
=
$?T!!' jqy('. rf rjf (jr: ('( !!:.qi (' ( li(.' ' )'!.('( rqr: . q.'' r' jhql ' lrqlrllqppillllhjllci j'( i (pp! ) .r.:. r( (..(..i.( ( ('.,:j,', . .j. .. .. ..jy . ygyy.glgyyj:jj.jqylj.j.gyyjjqjjj..t jt(:yj.. j( ... y.jg. yjt.y ygt kjjjj jtjttttv.lljlhlsryj.it.qijjj)jii k.)r! ry ,1::2,5;:Ir..!k ,:::1; .jj: jj tjyy jj,,yj jjjy..yj jjtlrpz),-/-' tjgy Ii..,l i,tIk:::,. rl:: . Ir.. 1E:::,.1.:. iies lk..llii1ii.1Ik:(!,-/,. iiIiiL 12::,. e Ik..,l a e iiIEII 9t)--:#1-.f....,.......i...-...)i--.i..?-'-..........'.----.?.i..)-.-.)kt.'-...'..:-...-.-z. .-t-y-yr)r.?77I1E77....,.....:.-.'$p!t.-.... .LLkiik..-. tk:;kkl)-.---..-.-. of economy. and the low h states hi9
pricing, we obtain
r0.49025 x maxto, $180 $130) + (1 0.49025) x maxto, $30 1.08366 -
C =
=
= $22.62
Using equation (11.22)the plice of
real probabllities
uslng
Plicing. Suppose the call has a strike of $130. Ten probabilities and utility weights is
C
1 = 8.366%
-
us'll 0.4524 x $180+ 0.4704 x $30 4)7* !' C' :(1* '('(!j'' ('' lilt!f ''('r/f i';''frr)f 7* )q' 91* j'(' r'' ;1* r'r)f !' T' r'
= $95.544
.
.
QL.
$0.4524 $0.4524 + $0.4704
*=
= 49.025% Now we can value the stock using p* instead of the true probabilities, and discount at the risk-fTee rate: 0.49025 x $180 + (1 0.49025) x $30 un 1.08366
'''
''
probability is
lisk-neutral
=
bond pays $1 in each state. The lisk-free valuins the risk-free bond is price, bond riskrfree the Bfj, equation (11.23) =
=
The
of the stotk
valuation
=
u
-
-
$130)1
.
= $22.62
'''
'
:
:
i .
'j/Ejyji ..t .
' E IE EEE E ...
'
:
'
' '
'
q-tr
'' ' ' ..
'
::
' ' '
.'
'
.
:
:
. . . . : .. .
:
.
:(....
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As with th binomial model, there are six inputs to the Black-scholes formula: S, the current price of the stock; K, the stn'ke price of the option', o', the volatility of the stock; t5, ?', the continuously compounded risk-free interest rate; F, the time to expiration; and the dividend yield on te stock. Nx) in the Black-scholes formula is the cumulative normal distribution funci.hat a num b er ran donily dr?n ffom a standard nonnal tion, which is the probabil distribution (i.e.,a nonnal distribution with rpean 0 and variance 1) will be less than x. Most spreadsheets have abuilt-in function for computing Nx). In Excel, the function is ttNormsDist'' The normal and cumulgtive normal distributipns are illustrated in Figure
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Let S $41, K $40, o' 0.3, r 8%, F 0.25 (3 months), r Example 12.1 ' call price, we obtain l the Black-scholes and J 0. Compung f'j jl lnt!l + (0.08 0 + () aa) x 0.25 td oxo.:?.sx N e$41 ',@ x 0.3 0.25 tt t ().32 1l) /..!. + (0.08 0 ln(4n) :2 ) x 0.25 li $3.399 $40 x e x x t) 0.3 0.25 ki j, =
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6.969 6.966
lThecilpHceherecubp-omputedusingieBlack-scholes
BSCaIIIS',X', c, r, t, J)
=
formulacall spreadsheetformula,
BSCall(41, 40, 0.3, 0.08, 0.25, 0)
=
$3.399
BSCZI:
378
% TH E Bl-Acu-scHol-Es
FonM
$
APPLYI N G TH E FO RM U LA TO OTH ER ASSETS
uLA
There is one input which does not appearin the Black-scholes formula, namely the expeeted ret'urn on the stock. You might guess that stocks with a high beta would have a higher expeeted returfl; hence, options on these stocks would have a higher probability of settlement in-the-money. The ltigher expected return would seem to imply a higher option price. HoWever, as we saw in Section 11.2, a high stockbeta implies a high option beta, so the discount rate for the expected payoff to such an option is correspondingly of the key insights from the Black-scholes analysis-is that net result-one greater. larger intlevant: is The average payoff to options on high beta stocks is exactly beta 'l'he
offset by the larger discount rate.
379
'When Is the Black-scholes Formula Valid? Derivations of the Black-scholes formula make a npmber of assumptions that can be sorted into two groups: assumptions about how the stock price is distributed, and assumptions about the economic environment. For the version of the formula we have presented, assumptions about the distribution of the stock price include the follow-
ing: Continuously compounded returns on te stock are normally distzibuted and independent over t'ime. (As discussed in Chapter 11, we assume there are no in the stock price.) Ejumps''
* The volatility of continuously compounded remrns is ltnown and constant.
Put Options
@Fumre dividends >re known, either as a dollar amount or as a tixeddividend yield.
The Black-scholes formula for a European put option is
ps zt7
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o. r '
1
J)
r,
=
Ke-rTx-dz)
se-sTN-dt)
-
(12.3)
where #1 and dz are given by equations (12.2a)and (12.2b). Since the Black-scholes call and put prices, equations (12.1) and (12.3),are for European options, put-call parity must hold: P (S, .J(', c',
l',
F
)
,
CS K
=
.
(7'
,
,
1*,
T
)+
,
Ke-c
X
Se-&T
-
(124) .
This version of the formula follows from equations (12.1)and (12.3),together with the (This equation says that the probability of a fact that for any x, 1 Nx) = N-x). normitl being above x, 1 Nx), equals the distribution standard from the random draw #(-x).) below of being probability a draw -
-
--:,
Example 12.2 Using the same inputs as in Example 12.1, the put price is $1.607. We can compute the put price in two ways. First, computing it using equation (12.3), we obtninz
$4:4:-0.08x0.25x $4j.-ox0.25x
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-
.
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Assumptions about the econornic environment include these: 'l'he
@
risk-ft'ee rate is known and constant.
* There are no transaction costs or taxes. It is possible to short-sell costlessly and to borrow at the
lisk-free
rate.
Many of these assumptions can easily be relaxed. For example, with a small change in the formula, we can permlt' t.h vo l a tility and interest rate to tal'y over time in a known way. In Appendix 10.A we discussed why, even though t ere are tues, tax rates do not appear in the binomial formula; the same argument applies to the Black-scholes
fonnula.
about te stock price As a practical matter, te flrst set of assumptions-those disMbution-are the most crucial. Most academic and practioner research oh option pricing concenates on relaxing these assumptions. They will also be our foctts when we discuss empirical evidence. You should keep in mind that almost any valuation procedure, including ordinary discounted cash flom is based on assumptions that appear song; the interesting question is how well the procedure works in practice.
.
0.3 0.25
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(0.08 0 + -
0.32)0 2
.
:5
12.2 APPLYING THt FORMULA
:;:z$j.6(c
0.3 0.25
Computing the price using put-call parity, equation
#(41, 40, 0.3, 0.08, 0.25, 0)
=
(12.4),we have
3.339 + 40c -0.08x0.25
-
41
k
= $1.607
2The put price here can be computed using the Black-scholes
BSPut(5', K, (r, r, t, d)
=
put spreadsheet fonnula, BSPut:
BSPut(4l, 40, 0.3, 0.08, 0.25, 0)
=
$1.607
TO OTHER
ASSETS
The Black-scholes formula is often thought of as a formulafor pricing European options on stocks. Specifcally,-quations (12.1)and (12.3)provide the priee of a call and put option, respectively, on a stock paying continuous dividends. ln practice, we also want to be able to price European options on stocks paying discrete dividends, optipns on We have already seen in Chapter 10, Table 10.1, f'utwes, and options on c=encies. adapted that the binomial model can be to different underlying assets by adjusting the adjustments work in tlle Black-scholes formula. dividend yield. The snme Black-scholes the We can rewrite db in formula, equation (12.2a).as 3nSe-&T(Ke-rT)
dj
=
cJF
+ 1(r2r 2
'
%.THE
380
APPLYI N G TH E FORM U LA TO OTH ER ASSETS
FORMULA
BLACK-SCHOLES
When #1 is rewritten in this way, it is apparent that the dividend yield enters the fonuula Se-&T, and the interest rate enters the formula only only to discount the stock price, as Ke-rl-. Notice also that volatility enters only as c'2F. to discount the strike price, as Se-&T The prepaid fonvard prices for the stock and strike asset are Foppuj Ke-r-. Then we can write the Black-scholes formula, equaion (12.1), andFoPvK) entirely in terms of prepaid forward prices and trVF:3 =
$
381
Compared to the $3.399plice computed in Example 12.1, the dividend reduces price by about $1.64,or over half the amount of the dividend. Note that this option the of a European option. An American option might be exercisedjust prior to price is te dividend. and hence would have a greater plice. the
=
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c'(F() w(s)
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We can price an option on a currency by replacing the dividend yield with the foreign interest rate. If the spot exchange rate is x (expressed as domestic currency per unit of foreign currency), and the foreign cuaency interest rate is ?-y, the prepaid forward price
1c2r 2
cJ-r
for the cun-ency is
This version of the formula is interesting because the dividend yield and the interest rate do not appear explicitly; they are implicitly inoporated into te prepaid forwgrd pdces. To price options on underlying assets other t.11%stocks, we can use equation (12.5) in conjunction with the forward price formulas f'rom Chapters 5 and 6. For all of the examples in this chapter, we will have a sfrike plice denominated in cash, so that FP
0.F
(X')
=
Options on Cttrencies
sz'0,
#
C@ ,
,
c'
-
,
?.
F
-
0.25 (3 8%, and r $40, tr 0.3, r Suppose S $41, K r Example 12.3 't:!1months). The stock will other makes month, but 1 payouts in over dividend no ))t pay a $3 tt .the dividend is vdue of l) i the life of the option (hence,8 0). The present .l( =
=
=
=
yt'g tt
=
$3/-
()gg x j / j :t '
=
jnxjKj
$2.98
lj?settingthe stockprice in the Black-scholes formula equal to t)tBlack-scholes ca.ll price is (1 . $1.763.
$41 $2.98 -
=
$38.02,the
$
P (x, K, o- ?' F r.f ) ,
,
(r G
d1
2
(126)
)
.
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-
WF
o.V-f
-
,
=
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,
Ke-rT
-
xc-rzF
,
$0.92/6, K $0.9, o' Suppose the spot exchange rate is x )1 Example 12.4 ).) ' ' euro-denominated rate), F 1, and dollar 3.2% interest 6% 0.10, (the r ltlze rr t!q price of a dollar-denominated ekuo call is $0.0606,and the price of a tt. imerest rate). t- donar-denominad etuo put is $0.0172. % =
'1
=
=
=
=
=
'r'he
Options on Futttres The prepaid forward pri.e for a futures conlact is just the present value of the f'utures . .' . . . .. cokttat option f'utur:j European Thus, by . sing the futures price We l4ce pric on a a P q r as the stock price and setting the dividend yield equal t te tk-free rat. The resulting formula is also known as the Black formula: .
C(F about the relative time-r values 3We can also 1etV(T') o' T represent total volatility-uncertainty of the underlying and stlike assets--over the life of the option. The option price can then be written solely in tenns of F1vS), FOPpLK), and 7(F). This gives us a minimalistversion of the Black-scholes fonnula: To plice an option you need to lnow the prepaid fonvard plices of the underlying asset and the strike asset, and the relative volatility of tlte two.
+
Ke-r-Nd
-
'
where PV0,w(Div) is the present value of dividends payable over the life of the option. Thus, using equation (12.5),we can price a European option with discrete dividends by subacting the present value of dividends from the stock price, and entering the result into the formula in place of the stock price. The use of the prepaid forward plice here should remind you of the approach to plicing options on dividend-paying stocks in Section 11.5.
Pvmiv)
=
)
1
This formula for the price of a European call on currencies is called the GnrmanKohlhagen model, after Gnrman and Kolllhagen (1983). The price of a European currency put is obtained using patity:
,%
'.i
d3
xe-rf-Nd
=
W en a stoc k m ales discrete dividend payments, the prepaid'forward plice is Fvlj = PVo,w(Div)
=
=
./
dg
Options on Stocks with Discrete Dividends
=
?
.70d
=
)
-
,
Ke-rT
(&
.-ryw
w@) Using equation (12.5),the Black-schole formula becomes
K c r F
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=
#1
=
ln(F/#)
=
d,
Fe-rTNdkj
=
-
+ 1o'2F
clF d,
-
c,/f
Ke-rl-Ndzj
2
(12.7)
%.THE
382 &'
op-rlolqGREEKS %
FORMULA
BLAcx-scHoLEs
Theta 04 measures the change in the option price when there is a decrease in the time to maturity of 1 day.
The put price is obtained using the parity relationship for options on futures; Ke-rT Fe-rl#(F K tr ?' F ?') = C(F, K, (r, r, F, r) + -
3
'
?
'
lkho (p) measures the change in the option price when there is an increase in the interest rate of 1 percentage point (100basis points).
jk'i 1)*
tu and Suppose the l-yearfutures pricefornatural gas is $2.10 q Example 12.5 1, 0.055, T 0.25, r $2.10, K $2.10, tr the volatility is 0.25. We have F ltl and 0.055 tthedividend yield is set to equal the interest rate). The Black-scholes call Q 0 plice and put price are both $0.197721.
Psi (T) measures the change in the option price when there is an increase in the continlpus dividend yield of 1 percentage point (100basis points).
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=
=
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=
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12.3 OPTION
383
GREEKS
Option Greeks are fonuulas that express the change in the option price when an input inputs/ One importpnt use of to the fonnula changes, tnking aj :xd all the other lisk exmuple, bnnk with market-mnking For a Greek measures is to assess exposure. porifolio of options wold Want to understand its exposure to stock price changes, a interest rates, volatility, etc. A portfolio manager wants to know what happens to the vlue of a jrtfol'i of stock index options if there is a change in the level of the stock index. An options investor would like to lnow how interest rate changes and volatility changes affect prot and loss. Keep in mind that the Greek measures by assumption change only one input at a time. In real life, we would expect interestrates and stock prices, for exaple, to change 012% one input together. The Greeks answer the questfon, what happens when one t7?7t changes? The actual fonnulas for the Greeks appear inAppendix IZ.B. Greek measures can be computed for options on any kind of underlying asset, but we will ftus here on stock Options.
Etvolatility''
and Ausef'ul m emonic device forremembering some of these is that tttime.'' often and Also used letter, do is frst to denote the share the snme as tlrho.'' and is first letter in the interest rate We will discuss each Greek measure in t'urn, assuming for simplicity that we are tnlking about the Greekfor aptlrchased option. The Greekfor a wlitten option is opposite in sign to tat for the same purchased option. E'vega''
Gr''
Gtheta''
We have already encountered delta in Chapter 10, where we defined it as the number of shares in the portfolio that replicates the option. For a call option, delta is positive: As the stockprice increases, the call price increases. Deltais also the sensitivity of the option price to a change in the stockprice: 1f:.14option is replicated wit.h 50 shares, the option should exhibit the pric sensitivity of approximately 50 shares. You can tink of delta as the share-equivalent of the option. Figure 12.1 represents the behavior of delta for three options with different times illustrates that an in-the-money option will be more sensitive to to expiration. 'I'he sgure the syock price t.11m1 an out-of-te-money option. tf an option is deep in-the-money (i.e., t.he stoclprice is Mghrelative to te strike price), itis likely to be exercised and hence the option should behave much like a leveraged position in a f'ull share. Delta approaches 1 in this case apd the share-equivalent of the option is 1. If the option is out-of-the mopey, itis unlikely to be exercised and the option has alow pric, beh:ving like aposition wit case delta is approximately 0 and the share-equivzent is 0. A.n very few shares. In at-the-money opuon may or may not be exerised and, hene, behaves like a position with betwen 0 and 1 share. This behavior of delta can be Feen in Figure 12.1. Note that as time to expiration increases, delta is less at ltigh stock prices and greater at low stock prices. This behavior.of delta reflects the fact tat, for ihe depicted options that have greater time to expiration, the lilelihood is gregter that arl out-of-te money option will evenmally become in-the-money, and the likelihood is greater that an in-the-money opion will eventually become out-of-the-money. We can use the inyepretation of delta as a shgre-equivalent to intrpret the BlackScholes price. The fonnula both pries the option nd also tells us what position in the stock and borrowing is equivalent to the option. The formula for the call delta is Delta
'
'tis
De6nl'tion of the Greeks measured are a matter of convention. Thus, when we The units in which changes will also provide the assumed unit of change. desne a Greek measure, we 'are
Delta (A) measures the option price change when the stock price increases by
Gamma (r) measures the change in delta when the stock price incress
$1.
by $1.
Vega measures the chanye in the ootion orice when there is an increas in vplatility pint.D of one percentage ..1$.
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ttlappa''
1lS Vega.
xlambda''
If we hold e-tNdqj
e
=
-spyy
1
;
shares and borrow Ke-rTNdz)
dollars, the cost of tlzis portfolio
is
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)
-
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op-rlox GREEIIS k
FORMULA
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at-the-money 4o-strike options wth diferent times to expiraton.
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a mqrtths + 1 year . 3 years -s
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3 months -+. 1 year . a y ears
0.1 () 20
25
30
35
40 Stock Prie
45
50
55
-0.9 -1.0
20
25
30
35
60
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50
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Gnmma-the change in delta as the stock price changes-is always positive purchased call or put. As te stock price increases, delta increases. This behavior for a both be Figures 12.1 and 12.2. For a call, delta approaches 1 as the stock seen in can For increasest price a put, delta approaches 0 as the stock price increases. Because of parity, put-call gnmma is the same for a European call and put with the same strike price time expiration. to and Figure 12.3 graphs call gnmmas for options with three different expirations. Deep in-the-money options have a delta of about 1, and, hence, a gamma of about zero. (lf delta is 1, it cannot change much as te stock price changes.) Similarly deep out-ofthe-money options have a delta of about 0 and, hence, a gamma of about 0. The large gamma for the 3-month optio in Figure 12.3 corresponds to the sieep increase in delta for the same option in Figure 12.1.
i
:
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I'his is tl)e Black-scholes pzice. Thus, the pieces of the formula tell us what psition in the stock and borrowing syntheticdly recreates the call. Figure 12.1 shois that delta changes with the stock pdcek so as the stock price moves, the replicating portfolio changes and must be adjusted dynmically. We also saw this in Chkter 10. Delta for a put option is negative, so a stock price increase reduces the put price. relaiionship Thls afl be seell in Fiure 12.2. Since the put delta is jut te call delta e-&T minus (fromput-call parityl, Figure 12.2 behaves similarly to Figfe 12.1.
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0.03 0.02
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20
25
30
35
40 Stock Price
45
($)
50
55
60
% THE
386
BLACK-SCHOLES
op'rlolq GREEKS %
FORMULA
Options generally-but not always-become less valuable as time to expiration decreases. Figure 12.5 depicts the call price for out-of-the-money, at-the-money, and inthe-money options as a function of the time to expiration. For the at-the-money (strike = $40) option, time decay is most rapid at expiration. For the others, time decay is steady. Figure 12.6 graphs theta explicitly for three different times to expiration, more
Theta
A delivative for which gamma is always positive is said to be convex. If gnmma is positive, then delta is always increasing, and a graph of the plice f'uncdon will have curvattlre hk'e that of the cross section of a bowl. An increase'in volatility raises the price of a call or put option. Vega measures sensitivity of the option price to volatility. Figure 12.4 shows that vega tends to be the options, and greater for options wit moderate ta wit short at-the-money for g'reater expiration.6 of put-call parity, vega, lilce gnmma, is the same for calls Because times to price and time to expiration. with strike the artd puts same When calculating vega, it is important to be clear about units: How large is the assumed change in volatility? lt is common to expreqs vega as the change in option price for a one percentage #t?f?ll (0.01)change in volatilit.v Figure 12.4 follows this
Vega
shoWing that time decay is greatest for the at-the-money short-term option. Time decay can be positive for European option! in some special cases. Deep-inthe-money kall optios on an asset with a high dividend yield and deep-in-te-money puts are two examples. ln both cases we would want to exercise the options early if possible. Since we cnnot, the option effectively becomes a T-bill, appreciating as it gets close to expiration. This efect is evident in Figure 12.7, in which the in-the-money tso-strikel put becorizes more valuable, other things equal, as expiration approaches. Figure 12.8 on page 390 graphs the put theta explicitly, illustrating the positive theta. When interpreting theta we need to know how long is the assumed change in time. Figures 12.6 and 12.8 are computed assuming a l-day change in time to expiration. It is also common in practice to compute theta over longer peliods, such as 10 days.
convention.
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Rho is positive for an ordinary stock call option. Exercising a call entails paying
Rho
the fixed strike price tti receive te stock; a higher intrest rate reduces the present value of the strike. Similrly, for a put, rho is negative, ince the put entitles the owner to
3 montlu 1 year -+.a years
-F-
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50
55
60
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2 6Be aware that neitlzer result is true for very long-lived options. Wit.h a 20-year opton, for example, of prices vega is greatest for out-of-the-money calls and lower than that for a 3-year call for the range
in the hgure. ?vega is the derivative of the option price with respectto o'. lt is expressed as the result of a percentage point change in volatility by dividing the derivative by 100.
0
0
0.5
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2.0
(years)
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=
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=
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=
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$
388
'..' . .'.'i ---;2;:;!jj('. ;'j :')'j(yy. ;'-' yi' jjr!jg!f jj:y:y:j,f ''. ''2* :''. j'F' 7.* ;';L';X :''t'LL'' y' 'sr'E ' '' Cl :(E' : q
y' yjy'.yjjjyt'. yy('yyjyyy'. :''L'' ktf j'1* ;'yjjy.y)jyy::jj.f r' @t' .i)' t'j;f s.r;llf lf 'j' f' 7777771: kyjly,f l'y y'; jy'. p' l')f (' (yj' ;'5* 1* r' q' r'.:l y' t'Cyf ;q;7(7' i'!'ll''qpl'l $)'(! qt'lr j)rqi7f 7* 'L (;'FY (5r7r:: f'( C;'t' E (Ei iiE ( i .!( ('.'.i.('.(' '' (' E''E7:q:4* jjjjjyrjjf rjjyyrjj:yr:jr.f :rjjy;;f (' jjjjtf ((j.y;y'. ryf p)' )' )(. 'E;. : T'(: 1*. ).jjjf ;'' j'yjf (tf j': (i' ; i()'' '(i' .rrr7rq7l j;jjf )' pjIj:qy'. y'. ptjjk(-,jjjr'. q' (' y' r' jljj!!jj,f )5$* ttf t'.yyyk'. t'. tk;:)j'.r(j'y;;;jj!rrry'.. r'' !i : r. E : ;. E I jI'.i ! l . ;. E !i Ey). jj ;.jy;(.y.j(y.. ((i((j..(. rr'i qj..ji ' . r .j. y (. .;. .j-( .4Ij1::::;;;j.. ytj . . . j )j. ( y. ...y. y )).. )tjjr ! i. jjjjj:;jj:j, y));. j . :j,)L,. j . j r EEilrr.li.rl rk;jjjjy. jjyjy.!...jjyy yy E@! y.(). ii( ig. . y .. liyi ..jy;y(jj;jyr. . . ).. . . .... . 'q. . . . . . Put prices for options with different strikes at .
,(jy'. -' .',,yy'(j;'j),-. ........' ''' '
E ;';'' t'jrf (' 1* 4);* (k'E j)' ' ' ' ' 1! E F'' f' ()'!i!7q7'7:T!(rq'. 1* !' )(' (@' d!)' 'q;d 71!: ( C'lqIP'E E j'jjf y' ''q' )' jjf '1!'l)' ;'11* (' yf t'kjd ((' i'g': ' ' E'EE i! (;jE' E((i' E'.. .' E;(( !..(.i'.'!E'tjyjj '';'EEE'i(iE!E(i( y' k' (k)'. E ;I!..(.EEE;.E(. i.E1. yytj(r'. yy': y))f ttrrf /')';' k--)'. jj'. y' j')j'jjy'yyytj r' q! t/j.(:.'q;i(t':q..jiE(tl. !:(ilqrliqilifi IIIIF:PiII!)X 4111:2-:1.1f* 11114* jI1I:!'yjf jg.' jj'. (' yr;td y'. . . y jjy . j. .. . . ''.'(' ( . ( ' @ )' t'()' r' y' ...j'. .g' .r( g .. EE y.. iE gygg. .j..g. jjyjjjyjj jjj (yyygj y.yq.y y jj ( ;.j @j.. EE i E' E' ElE((. @;qjj ' . k;iII ;. tk k yj jj jjjjjj g . tyyg: jj.. jjllii!tl i':iiIiT. j.;..jgE.qj.jy. g;jEj. 1)2)7). E ,444,j;1111:. tgyj :.. .;;E . !i1I::::;;k,q. .. j.E.g E (q(.. .. j). .y qq! y t.:q.. ..;E yt.tj, ; yqrt- ....:... . .. . t...... . ; ; . . . . .g.t. .j1krr:,!:' -'' -yry-'.
-j'; ---:::k;!!!:''''q biq' 'j' y' f'. E' yyf ';')':' jtdjtE.
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. -
- -
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.. . . . Thta for call options with different expirations at different stock prices. Assumes /( = $40, o' = 3091, 0. r = 8%, and
........:
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=
-0.004
Strike
=
strjxe
+
=
wstrike
j)
..
. '+-
10
=
30 4:
so
8
=
=0.
=
=
.
jj:yj .jjyyyyjjk, yjty.yjjr, .yj.,kjj, jjjyy:),,tjjjs
diferent times to expiration. Assumes $40, tr 3094, . r 8% and
-().c(s
389
' E' . ' . E ' i E ;(E . : . . :
-
. yy . y yy. . E .... ..y
.
...
k.
GREEKS
opTlox
FORMU LA
BLACK-SCHOLES
THE
.
7
-0.006
6
-0.008
5
-0.01
4
-0.012
3
-0.014
-+. 3 months + 1 year .. 3 years
-0.016 -0.018
20
25
1 30
35
45 40 Stock Pdce ($)
50
55
0
60
receive cash, and the present value of this is lower with a higher interest rate. Figtlre 12.9 shows that as the time to expiration increases and as a call option becomes more in-the-money, rho is greater. interest Figure 12.9 assumes a one percentage point (100basis point) change in te
0
0.5
optionposiuons is easy to evaluate. stock, where th quantity underlying
expiratipn. Figure 12.10 shows that the absolute value of psi increases with time to matttrity, but time with short effect to little a An increase in the dividend yield has that Figure mattlrity. Note with time to dividends lost by not owning the stock increase 12.10 is a mirror image of Figure 12.9.
Greek Measures for Portfolios l/7e individual polfolio The Grcck measure of a porlfolio is l/?d stml of /d Greeks of lisk of complicated This relationship is important because it mans that the ct/?llpa?ld?yr..
2.0
2.5
3.0
(years)
11
opuons with a single we have
)f ,
tt
ati Af i= 1
The same relaon
Psi
up, is positive.
For a portfolio containing of each option is given by
Aportfolio=
rate.
Psi is negative for an ordinary syock call option. Acall entitles the holder to receive exercising the opion. stock, but withoutreceiving the dividends paid on the stockpriorto the received is lower, greater the dividend yield. Thus, the present value of the stock to be in exchange forcash. stockin f'uture te Owning aput entitles an obligation to deliver the dividend yield goes the when down dlivered goes The present value of the stock to be psi for a put yield is Hence, greater. so the put is more valuable When the dividend
1.0 1.5 Time to Fxpiration
q Example 12.6
j'lt
*.
.
holds
eue for te other Greeks as well.
Table 12.2 onpage 392 lists Grekmeasttres for a40-45bu1l spread. zm-stnke call less those for the 45-stnke cll. '
'
.
.
.
t Greeks for the spread are Greeks for the ')) . E(
.
.
.
.
.
%.
Option Elasticity An option is an alternative to investing in the stock. Delta tells us te dolla.r risk of the option relative to te stock: If the stock price changes by $1, by how much does the option price change? The option elasticity, by comparison, tells us the risk of the option relative to the stock in percentage terms: lf the stock price changes by 1%, what is the percentage change in the value of the option?
%.THE
390
op-rlolqGREEKS %.
FORMULA
BLACK-SCHOLES
''.' ' '' ...'
l!)'q?lf (rjjf (' j((q' ;')' f' JFY @' !' 1* (isif i'('( j:' (k!)@i (' (r' pir!lE lijllrii'Ii' E( ( ' !'' E(:! ( ?' tqf ;l(: (' ( (' (( E!''(fkl)( $::. (' JE( 'pi 1I!1:pFq'. ;)-)))t(''111L1'-i1Il'. ))' lili:'iiittf @q' t';'. .y q))'' . (';'' . !.. ..'i j . (: i jy.' .y ' . .y . y.... ... @ ' y ijt. ..)(.;.tr@;)/ry)!kt.j)/.Fii':)
:''' ----' .-it-'. -' IFX -'. t'!'!il:::::;'E;'''T' 69* '5* iyjEE ' ' -l:::::.,f 'E' fr''f 7* jjf f' -.'E kf E E :I' l);E i 'l.'jjE.
391
'.'''
:'
y' y.yjyy'. (' :'j'' k' jjf 'q s' Eiilr'ii.Ep' 87*:.71 .'(. tjrjrf y'. yyyjy'jtyjy,jytyj'y.yjjjj.. yt'. qn (' q' p''' !' lrf )'))#' :?'q77:!77!: ! )'1 i i'' ! ! qrpr!: i F' i j)y'. jjtf ltkttf .yry:' )' r' r-r!:f tllf TE ' prq jjjjjf rf t')'j'7* j'1* )'yy(y'.r'' ykyyf tjjyy,r,j.f y' '!qf jl' t'y'-' )' y'y .r.' 'r' .q':' (l.'. i(ti:. ,i' ! !(-...r q.qii'; (:k i 'j ..','... r' tt'. ;.. ,jij(jkjjjjj. jg ( y .. jjjjj:tjjjj, r .yjj,j yj),...yrj 'yjj;..:j (yjy j,.j;j . t.!j:j:jj:;r;jk. ) ). jy ' E!Ijjjqr; .. (: . j. . jj. y j . t)q . . jjjij!jjj j . y jg. . :.L-,....,.;. . ..,...........LL.L y yj. yy.yy p'ltq.. . y.tj. . ... ....... Theta for put options --' g'-:jjjjj;;k,,. --'' '''rtf ?'r' :'yjgyjjy'. Eyf y'ljq-yyfjt'l'yjjjf y'. f:)' l''tE :'C' F' y' q' ;'(' tFf ?' jE i ': -',,.yjjyy,,' ' -:;;;;jjjjj'!.':''
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i
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. ..
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-
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a
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().01
with different expirations Mtdifferent stock prices. Assumes 309% # 40, (r r 8%, and & = 0.
.
() 008
-+-
w.
0.006
=
=
.
--
- E . .. -.. - - -EE .y -E E qy .. -y y ky. jy. .g.- y.y;( y - yyyj. . , . ... .. .. . -.
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a months 1 year 3 years
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:' ' ' ' : : ' : ' ' E:E ' E' E :: ;' (Ei.ii ' f(i: '' '''' .E . .: E : . E Ei EE E E .. EE . .. . . .. . . . .j.E.ttj(.(j.j..Eg.rj(q.j.j'.j.y(.,.;:..jE().yj!.t.r.;y'(;y,yt.'jt.yyy(',.()j ...'. ..E.;i.)''i''!(.j(.E.jy.kE:E;Eji!'t;E.i:(jyy!,,r.EF.Er.),';:j.'.y(.y'#.itk.Ett).E,t)yyj(@.yj.yk...
liliiiii. r'.iiI!: y. ''.! lr ..14444. -
-@(. -..-(--. - . . -. . . - - -. . .. .
' . . . . . . .. . Rho for call options with different maturities at difrent stock prices. Msumes # $40, o' 3096, r = 8%, and 8 0. .
.
'u
E
' E(E 'E E. ..
--- .
. ..
. -
..
.. .
.
. .-
.
.
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-
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--
.
.
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0.8 0.7
=
.'
...
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.
a months
1 year 3 vears
=
0.6
=
=
..+-
.
..:
)lEq:i:-.lllg.:l,..
0.004 0.002
0.5
0
0.4
-0.002
0.3
-0.004
' -0.006
0.2
-0.008
0.1
-0.01
O
20
.'
.
c ltange
35
i n op tion price
changein stock
=
50
40 45 Stock Piice ($)
If the stock price changes by
Dollar risk of the option Pjj js ' . .
30
25
)' t')'.
6,
the
change
in the option
. . .. . price x option
tlelta
.
.
,r
.
.
.
.
.
.
.
The option elasticity, dnoted tjj'jjj
.
11 $6.961. Deltayis 0.6911. lf we own optlon; io buy 1000 shares of stock, th delta of the . . . ..j') ... .. . . . . . is l position . t 1000 x A 691.1 shares of stock ). . of 691 shares. l.f '' Thus the option position at this stock price has a ).;t t)t :
. .
=
-
-;
I'share-equivalent''
,,
jrjlthe
stock pzice changes by
)) l),i
$0.50,we expect an option price ch>nge of a 1000 x A x
$0.50
=
$345.55
k
50
.55
60
(5)
Q
C by f), is the ratio of these two:
k % change in option pnce jyygyggggy ............ % change in stock price
.l
r;:;:;::::yc .%
S
:;;;;;;;;;
SL
C
The elasticity tells us the percentage change in the option for a 1% change in the stock. lt is effectively a mepsure of the leverag implicit in the option. For a call, f2 k: 1. We saF in Chapter 10 that a call option is replicated by a levered investment in the stck. A leverd position i asset is always rislier than the underlying ajst> r. Also, th implicit levefge in the fition becomes greater as the option is mor out-o'f-the-money. Thus, f2 decfeases as the strike price decreases. For a put f2 S 0. This ocurs because the replicating position for a put option involves shorting the stock. .
'
and gamma. This accurate measure of the option price c hange is obtained by using both delta 'delta-gamma approximaton'' is discussed in Chapter 13.
8A more
45
rislt of the option The optlon elasticity computes the percentage change option plice relative in the to the percentage change in the stock plice. Th percentage in plice stock change the is simply ejS. I'he percentage change in the option price is dollar hange the option price, cA, divided by the option plice, C'. the in
=
=
.
.
40 Stock Price
Percentage
=
E
3
6A
=
j)
30
25
@
ttd Example 12.7 thatthe stockprice is S $41,the strike pdce is K $40, suppose r lisk-free 1, the time to expiration is F 1* 71) 0.08, rate is 0.30, the C)tVOlatility is o' option price the chapter, is in the d the dividend yield is & 0. As we saw ellier li .j an =
20
60
55
gMathematically,
this follows since S
-
Se-'t N(#l )
>
C(J).
k
392
oraTlox
FORMULA
THE BLAClt-SCHOLES
%.
GREEKS
393
('E
.'''.';'
1, and ) Example 12.8 Suppose S 0.30, r 0.08, F $40, ty $41, K bjtt8 0. The option price is $6.961and A 0.6911. Hence, the all eljticity ij
.'-.-:t;;;1!!::'-' -'..' .*1* :''''
;':i!ii'-' -' -'E(-''' ;')E-''!!1::::;kiE. 'j'y'E .111:::11..--* y' -'E E.'Ef 1* '.F'i !' ;'S' :'fqf f!' )' :!i!;;f rf q' 7* J' j)' r' jld y' (' (7* p' fL,'.Lf'. jj':ti Jj-j).lg(' t'!' j')''j' y' (' lryd )''i'E )' tk.' ))' Err':.(: 7741* )t!-rl' !'' rrrf '?' j)f ',-'' t'' t'ryf i'jqlE!ElE:srrtrlsf rq!prf ;':('ptf 5!(?' ( (E (E E lq !!q('g.(!l' i'Eqq !E(I(;'y((( itg g' !lr' ' ' ''E 'EE(; !' r'('.()('.). . .((.(;g..( (kiE....' ((T' q! ()17)! ;EE'EE.E'E.CE .. . . . ! .. . .. .i ! ll)..E i . EE Ey (q .y ;E; !iE.EEE IE ;, !. )!(@. E.EE.E.EE.i; EE.;E.EE'.E(. .. q.k )p ;-!!E:, gi ;. i! r! '):ktjt:.jy)y. . ;.y(.j .. 5:-. y; ((j. (y y: ;.q . .. -' j j)!'iiE. ; . y p @ E' t':'--:'qi ;ti.. .. ..p - ii...;.. l :- ' ;; ;.; ' . . .. . . . . . -'. ----:;-.ik(t?'. --' '
(21(::,-* jjf j't)t.tjk))y()yt'..' yrf li-.llf )' :2::-*-))r*. ))---('-' )-tk(i'k7'7I!It'. 1111k,-k1111*. ()' y' r'rqrq!qd r)' y)jr-' ('yj. 1* (')r t'!!$1iqiqq,.' r r r!.jyj.. l ' '. (jy j(j.y. .tj:. )y;.k. .y. j j.ty .t'-t' r;ty .rk. . y . 1IIE:1iilt:.. . . 11115E1!:.. . ' '-).'. ).; i '. ... '. ri pi.1111!.. .i. ))--. . .-.. )).. .
ilf .' '
'
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.
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-. -
.
-
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-
-
'-
.
-
.
.. . ..
E. -
-
'
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--
:
:
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- -
'
.
.
-
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=
...,.......z
..
;:,
' :
' ' :... : : : .
: .
.. :
:
: ---'-
.
. . . ...
:.
:
- .
.
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-.
. . ; -. : . .
- -. . . .
'i ' '
yE tjE E
:
.
-
.
:
- -
:.
-.
.
tsi.
. -. .-
-
.
.. .
- -
. - .
.
Psi for call options with differet maturities at different stock prices. Msumes # = $40, = c 309% r 8%, and = 0
.
=
. .
--...
n 4
=
.
=
.''''
f2 't'he
-0.6
$41 x 0.69 1l $6.961
=
=
$2.886and 2 of $41 x f) $2.886
.
4.071
=
-0.3089.,
put has a price of
t). ti) #t jil j
z
=
=
.()
)i' (t 'j,. ;. tji
() n
=
=
'
'
'
hence, the elasticity
is
.
-0.3089
%.
-4.389
=
=
Figure 12.11 shows thebehavior of elasticity for a call, varyingboth the stockprice andtime to expiration. The 3-monthout-of-the-money calls haveelasticities exceeding 8.
-0.8
For longer time-to-expiration of the option.
-1 -1.2
The volatility
options, elasticity is much less sensitive to the moneyness
The volatility
of an option
qf an
volatility of te stock:
-1.4
-+. 3 ponths .+- 1 year u- 3 years
-1.6
.
Joption
j'((y'. y' j)'(.j'i-i', .j' yyd (' jjjLj)jj,j'. jyyyjj'. jj;jjjjj'. t')'
E'yE ()' yjd (' ;(j' j)d t';' y' ;')(' yjd j'. g' :'j')' yyjgyj,d ggjyy;d f' ?' jjj'. y' )' p' j'(' i'@' q' ljd .yjjg;gjjj.' yy'g (.k(@14*;.1415((((4!* ll'ir'qd 'itd j'( t'()ky' ir;d )' yy'yj'. jjydyyjyjjy'j jgj.d jgjjjjjgyd (jj.' jyjy', (rt'tlli j'. y.jjd j'yjjj'.
Jstock
X
lQl
(12.9)
y' kygygjjjjjyytk,yd
-1.8
35
30
25
20
50
45 40 Stock Price ($)
55
60
.'.
=
option is the elasticity times the
.
' !' i'E )(i :E ( E ((' !'' !E(' ' iE(7*i i 7Ei ' (''('!' i' ('' :' ('qq'qr'q!?:?l'p'l'?'!!lq:rqqq;:'q):'' '(' (q' f.' ' )) ti ;@.: . jyr.j('j j(.q'(..;...(.(. .r'. (..g.. .k.y.(jjyyj.r'(.jj(jj.y(..j.(.j (.y.y..g.jy.jy....' t:.y '(..j,.tyg y j j j yjy y . j (yr gyg... g..g. g -tr.-tEi). .jj.g..jgjj.g.yj...j.ygjj.. j ..tyjjyyy yggy..yy .j !tq.?.-.r.......--p.i..-)4.-..-....'.'.'.'.!:.b. tjttj ...IitipI:.. . . . ..E--i.(.... yy y j ...j.yyjjjj.gyy jg jjjjjjjjjjjj iit5it.. . j;g. jyyjyjj .. y..y.yjyy . ....yj :E 1:(:E
(E
,
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:
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45
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k. THE
394
BLAcx-scHouEs
FORMULA
PROFIT
whre If)I is the absolute value of f2. Since elastiity is a measure of leverage, this calculation is analogous to the computation of the standard deviation of levered equity valu to quity. Based on Figure by multiplying the unlekered beta by the ratio of f11.111 option could easily have a at-the-money volatility, stock with 30% 12.11, for a an a of 120% volatility or more.
of an option Since elasticity measures the percentage sensitivity relative stock, it tells the option us how the risk premium of the option compares the to of lhat of the stock. In Section 11.2, we computed the discount rate for an option. We to wereimplicitly using option elasticity to do this. At a point in time, the option is equivalent to a position in the stock and in bonds; hence,the return on the option is a weighted average of the rell'n on te stock and tlze lisk-freerate. Let a denote the expected rate of rettlrn on the stock, y the expected return the option, and r the lisk-free rate. We have
The risk premium
on
'
y
LS =
ct5'l
)
a +
AS
CSl
The elasticty
=
Qct +
(1
risk premium
and
395
se does not change
.
=
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(12.13)
i= 1
'I'he elasticity of the portfolio is the percentage chapge in the portfolio divided by the P ercentage change in the stock, or Qportolio
=
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,, =
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'Af
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i= l
c?
,,
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Using equation (12.10),the lisk premium of the portfolio, y elasticity times the risk premium on the stock, a r:
or
$
The elasticity of a portfolio is the
of a portfolio
oh d.f )(27,-,1
Q)?*
-
MATURITY
veighted twcmge of the elasticities of the portfolio components. This is in contrast to the Greeks expressed in dollar tenns (delta,gamma, etc.), for which te portfolio Greek ' is the .If?!; of the component Greeks. 'fo understand this, suppose there is a portfolio of lt calls with the same underlying stock, where the th call has value C and delta Li, and where oh is the fraction of the portfolio invested in the th call. The portfolio value is then S)' l ah Q For a $1 change in the stock price, the change in the portfolio value is
Since LS/CCS) is elasticity, this can also be wlitten y
BEFORE
equivalent to a levered position in the stock, and that leverage per jo the Sharpe ratio.
).
-
DIAGRAMS
(12.14)
j.j -
r, is
just the portfolio
-
y
'
-
1--=
(a
r) x f2
-
(12.10)
premium on the stock tnes f2. Thus, the risk premium on the option equals the Using our earlier facts about elasticity, we conclude that ifthe stock has a positive risk Premium then a call always has an expected rettzrn at least as great as the stock and that, other things equal, the expected ret'urn on an option goes down as the stock price goes up. ln terms of the capital asset pricing model. we would say that tlle option beta goes down as the option becomes more in-the-money. For puts, we conclude that the put alwys has an expected ret'urn lss than that of the stock.
y
-
r
=
Qportfoliotl
-
r)
(12.15)
lisk
,
The
sharpe ratio
premium to
of an option volatility:
The Sharpe ratio for any asset is the ratio of the risk Sharpe ratio
Using equations
(12.9) nd (12.10),the
J
..-'
)*
?. o (a ?.) a (12.12) f)c o' Thus, the Sharpe ratio for a call equals the Sharpe ratio for the underlying stock. This equivalence of the Sharpe ratios is obvious once we realize that. the opdon is always -
-
=
DIAGRAMS
BEFORE
MATURITY
In order to evaluate investment strategies using options, we would lile to be able to answer questions such as: If the stock price in 1 week is $5 greater than it is today, what will be the change in the price of a call option? What is the prost diagram for an' optiori position in which the options have different times to expiration? Our previous discussion of option strategies in Chapter 3 examined only expiration values. Now we will examine the behavior of option prices prior to expiration. To do tllis we need to use an option pricing formula.
(12.11)
=
Sharpe ratio for a call is
Shatpe ratio for call
12.4 PROFIT
=
loerhereis one subtlety: While the Sharpe ratio for the stock and option is the same at every point in time, it is not necessnn'ly the sam'e when measured using realized remrns. For example, suppose you perform the experiment of buying a call and holding it for a year, and then evaluate the after-the-fact risk premium and standard deviation using historical returns. A standard way to do this Nkould be to compute the average lisk premium on the option and the average volatility and then divide them to create the Sharpe ratio. You would find that the call will have a lower Sharpe ratio than the stock. This is purely a result of dividing one estilnated statistic by another.
k. TH E BLACK-SCHOLES
396
PROFIT
FORMULA hf F' p.jjjj(,kjjjj('i'(' q' pr';r qff f' t)' @' (ll:y::ltr.:l..f r' (ll:rr:llr:,..f q' !1!7lqi(' El' )' )'1l';$li' ')j(' ;'y' ))#,;-..-.).;'. j'k' 7::71( T' J'T'))EITT i (!' i.!(('(: ( !fi (!;( i ( !'' (!:(:7$q* (ip)f r' (j)!' t'll' y' ,411:::::2..*. (ii.' '(.(jl )i.'. Ilii!j!!-f .-' (' l'y 11111*. j'jjf ;'y')'(-' )y' ,jj:Ljjjj', ('; jytt'-iiggyjjjrt.yf j:y,;gjjjj';i y'. )' 'f1*77* yjyjyt'j. t'.r' (rlf .(i'(): i . ' ' (': 'Ii.'(.' l ..' . i )(.k' ('' ' '. '. (()( ' rrt('. .y k(y jjjk::jjjrj,: y, .:. p..jy (yy(..y( yy(yy...yrjy.jy(.j(.yy y. .y .(i'i j.y.jjjjrjj:ryg. ( (. y .! .( y)')' j yy ;. j E .y . . . .j. . jk;-;. t''l' ' ).rr .. jLjL. . ( 7* ''q' :'tr'rf ?' -' ---.'.-' ;'':' j'y -'. '!' (' ),' ?' -..r' (y'E -' !'' ''5* (6f' : : i;iE E fiE (' )' ptf ()' y' 41::22:2::,,* (( ; . ----' ---,)')t--'.. ..' .'' -'
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what the Consider the purehase of a call option. Just as with expiring options, we can price. stock Table particular in time point >ndfor a value of the option is at a particular four prices five at stock for option different call of 12.3 shows the Black-scholes value a expiration, time given to price stock for the varying a different times to expiration. By keeping everything else the same, we are able io graph the value of the call. Fieure 12.12 olots Black-scholes call ootion prices for stock prices ranging from $20 to $60, including the values in Table 12.3. Notiee that the value of tlle option prior the option at expiration. to exp irat ion is a smoothed version of the value of 12.12 does not show us how the value of Figure depicted in The payoff diagram order original to dt that, we can subtract the cost of cost. In the option compares to its l interest.l the option, plus In order to determine proftability, we need to answer two questions that were initial cost of the option position, and unnecessal'y for the jayoffdiagram: What is the proht, the we take the valu of the position and what is.the holding peliod? To compute interest. including subtract the cost of the position,
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12.3 costs $6.285at a stockprice of $40. q)Example 12.9 The l-year option in Table will have fallen to ll 1f after 1 day the stock price is still $40, the value of the option 0.08/365 '))( 'i.E'. = -$O 012 $6 274 and the l-day holding period prost is $6.274 $6.285 x e iiE'j' .'rhis loss reflects the theta of the option. If the stock price were to increase to $42, the ! li option premium would increase to $7.655,and the l-day holding period prolit would be
$
,
Payoff diagram for a call
=
MATURITY
-y-,
option for diferent stock prices and times to expiration. Msumes # $40, r 8%, & 0, and c 30%. =
BEFORE
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40
45
Stock Price
($)
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--
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c0.08/365=
-
$1.369.
FCX
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the holding period prot at a price of $40 would be $4.t55 $2 a86 svenif the stock price had risen to $42, the holding -pj l;iperiod ret'urn would still be negative Thege' prbst calculations are illuseated a jrj (11 in Fir 12.13t % After 6 months, ); 0sxa.5 'ili lij $6.285 x e ?-
-
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discussed in Chapter 2, this is like assuming the option is financed by bonowing.
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The option premium g'raphs in Figures 12.12 and' 12.13 can help us understand the behavior of delta and gamma discussed in Section 12.3: In a1l cases the slope of the call option graph is positive. This corresponds to a positive delta. In addition, the slope becomes greater as the stock price increases. Delta increasing with the stock plice corresponds to a positive gamma. 'Ie fact tat gnmma is always positive implies that the graphs will be curved like the cross section of a bowl, i.ek, te option price is convex. A positive gamma implies convex curvature. A negative gamma implies the opposite ! (concave) curvamre. ,
Calendar
y
#rzau
We saw in Chapter 3 that there are a numberof option spreads thatpermit you to speculate on the volatility of a stock, including straddle, strangle, and butterlly spreads. These
'
Q Tl4E
398
PROFIT
FORMULA
Bl-Aclt-scldonEs
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MATURITY
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Assumes we sell p 91-da# Mo-trike call with premium of $2.78, and buy a 365-day 4o-strike call with premium of $6.28. Msumes $40, 30%, 8%, and r o' & 0. .
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1 day 6 months 9 months Expiration
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DIAGRAMS
-4 20
25
30
35
45 40 Stock Price (5)
50
55
60
-+-
20
25
30
.
spreads all contain options with the same time to expiration and different strikes. To speculate on volatility you could also enter into a calendar spread, in which the options dates. you buy and sell have different expiration X'YZ'S stock price will be unchanged ver the speculate that Suppose yo want to seaddle written alternative mortths. An 3 or a written butterlly spread is simply to a next will remain unchanged and you will stock price the that hbpe call in'the sell or put, a to option does move into ihemoney, you that the ptential if is premium. The the cost earn can have a large loss. To protct against astockpriceincrease when selling acall, you can simultaneously buy a call option with the snme strike and greater time to expiration. This purchased calendar spreadexploits thefactthatthe written near-to-expiration option exhibits greater time decay than te purchased far-to-expiration option, and therefote is profitable if the zm-strike call with 91 days stock price does not mvek For ekample, suppose you sell a price of $40, stock expiration. zm-sfrike At with eall to 1 expiration and a buy year a to call. l-year the The profit call and for $6.28 91-day premiums for te the are $2.78 diagram for this position for holding peliods of 1 day, 45 dgys, apd 91 days is displayel in Figure 12.14. You can see tat you enrn maximum pr flt okt 91 daf if the stock price does not change.
i10
45
Stock Price
($)
35
'
.
50
60
55
'
''
We can understqnd the behavior of prost for tlzis position by considering the theta of the two options. Figure 12.6 shows tat theta is more negativ for te 91ray call if the stck price does nt change (-0.0173) than for the l-year call (-0.0104). over the course of 1 dayi the position will make money since the written option loses the purchased option. Over 91 days, the written 91-day option will lose more viue t-1::.r1 . its full value (its price declines from $2.78to 0), while the l-year option will lose only about $1 (itsprice declines from $6.28to $5.28)if the stock price does not change. The difference in the rates of time decay generates proft of approximately $1.78. The prot diag'ram also illustrates tat at a stock price of $40,delta for the position is initially positive. Over 1 day,the pnximum prpst pcurs if the stpck price rises by a reflects tl,e fct thatthe delta ofie tten l-ak catl if 0.525 and amount. ' This ' . o? the purchased .. ...l-year . call 0.66t5, for a nei ppsttive deltg of Withthe that .'. ' .. .. .. of the graph . . the purchased below 40 reqetd l-year holding period, the 91-day portlon stock unprfitable 40 increasingly price falls. Above wh h te optiop, ic becoms as the gain on the purchgsed l-year option is offset by the loss on the expiring 91-day call. Since it is expiring, the delta of the 91-day call is = 1 for stock plices above 40, which results in the graph ttlrning back down to a negative slope above 40. As the stock price continues to increase, however, the delta of the purchased l-year call increases toward 1, so the slope of the net position approaches zero. rfhus,
small
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ts
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$
400
THE
Bl-Aclt-sclqoLEs
IMPLIED
FORMIJLA
12.5 IMPLIED VOLATILITY
t(.
and Figure 12.4 shows that option prices, particulady for Volatlity is unobservable, near-the-money options, can be quite sensitive to volatility. Thus, choosing a volatility to use in pricing an option is difficult but also quite important.lz O ne approac h to obtaining a volatility is to use the history of retllrns to compute historical volatility (seeSection 11.4). A problem with historical volatility is that histl'y is not a reliable guide to the future: Markets have quiet and turbulent periods and predictable events such as Federal Reserve Board Open Market Committe meetings can create periods of greater than normal uncertainty. There are sophisticated statistical models designed to improve upon simple volatility estimates, but no matter what you do, you cannot count on history to provide you with a reliable estimate ofhltltre volality. In many cases we can observe option prices for an asset. We can then invert the question: Instead of asking what volatility we should use to price an option, we can compute an option's implied volatilit'y, which is the volality that would explain the observed option price. Assuming that we observe the stock price S, strike price K, intrest rate r, dividend yield J, and time to expiration F, the implied call volatility is the that solves
jitjBy tlial and error (or by using a tool such as Excel's Goalseek), t,t ja t, o' 28.7% gives us a call price of $8.07.
=
CS, #,
,
r, F,
)
Table 12.4 lists ask prices of calls and puts on the S&P 500 index, along with using the Black-scholes formula. These S&P options are Europan style, so the Black-scholes model is a/propriate. Notice that, although the implied volatilities in the table are not all equal, they are all in a range between l3% and 16%. We could describe the general level of S&P option prices by saying that the options are trading at about a 15% volatility level. There are typically numerous ptions on a given asset', implied volatility can be used to succinctly describe'the general level of option prices for a given underlying asset. There is one clear pattern in the implied volatilities in Table 12.4. For a given expiration month, volatilitis decline as the strike price increases. This occurs with both the calls and puts. Fof example, the volatility for the November calls declines from 16.3% to 12.84% as the strike rises from 1100 to 1150. The decline is evident but smaller for the later expiration months. This systematic change in implied volatility across strike
implied volatilities computed
:'-' 7j'('* j'';'1* T'qf C' :IIr:.:Il,'' 1iI:.--i'' gjtf ()-' p' k.))j'. k' t'. '1/*(' E( (!jl .! i1! kliid (iiiqd t'tt;f t,f 4112Ej,,)* f' )' (' i':'q'' !' ,)' ,;'
5' 'i q !E E(' ('''(E'i :' ' '!Cll'7pq !)?1T'q(7':F!'l' E.-y...r.---...--t(jj.jyt:.;..yy!yy.'')tjy.y.yy...y...g ((r !..''' (( !!' (.'i i (''( ( !j t. )5 ' ( ri r' )t;' :11I:r' i1IiL11-''.' iiliiEi@l.'' r'. (@:-7Iiiiiii:': '..'. (1II..1iii.' (111).* tqjllElE:'l.-d ;:!(ijiI;' )' .:'iiiIi'. yrf )'. ('t'iq'tt't'i'. )'rj-.t-(jjjjjj;.... '..t . ..!.-yy ))).' kllijillltd tl;i!!:d '.' iiiii@!2ik'. ttr.ttfplittyqlllit.tl7l:tt.' ('. i'yjytrty.lrtjj yy.jj)j.)j.y;yy..q.y,q.y.y-ryy;yy.y..-y.(yyq ' ' i . d:::;p tl::d::::,kpij;l:E 4(:2),. d:::,, lC. jjjj;g,r ij,l,1E: ,!!;.I: Ir..1!14; k j::gjjj ,:::,.11 ii lr..lk .t;tl!r i;Iiiq:1r:.i(,lE!!t 1E::,. d:El)p cl. jyry,j '.r'.-,'LikiiijL-. s Ig::;,. .'-'-.'''-' .'' ). ... r. ).-. -.--. . . . ; -.-. :. j;jrs.,r..,j 'li:tt'i . . Option prices (ask)from www.cboe.com; . .. .. 1 $1 127.44, r 2%. '.111r.*. .-'
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t) y ield is zero. We can compute the option price as i!
$8.07 BSCa1l(50, 45, c, 0.08, 0.5, 0). =
I2of the five other inputs to the Black-scholes formula, we also cannot observe dividends. Over short horizons, howeveq dividends are typically stable, so past dividends can be gsed to forecast f'uttlre dividends.
.
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assumes
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=
iII(!r,).' ' ' '. .1112,. :iikj!llg. $1Ij!,' ' (jillEEF?!i,. .11;..:11r..1111. '' ,I1j::!!. kkl:tillt.t .' ' (jII).tziirj-,. .. '. jiii.y!llt:.r;ztg ..'.. '...'.. ()114. ..('. ... .. ''. tlli!ir: . ;. '. ). ....-.-.tt ly'.'.. rjilj--. '.'..(y ..'.yy.i ... ... ..;... i......( .. . .. . .;' . . . ..iy .....-. )..) y.. .... . . . .(.:-...y...;.( .). )). .(...t . ..yt . ..-..;. .. r--.z.. .;..-.;,.... ;.-. .-.. ti.. ... r ....t t ...t .. ..y.. . tq.....41:11k. . .... .. .) . j .. )...... ..y :,.. . t. ' ....t. . :iiri1l.. ., ..t..-.........t L?'b-.-L' :11::112. iIEEll.. 11!5:112... ..)'.. 41::,,.. ;........ ltlii;. . 1(L) ..:iEjli(. ....L.. IIEi!(I;..:!iij2jtIjjj!;,. tIi2t.... .-.. 111.. .jii.....11l2(:!j/'.t. .)-... 111.. 11L.. tii.. . .)..-.4(L. ..1:). t!tli;-. t!tii;*''-*.*,';)4.. .-r......). )...t.. ) ... d1:Ei,-. )...(.......... .. .-.. ...111..... ...... tii:(it:''. . ..... ( t ....t-..... .. . .... . ... . . .. . . . . . . . 0.1630 6.80 0.1575 1100 11/20/2j04 34.80 '
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.)*)' lq Example 12.10 Suppose we observe a4s-sfrike 6-month European call option with of $8.07. stock price is $50, the interest rate is 8%, and tlle dividend premiuin a jj
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Computing an implied volatility requires that we (1)obsel've a marketprice for an option and (2) have an option pricing model with wlzich to infer volatility. Equation (12.16) so it is necessary to use an iterative cannot be solved directly for the implied volatility, prpcedure to solve the equation. Any pricing model can be used to calculate an implied volatility, but Black-scholes implied volatilities are f'requently used as benchmarks.
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Computing
o. w.
(12.16)
By desnition, if we use implied volatility to price an option, we obtain the market price of the option. Thus, we cannot use implied volatility to gssess whether an option price is correct, but implied volatility does tell us the market's assessment of volatility.
Implied
401
we ;nd that setting
=
'
Market option price
%.
VOLATILITY
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11/20/2004
5.80
0.1434 0.1284
12/18/2004
41.70
0.1559
13.80
0.1389 0.1539
12/18/2004
24.50
22.50
0.1436
12/18/2004
13.00
0.1396 0.1336
35.50
0.1351
1/22/2005
49.10
0.1567
20.40
1/22/2005
33.00
0.1518 0.1427
1/22/2005
20.00
0.1463 0.1363
29.40 41.50
0.1337
11/20/2004
17.10
14.70
0.1447
29.20
l3An implied volatility function is available with the spreadsheets accompanyin.g this book.
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402
k
THE
BLACK-SCHOLES
PERPETUAL
FORMULA
kesv.
14
is cled volatilit'y S prices oecurs generally for different underlying assets and the resulting line can take price, strike When you graph implied volatilit'y against the smirks.'' Explaining these and 'Trowns,'' different shapes, often described as is a challenge for option for them account patterns and modifying pricing models to in Chapter 23. f'urther issue pricing theoly We will discuss this important ppt-call pality in mind. keep helpful to 'When examining implied volatilities, it is strike t'??/Jtime to evkpiratiolt IWr/lthe w??ld If options are European, then plfr. and calls prices of European puts and because is p?7lt./have l/?c same Ibw/fW volatilty. This true opporttmity. Thus, arbitrage there is relationship or else an cal1s must satisfy the parity rather but cqll, to differences in skew is not related to whether an option is a put or a volatilities and re not exactly call Although put the strike price and time to expiration. not be profitable would arbitrage enough that parity equal in Table 12.4, they are close options in-the-money for example, the after transaction costs are taken into acount. For dtfference for that more pccount can in Table 12.4, bid-ask spreas are as wide as $2, a alone. option than percentage point o implied vo 1a tility on one tsmiless''
a
12.6 PERPETUAL
OPTI ONS
AM ERICAN
$
403
OPTIONS
AMERICAN
formula prices options that are oply exercised at expiration. ln this The Blackcholes based on Merton (1973b), for the prices of calls and puts fonnulas, section we present will call such options perpetual options. expire. We are also ltnown as that never options. expirationless American options are harder toprice than European options because it is difcult to characterize the optimal exercise strategy. Using the binomial model, we saw in Section 11.1 tiat for a finitely lived call option on a dividend-paying stock, the stock price at which it is optimal to exercise the option declines as the option approaches expiration. It is this changing optimal exercise price that makes it hard to derive a vqluation formula. With perpetual Amelican options it is possible to dedve a valuation formula beinlinity. Since time to cause such an option always has the same time to expiraion; option problem will look exercise the the is expiration constant, same today, tomorrow, exercise the option is constant. price optimgl which it is Thus, the forever. at to and The optimal exercise strategy entails picking the right exercise barrier and exercising time the stock price reaches that bnrrier. the option the 'rhey
srst
Using lmplied Volatility
need to price an if lmplied volatility is important for a number of reasons. First, you price, you can use implied volatility option for which you canllot obsel've a market with the prices of aded options. Market-makers, for to generate a price consistent of similar options. Second, as we example, will price options consistently with prices used volatility often is as a quick way to describe the level implied previously discussed, prices are sometimes quoted in tenns of option prices on agiven underlying asset. Option volatility skew provides a measure of of volatility, rathef than as a dollar price. Third, volatility is unbservable, a standard how well option pricing models work. Because whether, for a given is approach to testing the validity of the Black-scholes model to see The existence of prices. al1 strike underlying asset, implied volatilities are the same at assumptions tre not a perfect and volatility skew suggests that the Black-scholes model models in Chapter 23. description of the world. We will discuss alternative pricing that informarecognize volatility To better understand the importance of implied markets. You can think of options tion about futtlre volatility is uniquely provided by elsewhere. available volatility that is not option markets as providing information about based on an index of implied In fact, there is now an exchange-laded futtlres contracts about stock priees volatility for the S&P 500. Just as stock markets provide information volatility, and, in about information and permit trading stocks, option markets provide perspective, we should expect effect, permit the tading of volatility. Viewed from this learn from information about volatility to be 0ne of the most important things we can depth in Chapter 23. option prices. We will discuss volatility in more
Barrier Present Values As a prelude to valuing a perpetual option, consider computing the present value of $1 payable when the stock plice reaches a level, S. We call H the barrier level and call th ttbarlier present yalue.'' lt value today of $1 paid when te stock plice reaches H the depending this, simple formula which differs there for is upon whether H is a t'urns out value, S. stock below the current above or tf S is above S (i.e., S has to rise to reach S), the value today of $1 received when bnrrier present value-is S reaches S-the Value of '
$1 received when S first reaches H from below
S =
;
(12.17)
S
where /? j
1 =
-
?'
8
-
-
0-2
2
+
?'
1 2
8
-
-
-
c2
lf H is below S (i.e., S has to fall to reach S), the
2 value of
2? .
+
w
o'z
$1 received
when
S reaches
H is Itz
Value of
$1 received when S first reaches H from above
=
-
S
where strikes. Thus, 14The Black-scholes formula is defived assuming that volatility is constant across volatility. implied in changes track model to it is intemally inconsistent. to use the Black-scholes benchmark. Nevertheless, it is a widely used
ltl
-
hz
=
-
1 2
1-
-
-
-
c2
r
1 2
-
-
&2
-
2
2? .
+
=
c2
(12.18)
404
%.THE
BLAcK-scHoLEs
FURTHER
FORMULA
Suppose we have a perpemal American call with strike K. If we decide to exercise the K. From option whenever S reaches the banier S, then at exercise we receive H equation (12.17),te value of receiving H K when S reaches S is -
-
K4
-
=
K
-
Price of perpetual put
-
H
ln order to finish computing the value of the call we need to specify S, the price at which the ca.ll should be exercised. We simply need to pick a value for H that makes the value of the call as great as possible. If we make H too small, then we prematurely throw away option value (i.e., protection against a subsequent price decline). If we make H too large, the we forgo dividends for to long while waiting to exercise. lt is possible to show that the exercise level S* that muimizes the vlue of the call isl5 H* 1, we have S*
=
11l
K
=
hl
1
-
K. Maling this substittltion, the value of the perpetual
>
Price of perpetual call II'J = 0, then H* Paying stock.
405
h2 /?2 1 which implies that the price of the perpetual put is H*
lI
(S
>
%.
Again selecting the exercise level S* to maximize the value of the put, we get
Perpemal Calls
. Since 111 call is
READING
=
/7I
K
121
l
-
/r;l
1 S 11l K -
(12.19)
-
x; i.e., itis neveroptimal to exercise acall option on anondividend-
Perpetual Puts For a perpet'ual put, using equation given by
(12.18),the value if we exercise when S
CK
S)
-
=
Ita
S -
S
where h
z
=
-
1 9
?-
?-
-
-
1 2
-
-
-
(7'2
ls-rhisis accomplished by differentiating the expression tozero, and solving for H.
t:r
2
-
2
e?
-?
+
(y
-
2
wth respect to H, setting the derivative equal
H is
=
K
.
112
1 S
-
-
1
-
hz
2
K
:t 11
(12.20)
Under certain assumptions, the Black-scholes formula p/ovides an exact formulaapproximated by tlle binomial formula-for pricing European options. The inputs to the Black-scholes formula are the same as for tlze binomial formula: the stock price, strike price, volatility interest rate, time to expiration, and dividend yild. As with the binomi formula, the Black-scholes formula accommodates different uderlying assets by chanjig the dividend yield (seeTable 10.1 fr a summary). Option Greeks measure the change in the option price (or other option characteristic) for a change in an option input. Delta, gamma, vega, theta, rho, and psi are W idely used in practice to assess the risk of an option position. The option elasticity is th percentage change in the option's price for a 1% change in the stock price. Th volatility and beta of an option are the volatility and beta of the stock times the option elasticity. Thus, an option and the underlying stock have the same Sharp ratio. Of te inputs to the Black-scholes formula, volatility is hardest to estimate. ln inferthe market's stimate practiceitis common to usethefo=ulainbackwrdfasMonto of volatility from the option price. This implied volqtility is computed by fpding the volatility for which the fonuula matches observed market prices for options. In theory, all options of a given matgrity should have the same implid volatility. In practice, they' do not, a phenomenon known as volatility skew. Mthough there is no simple formula for valuing a fnitely lived American option, there are simple fprmulas in the special case of perpemal puts and calls.
FURTHER
READING
Chapter 13 will explore in more detail the market-maker's perspective on options, including how a market-maler uses delta to hedge option positions and the circumstances under which market-makers earn prcts or make losses. Chapter 14 extends the discussion in this chapter to include exotic options. In Chapters 15, 16, and 17, we will use option pricing to explore applications option pricing, including the creation of strucmred products, issues in compensation of capital options, structure, tax management with options, and real options.
406
% THE
of the Chapters 18 through 21 delve more into the mathematical underpinrngs again discussed in Black-scholes model. The barrier present value calculations will be detail in Chapter 23. much in volatility more will discuss We 22. Chapter The classic early papers on option pricing areBlackand Scholes (1973)andMerton detils of how the binomial model converges to the Black-scholes model (1973b). derived inMerton (1973b).The IiIIIC arein Cox et al. (1979).Theperpemalputformulais and Siegel (1986). between theperpet'uf catl and putfonuulas is discussedby McDonald
J
=
0, and r
=
$120,c
=
30%,
?'
=
0.08, and
t5
=
0.
price for 1 year to mamrity and for a a. Compute the Black-scholes call maturity. What happens to the option price times to variety of very long x'? as F O.OOI. Repeat (a). Now what happens to the option plice? What b. Set accounts for the difference? -->
$120, K
=
=
$100.c
=
30%, 1-
=
0, and
=
0.08.
call price for 1 year to matulity and for a a. Compute the Black-scholes variety of very long times to maturity. What happens to the price as
F
-->
(x)?
O.OOI. Repeat b. Set r difference? =
(a). Now what
happens? What accounts for the
12.5. The exehange rate is Y95/E, the yen-denominated interest rate is 1.5%, the volatility is 10%. euro-denominated interest rate is 3.5%, and the exchange rate yen-denominated euro put with 6 months a. What ij the price of a go-strike to expiration? call with 6 months b. Whatis theprice of a l/go-stlike euro-denominatedyen
to expiration? and your answer to (b), c. What is the link between your answer to (a) convrted to yen?
=
30%, r
=
0.08, &
0.03, and F
=
=
=
.,
,
0.75.
=
=
(
,?
=
$100 x /.
=
g.
12.8. Make the same assumptions as in the previous problem. a. What is the g-month forward price for the stock? b. Compute the price of a 95-stlike g-month call option on a flatures ccmtract. c. What is the relationship between your answef to (b) and the price you computed in the previous question? Why? 30%, 1, 0.08, F 0.5, and the stock is to pay a single 12.9. Assume K $40,c dividend of $2 tomorrow, with no dividends thereafter. . #' option? IIPPOSe price of call What the European Consider S is a. $50 a an otherwise identical American call. 'What is its price? =
=
=
b. Repeat,
=
12.4. Let S
$95, c
=
=
=
$100, K
$100, K
=
a. Compute the Black-scholes price of a call. b. Compute the Black-scholes price of a call for wlch S ,-0.08x0.75 .-0.03x0.75 ,G () r K Lj.y5 $9,5x , that for How does your answer compare to (a)?
approxi12.2. Using the Binomcall and Binolnput functions, compute the binomial rpations for th options in Examples 12.1 and 12.2. Be sure to compute prices 8, 9, 10, l 1, apd 12. What do you observe about the behavior of the for n binomial approximation? =
40%,
=
0.06.
=
c. What is the price of a l-year los-stke option, wflere the underlying asset is a futtlres contract maturing at the same time as the option?
PROBLEMS
a
$100, o'
=
v
407
a. What is the price of a los-strike call option with 1 year to expiration? b. What is the l-year forward plice for te stock?
12.7.' Suppose S BSCaII, Bsput, Callpel'ln answering many of these problems you can use the f'tmctions forthe accomjuyingfunctions Greeks (seethe spreadpetual, and Putpelvetttal and the sheets on the CD-ROM aecompanying this book). 12.1. Use spreadsheet to vel'ify the option prices in Examples 12.1 and 12.2.
stock. Suppose
12.6. Suppose XYZ is a nondividend-paying
'l'he
12.3. Let S
%.
PRo BLEMS
FORMULA
BLACK-SCHOLES
=
=
.
t y suppose S
=
$60.
c. Under what circumstance would you pot exercise the option today? 12.10.
decay is greatest fpr an option close to expiration.'' Use the spreadsheel f'unctions to evaluate this statement. Consider both the dollar change in the option value and the percentage change in the option value, and examine both in-the-money and out-of-the-money options. EErfime
12.11. ln the absence of an explicit formula, we ean estimate the change in the option price due to a chage in an input-such as o'-by computing the following for a small value of c: BSCaIICS,K, o' + 6, ?', /, ) BSCaIIIS, K. ty e, r, t, J Vega 26 -
-
'
-
'
'
=
.
a. What is the logic behind this calculation? Why does 6 need to be small? b. Compare the results of this calculation with results obtainedfromfsfc//kkgtz. 0.03, and F 0.08, & 30%, ?' 12.12. Suppose S $95, o' $100, K 0.75. Using the technique in the previous problem compute the Greek measure =
=
=
=
=
=
408
% THE
BLACK-SCHOLES
c. Suppose ?- 0.07 with al1 other inputs the same. What happens to the price and exercise barrier? Why? . =
$40. What are delta, gamma, vega, theta, and rho? $45. What are delta, gamma, vega, theta, and rho? different? ((f so, why? c. Are any of your answers to (a)and (b)
a. Suppose S b. Suppose S
50% with all other inputs the same. What happens to the d. Suppose o' exercise barlier? Why? and price =
=
=
=
=
=
12.19. Consider a perpetual put option with S 0.03. and
$45. What are delta, gnmma, vega, theta, and rho? different? lf so, why? c. Are any of your answers to (a)and (b)
= 0.07 with all other inputs the same. What happens to the plice and exercise banier? Why?
=
as it does.
0. Using l-year-to-expiration European 30%, $ 8%, c 12.17. Assume 1' sell two Bo-stn'ke puts, buy one 95oPtions, construct a position where you strike put, buy one los-stHke call, and sell two lzo-strike calls. For a range of stock prices from $60 to $140, compute delta, vega, theta, and rho of this position. As best you can, explain inmitively the signs of the Greeks. =
=
12.20. Let S
=
$100, K
$90, (7'
=
=
$50, K
=
$60. r
=
0.06, tz
=
0.40,
=
30%, 1-
8%,
=
=
5%, and
a. What is the Black-scholes call price? b. No* price aputwhere S $90, K $100,(7' 1. and F =
=
=
r
=
30%, r
1.
=
5%,
=
8%,
=
c. What is the link between your answers to (a) and (b)? Why? 12.21. Repeat the previous problem, but this time for perpet'ual options. What do you notice about the prices? What do you notice about the exercise barriers? APPENDIX 12.A: THE STANDARD NORMAL DISTRIBUTION T he standard
p7t/?7n/l
probability densityhtnction is given by
4@)
=
12.18. Consider a perpetual call option witll S 0.03. and =
50% with all other inputs the same. What happens to the d. Suppose c price and exercise bnrrier? Why? =
30%, 0. In doing the following calculations, use a 8%, o' 12.16. Assume r stock price range of $60-$140,stock priee increments of $5, and two diferent times to expiration: 1 year and 1 day. Consider purchasing a loo-strile straddle, i.e., buying one loo-strike put and one loo-strike call. call and put separately, for the of a. Compute delta, vega, theta, and rho the expiration. and times to different stock prices rho and of the purchased straddle (dothis by theta, b. Compute deltg, vega, options). As bst you can, explain individual of the adding the Greeks straddle Greeks. of the intuitively the signs and rho of the straddle with 1 year to expiration theta, c. Graph delta vega, stock of the price. In each case explain why the graph looks as a f'unction =
0.40,
=
c. Suppose r .
=
d. Are any of your answers different in this problem from those ip Problem 12.142 If so, why? =
0.06, cr
=
=
=
b. Suppose S
$60, 1,
b. Suppose & 0.04 with a1l other inputs the same. What happens to the price and exercise barlier? Why?
=
=
=
a. What is the price of the option and at what stock price should it be exercised?
12.15. Consider a bull spread where you buy a 4o-strike put and sell a 45-strike put. 0.08, J 0, and F 0.5. ' Suppose o' 0.30, r and rho? a. Suppos S $40. What are delta, gamma, vega, theta, =
$50, K
=
=
=
=
409
=
.
12.14. Consider a bull spread where you buy a zm-strile call and sell a 45-strike call. 0.5. 0.30, 1, 0.08, $ 0. and r Suppose o'
$
0.04 with a1l other inpts the same. What happens to the b. Suppose exercise bnrrier? Why? and price
=
=
=
=
DISTRI BUTI ON
a. What is the price of the option and at what stock price should it be exercised?
corresponding to a change in the dividend yield. What is the predicted effect of dividend yield? a change of 1 percentage point in the 12.13. Consider a bull spread where you buy a 4o-strike call and sell a 45-strike call. 0.5. Draw a graph with Suppose S $40, (7' 0.30, 1. 0.08, J 0, and F the depicting proft on the bull spread after ranging $60 from $20 to stock prices months. and months, 6 1 day, 3 =
N O RMAL
TH E STAN DARD
12.A:
APPEN D IX
FORMULA
l
HB
lx
e-
Jra
(12.21)
'
distl-ibtltion d/zc/cpl, evaluated at a point x, fOr eXThe cumulative stalldard llorlnal nmple, tells us the probability that a number randomly drawn from the standard normal distlibution will fall below x, or -$.-
N @)
-!.-
$ (x)#-'r
jg
>
-
-x
-x
2x
-Y e -J.v2#
(jr,!p,:) .
410
%.TH E BLACK-SCHOLES
APPEN DlX
FORMULA
Excel computes the cumulative distribution using the built-in f'unction NORMSDIST. Note that #/(x1) = /la'j ).
Call theta
OCCS, f('. o' r, F Pt
-((T-l) N(# I ) = &Se
In this section we present formulas for the Greeks for an option on a stock paying Greek measures in the binomial model are discussed in continuous dividends.l6 Appendix I3.B.
Delta measuz'es the change in the option price for a $1 change in the stock price: DCCS, K, c', ?', F t, ) = Call dblta = e -
p
Put delta
,
).P(5', #, c,
=
-j(,z.-?)s(#j;
?-,
F
1,
-
8)
put '
ye-rv-tj
ytfygjg
2
r
t
-
)
t,
-
C a 11theta +
=
J5'd-tlT'-''
-
Vega measures the change in the option price when volatility also use the terms Ianlbda or kappa to refer to this measure:
;
Call vega
(r) Call gamma Put gamma
=
pzctu
'
t, J)
-
'
0SI
p2#(5', K, =
t7', r,
F
t,
a
=
)
=
F
-
Rho
Call gamma
Theta measures the change in the option priee with respect to calendar time tixed time to expiration (F):
N (-a-)
With some effort, the second can be verilied algebraically:
Se
-Jrx?
(d l )
=
Ke-rvx'otzj
p.P4u$',K,
r, F
F
tr, r,
-
=
-
-
t,
()
t,
-sv-tjytyj
=
Se
=
Call vega
'
; p
.j
chang
(p) -
Call rho
=
jjj:)i jgkjlr.jjy)t jgjj. jyy)j:::::::2
(1),holding
l61f you wish to derive any of these formulas for yourself, or if you lind that different authors use formulas that appear different, here are two useful things to know- The lirst is a result of the normal distribution being symmetric around 0: -
t7',
Rho is the partial derivative of the option price with rspect
(p)
1
DCLS, K, =
t
The second equation follows from put-call parity.
N (a-)=
in the
Some writers
changes.
It is common to report vega as the change in the option p6ceiael'pel'celtagepoillt in the volatility. This requires dividing the vega fonuula above by 100.
e---ttN'dL) us'ty
-
change
Vega
Put vega K o-, r, T
411
-
,
pz'(,s,K, c., ?-, T t prKe-''T-''
=
$
-(5(w-,)x(-#j
-d
=
Gamma measures the change in delta when the stock price changes:
Theta
theta
GREEKS
If time to expiration is measured in years, theta will be the amlltalized option value. To obtain a per-day theta, divide by 365.
Delta (A)
Gamma
2) # (y.j
).Ke-rT-l)
-
GREEKS
FOR OPTION
12.B: FORMULAS
APPENDIX
FOR OPTION
/, &)
-
,
=
FORMULAS
12.B:
pC S, K, c'
,
?-,
F
t, J)
-
pr
0 P S, f('. c'
,
r, F
. ...............................
0r
t, $j
-
.
to the interest rate: -r(w-,)
=
(F
-
yyjy)
t) Ke
-vv-tj
::::::::
.......
jgk (jlr'' .......
.S ()j
jrk
jsjr
x (.tyz)
These expressions for rho assume a change in 1, of 1.0. We are typicatly interested in evaluating the effect of a change of 0.01 (100basis points) or 0.0001 (1 basis point). To report rho as a change per percentage point in the interest rate, divide this measure by 100. To interpret it as a change per basis point, divide by 10,000.
Psi
(#)
Psi is ihe partial derivative of the option price with respect to the continuous dividend
yield:
'j)d ,'Ejl' jj:yry:jjd ,j:yyr::' ijd y' :'1* E' 11:::(j1(.* Ejd k' ;'f' E;' j''''#' yjjd 11118* i'yr'.' 'jk(-,' ,')' j'qgjjltry'..rjyt'.,qd E;' ;-' J' ?' ljdepldjjjld.ttltll.lEjttd y' @E.' F' ;',' ld (,' :'',' q$' Cttd (' 5* ''').' jii' j-,,' ')' h' k' E-' i'Ep' qfililiijjjjjlii'j' .E' F' ;',jj.jj)jjjjjyyjjjyjjyyj(;.' qqt.gld jyjqtqjEjyjy'.!ytyrrj)rtd #' q.?(jjhy-:,t@jy!!L' i'j'lli'!:d :jjky' .4* ;'(' 11::(::)j* 112::::11* j'1* q' 41::2::1,* 41::22:1* kd y' jjjj!jjjjjjjjj!q'd (' t';'))' jy'y' i'Ejyi.!.d '.' $' )' E' ?' ,y--.:j' f' C' ''kkd ;,' tti:d j'k' q' itE..d jrd j..$5:*:21y.,--,j,*-,?..-,2,.*;:j--4(4,,.4,1,,,1,,,,,,..* k);;d ,'' (t' Ejd itd :''j;' g' ;)(y' !' ()' yrd 'jtd jl.kttjisllttl'sd )y' ;;;t);,' ;';'?' 'jd ljd jyd j'.jd y' jtrjrjg').d )#(-'.:' )jk..' 4:!)* r' .j' j;d ?' k14,1E1!.* ''i:.' ;kr:.' jkd kplti'd ,jjjd y' 7* py')kkd frd i'';' k' j?,3Lit:L-LL$j;L,q;L.',,3b-',LL' 4* ;)'-' jid ;.jq)' C' yyryyd j(' jyd j?d p;)d j'yjd lytyj'j' .j' (' j(' 1* g' )jj' jytjigr-'jlt'.d )' jygd 1;* .jjyr' t'?'p' rd /''jr' ;'.;''(' k' l'td 'y' )'q' )-*-* j';)')'('ld $' )' t'jjjllr::,id ()lrq:51(.' (iit.d rl!ll.llt,;rll.d (qI12qjI::''' y(' j!' jyd gyd jpd (',j' jjd (j'.' j'p'gjd )))' j.jyt' yyd q' r)' .y' f' !!.' jt)!yjpd )y'y@qq.' td ;f' (* ly'p;d $' y!.y' (' p' pt'lll!?d q.)kjzqtd :i#l('k)#)t-.@Ttr'!r' (((' .,' l'd );' r.p)k)'-j((' ;p'(k..' @' t')kp;!-' it'ljd tE..r.;$It' ('(' p.''-?j-'-'' .-' )))!@j(r:-' qqjd trd i'yrd )?' j.)d pgppd .'' jtd .))' ;';' t,j.'-kj'fq.'jqLi,.,' (k.;' tktd 4* i)' ?' ktd lylj:--d j.yd yjd ;'-jt' y)' j:r.:r..d jlyyiij,j'.)d ');'')'@';E,,,.kj;.'.jjj)' ' yd 11)* i.ld jd 1* ljd )' j'(ijtd gydyy'kjt'jtd (jd .y' ;yjgt'yrtj(y..' qrd j;d piljyd Tjt)jjd jj;.yq;)jy'.' S!yd j-' j)'t))tjjp,' g' C'-' .:' ytd ysjsgikt.jd 1* )jjtj;',' ))' .y' y' (' lgd )' f' 1* J' /'i'(qlIIEEqElk:-..' (' p' y' 'y' )' ygd yyd r' k' qtd y.')' ;'!'r' p))r'..' y)d ):r*:)* .;;' y' t);tj).).(jygj' ljd kiltd ):k)d 4)* (j' 2* 1* .r' )d t.;td ty'-)d )(' tritgdyrjy.ykjdyjydtjrrysjjyyryyjjr--lk-jjjjd (' ttd tkd ttyd q' yyj)d $' pjd y-':'-' 4* .(jryj)jjj(j,.' qj'jyyd tryd ykjjttjjjjjyly:.rrjpd (;jjjyj(();'.' rits.lpylqqqiytjpppy.yytsqljjqidirypplpptqkryt't)j,!d 4/2.4* )' ((' 4()* kttd yqjd t'yjjk.pd rid .jjjy,rjj.' t'i''J'ltd rjd td r(.:jli'i' tjd t.pd t't' t.(j)(y' lllfl'-ip''d t;()* r:.yy..' t'?d yjjy--;d ;)f'j');jL(;:j',.' t)d rqd t'tpd 'jirk'lpd ttd rrd r'jjd r' .,I'IIIIIII'-,.'''''II)IIIIF'''X (jjgl;:lpd (::j),;,,* 41::(j11L.* .(' tjd jjd r' rylltji.)rtd
ryjd ''''(q!II((q'''' 111:::E(q(2::1,1* (E1IIEiF;ll:-..' ((IIEiiEi(:,' r' rid rjtd rd .((i!!!1'd1.,-''!:!IIi((.' ((111(((((((::)11.* ,:E1E11!11,* rqlllll...d
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412
%.THE
Bl-Acl-scldol-Es
Call psi
=
FORM uLA
OCLS, ff, o-,
?-,
F
t,
-
()
-j(w-,)s(#j;
=
-(F
-
tj'e
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t least as lm' portant as the Black-scholes fonnttlais the Black-scholes technique for denvmg the formula, In thls chapter we explore the Black-scholes techmque by Cortfront the market cons jdering tlle market-maker perspectlve On Opt1OnS. What iSSIHS .
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0. We will let F denote the expiration ?' 0.08 (continuouslycompounded), and time of the option and t te present, so time to expiration is F t. Let F t ct:c 91/365. The price, delta, gnmma, and theta for this call are listed in Table 13.1. Because te market-maker has written the optipn, the sign of the Greek measures for th position is opposite those of a purchased option. In particular, the wlitten option is like shorting shares of stock (negativedelta) and the option gains in value over time (positive theta). Because deltais negative, the risk for te market-maker who has written a call is that the stock price will rise. Suppose that the market-maker does not hedge the written option and the stock price rises to $40.75. We can measure the profit of the market-maker by marlking-tomarket the position. Marking-to-market answers the question: If we liquidated the position today, what would be the gain or loss? In the case of an option price increase, the market-maler would need to buy the option back at a higher price than that at which it was sold, and therefore would lose money. At a stock price of $40.75,the call price would increase to $3.2352,so the market-maker profit on a per-share basis would be $2.7804 $3.2352 Figure 13.1 graphs the overnight proit of the unhedged written call option as a function of the stock price, against the profit of the option at expiration. In computing overnightprost, we are varying the stockpriceholding fixed al1other inputs to the BlackScholes formula except for time to expiration, which decreases by 1 day. lt is apparent from the g'raph tat the lisk for the market-maler is a lise in the stock plice. Although it is not obvious from the graph, if the stock price does not change, the market-maker will profit because of time decay: It would be possible to liquidate the option positio by buying options at a lower price tlle next day than the price at which they were sold originally.
able to proft based on the markup. The store mantains an inventory, and the owner is Market-inakers immediately. television buy and wnlk who want in a to satisfy customers supply immediacy, permitting customers to trade whenever they wish. Proprietary trading, which is conceptually distinct from market-mnking, is tradtypically expect ing to express an inkestment strategy. Customers and proplietary traders market the whether depending up or down. In prostable goes be upon positions to their bid and selling at the ask. 'I'he position the buying profit mayket-makers at by contrast, of a market-maker is the result of whatever order flow arrives from customers. A difference between appliance sellers and financial market-makers is that an appliance store must possess a physical television in order to sell one. Alinancial marketgenerating inventory as makeq by contrast, can supply an asset by short-selling, thereby needed. ln some cases market-makers may trade as customers, but then the market-makr market-maker. is pay ing the bid-ajk spread and therefore not serking as a
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Without hedging, an active rparket-maker will hgve an arbitrary position geperated by tisk. An advepe price fulfllins customer orders. An arb 1trary po rtfolio has uncontrolled the market-maker. Consequently. market-makers movehas the potential ioofbnnkrupt tlaeir positions. zisk hedge aitempt to te . Market-makrs can control tisk by delta-hedging. As in Chapter 10, the marketmaker computes the option delta and takes an offsetting position in shares. We say that such a position is delta-hedged. ln general a delta-hedgd position is not a zerovalue position: The cost of the shares required to hedge is not the sarile as the cost of capitf to the options. Because of the cost difference, the market-maker must invest .
position. is that such a hedged position should earn ihe risk-free derivatives key in A idea it, and you have no risk rate: You have money tied up so you should earn a ret'ul'n on used this argument explicitly in our discussipn so you should earn the risk-free rate. We in binomial pricing in Chapter 10. The implicitly and 5, Chaptr in pricing of fonvard notion that a hedged position earns the risk-free rate is a linchpin of almost alldelivave thei.r pricing models. lt was the ftmdamental idea exploited by Black and Scholes in derivation of the option pricing model. With the help of a simple numerical example, we can understand not only the intuition of the Black-scholes model, but the mathematics as well. Delta-hedging is key lisk of an option position. If we to pricing because it is the technique for offsetting the provides us with delta-hedging think of option producers as selling options at cost, then is Delta-hedging replicated. option is when it is an understanding of what the cost of the plicing. option understanding thus both a technique widely used in practice, and a key to
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Suppose a market-maker sells one call option and hedges the position Withshares. With the sale of a call, the mrket-maler is short delta shares. To hedge this position, the market-maker can buy delta shares to delta-hedge the position.
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We now will consider the risk of a delta-hedged position by assuming that the market-maker delta-hedges and marks-to-market daily. We srstlook at numerical examples and then in Section 13.4 explain the results algebraically.
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-20 20
25
30
An Example of Delta-l-ledging for 2 Days
4'0
45
Stock Price
($)
35
Consider the 4o-stn'ke call option described aboke, written on 100 shares of stock. The market-mker sells the option and receives $278.04.Since A 0.5824, the market-makr also buys 58.24 shats. (We will pftit fractional share purchases in thij example.) The net investment is Day ()
55
50
60
=
(58.24 x $40)
$278.04::Fz$2051.56
-
At an 8% interestrate, the market-maker has an overnight financing charge of $2051.56x
Delta and Gamma as Measures of Exposttre Since delta tells us the price sensitivity of the option, it also measures the market-mAer's which sggests that a exposure. I'he delta of the cll at a stock price of $40 is 0.5824, by approximately option of the value increse should the $1 incrase in the stock price terefore the option plice would increase Jirice $0.5824. A $0.75increase in the tock option's value is the act'ual increase = in However, the 0.5824 $0.4368. by $0.75 x
$0.4548, greater by $0.0180. This discrepapcy occurs because delta varies wit.h the stock price: As the stpck price increases and the option moves more into the money, delta also increases. At a stock price of $40.75,delta is 0.6301. Thus, the delta at $40 will ullderstate the act'ual change in the value of the option due to a price increase. Similarly, delta will overstate the decline in the value of the option due to a stock price decrease. tf the stock price had fallen $0.75 to $39.25, the option price would have declined to $2.3622,wlaich would result in a gain of $0.4182to the market-maker. x 0.5824 Using delta, we would have predicted a plice decline of option delta decreases the because decline. actual This than the occurs which is greater 0.5326: price this is The delta at the stock new plice declines. as Gamma measurs the change in deltawhen te stockprice changs. In the example above, the gamma of 0.0652 means that delta will change by approximately 0.0652 if
(c0.08/365
j)
-
=
$0.45.
Without at Erst worrying about rebalancing the portfolio' : Marlting-to-marltet to maintain delta-neueality, we can ask whether the market-maker made money or lost money ovemight. Suppose the new stockplice is $40.50.The new call option price with 1 day less to expiration and at the new stock price is $3.0621.Overnight mark-to-market proft is a gain of $0.50.computed as follows: Day
,1
58.24 x ($40.50 $40)
Gain on 58.24 shares
-
Gain on wlitten call option lnterest
=
$278.04 $306.21 1) x $2051.56
(j(syato
-(c
'
-$28.17
-$0.45
-
$0.50
Overnight profit '
-$0.4368,
-$0.75
$29.12
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the portolio new delta is 0.6142. Since deltahas increased, = additional shares. This transaction requires an 58.24 3.18 buy 61.42 must we investment of $40.50x 3.18 = $128.79.Since the readjustment in the number of shares 'l''he
Day 1: Rebalanclng
-
.' 418
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% MARKET-MAKI
N G AN D
DELTA-HEDGI
DELTA-HEDGI
lq G
profits entails buying at the current market price. it does not affect the mark-to-market for that day. .
E
Thus, we peed
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.
The stock price now falls to $39.25. The market-maker 2: Marldns-to-market , pay th'e written option andloses money on the 61.42 shares. lnterestexpense makes money on required for the has increased over the previous day because additional investment was of loss is marldng-to-market a extra shares. The net result from -$3.87:
61.42 x ($39.25 $40.50)
Gain on 61.42 shares
-
call option
Gain on written
$306.21 $232.82 1) x $2181.30
-$76.78
=
$73.39
-
-(c0..08/365
Interest
-
.
=$0
48
= -$3.87
!
.
Overnight proht
$128.79to buy more shares and $0.45to pay interest expnse. The
$129.74
$128.79 $0.45
-
$0.50
=
-
Since the bnnk is willing to lend us the value of our securities, we are free to pocket the $0.50 tat is lef't over. 'fhis exnmple demonstrates that the mark-to-market prolit equals the net cash flow genepted by always bon-owinjto fully fundtheposition. Another way to see the equality of mark-to-marketprost and net cash flow is by examining the sources and pses of funds, and the extent to wllich it is necessary to injct additional cash into the position in order to maintain the delta-neutral hedge. We can calculate the net cash flow frpm the portfolio aS
Interpreting the Prolh Calculation ConcepAt the end of day 1, we show a$0.50 protitfrom the mark-to-marketcalculqtipn. p' ortfolio whieh the the measuring extent loss proht to of the think as or tually, we can show a When hedge. delta-neutral maintain we order infusions a in to requires cash portfolio. of the cash take this out positive proft, as in case, we can To see that mark-to-market proft measures the net cash ipfusions required to maintain the delta-neutral position, suppose that a lender is willing t a1l times to lend buy 58.24 shares of stock, wlzich us the value of securities in the portfolio. lnitially, we option premium, so the net cash the by offset is $278.04 costs $2329.60,but this amount portfolio of value the also is (stockless the option), our net This we require is $2051.56. amount.z this so we can bon'ow As time passes, there are three sources of cash flow into and out of the portfolio: 1. Borrowing: Our borrowing capacity equals the market value of securities in the portfolio; hence, borrowingcajacity changes as thenetvalueof thepositionchanges. On day 0, the net value of our securities was $2051.56. On day 1, te share price rose and we bought additional shares; the market value of the position was 61.42 x $40.50 $306.21 $2181.30.Thus our borrowing capacity increased by capacity, but there is $129.74. The change in the option value changes boaowing options. number of changing the no cash flow since we m.e not 2. Purchase or sale of shares: We buy or sell shares as necessary to maintain deltaneutrality. In the above example, we increased shares in our polrtfolio from 58.24 to 61.42. The price at the time was $40.50,so we spent 3.18 x $40.50 $128.79. =
=
3. Interest:
419
extra amount the bank will lend us-is $129.74. change in our bonowing capacity-the 'l''he difference between what the bank will lend us on the one hand and the cost of addidonal shares plus interest on the other is
Net cash flow
-
%
NG
We pay interest on the borrowed amount. On day 1 we owed
$0.45.
2In practice the market-maker would be able to borrow only part of the funds required to buy securities, with market-maker capital mald ng up the difference. .
Change in borrowing capacity additional - cash used to purchase interest
=
shares
Let Af denote the option delta on day i, Si the stock price, Ci the option price, and
Mjq the markefvalue of the portfolio. Borrowing capacity on day i is MY. hence, the change in borrowing capacity is MVf
MVf-l
-
=F:
2; Si
-
Ci
Si-j
(Af-I
-
The cost of purchasing additional shares is Si(Af depends on the previous ay's borrowing, rMVf-1
-
MV/-I
-
-
Si(A?
-
.
-
-
Ci ;
Ci-k )
-
-
-
-
-
), and interest owed on day i Thus, on day i we have
Af-l
-
A/ Si
Af-I
-
) rMVf-l LLCSL Ci (Af-l Si-q Ci-k ) Si(Af = (Cf Ci-j Si-j Af-l Si ) rMVf-l ) =
Net cash flow = MVf
=
-
-
Af-I)
-
rMVf-1
-
The last expression is the overnight gain on shares, less the overnight gain on the option, less interest-,this result is identical to the proft calculation we pedbrmed above. ln the numerical example, we have
11-71 -
MVa
-
5'l(AI
-
A0)
-
?'MV:
$2181.30 $2051.56 $128.79 $0.45 $0.50 = =
-
-
-
This value is equal to the ovenzight proft we calculated between day 0 and day 1. Thus, we can interpret the daily mark-to-market protit or loss as the amount of cash tat we can pocket (if there is a proft) or that we mst pay (if there is a loss) in order to fund required purchases of new shares and to continue borrowing exactly the ainount of our securities. When we have a positive proft, as on day 1, we can take money out of the portfolio, and when we have a negative prost, as on day 2, we must put money into the portfolio. A hedged portfolio that never requires additional cash investments to remain hedgedis self-financing. One of the questions we will answer is under what conditions delta-hedged portfolio is self-fnancing.
420
%.MARKET-MAKING
DELTA-HEDGI
AND DELTA-HEDGING
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Figure 13.2 graphs overnight market-maker prost on day 1 as a function of the
Delta-Hedging for Several Days
stock price on day 1. At a stock price of $40.50,for example, the profit is $0.50,just as g'aph is generated by recomputing the lirst day's prost per share for a in the table. variet.yof stock prices between $37 and $43. The gpph veres what is evident in the delta-hedging market-maker who has written a call wants small stock price table: and can suffer a substanti loss with a big move. ln fact, should the stock price moves move to $37.50,for exaple, the market-maker would lose $20. If the market-maker had purchased a call and shorted delta shares, the prost calculation would be reversed. The market-maker would lose money for small stock prtce moves and make money with large moves. The profit diagram for such a position would be a mirror image of Figure 13.2.
We ean continue the example by letting the market-maker rebalance the portfolio each lay. Table 13.2 summarizes delta, the net investment, and proft each day for 5 days. The proft line in the table is daily prost, not cumulative profit. What determines the pattern of gain and loss in the table? There are three effects, attributable to gamma, theta, and the carrying cost of the position. 1. Gamma: For the largest moves in the stock, the market-maker losej money. For small moves in the stock price, the market-maker makes money. The loss for large moves results from gamma: If the stock price changes, the positi becomes unhedged. In this case, since the market-maker is short the option, a larj move lises, the delta of the call incregses and the call generates a loss. As the stock price loses money faster than the stock makes money. As th stck pfie falls, the delta of the eall dereases and the call makes money more slowly than the fixed stock position loses money. In efect, the market-maker becomes unhedged net long as losses on the stock price falls and unhedged net short as the stock price rises. days 2 and 4 are attributable to gamma. For a11of the entries in Table 13.2, the per share. garnma of the written call is about
'l'he
'l'he
rfhe
-0.06
2. Theta: Ifaday passes withno change in the stockprice, the optionbecomes heaper. This time decay works to the beneht of the market-maker Fho could unFind te position more cheaply. Time decay is especially evident in the proft on day 5, but is also responsible for the proht op days 1 and 3. 3. Interest Cost: tn order to hedge, the market-maker must purade carrying cost is a component of the overall cost.
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Using Gnmma to Better Approximate the Change in the Option Price
:
-'
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TH E MATH EMATI CS O F
0.51
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0.42
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Delta alone is n inaccurate predictor of the change in the option price because delta changes with the stock price. When delta is very sensitive to the stock price (gammais large), the inaccuracy will be relativety great. When delta is not sensitive to the stock price (gammais small), the inaccuracy will be relatively small. Since gamma measures the chqge in delta, w can use gamma to develop a better approximation for the change in the option price. If the stock price were $40.75instead of $40, the option price would be $3.2352 instead of $2.7804. For the purpose of computing the change in the option price, we want to know the average rate of price increase between $40 and $40.75,which we can approximate by averaging the deltas at $40 and $40.75: AAverage
ln the previous example, the stock price changes by varying amounts and our daily proht varies substantially. However, in Figure 13.2, there is an up move and a down move for the stock such that the market-maker exactly breaks even in our proft calculations. If the stock always moved by this amount, the portfolio would be self-financing: No cash in:ows are required to maintain delta-neutrality. lt turns out that the portfolio is self-snancing if the stock moves by one standard deviation. In the binomial option pricing model in Chapter 10, we assumed that the stock Serh-''ll where o'X-h is the standard deviation per move d up to or down to interval of the rate of remrn on the stock. Suppose we assume the stock moves up or down according to this binomial model. Table 13.3, which is otherwise the same as the previous example, shows the results of the stock moving up, down three times, and ten each day. I. the up. You can see that the market-maker comes close to brealcing even approximately portfolio is stock moves according to the binomial model, therefore, the self-fnancing.
=
$40.75 y computing
C($40) + 0.75 x Aywerage
(13.1)
When we average the deltas at$40 and $40.75,wehave to compute deltas at two different stock prices. A different approach is to approximate the average delta by using only the delta evaluated at $40together with gamma. Since gamma measures the change in delta, we can approximate the delta at $40.75by adding 0.75 x F' to Ag0: A40.75=
+ 0.75 x 1-'
40
Using this relationship, the average delta is A4() + (A4() + 0.75 x AAverage =
2
r)
j
= A40 +
Serh-bTvl
,
2
We could then approximate the option price at
C($40.75)
A Self-Financing Portfolio: The Stock Moves One o'
A40 + A40.75
=
x 0.75 x
(13.1),we can then approximate the
Using'equation
C($40.75)
=
C($40) + 0.75 x
= C($40) + 0.75 x
r
call price as
AAvecage
1 r. x 0.75 x F'
A4o +
= C($40) + 0.75 x A4: +
1
j
x 0.75
2
x In
(13.2)
The use of delta and gammato approximate the new option price is called adelta-gamma
13.4 THE MATHEMATICS
OF DELTA-HEDGING
Clearly, delta, gamma, alad theta :11 play a role in determining the prolit on a deltahedged position. ln this section we exnmine these relationships more closely in order to better understand the numerical example above. What we do hely is a kind of financial forensics: Once we larn how the stock price changed, we seek to discover why we earned the profit we did. .
approximation/
3You may recognize that we have already encountered the idea of a delta-gamma approximation in Chapter 7, when we used duration (delta)and convexity (gamma)to approximate the price change of a bond.
k. MARKET-MAKI
424
lq G AN D
D ELTA-HEDG
TH E MATH EMATI CS O F
l NG
?' )t@' )' 7* )j;lfEkample 13.1 lf the stock price liss from $40 to $40.75, the option price injyj )) creases from $2.7804to $3.2352.Using a delta approximation alone, we would estimate. .)(jC($40.75) ) as .1(.i .p.tt . C($40.75) l) C($40) + 0.75 x 0.5824 $3.2172 l.!j '. lj) j i))I delta-gamma approximation, we obtain Using a j J'jE
=
=
.
L (&) + 8
(SJ+)
= L$)
2
=
=
-
'
$
425
+
1
(13.4)
-E:1-'(u%)
2
The olkion price at te new stock plice is the initial option price, C&), plus the average delta times the change in the stock price, or
CS+l,4
.
1 )(t C($40.75) C(740) + 0.75 x 0.5824 + $3.2355 x 0.752 x 0.0652 li 2 tti li fhe delta-gamma approximation is signifcantly closer to the trtle option price at $40.75 t,, than is the dlta approximation. y Similarly, for a stock price decline to $39.25,the true option price is $2.3622.The t! i(' delta approximation gives
NG
simply the average of LS:) and A(u%+/,), or (usingequation 13.3)
E
'
DELTA-HEDGI
Using equation
Cst
=
)+
6
A(&) + A(&+?,)
2
(13.4),we can rewrite this to express A(&+/,) in terms of r(&): Cstwllt
gi.
=
C'tj
+ E7A(q%)+
l a j.6 P(u%)
(13.5)
..'
.
q. . j: E .
:
:.
k..)
.
E E .
f'' C($40) C($39.25) j 'l tiy ) The delta-gamma approximation gives ,y).
=
). iy j'!
' iltii 'j
.
c($39.25) C($40) =
-
0.75 x 0.5824
-
0.75 x 0.5824 +
=
$2.3436
prediction.
1
0.752 x 0.0652 :5- x
=
$2.3619
-
. approximation is more accurate. )i Again, the delta-gamma
Delta-Gamma Approximations We now repeat the previous arguments using algebra. For a move in the stock price. we lnow that the rate at which delta chlnges is given by gamma. Thus, if over a time interval of length /? the stock price change is 'Gsmall''
e
=
-,+/,
The gamma correction is independent of the direction of the change in the stock price 62, which is always positive. When the stock price goes because gamma is multiplied by change predicts little call price, and we have to add something alne the in too a up, delta the stock price prediction. When the goes down, delta alone predicts too much to correct and the option plice, again have to add something to correct the of a decrease in we
&
-
The new predicted call plice is not perfect because gamma chgnges as the stock price changes. We could add a tel'm correcting for the change irt gamma, and a term correcting forthe change in the change in gamma, and so forth, but the gamma correction alone dramatically improves the accuracy of the approximation. You might recognize equation (13.5)as a second order (becauseit uses the frst and second derivatives, delta' for the change in the call price. Taylor series and gnmma) Taylor series (?pp?'tW???t#ft??? Appendix discussed in I3.A. approximations are Figure 13.3 shows the result of approximating the option plice using the delta and delta-gnmma approximations. The delta approximation is a straight line tangent to the option ptice culwe and is always below the option plice culwe. Because of this, the delta approximation understates the option price, whether the stock price lises or falls. The delta-gamma approximation uses the squared stock plice change, which generates a curve more closely approximating the option price curve.
then gamma is the change in delta per dollar of stock price change, or r(&)
=
Lvt-hl
-
A('%)
Rewriting this expression, delta will change by approximately the magnitude of the price change, 6, times gamma, cr: A(u%+/,) = A(.%) + 6P(u%)
lf the rate at which delta changes is constant (meaningthat gamma is constant), this calculation is exact. How does equation (13.3)help us compute the option price change? 1-fthe stock price changes by EF, we can compute the option price change if we know the average delta over the range 'St+;, to St lf r is approximately constant, the average delta is .
Theta: Accounting for Time 'Fhe preceding calculations measured the option lisk that arises from plice changes alone. Of course, as te price changes, time is passing and the option is approaching maturity. The option's theta 0) measures the option's change in price due to time passing, holding the stock price lixed. For a period o length /?, the change in the option price will be 011. For example, consider the 91-day option in Section 11 3 ad consider the effect of a day passing, with no change in the stock price. Since the variables in the option pricing formula are expressed as annual values, a one-unit change in F t Since h is 1/365, the implied daily option price change is implies a 0 of .
-
-6.33251.
-6.33251/365
-0.01735.
=
$
426
MARKET-MAKI
N G AN D
DELTA-HEDGI
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-
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The calculations in Tables 13.2 and 13.3 use actual option price changes over 1 day, not approximations. However, by using equation (13.6) to approximate the change in the option price we can better understand the profit results in those tables. Theviue of the market-maker's investmeptvlong deltashares and shortacall-i.s the value of delta shares of stock less the value of the call, or
(5)
Lt St
t)
(13.6)
The experiment re:ected in this equation is t periods until expiration and a stock price of St, we want to express the new option price as a function of the old plice and the delta, gnmma, and theta evaluated at the original price and time to expiration. This formula is quite accurate for relatively smil changes in time and in the stock price. -
EC(&+)
(q%+/,&)
Lt
-
-
&)
-
-
l-lLLt
.% -
C(&)1
Interest expense
(13.6),in which we characterized te change in the option price. C't) in this expression tells us (13.6)for CSt+) (Ar &+lt SJ) + 1 &+h S?): F'J + 0hj l-lLhtst Cvttj 2
Now recall equation Substimting equation Ltst-l'
Cstjj
-
Change in value of option
changein value of stock
-
-
-
-
= Example 15.2 Table 13.4 shows the results from using equation (13.6)to predict the option price change when prices move by $0.75 (6 = +0.75). ln this exnmple, Note that in each case the formula L = 0.5824, In = 0.062, and ho = predicted the plice the change, is close to the actual price. but slightly overstates %
CSt)
-
Suppose tat over the time interval h, the stock price changes from St to St+. The change in the v/ue of the portfolio is te change in the value of the stock and option POS it.iOIIS, less interest expense:
this: Starting with an option with F
-0.0173.
-
. . . . .
.
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Adding theta to otlr previous option price prediction equation, we have C CSJ->
=
-
.
:
..' '
-
approximation
=
-
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.
DELTA-HEDG
TH E MATH EMATI CS O F
NG
1 2 F'r + 0th + 2
-6 -
-
-
?-
(A?St
-
Cst
)q
(13.7)
'l'his expression gives us the prot of the market-maker when the stock price changes by E over an intelwal of length h. lt is quite important and variants of it will appear later in the book. On the right-hand side of equation (13.7)we see the effects of grlmma, theta, and interest:
k. MARKET-MAKING
428
AND DELTA-HEDGI
NG
TH E BLACK-SCH -1czrJ.
o
Since the gamma of the 1t Gamma: The effect of gamma is measured by 2 writing will market-maker lose positive, the call the by call is money in proportion The larger the stock the of the stock price change. move, te greater is the to square loss. 2. Theta: The effect of theta is measured by the option writer benefits from theta.
Since theta for the call is negative,
-0th.
If
ty
length 11is
is measured annually, then a one-standard-deviation Therefore a squared one-standard-deviation CstX-h.
2
E7
Market-maker profit
(Al St Cstjj. The option writer has 3. lnterest cost: lnterest is measured by investment because shares Lt are more expensive than one option. Hence, a net interest is a net cost. -
Since 0 is negative, time decay beneits the market-maker, whereas interst and gamma work against the market-maker. Let's examine how 6, the change in the stock plice, enters equation (13.7). Note first that because we have delta-hedged, E itself does not affect market-maker prot. 62. Consequently, However, prolit does depend on the sqttared change in the stockplice, as w saw in Table 13.2, it is the magnimde-not the direction of the stock price movethat determines prost. Table 13.5 calculates equation (13.7)for various moves in te stock pzce. Because equation (13.7) depends only on the squared stock price move, the calculation is the same for moves up and down. If the stock price moves $0.6281,equation (13.7)is exactly zero. We have already seen in Table 13.3 that the market-maker approximately breaks even for a one-standarddeviation move in the stock. Here we at the same result. Let's explore this idea further. '
-nrrive
$' ;'ryjyyit'. E' )' k...k;-' ;r:)' 7*():;E-' j'jg,,,f itf ''rj-.jijjrkir.'' jrrlf ?'. y' 577'5* j'qf 7* 1* jqf ,')' k(' (' jj;.,.:,;f jjy,..rjjf ('' (' r'.,;jjjjj'q' y' jyf $4* lq' )(' y'. )' ,rjE;;' .jjyyy' ,j,..,jj' :---'. r' j;' jjg:;:,jf ;')' p' )#'. t'1*rrlEE.E;'qll(1:1. ..' k' yyjjg;,f !Ei ( jg,--j'yyyf j'1*. 1iIt-k-2'' p' 1* y' yjtyyf 1$* (1( Eq;i)i'(i!
y' jjyt'-. ttf ty'y(j'. t't1*!''.iI. 1! ' r (;;'y ' ;.'..(' ( (.(.i'@''' tjyt'-
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.
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i
:
! E r.Ftrr''((.q(i.i.'i(@;EE!; .r!'lrpl''rql' ((((. y.E.. yjy. jy , ;q.... .(y .yy..q'.yj.y . j y g k . j yj y.j j y y y . y Ey. .. -. j...j. ; . . . .. (.. . . i .-...(.)- .:i-.:..- (
' ' :. E'!.' EE' EEE' ''' E. .. : . '' ' ' .: '' .' ' ' E E :' E '.' (E . E. . E : E E . . :( . (. (' ' . '. - .. .- -.. E, . . . -- - . . . -... ! E ! q-. . - . y. '
'
!.. .
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:
:
:
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' $4(7')(
-E ; - -! ; ! .
.
.-,b;b3tib-..
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'
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..
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.
'.
.
-.
.
:: E' .
:
.
yyy . .,. y. ;''.... ''i ' i..' r ..pjyri .y 1iiIii!k.. yt... qri . ..j ... )(k))).. '.'. .r. yjjjjjri, . '! (' .. . E..:.) . ..... .... . . . . ... .
..
j.
. --
.
-.
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. .. .. . . . . . . . . . . . . . .
.
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-
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-y
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..
.
. . - -. .
-
. . .. . .
-
$
429
=
ty
2
2
h /1
we can rewrite equation
1 2
-c -
=
2
move over a period of move is
(13.8)
(13.7)as
St2 L + 0t'+ r (A?,%
-
Cstl
/7
(13.9)
This expression gives us market-maker pront when the stock moves one standard deviacs-h = $0.6281.From Table tion. As an example, let 11 1/365 and o, 0.3. Then equation stock plice exactly zero! (Prblem 13.13 asks 13.5, with this (13.9)is move accident equation that is verify this.) lt is not an (13.9) zero for thij price move. you to Section 13.5. explain tis result in We In Table 13.5, the loss from a $ 1 move is substantially larger in absolpte valpe than gain from no move. However, small moves are more probable tn bt: moves. 1.f the we think of returns as being approximately nonnally distributed, then siok pricemoves greter than one standard deviation occur about one-third of the time. The market-maker thus expects to make small profits about two-thirds of the time and larger losses about one-third of the time. On average, tllp market-maker will break even. =
=
13.5 THE BLACK-SCHOLES
ANALYSIS
We have discussed how a market-maker can measure and manage the risk of a portfoli6 containing options. What is the link to pricing an option?
The Black-scholes Argument
;'.'i(' ',jjyyy' ..' .,,,' ,,,jj' .,,' -jjjjjjjj;j'; --.----' -'' -'-'.' --' y'j'.jtIjj;jj;yE j':'r'L' )Fy'. j,yyg,,f j,::yyrjjf jj::rygyf j,yyjgr,f j,::yyrj;f jjyjjy,,f -' j,:ryy:j;f jj:j,,f jj:yyr:,jf jj::yyf
.. . .
62,
Substituting this expression for
-1'11
'
O LES ANALYSIS
'
. .
-
..
-
.
. -
jjg.,jjyy,
jj:y:y:ys jjyyyg,,s
.!jy::.jj:::,j,,,,,jy jjj ,,,,
tjggjjgjj,jjy
jj:y:g:,,, k;yjyj,g
dffqrent-sized stock price moves on the profit of a delta-hedged market-maker.
1.283 1.080 0.470 0.6281 0.7500 1.0000 1.5000
y;;;;;,,, jy;jj.rjj::y
0.000 -0.546
#' a stock moves one standard deviation, then a delta-hedged position will exactty break even, taling into account the cost of funding the position. This dndingis not a coincidence', it reflects the arguments Black and Scholes used to derive te option pricing; formula. Imagin :e for example, that the stock always moves exactly one standard deviation evel'y 'minute; A market-maker hedging every minute will be hedged over an hour or over any period of time. Blackand Scholes argued that the money invested in this hedged position should earn the risk-free rate since the resulting income stmam is risk-free. Equation (13.9) gives us an expression for market-maker proft when the stock moves one standard deviation. Setting this expression to zero gives 1 2 z 1C(u%)) I 0 - 2 St F'; + 0 + LLt't -o'
-
=
-1.970
-6.036
'lThere are 365 x 24 x 60 525, 600 minutes in a year. Thus, if the stock's annual standard deviation is30%, the per-minute standard deviation s 0.3/ 525,600 0.04%, or $0.016 for a $40 stock. =
=
430
% MARKET-MAKING
TH E BLACK-SCHOLES
AND DELTA-HEDGING
f
+ P
=
(13.10)
l'cst)
to characterize the lldzWt??- of :7,7 option. This is l/2c equation Black J??# Scholes II-EW is the The Greeks F', A, and 0 are partial derivatives of the option price. Equation (13.10) equation Black-scholes partial differential equation, orjust Black-scholes well-lnown . . .. ll). We Fill see (s opppse to the Black-scbesfonnula for the price of a Eurpean in-later chatrs that this relationship among the Greeks is d fundamental in valuing risky cash flows qs is c-rr when valuing risk-free cash flows. that the underlyquatin (13.10)embodies numerous atsumptions, among thin uzviuend, O))ti0Z ZC the interest iZg ZSSG UOCS JROt itsclf does hot liy a PZJ! t dividcfld: devitio standar over a small rate d vltility are constant, and the stock moves one holdj for calls, pu, Ariierican time itefkl. Wit these assumptions, equation (13.10) will consider in hapter options, Europah options, and most of the exotic variats we j wi 14. With simple modifcations, an equation like (13.10) 1 also hold fof ftionson ('jivtdnd-paylng' stocks, currencies, fumres, bonds, etc. The link between delta-hedging and pricing is one of the most important ideas in financ. -
.''''
.'x
'
.
.
.
Equation (13.10)holds forAmerican options as well as for European options, but it does early-exercise the option. Consider a deep-innot hold at times when it is optimal to the-money American put option and suppose the option should be exercised early. Since = r = 0, and 0 = 0. early exercise is optimal, the option price is K S',hence, A. In this case, equation (13.10)becomes -1,
-
(?' x (-
-
431
In practice. because of ansaction costs, it is expepsive for a market-maler to trade shares for every change in an option delta. Instea, a delta-hedger will wait for the positio to become somewhat unhedged before trading to reestablish delta-neueality. In the binomial model in Chapter l0, and in the preceding discussion, we assumed that market-makers mintain their hedged position and that stock prices move exactly one standard deviation. In real life the stock price will rarely move exactly one standard deviation over yhe course of a day. What does the market-maker lose by hedging less
equently?
Boyle and Emanuel (1980)considered a market-maker who delya-hedges at set intervals, rather than evel'y time the stock price changes. Let xi denote the number of standard deviations the stock price moves-we can think of xi as being drawn randomly from the standard normal distribution. Also 1et Rvt. denote the peliod-/ ret'urn to a deltahedged market-maker who. as in our eadier example, has wlitten a cll. Bo#le and Emanuel show that this ret'urn can be written as5
1 2 2 1)/7 S tr In xi2 (13. 11) 2 where r is the option's gamma and, is the time interval between hedge readjustments. 1 is 2', hence, From Boyle and Emanuel, the variance of x/ Rki
=
-
-
-
MaR,il
Delta-l-ledoinog of American Options
1) x 5'/1 +
$
What ls the Advantage to Frequent Re-l-ledgm'g?
lf we divide by 11and rearrange terms, we get 1 2 a St rf + t'St -t7' 2
ANALYSIS
1 x 0 + 0 2
=
1,
x CK
-
&)
Note that -rSt appears on oth sides of the equation. Thus, we can rewrite the equation
2
2
(5'c. r//)
2
(13.12)
=
-
1 (S ac a r/365)2 2
: The dally remrfl of the market-maker who hedges hourly is the sum of the hourly returns. Assuming for the sale of simplicity that S and r do not change much, that variance is
as
var Since this equation is false, something is wrong. We began by assuming that the put early-exercised. From the discussion of early was so far in-the-money that it should be received on the strike exceeds the loss interest that exercise in Chapter 10, this means should be exercised but you own it, option the if Thus, of the implicit call option. lose interest on the sfn'ke you are not then exercise it, you delta-hedge it, and do not option and delta-hedged, and the owner the written have receiving. Similarly, if you of rK. arbitrage earning prct does not exercise, then you are where early exercise is not optimal. region valid in only Thus, equation (13.10)is a is irrational and there behavior then exercised, exercised is but not lf an option should bq risk-free and thus no reason the should position rate delta-hedged earn is no reason why a hold. should that equation (13.10)
1 2
We assume-as in the binomial model-that the stockreturn is uncorrelated across time, so that xi is uncorrelated across time. N ow l e t's compare hedging once a day against hedging hourly (supposetrading occurs around the clock). The daily variance of the return arned by the market-maker who hedges once a day is given by Var(.RI/365,l)
=
-
24
24
J''l i= 1
R;,,i
=
1
J'''l2 g.ao. ar'/(24x 365)j -
2
izzzz 1
1 = 24 x Var(Alp65, I ) -
Thus, by hedging hourly instead of dily reduced by a factor of 24.
the market-maker's
serhisexpression can be derived by assuming that the stock price variance /,, and subtracting equation (13.9)from (13.7). vc
move,
6,
total return variance is
is nonnally distributed with
% MARKET-MAKING
432
Arqo
TH E
DELTA-HEDGING
Whateverthe hedgingintelwal, abouttwo-thirds Hereis teinmitionforthisresult. of the price moves will be less than a single standard deviation, whereas one-third will be greater. Frequent re-hedging does not avoid these large or small moves,since they can occur over any intervi. However, frequent hedging does permit better averaging of the effects of these moves. Whether you hedge once a day or once an hour, the typici stock price move you encounter will likely not be close to one standard deviation. However, if you hedge every hour, over the course of a day you will have 24 moves and 24 opportunities to re-hedge. The average move over this period is likelier to be close to gins from small moves and losses f'rom large moves witl one standard deviation. tnd to average over the course of a day. ln effect, the more f'requent hedger benests f'romdiversiEcation over time.6 'l'he
''
j'
q Example 15.3. Using Boyle and Emanuel's formulas to study the market-maker (4.1 ' tty problem in Section 13.3, the standard deviation of profit is abotg $0.075for a market) mafter who hedges hourly. Since hedging errors are independent from hour to hour, the j.. daily standard deviation for hourly hedger would be $0.075 VV $0.37. If the x an g) 'lrjt.j mr ket-maker were to hedge only daily as in otlr example, the daily standard deviation %. k' would be about $1.82. '
'
f.
..
E
E
=
E
.
As you would expect, the mean return on a delta-hedged position is zero, even if
the hedge is not frequently readjusted.
Delta-lqedglg
in Practice
The Black-scholes analysis outlined here is the linchpin of modern optiop pricing theory alld practice. Market-makers use equation (13.10)to price options, subject to qualiscations mentioned above. . We have seen, howeker, that delta-hedging does not eliminate lisk. One problem, wllich we emphasized above, is that a delta-hedged portfolio wi th negat ive amma can sustain losses due to large moves in the price of the underlying asset. Consequently, a delta-hedging market-maker needs to wort'y about gamma. Anoter problem, discussed in the box on page 434, is that firms can unexpectedly change their dividend payments. There are at least four ways a market-maker can try to reduce the risk of exeme price moves. Note that some of these seategies require the market-maker to acquire :
6-fhis resembles the problem faced by an insurance company. lf the company insures one large asset, the standard deviation of the loss is /1 greater than if it insures ?; small assets, with the same total insured value in each case. Similarly, we can view the return over each hedging interval as being an independent draw from a p'robabilit'ydistribution.
BLACK-SCH
O LE5 ANALYSIS
$
433
specific option positions, which means that the market-maker may have to pay the bidask spread. Since the bid-ask spread is revenue for the market-maker, paying the spread is undesirable. Firstjust as market-makers can adopt a delta-nutral position, they can also adopt position. This position cnnnot be achieved with the stock alone, since gnmma-neutral a the gamma of the stock is zero. Thus, to be gnmma-neutral the market-maker must buy or sell options so as to offsetthe gamma of the existingposition.? We provide an example of gnmma-neutrality below. Second, in a related strategy, market-makers can use static option replication, a strategy in which options are used to hedge options. In our delta-hedging example, the market-maker might not be able to buy an exactly ofsetting call option to hedge the written catl, but by selectively setting the bid and ask prices for related options, might be able to acquire an opion position requiring only infrequent rebalancing. To take a simple example, if the market-maker were able to buy a put with the same strike price and maturity as the written call (e.g.,by setting the bid price to attract any sellf of that op:on), then by buying 100 shares to offset the risk of the position, tlze inarket-maker would have used put-call parity to create a hedge that is both gamma- arld deltauneual for the life of the options. Third, a market-maker can buy out-of-the-money options as instlranck In our example of delta-neual hedging of a written option, the market-malcer could buy a deep-out-of-the-money put and a deep-out-of-the-money call. The two options would be relatively inexpensive and would protect against large moves in the stock. One problem with this solution is tat, since option positions in the aggregate sum to zerd, the market-maling community as a whole can buy protective options only if investors in the aggregate are witling to sell them. hwestors, however, are usually thought to be insurance buyers rather than inslzrance sellers. Fourth, a market-maker can create a financial product by selling tlle hedging error, by Carr and Madan (1998). For exnmple, to hedge a negativ-gnmma, discussed as delta-neutral position, the market-maker would make a payment to a counterparty if the stockmakes a small move in either direction, and receive payment from the counterpm't)/ if the tock makes a large move in either direction. This is effectively a variance swap, which we will discuss in more detail in Chapter 23. The point is that the market-maker can potentially solve the delta-hedging problem by creating a new product.
Gamma-Neutrali'ty Let's explore gnmma-hedging in more depth. Suppose we wish to both delta-hedge and gamma-hedge the written option described in Table 13.1. We cnnnot do this using just
RMarket-makers sometimes buy and hold over-the-counter options issued by a firm. For example, in the late 1990s some finns sold put options on their own stock. These options were reportedly held by dealers and delta-hedged. One possible motivation for dealers to undertake such a transaction would be to acquire positive gnmma. See McDonald (2004)for a discussion.
%
434
MARKET-MAKI
D ELTA-HEDG
NG AN D
TH E BLACK-SCHO
IN G
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options forevery zm-strike 3-month Thus, we need to buy 1.2408 of the 4s-strikezmonth position are in the last column of from resulting the option we have sold. The Greeks 17.49 shares of stock to be both buy need 1749, we to Table 13.6. Since delta is delta- and gamma-hedged. Figure 13.4 compares this delta- and gamma-hedged position to the delta-hedged delta-hedgedposition position, discussed earlier, in which the same eall was written. The position gamma-hedged The losses. always cause has the problem that large moves stockprice increases. the if male and money can loses less if there is a large move down, positive vega. Why Moreover, as Table 13.6 shows, the gamma-hedged position has a ltot gamma-hedge? would anyone
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44
46
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436
%.MARKET-MAKING
MARKET-MAKI
AND DELTA-HEDGING
ganuna-hrdging requires the use of There are two reasons. First, as noted already, the required option position obtain additional options. The market-maker will have to example, a11profits enrned this In spread. bid-ask from anothermarket-maker, paying the who sells the 45-stIike market-maker zjo-strike the call will go to pay from writing the with many options bought call used in gamma-hedging. In a large portfolio, however, the net exposure would not gamma-hedging and sold, naturally offsetting one another, require many option transactions. and calls, then in The second reason is that if end-users on average buy puts gamma,'' end-users to use marketthe aggregate they have positve gamma (the negative aggregatemustthenhave in market-makers the makingparlance). By definition. gamma-hedge, could not a11 market-maker gamma. Thus, while in plinciple any one will not insurance, they buy lf want investors to market-rakers could be gamma-neutral. theaggregatebegnmma-neutral. cannotin market-makers begamma-neutral, and, hence, in ln addition to gamma, other risks, such as vega and rho, can be hedged the same fashion as delta. 'tbuy
13.6 MARKET-MAKING
AS INSURANCE
write options can sustlin The preceding discussion suggests that market-makers who that option marketlarge losses, even if they delta-hedge. This conclusion suggests with making (and delivatives market-maling more generally) has more in common insurance than you might at first think.
lnsurance individuals pay an lnsurance companies attempt to pool diversisable risks. All insured who suffer lotses, those used compensate then prerniums to are insurance premium. The spreads the pain rather thereby Insurance premium. while those without losses lose the losses. of inevitable the bear brunt individuals to than forcing a few unlucky ln the classic model Of insurance company, risks are independkt. Suppose an
an households each Prfivides ZSUFRIRCC fOr a lafge nllmber Of identical in ftre 'Fhe of which has an independent 1% chance in any year of losing $100,000 a = fa''' the is 000. This 000 1% $1, x $100, expected loss for each house is this from collects amount the insurance company that if insurance premium, in the sense annual insurance daims. In general each household, it will on average be able' to pay of doing business, less insurance will be priced to cover the expected loss, plus costs premium. interest earned on tle claims However, evn with diversication, there is a chance that actual insurance which insurance the in household, insured case exceed $1000per in a particular year will promises unless it has access to additional f'unds. company will not be able to fulfill its lisklisk-management insurance company-has product-the a Thus, the seller of a its obligations to management problem of its own-namely to be sure that it can meet conscience management; for of notjust matter obligations is such a customers. Meetng
irlstlrlmce
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NG AS INSU RANCE
$
437
if there is a signifcant chance that the insurance company will be bankrupted by claims, there will be no customers in the srstplace! Insurance companies
have two plimal'y ways to ensure that they meet claims:
1. Capital: Insurance companies hold capital, i.e., buffer fund in case there is an unusually large number of claims. Because of diversifcation, for any given bankruptcy probability insurers can use a smaller reserve fund per insured house as 4henumber of insurd houses groFs. Capital in the form of reserves has traditionglly been an important buffer for insurance companies against unexpectedly large claims. 2. Reinsurance: There is always the possibility of a loss that can exceed any flxed amount of capital. Am insurance company can in ttll'n buy insurarice against large losses, in order to be able to make large payouts if necessaly lnsurance for insurance companies is called reinsurance. lnsurance companies buy insurane from reinsurance firms to over the vent that claims exceed a certai amount. Reinsurance is a put option'. The reinsurance claim gives the insurance company the right to sell to the reinsurers claims that have lost money. Reinsurance does not change the aggrejate need for capital, but it does permit fimher diversihcation.
Market-Makers Now consider again the role of market-makers. Suppose that investors, fearful of a market crash, wish to buy out-of-the-money puts. Out-of-the-money put writers are martiet moves. lt is precisely when large market moves selling insurance against large delta-hedging Qccur that breaks down. Just lile an insurance company, a market-maker requires capital as a cus hion ag ainst losses. Since capital has a cost, markt-mkers may also raise the cost of writien options that require a disproportionately large commitment of capital per dollar of premium. buying out-of-the-money plk options Reinsurance foramarkey-makerwouldentail lisk market-maker, but ultimately if the nancial industry is a: another to to move some be capital in the of losses. writer of insurance, event tere must net of capital and the analogy insurance becomes more obvious The importance to derivatives markets. For example, consider think about weather delivawhen we new sld-resort institutions have hedged Financial operators against warm winters, softtives. (II-iIIk manufacttzrers againstcold andlawn sprinklermanufpcturers against wet summers, Ultimately, the bank must find a counterparty willing to absorb the risk. tf summers. weather insurance, on a global basis they are like tradithe risks think about in you in the United States can be diversified with weather Weather insurlmce. contracts tional capital committed to insurance absorbs the ultimately the global Asia, and conacts in defined are thus the npttlral party to capital markets broadly risk. Global reinsurance lisks. these absorb Some risk, however, is not globally diversisable. Consider writing puts on the S&P 500. lf the U.S. stock market suffers a large decline, other markets around the world are likely to follow. Ultimately, it is capital that safeguards the financial industly
438
%.MARKET-MAKING
PROB LEMS
AND DELTA-HEDGING
Delta-hedging plays a key role, but in the end there is always risk that must be absorbed by capital.
Market-makers buy and sell to satisfy customer demand. A derivatives market-maker resulting position. By desnition, the can use the underlying stock to delta-hedge the depend does position not on the directiol; in which the stock cettlrn on a delta-hedged lnagnitltde the stock prtce move. A position of the Price moves, but it does depend on small stock price moves and loses mak. for es money with zero delta and negative gamma is positive, it makes position of the the gamma money for large stock price moves. l.f small Either loses moves. way, the lelta-hedged money for money for large moves and deviation. standard stock position breaks even if the moves one Using a delta-gamma-theta approximation to charactelize te change in the value . of the delta-hedged portfolio, we can demonseate that there are three factors that explain the profitability of the portfolio. First, gamma measures the tendenc? of the portfolio to become unhedged as the stock price moves. Second, theta measures the gain or loss market-maler will have on the portfolio due to the passage of time alone. Third, te portfolio. interest income or expense on the If we assume that the stock price moves one standard deviation and impose the sattsfies a parcondition that the market-maker earns zero profit, then a fai.r option price relationship, equation theta. and relationship This (13.10), ticular among delta, gamma, is the foundation of the Black-scholes option plicing aniysis ad applies to derivatives in general, notjust to calls and puts. Ultimately, market-maldng is risky and requires capital. If customrs on average buy puts and cftlls, and if we think of options as insurance, then market-makers are in the capital, since if arl exeeme event same business as insuranc companies. This requires will fail. occurs, delta-hedging
FURTHER
439
call with 91 days to expiration. What is delta? lf the 13.1. Suppose you sell a 45-stIike option is on 100 shares, what investment is required for a delta-hedged portfolio? What is your overnight prost if the stock toronow is $39? What if the stock price is $40.502
13.2. Suppose you sell a m-strike put with 91 days to expiration. What is delta? Ifthe option is on 100 shares, what investment is required for a delta-hedged portfolio? What is your overnight prost if the stock price tomorrow is $392 What if it is $40.50? 13.3. Suppose you buy a 40-45 bull spread wit.h 91 days to expiration. 1.fyou deltahedge this position, what investment is required? What is your bvernight prost if the stock toinon'ow is $392 What if the stock is $40.50? 13.4. Suppose you enter into a put ratio spread where you buy a 45-strike put and sell two zm-stlike puts. Ifyou delta-hedge this position, what investment is required? What is your overnight profit if the stock tomorrow is
$40.50?
$39? What if the stock is
13.5. Reproduce the analysis in Table 13.2, assuming that instead of selling sell a 4o-strike put.
a call you
13.6. Reproduce the analysis in Table 13.3, assuming that instead of slling a call you sell a 4o-strike put. 13.7. Consider a zm-strike 180-day call with S $40. Compute a delta-gamma-theta approximation for the value of the call after 1, 5, and 25 days. For each day, consider stock prices of $36 to $44.00 in $0.25 increments and compare the acmal option premium at each stock price with th predicted premium. Where are the two the same? =
13.8. Repeat the previous problem for a 4o-strile lo-day
put.
13.9. Consider a 4o-stlike call with 91 days days to expiration. Graph the results from the following calculations. .
a. Compute the actual price wit.h 90 days to eypiration at $1 intervals from $30 to $50.
READING
The main example in this chapter assumed that the Black-scholes formula provided the correct option price and illusated the behavior of the fonnula, viewed from the perspective of a delta-hedging market-maker. ln Chapters 20 ad 21 we will start by building a model of how stock plices behave, artd see how the Black-scholes fonnula is derived. As in this chater, we willconclude thatequa:on (13.10)iskey to understanding option pricing.
b. Compute the estimated plice with 90 days to expiration using a delta approxlmation. c. Compute the estimated price with 90 days to expiration using a deltagamma applpximation. d. Compute the estimated price with 90 days to expiration using a deltagamma-theta approximation. 13.10. Consider a 4o-strike call with 365 days to expiration. Graph te results from the following calculations.
PROBLEMS ln the following problems assume, unless otherwise stated, tat S 1' 8%, and J 0. =
%.
=
=
$40, o'
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30%,
a. Compute the actual price with 360 days to expiration at $1 intervals from $30 to $50.
440
%.MARKET-MARING
AND DELTA-HEDGI
b. Compute the estimated price with 360 days to expiration using a delta approximation. expiration using a deltac. Compute the estimated price with 360 days to ganuna approximation. d. Compute the estimated price with 360 days to expiration using a deltagamma-theta approximation.
13.11. Repeat Problem 13.9 for a 91-day 4o-strike put. 13.12. Repeat Problem 13.10 for a 365-day
=
APPENDIX 13.A: TAYLOR APPROXIMATIONS
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I N TH E BINOM lAL MODEL
441
SERIES
We have seen that the change in the option price can be expressed in terms of delta, gamma, and theta. The resulting expression is really just a partiular approximation to the option price, called a Taylor selies approximation. Let Gx, y) be a function of two valiables. Taylor's tleorem says that the value of the fttnction at the point Gx + 6.v, y + e'y) may be approximated using derivatives of the f'unction, as follows:
zm-strike put.
13.13. Using the parameters in Table 13.1, verify that equation 13.14. Consider a put for which F that equation (13.9)is zero.
13.B:
APPENDIX
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$45. Compute the Greeks and verify
13'.15.You own one 45-strike call with 180 days to expiration. Compute and graph the l-dayholding period prot if you delta- and gamma-hedge this position using a 4o-strikecall with l80 days to expiration. 13.16. You have sold one 45-strike put with 180 days to expiration. Compute and graph the l-day holdingpeziod prctif you delta- and gamma-hedge this position using the stock and a 4o-strAe call with 180 days to ekpiration. 13.17. You have written a 35.-40.-45 butterlly spread with 91 days to expiration. Compute and graph the l-day holding period prolit if you delta- and gamma-hedgethis position using the stoct and a zm-strike catl with 180 days to expiration. 13.18. Suppose you enter into a put ratio spread where you buy a 45-strike put and sell two 4o-st14ke puts, both with 91 days to expiration. Compute and graph the l-day holding period proit if you delta- and gamma-hedge this position using the stock and a 4o-strike call with 180 days to expiration. 13.19. You have purchased a 4o-strike call with 91 days to expiration. You wish to delta-hedge, but you are also concerned about changes in volatility; thus, you want to vega-hedge your position as well. and vegqa. Compute and graph the l-day holdingperiod pmht if you deltazm-sirike call with 180 days to and stock the using position this a hedge expiration. b. Compute and graph the l-day holding period proft if you delta-, gamma-, and vega-hedge this position using the stock, a zm-stlike call with 180 days to expiration, and a 4s-st1ike put with 365 days to expiration. 13.20. Repeat the previous problem, except that instead of hedging volatility risk, you wish to hedge interest rate risk, i.e., to rho-hedge. In addition to delta-, gamma-, and rho-hedging, can you delta-gamma-rho-vega hedge?
(13.14) The approximation may b extended indenitely, tives. The nt.h term in the expansion is 11
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444
% Exo-rlc
OPTIONS:
AslAN
l
opiions. 'I'he challenge is to invent new options It is not hard to invent new kinds of did in the preceding example) n that that are potentially attractive to buyers (whichwe without too much difculty. In Chapters 10 and 13 we saw can be p riced and hedged That analysis 1edus to see how how a market-maker can delta-hedge an option position. of synthetically manufactudng the option. the priee of an option is equivalent to tl cost there is a certain relationship among the In particular, an option is fairly priced when Greeks of the option. and delta-hedged in the same Options with exotic feamres can generally be priced options.' As a consequence, exotic derivative products aze quite comway as ordinary well understood. Iiz - in practice and the technology for pricing and hedging them is mon is an anachronism. fact, since many such options are in common use, the term We will coniinue to use it, however. mathematical details of particular The goal in this chapter is not to master the understanding of the trad-offs in dsign and products, but rater to gain an intuitive of the formulas appearin the chapter apedixt Priqing. Consequently, most ordinary ojtions in minor Since exotic options ar oen constructed by tweaking exotic useful as benchmarks for exotics. To understand ways, ordinary options are options you should ask questions like these: exotic compare to that of a standard option? * How does the payof'f of the approximated by some portf.olio of other options? * Can the exotic option be exppnsive relative to standard options? Understanding . Is the exotic option cheap or understanding its pricing and use. the economics of the option is a critical step in the use of the exotic option? * What is the rationale for option be hedged? An option may be desirable to a o How easily can the exotic unless the risk arising from market-maliing can be customer, but it will not be sold controlled. ,
op-rloxs k
445
There are many practical applications in which we average prices. In addition to the 5rm cares about the average exchange rate (aswith XYZ), averaging is where cases also used when a single price at a point in time migt be subject to manipulatin or price swings induced by thin markets. Bonds convertible into stock, for example, often base te terms of conversion on the average stock plice over a 20-day period at the nd of the ond's life. Settlement based on the average is called an Asian tail, since the averaging occurs pnly at the termination of the contract. As we will see, Asian options are worth less at issuance than otherwise equivalent ordinary options. The reason is that the averaged price of the underlying asset is less volatile than the asset price itselt and an optio on a lower volatitity asset is worth less. '
'Iexotic''
'
14.2
ASIAN
OPTIONS
price over some period of An Asian option has a payoff that is based on the average which means that the option, path-dependent time. An Asian option is an example of a stock arrived at its the which by the path value of the option at expiration depends upon XYZ'S problem. hedging linal price.z Such an option has the potential to solve
diflicult to hedge even thaugh l However, as we will see in Chapter 22, there are options that are quite they are easy to price. pricing model. ln the binomial model 2Youcan think of path dependence in the context of a bnomial stock price moves paths--occurring irl a and down of ltdlt series up and dtttt m.e a of Chapter 10. stock price. Thus, both yeld the sarne payoff for a fnal the which lead order same different butHowev'er, to path-dependent option, these two pats w u1d yield different nal with a European option. intermediate stock pfices were different. optionpayoffs because the
XYZ'S
Hedging Problem
Let's think more about XYZ'S currency hedging problem. Suppose that XYZ has a monthty euro inqow of C100m, reiecting revenue from selling products in Europe. Its costs, however, are primarily sxedin dollars. Let xi dnote the dollar ptice of a euro in month i. At the end of te year, the converted amount in dollars is 12
C100m x
xie
r( 12-f
)/ l 2
i= 1
We have numerous strategies available for hedging the end-of-year cash flow. Here are a few obvious ones:
* Strip of forward contracts: Sell euro forward contracts maturing each month over the year. The premium of this strategy is zero. * Euro swap: Swap euros for dollars. We saw i.n Chapter 8 tat, except for the timing of cash flows, a swap produces te same result as hedging with the strip of forwards. A swap also has a zero premium.
i
@ Strip of puts: Buy 12 put options on C100m, each maturing atthe end of a different month. The cost is the 12 option premiums. As we saw in Chapter 2, the difference between the fonvard and option strategies the ability is to profit from a euro appreciation, but we pay a premium for the possibility enrning prost. You can probably think of other strteis that of as well. The idea of an Asian option stems from expression (14.1): What we really care about is the future value of the sunl of the converted cash Qows. This in t'urn depends on the sum of the month-end exchange rates. l.f for simplicity we ignore interest, what we are trying to hedg is 12
12
-q i= 1
=
12 x
Ef-1x 12
(14.2)
The expression in parentheses is the month-end arithmetic average exchange rate, which motivates the idea of an option on the average.
446
% Exo-rlc
AslAlq
Op-rlorqs: I
Options on the Average As a logical matter there am ight basic kinds of Asian options, depending upon Fhether ariihmetic the otion ij a put or a call, whether the average is computed as a geometric or underlying ' average, and whetherthe average assetpdce is psed i place of the pric rf the. . alternatives. asset or the stlike price. Here are details about some of these lt is most common in practice to defne the average as arithmetic Suppose we record the stoek price every /1 periods from time 0 cvcmkc. an is delined as to F; there are then N = F/ periods. The arithmetic average
The definition
of the average
A(F)
=
-
N
E
..
$55 + $72 + $61 + $85 )( sg ,5() ti4 4 j;E' )itT he g eometric average is illtlj. . $55 x $72 x $6l x $85)0*25 $6,7.315 'k, g:
=
=
vlt
.
.
.
x
SNlt)
.i-
The chapterappendix
averagein Chapier
The payoffat mattlrit.y average is used as the asset price or the strllte of the underlying asset plice the either stock price as computed using the average can be option is called an price, the used is asset the When the pric. strike the as average or as option is called the strlk' plice, the used is the qs e option. price When average average the geomeeic based options of variants the four optiolt. strike Here on are average an
average.
Geometric averag price call
=
Geometric average plice put
=
Geometric average strike call
=
Geometric average strike put
=
maxlo, G(F)
mlkxgo,K maxlo, Sr
-
f-j
G(F)j
-
-
maxlo, G(F)
G(F)) -
5'w)
(14.6) (14.7) (14.8)
strike'' refer to whether the average is used price'' and The tenus in place of the asset plice or the strike price. In each case the average could also be computed as an arithmetic average, giving us our eight basic linds of Asian options. The following example illustrates the difference between an arithmetic and geometric average. itaverage
taverage
SBecause the sum of lognormal variables is not lognormally distributed, there are no smple pricing formulas for options based on the arithmetic average.
simple) (relatively
options
formulas forpricingEropean
19.
Table 14.1 shpws values of geometric average plice calls and puts. If the number of avrages, N, is one, then the average is the Enal stock price. Irl that case the average price call is an ordinary call. Intuitively, averaging reduces the volatility of G(F) relative to the volatility of the price stock at expiration, Sp, and thus we should expect the value of an average price
There are easy pricing fonnulas for options based on the geometric average (see the chapter appendix).
Whether the
has
Comparing Asian Options
(14.4)
'v
k
=
based on the geometdc average. We further discuss options based on the arithmetic
i= 1
x Szh x
(
.
incpnvenient.3 It While arithmetic averages are typically used, they are mathematically gonietric tkv/rtzqj?c the in ue practice, to is omputationally easier, but less commori dehned as stockprice, which is G(F)
.
EE
(14.3)
Silt
447
r' t'(jjk.d Suppose thatwe compute the averagebased on quarterly stockprices ,q Example 14.1 )!!) stock prices of $55,$72,$61, and $85. The arithmetic average We observe 1 ) over year. )) jjE'k :riEj:i .... i. . ..y(.
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geoletric average strike calls nd puts, for diffrnt numbers of prices averaged, N. Th'e case N 1 fr the average price options is equivalent to Blackcholes values. Assumes .S' $40, # $40, r= 0.08, c 0.3, = 0, and t 1 )'lE.tE
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1
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3.209
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448
%.Exo-rlc
OPTIONS:
op-rloxs k.
BARRI ER
I
the ayerage. This is option to decrease with the number of stock prices used to compute price ojtion as evident in Table 14.1, which shows the decline in value of the average the frequency of averaging increases. of Table 14.1 also shows that, in contrast to average price calls, the price an average of stock prices The average strike call increses with the number of averaging periods. time F, Sp. lf price at between times 0 and F is positively correlated with the stock the makes averaging average G(T) is high, Sp is likely to be high as well. More frequent correlation between Sv and G(F). To the reduces valuable because it option strike more if the average is computed only using the final see this pattern, consider what happens stock price. The value of the call is
maxp,
uv
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.'
t.t'.)')' tf'')')) -'.-'.i:iiL'-'' t..,).).t).r,y.iy.qrjp,r,#,..('...... . . t !. 1Ii1i11-2l. ! ).t...t.f))..tl''r .1IE::.:!;; lliiii!!. :i;,r.' . !f!i;k Il'... ''r''''lr''':li Il..'' ll....,,. .:If::' lli(k! !i::r! ;i! s l,..'':'l a .1It:E. 11:::::.,. a l'''''iiq . a.,1k::: es e l...'' a.,Ik::. klE!s!,,r',....:ll .11t::. !.;-... .. '.'. '.'L-6''';''tib--.. ' ...!'.. '.' ii @))-' . . . . . ..... . . . . . . . .. for XYZ.The price in the second row is the sum of premiums for put.sexpiring after 1 month, 2 months, and so forth, out to 12 months. The first, third, and fourth row premiums are calculated assuming 1 year to by 12. Msumes the maturity, and then multiplied current exchange rate is $0.9/6, option strikes are 0.9, rs 6%, rq 3%, and dollar/euro volatility ij 10%. ..' '' '''f -' --.' .---' -..'
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profits are reduced if the euro euros and its costs are fxed in dollars, received for of number dollars a euro is lower. We could depreciates-that is, if the exchange rate received. construct an Asian put option that puts a Eoor, K, on the average would be option The per euro payoff of this
If XYZ receives
max 0, K
-
-
1 12
12
xi
:
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Put option expiring in 1 year Strip of monthly put options
zero the average strik option increases.
An Asian Solution for X'YZ
E'
E
'.
= S., and the value of the option is If only one stock price observation is used, G(T) is reduced and the value of con-elation averaging the for sure. With more frequent
off when When would an average strike option make sense? Such an option pays and the option of the the life price the asset over average there is a difference between where simation insurance could for in used option a be asset price at expiration. Such an accumulated entire sold the and time then of period we accumulated an asset over a position at one price.
:
. :.
G(F)q
-
449
Gepmetric average plice put Arithmetic average price put
..'...
rliiitifi
.
:'...)..... .
.E.:... :.
....
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...
..... ..'.'..
..
...... ; .E... .... :. ...
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....
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.
@,
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0.2753 0.2178
0.1796 0.1764
the lz-month average. The Asian put is cheaper since there will be situations in which some of the individual puts are valuable (for example, if the exchange rate takes a big swing in one month that is reversed subsequently), but the Asian put does not pay off.
The geometric option hedges less well than the arithmetic option since the quantity being hedged (equation14.1) is an arithmetic, not a geometric, average. Finally, be aware that this example ignores several subtleties. The option strikes, for example, might be made to val'y with the forwqrd curve for the exchange rate. The effect of interest in equation (14.1)could also be taken into account.
(14.9)
/=1
of $0.90per euro, we For example, if we wanted to guarantee an average exchange rate than ='z: less that, we would be paid wouldset k $0.9. lf the average exchange rate was repatriate e1.2bover the course thedifference between $0.9and the average. Since we 61.2b. of a yefmFe would buy contracts covering Do yourecqgnize the kind of option described by equation (14.9)? The average arithmeiic, the average is used in place of the asset price, and it is a put. Hence, it is is A,5-ft'Ill pttt. an al-ithltletic t'Ivc/wtc p-ice premiums for There are other hedging strategies XYZ could use. Table 14.2 lists expensive oplion. the year-end most is exping at several altenaatives. The single put could be quite exchange yegr-end rate the As discussed earlier, it has basis risk because the risk: basis less signfcantly strategies have different from the average. Two other of strip Asian puts The arithmetic put. and the strip of European puts expiring monthly whereas the Asian option protects month-by-month. exchange 1ow against rates protects
14.3 BARRIER
OPTIONS
A barrier option is an option with a payoff depending upon whether, over the life of the option, the price of the underlying asset reaches a specifed level, called the balwier. Barrier puts and calls either come into existence or go out of existence the first time the asset price reaches the barrier. If tey are in existence at expiration,' thy are equivalent to ordinary puts and calls. It can be tricky to desne what it means for the stock price to reach a barfier. See the box on page 450 for a discussion. Since barrier ppts and calls never pay more than standard puts and calls, they are expensive tan standard puts and calls. Bnrrier options are another example of more no option. path-dependent a Banier options are widely used in practice. One appeal of barrier options may be their lower premiums, although the lower premium of course reQects a lower average payoff at expiration.
'''. .'
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450
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The formulas for th various linds of bnmer options are discussed in Chapter 22. While we mention rebate options here for completeess, we witl discuss them in more detail in Chapter 22. The important parity relation for bnrrier options is '
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Figure 14.1 illustrates how a barrier option works. The stock price starts at around $ 100, ends at $80, and hits the barrier of $75 abouthalfway throughthe year. lf the option and pay $15($95- $80) at were a 95-strike down-and-in put, the option would lnock in would be worthless at expiration. expiration. lf the optiop were a down-and-out put, it would knock-in at $75 but still be worthless at lf the option were a down-and-in call, it price. strike expiration because the stock price is below the
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option
=
Ordialy option
(14.10)
options, we hve
Dowp-and-in call + Down-and-out call
=
Standard call
Since these option prenziums cannot be negative, tlzis equation demonstrates directly that barrier options have lower premiums than standard options.
Currency Hedging Consider once again XYZ. Here we will focus on hedging only the cash flow occurring in 6 months to see how banier puts compare to standard puts. What lcindsof barrier puts make sense in the context of XYZ'S hedging problem? We are hedging against a decline in the exchange rate, which makes certain possibilities less attractive. A down-and-out put would be worthless when we needed it. Sifnilarly,
an
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14.4 COMPOUND
0.0007 0.0188
option. Compound options are a little more complicated than ordinary options because there are two strikes and two expirations, one each for the underlying option and for the compound option. Suppose that the current time is to and that we have a compound option which at time rl will give us the right to pay x to buy a European call ption with strike K. This underlying call will expire at time F > 11 Figure 14.2 compares the timing of the exercise decisions for this compound option with the exercise decision for an ordinary call expiring at time F. Ifwe exercise the compound call at time rl then the prie of the option we receive is CS, K, T r1). At time F, this option will have the value maxto, Sp #), the same as an ordinary call with strike K. At time 11 when tlze compound option expires, th value of the compound option is .
=
,
-
=
-
,
exactly equal and which
0.8 are are you deduce which of te six premitms with K merely close? The option prices in Table 14.3 tell us something about the relative likelihood of different scenmios for the exchange rate. The ordinary put premium when the strike probability that the exchange rate will be below 0.8 at is 0.8 reqects the (risk-neutral) mamrity. Both of the own-and-ins, having strikes below the starting exchange rate of 0.9 and at least 0.8, will necessmily have lcnocked-in should the exchange rate fall below 0.8. Desclibed differently, a down-and-in put with a banier above the stlike is equivalent to an ordinary put. Therefore, the srstthree option premium in the K 0.8 row are identical. 0.8. The difference between the Now consider the knock-out puts with K barrier is that sometimes the exchange ordinary put and the up-and-out put with a 0.95 below and 0.8. ln this case, the ordinary put ten rate will drift from 0.9 to above 0.95, will not. will have a payoff but the knock-out put low premium of 0.0007 for the ordinary put tells How likely is this scenmio? drift from 0.9 to 0.8 over 6 months. us that it is relatively unlikely the exchange rate will 0.95 exchange hit in tlose cases when it does fall will rate lt is even less likely that the have likely, it but is rare to a knock-out occur ill those below 0.8. A knock-out may be vould tpr#f?7c?.y strike vith of 0.8 a pl)? o.ff Thus, the lnock-out put cases w/lcn l option. the the value This argument is even from o feature is not subtracting much 1.05. 1.0 of and Nevertheless, since tlere is a chance stronger for the knock-out barriers end in and then the money, the premiums are less than up these options will knock out =
OPTIONS
A compound option is an option to buy an option. If you think of an,ordinary option to a stock-then a compound option is similar to an ordinary as an asset-analogous
0.0869
upland-in put would provide insurance only if, prior to the exchange rate falling below lisen so the option could lnock-i. the sike, the exchange rate had Tlzis leaves down-and-ins and up-and-outs to consider. Table 14.3 presents prices of standard, down-and-in, and up-and-out puts with different strikes and different bar0.8. Notice that a11options appear to have the riers. Consider :rst the row where K understand why they same price. It is a useful exercise in the logic of barrier options to reading further: Can before solve appear equally priced. ln fact, here is an exercise to =
453
for the knock-in puts and are increasing with the barrier. Thus, the up-and-out prices in the K 0.8 row are slightly less than the price of an ordinary put. When the strike price is 1.0, the up-and-outs with barriers of 1.0 and 1.05 have substantially all the value of the ordinary put with the same strike. The interpretation is that most of the value of tlze puts comes from scenarios in which the option remains in-the-money; in those scenarios in which the option lnocks out, the exchange rate on average does not fall enough for the option to be valuable.
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q option
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Expiration of option (if compound option was exercised) underlying
454
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We only exercise te compound option if the stock price at time rl is suficiently glregt that the value of the call exceeds the compound option strike price, x. Let S* be the critical stock price above which the compound option is exercised. By definition, S*
satisses CS*, K, T
-
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(14.11)
a.
=
option is exercised for St3 > S*. Thus, in order for the compound call to ultimately be valuable, there are two events thatis, it mustbe worthwhile that musttake place. First, attime rl we musthave Stk> Sp > K; that is, it must have time F we must to exercise the compound call. Second, at underlying cfl. Because events must occur, the formula two be profitable to exercise the normal cumulative distribution, as opposed to for a compound call eontains a bivariate Black-scholes univariate formula. distlibution in the the Formulas for the four compound options-an option to buy a call CaIlOnCall), option an option to sell. a call putoncall), an option to buy a put callolpl+:nd an conpound is Valuing option appendix. chapter the in a to sell a put ptttollpu-e rathef thgn in mathemqtical for ln for option ordinary part condifferent from valuing ceptt!al repsons. The Black-scholes formula assumes that the stock price is lognormally distributed. However, the plice of an option-because there is jt signifcant probability that it will be worthless--cannot be lognormally distributed. Thus, while n option on it mathematically.. different. 4 an option is conceptually similar to an option on a stock, is The tlick in deriving a formula for the price of a compound option is to value the option based on the value of the stock, which is lognormally istributed, rather tan the plice of the undrlying option, which is not lognormally distributed. The compound
k*.,
Compound Option Parity among the compound option prices.
-t.
,
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tz,
) Putoncalltk, -
K, x,
t)- , r, /.1, tz,
= BSCall(5', K, c An analogous
455
We saw in Chapter 11 that it is possible to price Amelican options on dividend-paying . stocks using the binomial model. lt ttlrrls out thai the compound option model also pennits us to price an option on a stock that will pay a single discrete dividend prior to E
expiration.
Suppose that at time /1 the stock will pay.a dividend, D. We have a choice of exercising the option at the cum-dividend plice,D tk + D, or holding the call, which will have value reiecting the ex-dividend price, St, Thus, at 11 the value of the call option is the greater of its exercise value, Stk+ D K, and the option valued at the ex-dividend /1): Plice, Czt, F ,
.
-
-
,
max
gC(&I
T
,
St6 + D
/1),
-
#1
-
(14.13)
By put-call parity, at time tk we can write the value of the ex-dividend uneyercised call as Ke-rr-tt) ?1) = Pz rl) + I 1 F
cs
'.'
r
-
Maldng tlzis substitution in equation
v11 +
D
-
K + max
st,
-
-
(14.13)and rewriting the result, we obtain
tpgufl ,
F
-
-
/I)
gD
-
#(1
-
c-r(T-Jl))j -
,
0)
(14.14)
The value of the option is te present value of this expression. Equation (14.14)tells us that we can value a call option on a dividend-paying stock the sum of the following: as
1. The stock, with present value %. (& is the present value of Stj + D.) 2. Less the present value of the strike price, Ke-r
.
3. Plus the value of a compound option-a call option on a put option-with strike d-r(F-JI)) and maturity date fl permitting the owner to buy a put plice D #(1 maturity date F. with strike price K and option -
,
Suppose we buy a call on a call, and sell a put on a call, where both have the same strike, underlying option, and time to maturity. When the compound options expire, we will If the stock price is high, we acquire the underlying option by paying the strike plice compoun' d call, and if the stock price is low, the compound put will be will exercie the exercised and we will be forced to buy the call. Thus, the difference between the call on call and put on call premiums, plus the present value of x, must equal the premium to acquire the underlying option outlight. That is, x, c'
%.
Stocks
Options on Dividendaying
-
As you might guess, there are parity relationships
CallOnCall(5', K,
opTloxs
relationship holds for puts.
''lGeske (1979)was the hrst to derive the formula for a compound option.
,
J) + xe-rt ?-,
tz,
) (14.12)
In this interpretation, exercising the compound
option coaesponds
to keeping the
option on the stock unexercised. To see this, notice that if we exercise the compound optio in equation (14.14),we give up the dividend and gain interest on the strike in order to acquire the put. The total is Sl I + #(S
Jl
'.,
F
-
rI)
Ke-r'-t''
-
If we do not exercise the compound option, we receive the stock plus dividend, less the
strike:
St' + D
-
K
This valuation exercise provides a way to understand early exercise. We can view exercising an American call as not exercising the compound option to buy a put in
s'I'he stock is cltnt-dividetd if a purchaser of the stock will receive the dividend. Once the stock goes ex-dib'idelld,the purchaser will not receive the dividend.
k. Exo-rlc
456
equation
GAp
I
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(14.14). The cost of not exercising is that we lose the dividend, less interest
on the strike.
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457
14.5 GAP OPTIONS
Tlzisis exactly the intuition governing early exercise tat we developed in
K when S > K. 'Fhe sfrike price, K, here serves to determine A call option pays S of stock prices where the option makes a payoff (when S > K) and both the range payoff size of the CS K4. However, we could imagine separating these two the also price. Consider an option that pays S 90 when S > 100. Note of the strike f'unctions difference between the prices that govern when there is a payoff ($100) is there that a determine size of the payoff ($90). This difference creates a the price used the to and disconiinuity--or gap-in the payoff diagram, which is why the option is called a gap -
Chapters 9 and 11.
-
.' (': 4*
Suppose astockwith aprice of $100willpay a $5 dividendin 91 days p Example 14.2 (rI 0.249). An option with a strike price of $90 will expire in 152 days (F 0.416). y(j 0.08. The vglue of a European call on the stock is Assume o. 0.3 and r
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.)
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$11.6.78
option.
.
Figure 14.3 shows a gap call option with payoff S 90 when S >. 100. The gap in payoff the occurs when the option payoffjumps from $0 to $10 as a result of the stock changing from $99.99to $100.01. plice S when S < 100. This option Figure 14.4 depicts a gap put that pays 90 require, option be strucmred that for some stock prices, a to demonsates can a gap Figure 14.4 wit holder expiration. You should option compare from at the payout Figure 4.12-the gap put looks vel'y much lile a paylater strategy.6 Note that the owner of ihe put in Figure 14.4 is required to exercise the option when S < 100.7 The pricing formula for a gap call, which pays S- A-1when S > Kz, is obtained by modifcation of the Black-scholes formula. Let #1 be the strike price ttheprice simple a
value of an American call is computed as the present value of equation (14.14),with (0.416-0.2..1.9))ucza price for the compound option equal to 5 90(1 e exercise the i.;.g tt.. 3.805, and time to maturity 0.249 for the compound option and 0.416 for te underlying oPtion. The price of the compound option is
't)
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Callonput (100,90, 3.805, 0.30, 0.08, 0.249, 0.416, 0)
the value of the American option is
'
' + $:.999 $100 $90,-0.249x0.08 -
=
=
$0.999
$12.,7,74.
Moreover, the option should be exercised if the stock .price cum-dividend is above
k
),,)$89.988.
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Compound options provide yet another variation on possible currency-hedging strategies. lnstead of buying a 6-mont put option on the euro, we could buy a call option on wait arld a put option. In effect, this compound option is giving us the opportunity to see what happens. whether to buy the put option. Here is uPP ose that after 3 months we will decide 3-month one way to structure such a transaction. We could figure out what premium a BlackThe still at 0.9. put with a stlike of $0.9 would have, if the exchange rate were of strike $0.9would Scholes formula tells us that a 3-month at-the-money option with a have a premium of $0.0146.(This value compares with the premium of $0.0188for the 6-month option from Table 14.3.) Now we can use the compound pricing formula to plice a call on a put, setting the stlike to equal $0.0146.The price of this compound call is $0.0093. So by paying less than two-thirds the premium of te 6-month at-the-money optin, tve can buy an option that permits us to pay $0.0146for a 3-month option. By seleting this strike, we have constructed the option so that we will exercise it if the exchange rate is below 0.9. If the exchange l-ate goes up, we will not exercise the option and save the premium. lf te exchange rate goes down, we will acquire an in-the-money option for the plice of an at-the-money option. Mariy other structures are possible.
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14.6 EXCHANGE
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Suppose an exchage call maturing r periods from today provides the right to obtain l unit of risky ajset 1 i ekchange for l unit of risky asset 2. (We could tlzink of this as, for example, the right to obtain the Nikkei index by giving up the S&P 500.) Let St be the price of risky asset 1 and Kt the price of lisky 2 at time r, with dividend asset yields 8s and &K and volatilities o's and c K Let p denote the correlation between the k contlpuously compounded remrns on the twoassets. he payo g to tus oyttjon is K;) mllxto, The formula for the price of an exchange option (seeMargrabe, 1978) is .
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the option holder pays at expiration to acquire the stock) and Kz the paymet trigger (theprice at which payment on the option is triggered). The fonupla is ihen KLe-rrxdzj (14.15) Se -&1'Nd l ) #2, o-, n F, t) CS, nse-'l-jltqe-rl-) + 1o-2r 2 #
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Consider an option to receive (IBM shares by giving up Microsoft 'l' shares. Wecan view this as an IBM call wtth Microjft as the stlike aset. On Kvember lt.q ')ll $27.39.Thus, oneshareof VM 15, 2004, thepriceof IBMwassgs.gzandMicrosoftwas i :j;yj 3.5020 shares of Microsoft. For IBM and ltt had the same dollar value as 95.92/27.39
tl Example .
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=
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Because Microsoft is the strike asset, we replace the risk-free rate Fith Microsoft's dividend yield. Assuming a lisk-free rate of 2%, a plain l-year at-the-money call on IBM would be worth BSCZI
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=
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Problem 14.19 asks you to think about the circumstances under which XYZ might
hedge currency risk using exchange options.
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Suppose the option permits exchanging equal values of Microsoft for IBM, based on the November 15 prices. We could then exchange 3.5020 shares of Microsoft for l share of price of a l-year exchange call would be mM.
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22.27% for Microsoft, with a return s in ce January 2003 had ben 20.30% for IBM and .
An exotc option is created by altering the contract'ual yerms of a standard option. Exotic options permit hedging solutions tailored to specilic problems and specvlation tailored to particular views. Examples of exotic options include the following'.
@Asialt options asset over the tlle underlying average strike
have payoffs that life of the option. asset (an average optiol. Averages
are based on the average price of the underlying The average price can be used in place of either price t7rrft/nl or in place of the strike price (an can be arithmetic or geometlic.
have payoffs that depend upn whether the price of the underlying * Barrier (//7/*J4777, reached has asset a bnrrier over the life of the option. These options can come into when te existence knock-in option or go out of existence kllock-ollt option barrier is reached.
* Compound options are options on options: Put or call options with put or call options as te underlying asset. * Gap optiolls are options where the option payoffjumps at the pricewhere the option comes into the money. * Exchange options are options tat have risky assets as both the underlying asset and the strike asset. It is helpful in analyzing exotic options to compare them to standard options: ln
what ways does an exotic option resemble a standard option? How will its plice compare
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462
k
Exo-rlc
OpTloxs:
PRo BLEMS
I rnight someone use
to that of an ordinary option? When standard option?
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463
a. What are the possible geometric and arithmetic averages after 1 year? b. What is the price of an Asian arithmetic average plice call?
the exotic option instead of a
c. What is the price of an Asian geometric average price call? FURTHER
READING
14.6. Let S
ln Chapter 16 we will see some pore appliations of exotic options. ln Chapter 21 we will discuss the underlying logic of pricing exotic options and in Chaptr 22 we will discuss additional exotic options. General bboks covering exotic options include Brifs and Bellala (1998), Haug -(1998), Wihuott (1998), and Zhang (1998). Rubinstein (1991b)discusses exchange options, Rubinstein (1991a)discusses compound options, and Rubinstein lpd Reiner (1991a) discuss bacrier options. PROBLEMS To answer many of these questions you can use the exotic option functions in the spreadsheet accompanying this book.
14.1. Obtain monthly stockplices for 5 years for tllree stocks. Compute the arithmetic and geometric average month-end plice for each stock. Which is greater?
14.2. Suppose you observe the prices (5,4, 5, 6, 5l. What are the arithmetic and geometricaverages? Now you-observe (3,4, 5, 6, 7). What are the two averages?
What happens to the ditference between the two measures of the average as the standarddeviation of the observations increases?
1. 14.3. Suppose that S 0.30, J 0, and F 0.08, o' $100, r $100, K Constnlct a standard two-period binornial stock price tz'ee using the method in Chapter 10. =
=
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=
=
a. Consider stock price averages computed by averaging the 6-month and l-year prices. Whatare the possible alithmetic and geomeic averages after 1 year? b. Construct a binomial tree for the average. How many nodes does it have after 1 year? llint'. While the moves I4# and #If give the same year-l price, they do not give the same avergge in year 1.) c. What is the price of an Asian arithmetic average price call? d. What is the price of an Asian geometric average price call? 14.4. Using the information in the previous problem, compute th pries of a. An Asian arithmetic average strike call. b. An Asian geometric average strike call. binomial tree. Assume 14.5. Repeat Problem 14.3, except construct a lrcd-periol averaging the prices Asian options based that every 4 months. are on
$40, K
=
$45, c
=
0.30,
=
1.
=
0.08, F
=
1, and 8
=
0.
a. What is the price of a standard call? b. What is the plice of a knock-in call with a barrier of $44. Why? c. What is the price of a lnock-out call with a barrier of $44? Why? 14.7. Let S $40, K 4 5 100). =
9
$45, o'
=
0.30,
=
?'
=
0.08,
=
0, and F
(0.25,0.5, 1, 2, 3,
=
'
'
a. Compute the prices of knock-out calls with a barrier of $38. b. Compute the ratio of the knock-ut call prices to the plices of standard calls. Explain the pattern you see. puts assuming a barrier of $:I4.
14.8. Repeat te previous problem for up-and-out
14.9. Let S $45, o' 0.30, ?' 0.08, and 8 0. Compute the value $40, K with lnock-out calls of a bnrrier of $60 and times to expiration of 1 month, and 2 monts, so on, up to 1 year. As you increase time to expiration, what the price of the knock-out call? What happens to the price of the happens to call relative knock-out to the plice of an otherwise identical standard call? =
=
=
=
=
14.10. Examine the prices of up-and-outputs with strikes of $0.9and $1.0in Table 14.3. up-and-outs appear to have the same With barriers of $1 and $1.05,the premium as the prdinary put. However, with a strike of 1.0 and ihe samebnrriers, the up-and-outs have lower premiums than the ordinary put. Explain why. What would happen to this pattel'n if we increased the time to expiration? .go-strile
14.11. Suppose S
=
$40, K
=
$40, c
=
0.30, r
=
0.08, and J
=
0.
;
a. What is the price of a standard European call with 2 years to expiration? b. Suppose you have a compound call giving you the right to pay $2 1 year from today to buy the option in part (a). For what stock prices in 1 year will you exercise this option? c. What is the price of tllis compound call? d. What is the price of a compound option giving you the right to option in part (a) in 1 year for $2?
sell the
14.12. Male ttle same assumptions as in the previous problem. a. What is the price of a standard European put with 2 years to expiration? b. Suppose yo have a compound call giving you the right to pay $2 1 yea.r f'rom today to buy the option in (a). For what stock prices in 1 year will you exercise this option? c. What is the price of this compound
call?
464
k. Exo-rlc
OpTloxs:
l
PROBLEMS
d. What is the price of a compound option giving you the right to sell te option in part (a) in 1 year for $2? 14.13. Consider the hedging example using gap options, in particular the assumptions and prices in Table 14.4. a. Implement the gap plicing formula. Reproduce the numbers in Table 14.4.
$0.8 and Kz $1. 1.fvolatility were b. Consider the option with #1 of this option be? What do you thirlk will would the what price zero, happen to this premium if the volatility increases? Vel'ify your answer using your pricing model and explain why it happens. =
=
14.20. A chooser option (also known as an asryou-like-it option) becomes a put or call at the discretion of the owner. For example, consider a chooser on the S&R ' lndexfor which both the call, with value Cst K, T t4, and ihe put, with value l), have a stlike price of K. The index Pvt K, T pays no dividends. At the choice date, ,1 the payoff of the chooser is -
,
-
,
,
maxlct.%l K, T '
'to
,
,
,
,
=
=
=
=
=
=
=
=
=
=
=
=
=
=
30%, r 14.21. Suppose that S $100, cr contract which, 6 months from today, will at-themloney call option. (This is called a r, o', and are certain not to change in the =
a. Val'y &frpm 0 to 0.1. What happens to the price of the call? b. Vary 8c from 0 to 0.1. What happens to the price of the call?
=
=
=
=
=
=
=
=
a. What is the price of an exchange call with S as the
underlying
asset and
Q as te stlike price?
b. Now suppose c o
=
0.40. What is the price of the exchange call?
ca Explain your answers to
(a)and (b).
K, T
-
Jl))
=
=
= 8%, and 0. Today you buy a give you one 3-month to expiration forward start option.) Assume thjtt next 6 months.
c. What would you pay today for the forward start option in this example? d. How would your answer change if the option were to have a strike price that was 105% of the stock price?
-0.5
0.30, ?' 1, and 8 0. Also let Q 14.18. Let S 0.08, F $40, $40, o' 0.30, 8c 0, and p u:cq 1. Consider an exchange call with S as the price o'c of the underlying asset and Q as the price of the strike asset.
,
a. Six months from today, what will be the value of the option if the stock price is $1002$50? $200? (Use the Black-scholes formula to compute the answen) ln each case, what fraction of the stock price does the option cost? b. What investment today would guarantee that you had the money in 6 months to buy an at-the-money option?
=
to 0.5. What happens to te price of the call? c. Vary p from d. Explain your answers by drawing analogies to the effects of changing inputs in the Black-scholes call pricing formula.
P't,
.
=
=
/1),
b. Suppose that the chooser must be exercised at /1 and that the underlying options expire at r. Show that the chooser is equivalent to a call option ,-dtT'-Jl) with strike price K and maturity F plus put options with strike Ke-r-bl--tbb expiration and rice rl P
0.08, r 1, and $ 0. Also let Q 14.17. Let S 0.30, r $60, $40, o' this problem will and In 0.5. 0, 0.50, compute prices of we p o'c c exchange calls with S as the price of the underlying asset and Q as the plice of the stlike asset. =
-
a. lf the chooser option and the underlying optims expire simultaneously, what ordinary option position is this equivalent to?
,
0.08, F 1, nd & 0. Also let Q 14.16. Let S 0.30, r $60, $40, o' What is the price of a standard 4o-strike and 0.5. 0.04, 0.50, p o' c c call with S as the underlying asset? What is the plice of an exchange option with S as the underlying asset and 0.667 x Q as the strike price?
465
14.19. XYZ wants to hedge againstdepreciations of the euro and is also cpncenaed about the price of oil, which is a signiicant component of XYZ'S costs. However, there is a positive correlation between the euro and the price of oil: The euro appreciates when the price of oil rises. Explain how an exchange option based on oil and the euro might be used to hedge in this case.
14.14. Pfoblem 12.11 showed how compute approximate Greek measures for an technique Use this option. to compute delta for the gap option in Figure 14.3, ranging stock prices for from $90 to $110and for imes to expiration of 1 week, How months, and 1 3 easy do you think it would be to hedge a gap call? year. 14.15. Consider the gap put in Figure 14.4. Using tlze technique in Problem 12.11, compute Vega for this option at stock prices of $90 $95 $99 $101 $105 and $110, and for times to expiration of 1 week, 3 months, and 1 year. Explain the values you compute.
%
14.22. You wish to insure a portfolio for 1 year. Suppose that S $100, ty 30%, considering simple You strategies. The f/zlll?-t7/lcc 8%, and 0. two are ?price strike option maturity entails buying with l-year at put a one a strategy that is 95% of the stock price. The rolling pl-lf?wncc strategy entails buying one l-month put option each month, with the strike in ech case being 95% of the then-curren t stock plice. '=
=
=
=
E
.
'
a. 'What is the cost of the simple insurance strategy? b. 'What is the cost of the rolling insurance strategy? Hint.. See the previous problem.) c. lntuitively, what accounts for the cost difference?
466
%.ExoTlc
OlnTloxs:
14.A:
APPENDIX
I
PRICING
FORMULAS
F0R
EXOTIC
OPTIONS
$
467
where the correlation between 1n(&) and G(F) is given by
APPENDIX 14.A: PRICING FORMULAS FOR EXOTIC OPTIONS
1 6N + 1)
#=
discussed in this ln this appendix we present formulas for some of the options
chapter.
Asian Options Based on the Geometric Average (an average price option) or
The average can be used in place of either the asset price the strike price (an average strike option).
rate is 1- and the stock has a dividend Suppose the lisk-free price options computtthe average using N equally spaced prices from 0 yield and volatility o'. We geometric average plice option time to F, with the first observation at r/#. AEuropean formula for Black-scholes a call by setting the dividend can then be valued using the volaiility equal to yield and Averase
(14.18)
IN + 1 * Use the current stock price as the stlike price. * I'he dividend yield remains the same.
Compound Options Letting p denote the correlation coefcient between normally distribtged zl and z2, we denote the cumulativ bivariate standard normal distributiim as '
Probtzl
a,
<
z2 < b; p4
=
NNIcI,
b., p4
This function is implemented in the spreadsheets as BINORMSDIST Suppose we have a compound call option to buy a call ojtion. Let r1be the time maturity of the compound option, and tz the time to mat-tllit'r of tlie undlrlyinj option to (obviously, we require that tz > /1 ). Also let K be the strike price on the underlying option and the strike price on the compound option', i.e., we have the right on date 11 to pay to acquire a eall option with time to expiration za tk Deline S* as in equation (14.11).,that is, S* is the stock price at which the option is worth the strike that must be Paid to get it.9 The formula for the price of a ctll option on a call option is -x
and
.):
-
.
G
With continuous
:/
=
sampling. i.e., N
-'
tr
N =
t3* =
CN + 1)(2# + 1)
(14.19)
6 x, tle formulas reduce to 1 z 1 1' + J + 2 6 -(r
-
and *
G
=
G
-
1 3
Deriving these results is easier than you might guess, but requires some background cokered in Chapters 18 and 19. The derivation is in Appendix I9.A. In order to value the geometric average strike option, we need the terminal stock ptice, Sv. to know the col-relation between the average, G(F), and the is strike the asset average', hence, we value the option We also need to recognize that which 14.6), in we exchange tlze time-r stock plice like an exchange option (seeSection for its average. ln Appendix 19.A we show that the average sfrike option can be valued using the Black-scholes fonula, with the following substitutions: Average
where
options
stril
Iisk-free o Replace the
rate with the
yield,'' equation
ttdividend
''IC 't:E
4:::)4.. :::::::::: 4:::)F..
(jr. ....jh....
LN + l)(2N + 1) ..........
............
...
6N2
.........
(:;4!: jlr:;).
U
CN +
..............
.........
1)(2X + 1) 61
=
jnvb-jK) +
dz d l =
-
o'
(r
-
J + 0.5c2)/:?
x/-ra
and #2 are identical to the Black-scholes (/1 and #a, and relate to ultimate Notice that exercise of te underlying option, while (71 and az differ only in the stlike price and time :./1
(14.18)-
@Replace the volatility with .xkkjy/f'/''jj;:s;l'
:./1
ggyrj
gerhespreadsheetfunction to compute S* is called BSCaIIIlnpS,which is similarto the implied volatility except that it computes the stock jrice consistent with an option price, rather function BSCaIlInlpVoL the volatility. 01:111
468
k. Exo-rlc
Op-rloxs:
AIZPEN DlX
I
to expiration and relate to exercise of the compound option. The last term in equation (14.20) reoects payment of the compound option strike price and the condition under which it is paid. The sign on the correlation term, ,1 jtz, re:ects whether exercise of the compound option is associated with an increase or decrease in the likelihood of exercising the underlying option. (The correlation is positive for a call on a call. For a call on a put, an increase in the stock price reduces the value of the put and also reduces the value of the option to buy the put; hence, the correlation is again positive.) This discussion suggests that we can guess how the remaining compound option formulas will look. We would like to value puts on c/ls, calls on puts, and puts on puts. The put on the call requires a positive sign on Ke-rt and a negative sign pn Se-t - since the option if ultimately exercised will require the owner to be a call Writr. The underlying option is in-the-money if S > f('; hence, we want positive dj and #2. The compound option will be exercised and the stlike received if S < S*, which requires negative tzl and az and a positive sign on x. Finally, if the stock price gos up, this increases the value of the call and decreases the value of the ppt on the call; hence, the cpn-elation must be negatively signed. Thus, the fonnula is .:
Putoncallts',
K, a:, c
,
l
- 11 tz, J) ,
-Se-&hNN
=
,
+ Ke-rhNN
-a
l,
ch;
-tz:?,
dl
/.1 -
-
-
,
/1 /2
.-wj
+
-
tz
yi-agj
.ve
( jy-g.yj.;
Similar arguments give us the following formulas:
(14.22) P utonputls',
K,
-'t:
,
c r, 11 tz, ,
,
-
)
Ke
=
-rJ2
Se-hNN NN
ag
J'l
-#j
a
-.
,
;
j,
#a ;
--.
-
/'l tz
-
-
tz .-rtk
+
.y
e
As an exercise, we can check that as ,1 approaches 0, the compound simplifes to the greater of the value of the underlying option or zero.
y
Lag) (j4 za) .
option formula
ln6nitelyLived Exchange Option The logic of exchange options extends directly to the case of an insnitely livedAmelican option. A key insight is that the optimal exercise level H really depends on the ratio of the values of the asset being received to the asset being given up; the absolute level is unimportant. Thus, if it is optimal to exchange stock A for stock B when the price of A is 100 and the plice of B is 200, then it will be optimal to exchange A for B when their prices are 1 and 2. We therefore just need to 5nd the ratio of prices at which exercise is Optimal.
'
PRICI NG FORMU LAS FOR EXOTIC
14.A:
$
OPTIONS
469
The formula for the infinitely lived option to exchange stock 1 for stock 2 is CXISI
,
&, c' 1
.
t7'2.
p,
tl
,
t2)
@
=
1)&
.=
Sl /Sz lt
s
wher $i is the dividend yield on asset i, o-i is the volatility of asset i, p is the correlation between stock 1 and stock 2, and /2 J=
1
/?
-
the ratio of 5'l to Sz at which it is optimal to exercise the option. Let 2po' jcz and
is
I
=
-
1 2
J2
-
JI
8g
+
-
&2
1 2
Jl
-
-
-
(7.2
lt is possible to show that if we set &z r and call, equation (12.16),while if we set JI equation (12.17). =
=
(7'2
=
2 0N
+
G2
=
&2
l+
(7.2
2
-
2J2 (7'2
we get the formula for an insnite = 0, we get the put formula,
r and o' l
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bonds, and a put, and so forth. ln Chapter 10 we saw that an option could also be synthetically created from a position in the stock and borrowing or lending. If prices of act'ual claims differ from their synthetic equivalents, arbitrage is possible.
Financial engineering is an application of the Modigliani-Miller idea. We can combine claims such as stocks, bonds, fomards, and options and assemble them to create new claims. The price for this new security is the sum of the pieces combined to create it. When we create a new instrument in this fashion, as in the ModiglianiMiller analysis, value is neither created nor destroyed. Thus, financial engineering has no value in a pure Modigliani-Miller world. However, in real life, the new instrument may have different tax, regulatoly or accounting characteristics, or may provide a way for the issuer or buyer to obtain a particular payoff at lower transaction costs than the alternatives. Financial engineeling thus provides a way to create instrtlments that meet specifc needs of investors and issuers. As a starting point, you can ask the following questions when you confront new nancial instrments:
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PRI CI N G AN D DESI GN l N G STRUCTU RED NOTES
AN D SECU RITY DESIGN
ENGINEERING
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475
bond price #, So we will write .
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=
P is the current price of a r-periodzero-coupon bond. There are two important, equivalent interpretations of Pt. First, Pt is a discount factor, since it is the price today for $1 delivered at time r. Second, Pt is the prepaid forward prke for $1 delivered at time t. These re different ways of saying the same
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Zero-coupon bond price
=
Discount factor for
$1 c.:= Prepaid forward price for $1
Financial valuation entails discounting, which is why zero-coupon bonds are a basic
building block. The notion that prepaid forward prices are discount factors will play an impottant role in this chapter.
* What is the payoff of the instrument?
Coupon Bonds
e Is it possible to synthetically create the same payoffs using some combination of assets, bonds, and options?
Once we have a set of zero-coupon bonds, we can analyze other fixed payment instruments, such as ordinary coupon bonds. Consider a bond that pays the coupon c, 11 times over the life of the bond, makes the matulity payment M, and matures at time F. We will denote the price of this bond as /(0, F, c, ?l, M). The time between coupon payments is F/?;, and the fth coupon payment occurs at time ti i x F/n. We can value this bond by discounting its payments at the interest rate appropriate each for payment. This bond has the price
* Who might issue or buy such an instrument? * What problem does the instrument solve?
=
AND DESIGNING NOTES STRUCTURED
15.2 PRICING
11
We begin by examining sucttlred notes. An ordinary note (or bond) has interejt and maturity payments that are fixed at the time of issue. A structured note has interest or mattlrity payments that are notExed in dollars but are contingentin some way. Sucttlred notes can make payments based on stock prices, interest rates, commodiyies, or currencis, and they can hav optios embedded in tem. The equity-linked CD discussed in Chapter 2 is an example of a structured note, as it has a maturity payment based upon the performance of the S&P 500 index. ln tMs sectipn we discuss structlzred notes without options. In the next section we will introduce notes with options.
Zero-coupon
Bonds
The most basic tinancialinstrument is a zero-coupon bond. As in Chapter 7, 1etrstz, rj) represent the annual continuously compounded interest rate prevailing at time s S l0, for a loan from time tz to time 11 Similarly, the price of a zero-coupon bond purchased at time 1(j, mattlring at time r1 and quoted at time s is Pstf), rl ). Thus, we have .
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This valuation equation shows us how to price the bond and also how to replicate the bond using zero-coupon bonds. Suppose we buy c zero-coupon bonds mattiring in 1 year, c maturing in 2 years, and so on, and c + M zero-coupon bonds maturlng in F years. This set of zero-coupon bonds will pay c in 1 yar, c in 2 yeqrs, and c + M in F years. We can say tat the coupon bond is engineered from a set of zero-coupon bonds with the same maturities as the cash flows from the bond. In practice, bonds are usually issued at par, meaning that the bond sells today for its maturity value, M. We can structure te bond to make tllis happen by setting the M coupon so that the price of the bond is M. Using equation (15.1),1(0, F, c, ?), M) if the coupon is set so that ,k
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k. Fl NANCIAL
476
ENGI Iq EERI NG AND SECURITY
DESIGN
PRI Cl N G AN D DESI GN 1NG STRUCTU RED NOTES 11:(:)1*
t' rate is therefore 1.5%). From h' ,t'
Equity-lainlced Bonds ln this section we discuss pricing of various types of equity-linked bonds. Specihcally, le of XYZ we consider a bond that, instead of paying M in cash at maturity, pays uncertnl'n mattll'ityvalue. stock at mattlrity. With this change in terms, the bond has an for such Moreover, this change raises questions. What does it mean a bond to sell at paid in cash or in shares of XYZ? par? tf there are coupon payments, should they be stock For regulatory and tax purposes, is this instrument a or a bond? -/?tz?z
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We now add cash coupon payments to the bond. Represent coupons of c each and ashare atmaturity as #(0, F, c, ll, Sp). The valuation equation for such a note-the analog of equation (15.1)-is
equitplinld bond Zero-coupon Suppose an equity-linked bond pays the bondThere are no interim payments. What is a fair stock time F. share of holder at one price for this bond? this valuation problem is the same as Although the language is now that of valuing a prepaid forward contract, which we analyzed in Clpter 5. In both cases the investor pays today to receive a share of stock at time F. In te context of this chapter, we could also call this instrument a zero-coltpz ctylf/l/-llp/# bond. ecall from Chapter 5 that the prepaid forward price is the present value of the forward price, Fapw= #wFo,w. This relationship implies that for a nondividend-paying stock, -rT (r-1)F = St since J = 0. The prepaid fozward price is the stok price. F()P :.:= e &e w
cash coupon payments theplice of abondpayingn
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,
ln words: The price of the bond, B, will eqpal the stock price, %, as long s the present value of the bond's couons (the ftrst tenn on the right-hand side) equals the present value of the stock dividends (the third term on the right-hand side). ln general, if we wish to price an equitplinked note at par, from equation (15.4), price B will equal the stock price, %, if the coupon, c, is set so that bond the '
This example shows that if we issue a bond promising to pay one share of a nondividend-paying stock at maturity, and tle bond pays no coupon, then the bond will sell for the current stock price. In general, a bond is at par if the bond price equals the maturity payment of the bond. The bond in Example 15.1 is at par since the bond pays one share of stock at maturity and the price of the note equals the price of one share of stock today. Suppose the stock makes discrete dividend payments of Dt; Then we saw in Chapter 5 that the prepaid fomard price is
;
c
&
=
-
FP () v
(15.5)
'
,,
)Zf-,Pz
That is, the coupon must amortize te difference between the stock price and the prepaid forward plice.
.
lt
F'0,
,%
T
=
-
J'''llbi o,f
(15.3)
f= l
q Example (j
:' ..
Consider XYZ stock as in Example 15.2. If the note promised to note would sell for coupon equal to the stock dividend-the
15.3
'E
If the stock pays dividends and the bond makes no coupon payments, the bond will sell at less than par. Example Suppose the price of XYZ stock is 15.2 continuously compoundedinterestrate and the annual $1.20,
j'
$100, the quarterly dividend is is 6% (thequarterly interest
.
quarterly-a ltyjpay $1.20
t.).$100.
.
..
k
.
Notice that equation (15.5)is the same as the equation for a par coupon on a cash bond, equation (15.2). lnstead of 1 Pv in the numerator, we have s'tl Ff%. The former is the difference between the price of $1 and the prepaid forward price for $1 -
-
478
k
FINANCIAL
ENcl NEERING
PRI Cl N G AN D
DEslcx
AND sEcunl'ry
.
12
z:
c
=
F'aP;f +
zy i= l
=
ypv
-
,
=
1
'''
ynp
0,Jf
.
(15.6)
1
-
d
-
',
d-f
Sf=le
X -,Jf
(15-8)
,
Bonds
Now we repeat the analysis of the previous section, except that instead of paying a share the note pays one unit of a commodity. We ask the snme of stock at maturity, we questions about how to structure this note. We will see that the commodity lease rate 'suppose
rate is positive.
=
=
Fo%, we have
FX 0,F
-
;
E3i l P z:z
Comparing the equations for c and c* makes it apparent that the appropriate discount factor for a coupon is determined from the lease rate on the underlying asset. In the case of a bond denominated in cas the lease rate is the interest rate, while in the case of a bondcompletely denominated in shares, the lease rate is the dividend yield.
Commoditpl-inked
Since by desnition of the prepaid forward price, Pv F0,w &
becomes
1
.
Suppose we have a commodity with a current price of Sv and a forward cash interest price of Fo,z., and we have a commodity-linked note paying a cash coupon. For the note to sell at par, we need to set the coupon so that
C
jgjgpj e-sq i:zzz1
=
=
'
-T
We can compare this expression for c* with thatfor the coupon on an ordinary cash bond. ln the special case of a donstant interestrate and assuming a $1 par value, equation (15.2) C
4:) $455,-0.0625x3 lyyy :,m8 szx ))t ow y'i ). l.1This amount is less than the spot price of $400because . the lease =
becomes C=
Suppose the spot price of gold is $400/oz.,the 3-year forward price p Example 15.4 and the 3-year continuously compounded interest rate is 6.25%. Then a is $455/0z.9 (tt ) 17 t zero-coupon no te paying l ounce of gold in 3 years would sell for 'lt)
E ' .
When we pay coupons as shares rather than cash, the coupons have variable value. Thus, it is appropriate to use the prepaid forward for the stock as a discount factor rather tha.n the prepaid fonvard for cash. dividend yield, $, this equation In the specialcase of aconstantexpectedcontinuous +
479
commodity-linked Zero-coupon bonds Suppose we have a note that pays one unit of a commodity in the future, with no interim cash flows. What is the price of the note? Once again, the answer is, by definition, the present value of the folavard price, or the prepaid' forward price. As we saw in Chapter 6, the difference between the spot price and the prepaid fonvardplice is summarized by the lease rate. Thus, the discount from the spot price on a zero-coupon note reqects the lease rate.
-''
y uuf
$
t:(
P
z-nw
The number of f'ractional shares that must be paid each year for the note to be initially priced at par, i.e., for J/0 = %, is
cs
l N G STRUCTU RED HOTES
replaces the dividend yield. A commodity-linked note will pay a coupon if the lease rate is positive, and the present value of coupon payments on the note must equal the present value of the lease payments on the commodity.
delivered at time F. The latter is the difference between the price of one share and the prepaid forward plice for one share delivered at time F. ln praetice, dividends may change unexpectedly over the life of the note. The note issuer must decide: Should the dividend on the note change to match the dividend paid by the stock, or should the dividend on the note be sxedat the outset using equation (15.5)? The price should be the same in either case, but a different ptrty bears dividend risk. An alternative to paying interest in cash is to pay interest in fractioni Interest in-kind the coupon could be the value of 2% of a share at the time of For example, shares. payment, rather than a fxed $2. To price such a bond, we represent the number of fractionl shares received at each coupon payment as c*. The value at time 0 of a share Thus, the formula for the value of the note at time tfj, 4/0is received at time t is F
D ESI GN
exactl#as with a dividend-paying stock. The coupon serves to amortize the lease rate. Thus, the lse mrc plays /'/7c role of a lfW#e??tf yield ?l pricing a ct?/?/?nt/#i/)/-ff/l/cc# rfhe note. present value calculation treats the lease rate exactly as if it were a dividend yield', what matters is that there is a difference between the prepaid fonvard price and the current spot price.z Suppose the spot price of gold is $400/oz.,the 3-year fonvard price @ Example 15.5 lli) rate is tr is $455/0z.,the l-year continuously compounded interest rate is 5.5%, te z-year
2As we saw in Chapter 6, a lease rate can be negative if tere are storage costs. In this case, the holder of a commoditplinked note benefits by not having to pay storage costs associated with the physical commodity and will therefore pay a price above mattlrity value (in the case of a zero-coupon note) or else the note must carry a negative dividend, meaflng that the holder must make coupon payments to the issuer.
k. FINANCIAL
480
ENGI N EERI NG AN D SEcuRl'ry
PRI Cl N G AN D
DEslcrq
ltt 6%, and the 3-year rate is 6.25%. The annual coupon is then determined as j() jj $400 $3-77.208 $8.561 c )(q ,-0.0625x3 ,-0.055 tl + e-0.06x2 + 'h.) l11 The annual oupon on a 3-year gold-linked note is therefore about 2% of the spot jtq ri price. . . k ii.l'
-
D E5I GN
I N G STRU CTU RED NOTES
$
481
What if a bond pays one unit of the commodity per year, forever? We lnow that if it pays St in perpettlity it is worth %. Thus, if it pays $ it is wot'th
=
=
'E
.
f).)
A 2% yield in this example might seem like cheap financing, but this is illusory and stems from denominating the note in terms of gold. When the yield on gold (thelease rate) is less than the yield on cash (the interest rate), the yield on a gold-denominated note is lss than the yield on a dollar-denominated note. This effect is reversed in cases where the interest rate in a particular currency is below the lease rate of gtld. l Japan during the late 1990s, the yen-denominated interest rate was close to zer, d the coupon rate on a gold note would have been greater than the interest rate on a yen-denominated
note. ln-tind As wit stocks, we can pay fractional units of the commodity as a interest payment. The present value of the payment at time t is computed using periodic prepaid price, Fnp,. Thus the value of a commodity-lin ked note at par is forward te exactly the snme as for an equity-linked note paying interest in-lind: Interest
This is the commodity equivalent of a perpetuity. The cnclusion of this section is simple: Commodity-linked notes are fonnally like equity-linked notes, with the lease rate tnking the place of the dividend yield.
Currency-tainlced Bonds What happens if we change the currency of denomination of the bond? As you can probably guess by now, the foreign interest rate, being the lease rate on the foreign currency, takes the place of the dividend yield on the stock. Suppose that we want to compare issuing a par-coupon bond denominated entirely in dollars and a par-coupon bond denominated entirely in another currency. We will BF denote zero-coupon bond prices denominated in te foreign cuaency, ?'s(l) the 1et foreign interest rate, and PtF the price of a zero-coupon bond denominated in the foreign Ctlneency.
As you would expect, a bond completely denominated in a foreign currency
qa-
c
=
F() ,f + F(),z
(15.6).
'
-,>
.->
-r
I
=
-
PF T pp = i' 1 t -
''
Perpetuities A perpetuity is an innitely lived coupon bond. We can use equations (15.7) and (15.8)to consider two perpemities: one thatmakes annual payments in dollars and another that pays in units of a commodity. Suppose we want the dolltr perpettlity to have a price of M and the commodity perpemity to have a price of %. Using standard j, (x) in equation perpettlity calculations, if we let F (15.8) (thls also means that (x)), the coupon rate on the dpllar bond is ?) 1 Mer 1) 'M c M =
gy jy?J
C=
p
p
i= l
The fonuula for c* is givn by equation
1
F
lt .
will
have a coupon given by the formula
=
'-e-r
where is the effective annual interest rate. Similarly, for a perpetuity paying a unit of a commodity, equation (15.7)becomes
In other words, foreign interest rates are used to compute the coupon. What happens when the principal, M, is in the domestic currency and the interest payments are in the foreign currency? Once again wejust solve for the coupon payment tat makes the bond sell at par. There are two ways to do this. First, we can discount the foreign currency coupon payments using the foreign interest rate, and then translate their value into dollars using the cuaent exchange ratet of foreign currency). The value of te fth coupon is Pfc, ab (dnominatedas and the value of the bond is
s/unit
-p
?1
B (0, F, c F
,
?7.,
M4
=
xvcF
PF t;
.
Alternatively, we can translate the f'uturecoupon payment into dollars using the forward currency rate, Fp,?, and then discount back at the dollar-denominated interest rate, Pt. The value of the bond in this case is 11
BQ, T,
c
F ,
?l,
M4
=
cF
F flvt # p + MPp
f
=
where is te effective annual lease rate. Thus, in order for a commodity perpettlity to be worth one unit of the commodity, it must pay the lease rate in units of the commodity. (For example, if the leae rate is 2%, the bond pays 0.02 units. of the commodity per yealc)
+ MPv
i= 1
l
The two calculations give the same result since the currency forward rate, from equation (5.18) is given by
482
%.Fl NANCIAL
ENGIN EERI Nc AN D SEcu
BoN Ds WITH
Rl-ry DEslclq
Tllefotz3al-dpt-icefol-fol-eigll exchallge is set so tllat it plc/cd' no ttf/rtr/lct, vhether wc ?Ll l T7r?-l tlltr (llI?-?-l? 2t2)7Llll ff tI1tr7l (lisc t?tfll f, t7?* tlis (24)1ll1 t f7?l f tll tr?l ?L)ll 17t,?-l tIlt! (lI I1*1*3,1 ?(2)/. The coupon on a par bond with foreign interest and dollar principal is given by F C
=
p.f yg
1
Pp
-
s
i'=xl 0,h p
(j5
.
j(p
K
Ji
=
OPTIONS
Options in Coupon Bonds One common kind of equity-linked note has a stnlcture where, at matuhty, te holder can receive some fraction of the retul'n on the stock but does not suffer tt loss of principal if the stock declines. We obtain this seucture by embedding call options in the note. Let y denote the extent to wVc h the note participates in the appreciation of the underlying stock; we will call y the price participation of the note. ln general, the value J/0 pf a note with sxedmaturit.y payment M, coupon c, maturity F, strike price K, and price participation y can be written lt
J'')Pt
+ yBSCall(&,
483
.
These conditions the equation
imply that
&
JZ() =
Sj
=
M
K, o', r, F, 84
(15.11)
i= 1
Eqution
(15.11)assums that the principal payment is cash. lt could just as well be shares. Equation (15.11)also assumes that the note has a single embedded call option. Given equation (15.11),we could arbitrarily select M, F, c, K, and y and then value the note, but it is common to structure notes in particular ways. To take one example, suppose that the initial design goals are as follows: 1. The note's initial price should equal the price of a shar, i.e., 70 2. The note should guarantee a ret'ul'n of at least zero, i.e., M
=
=
A', and thus the price of the note satisses
=
3In addition to convertible bonds offered by firms, there are bonds offered under many names for different kinds of equity-linked notes-for example, DECS (Debt Exchangeable for Cornmon Stock), PEPS (Premium Equity Participating Shares), and PERCS (Preferred Equity Redeemable for Common Stock), a1l of which are effectively bonds plus some options position.
yBSCaIIIUV, %, c',
r, F, $)
(15.12)
J=zl
Given these constraints, equation (15.12)implies a relationship between the coupon, and price participation, y. Given a coupon, c, we can solve for y. and vice versa.
optionsin Equitpt-inked
c,
Notes.
With an equity-linked note, the maturity value is shares rather than a fixed number of dollars. The plice of a note at par paying qne unit of a share at expiration is '1
. =
P
c i= 1
Pz + Fa + y Bscallt5' w
,
K c' r ,
,
,
r
J)
,
(15.13)
Compare equations (15.12)and (15.13). lnstead of paying 'Sf)dollars at expiration, the equity-lirlkednote pays one share. lf the share pays no dividends, then (assuming y : 0) 0. To the th share extent the equitplinked note can sell at par only if c pays y dividnds, it is necessary for the note to ofer either coupons or options. =
=
Valuing and Sttuctuting an Equity-l-inked
CD
We have already described in Section 2.6 an example of an equitplinked CD, but we did not analyze the pricing. The CD we discussed has a 5.5-year mgturity and a retunz linked to the S&P 500 index. CD Suppose the S&P index at issue is Sft and is 55.5 at maturity. pays n6 coupons (c = 0), and it gives the investor 0.7 at-the-money calls ()z = 0.7 and Aftr 5.5 years the CD pays K = Pricins
'lnhe
the CD
'
,%).
Sj + 0.7 x max Using equation
-
Sj x P5.5 + 0.7 x BSCal1(u%, So, t)-,
where P5 5 +
=
(15.14)
(5'5.5 %, 0)
(15.11),the value of this payoff at time 0 is
%.
J/.
Pz + Szpv +
c
=
vz
4/0 MPp + c
$
lt
We now considerthe pricing of bonds with embedded options. Such bonds are common.3 Any option or combinatioh of options can be added to a bond. The option prernium (if a purchased option is added to the bond) is amortized and subtracted from the coupon. J.fthe option is written, the amortized premium is added to te coupon.
=
OPTIONS
3. The note should pay some fraction of stock appreciation above the initial price, i.e.,
l
The currency formula is the same as that for equities and commodities. tf we tlzink of the foreign interest rate as a dividend yield on the foreign currency, equation (15.10)is the same as our previous coupon expressions.
15.3 BONDS WITH EMBEDDED
EM BEDDED
?-,
5.5,
t)
(15.15)
d-rY5'5 *
To perform this valuation, we need to make assumptions about the interestrate, te volatility, and the dividend yield on the S&P 500 index. Suppose the 5.5-year interest rate is 6%, the s-yearindex volatility is 30%, the S&P index is 1300, and the dividend yield is 1.5%. We have two pieces to value. The zero-coupon bond paying $1300 is WO rl
$1300c -0.06x5.5
=
sjsy6: .
'
484
k
FI NANCIAL
ENGINEERI
NG AN D SECU RITY DESIGN
BoN Ds WITH
lf we
The 0.7 call options have a value of
0.7 x BSCall($1300, $1300,0.3, 0.06, 5.5, 0.015)
$309.01
=
=
$56.39less than the $1300initial investment. This difference suggests that the sellers enrn a 4.3% commission (56.39/1300)for selling the CD. This analysis makes it clear why the CD does not provide 100% of market appreciation. At 100%, the value of the CD would exceed $1300,and the bank would lose money by offering it. The bank is a retailer, offeling.the CD to the public in order to make a proft from it. The bank's position is that it has borrowed $934.60and written 0.7 call optins. You of eqation (15.15)as the vholesale cost of the CD-it is the theretical cost can tI-IiI'IIk to the bank of this payoff. As a retailer, an issuing bank typically does not accept the market lisk of issuing the CD. Banks offering products like this often hedge the option exposure by buying call options from an invstment bank or dealer. The bank itself need not have option expertise in order to offer this kind o t pro duct The CD is offered by a bank that wants to earn ommissions. Th originating bank will hedge the CD, and must either bear the cost and risks of deltachedging, or else buy the underlying option from another source. Retail custoprs may haye trouble comparing subtly different products offered by differeni bnnks. Customers who have not read this book might not understand option pricing, and hence will be unable io catculate the theoretical value of the CD. On balance, it seems reaspnable that we would find the value of the CD to be less than its retail cost by at least several percet. Here 're some other considerations:
Tlzis is
.
* It woul d have been cos tl y for retail customers to duplicate tl'lis payoff, particularly since s-yearoptions were not readily available to public ikestots at the time of issue. lnvestors buying this product are spared the need to learn as much about options and, for example, iaxeson options, as they would were they to yeplicate this payoff for themselves/ * The price we have just computed is a ballpark approximation: It is not obvious what the appropriate volatility and dividend inputs are for a 5.5-year horizon. Any specic valuation conclusion obviously depends entirely on the interest rate, volatility, and dividend assumptions. However, Baubonis et a1. (1993)suggest tat fees in this range are cofnmon for equity-linked CD products.
4It t'urnsout that the tax treatment in the United States of an equitplinked CD such as this one s fairly tcontingent interest complicated. A bond with a payment linked to a stock index is considered'to be debt-'' The bondholder must pay tax annually on imputed interest. and there is a settling up procedure at maturity. Issuers of such bonds frequently recommend that they be held in tax-exempt accounts.
$
485
allow
lt
1/(1
-
a)
=
MPv + c
J-')Pti + yBSCall(5',
K,
tr, ?',
(15.16)
F, J)
= 1
In the above example, a
$1243.61
OPTIONS
for issuer profit, a, as a fraction of the issue plice, a general expression for the value of a CD issued at par is
The two pieces together, assuming they could be purchased without fees or spreads in the open markt, would cost
$934.60 + $309.01
EM BEDDED
structuins example:
0.043 and J/
=
=
1300.
Many issues arise when designing an equity-linked CD. For
the product
@What index should we link the note to? In addition to the S&P, posgibilities include the Dow Jones lndustrials, the NYSE, the NASDAQ, sector indexs such as hightech, and foreign indexes, with or without currency exposure. How much participation in the market should the note provide? The CD we have been discussing provides 70% of the rettlnl (ifpositive) over the life of the CD.
@ Should the note make interest payments? (The example CD does not.) e How much of the original investment should be insured? (The example CD fully insures the investmento)
Alternative Structures Numerous other vadations in the structure of the CD are possible.
Sone examples
follow:
* Use Asian options injtead of ordinal'y options. * Cap the market participation rate, mrningthe product into a collar. * Incorporate a put instad of a call. * Male the promised payment different from the price. We will consider the
cover the other two.
srsttwo alternatives
in this section. Problems 15.9 and 15.11
The payoffdiscussed above depends on the simple over a period could 5.5 We the based the of instead comptge retul'n years. on avrage of year-end tlian an otherWise 14, As Asian option is worth less we saw in Chapter an P rices. ordinary option. Therefore, when Asian option used, the participation equivalent is an will be than with ordinary call. greater rate an Suppose we ;base the option on the geometric average price recorded five times ' . over the 5.5-yea.r hfe of the option, and set the strike price qual to tlie rrent index level. The value of this Asian call is $240.97as opposed to $441.44for an ordinary call. Assuming the equity-linked note pays no coupon and keeping the present value the same, the participation rate with this geometric-average Asian option is Asian options
'et'urn
0.7 x
44j.44 240.97
=
1.28
%.Fl NANCIAL
486
NG AND SECURITY
ENGINEERI
ENGI N EERED SOLUTIONS
DESIGN
If instead we base the option on the arithmetic average, the option price is giving us a participation rate of
441 273.12
=
Instead of shorting a forward contract, Golddiggers could issue a note promising to pay ap ounce of gold 1 year from now. Such a note is effectively debt collateralized by f'uture sales of gold. Ordinarily we would think a risk'y commodity like gold to be poor collateri for a debt issue. But if a gold-ltlillillg-firnt issues gold-linked debt, the lisk of the bond and the lisk of the collateral are the same.' Bondholders provide snancing as well as asorbing gold price risk. We begin with the information from Chapter 4: The current price of gold is $405/oz., the forward price is $420, and the effective annual interest rate is 5%. The effective annual lease rate is therefore 0.05 (420/405 1) = 1.296%. We wish to construct a debt contract that r atses $405 today (the cost of 1 ounce of gold), pays 1 ounce of gold 1 year from today, and if necessary, pays a coupon, c. We have already seen that the lease rate plays the role of a dividend. Thus, if the bond has a coupon equal to the lease payment on an ounce of gold, it should be priced fairly. Abond with these charactelistics should pay a coupon of 1.296% x $405 = $5.25. We can verify that such a bond is fairly pliced. The payoff to the bond in 1 year is $5.25 plus 1 ounce of gold. We lnow we can sell the gold in l year for $420 sine that is te folavardprice. The present value of the payoffis therefore the value of the coupon plus the prepaid forward price for gold:
1.13
The arithmetic Asian option has a higher price than one based on the geometric average, hence we get a lower participation rate. lncreasing the number of prices averaged would lower the price of either option, raising the participation rate.
-
Another way to raise the participation rate is to cgp the level participation of participation. For example, suppose we set a eap of k times the initial plice. Then the investor writesto the issuer a call with a strike of k.%, and th valuation equation for the
Capped
-
CD becomes
,ntl
-
a)
='
ste-rx' +
y x
pscalltsb,
u%,
c', r? t,
t)
Bscallt*(j,
-
kb, o-,
?-,
/,
))
j' Suppye we set a cap of a 100% retuw. Then the investor wdtes a !q' Example 15.6 t Ca 11with a strike of $2600to the issuer, and the valuation equation for th CD becomes tj ijr; 1300(1 0.043) 1300, + y x (sscalltlaoo 13()0 () 3 0.()6 5.5, 0.015) j' -BSCall(1300, 2.600, 0.3, 0.06, 5.5, 0.015)) ltr )) . .jjlj The value of the written 2600-stke call is $162.48.The participation rate implied by t(. . .. k k.. this equation is 1.11. -().(ox5.5
-
=
,
,
.
,
,
$5 25
-
-
SOLUTIONS
FOR GOLDDIGGERS
return to the Golddiggers example from Chapter 4 in order to see show how Golddiggers'could have used structured notes in place of forwards and options in the
we now
hedgingscenarios
we discussed.
Gold-tzinked Notes
FVPL
x /)1 +
=
-
$5.25+ $420
$405
=
Because the lease rate is paid as interest, the bond sells at pm We should verify that the bond serves as an appropriate hedge for Golddiggers. Table 15.1 summarizes the payoffs to Golddiggers and the bondholders at different gold prices in l year. The table assumes that Golddiggers invests the $405at s-this yields the $425.25that is labeled tFvtgross bond proceedsl.'' The net cash flow is deterinined by adding prohts without consideration of bond payfents (column2) to the difference between the invested bond proceeds (column3) and the payment to bond holders
t
15.4 ENGINEERED
487
contract linked to the hedging instrument, in particular the fact that it is azero-investment price of gold, meant to serve as a hedging instrument and not as a inancing instrument.
$273.12,
.44
0.7 x
$
FOR GOLD DIGGERS
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k: FINANCIAL
EERI Iqc AN D SECURITY
Excltq
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ENGI N EERED SOLUTIONS
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(column 4). ln this case, issuing the bond achieves the same result as selling a forward contract (compareTable 15.1 and Table 4.2), so Golddiggers is completely hedged. The chief difference between the gold-linked note and the forward contract is that former provides financing, the latter doesn't. lf Golddiggers seeks financing (in the order to construct the mine, for example), the issuance of a gold-linked note might be preferable to borrowing and hedging separately.
Notes with Embedded Options
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Solving for c gives c $11.04, which is a yield of 2.726%. Golddiggers thus issues a bond for $405, with a 2.726% coupon, with additional payments to bondholders if the price of gold exceeds $420. The difference between the 2.726% coupon and 5% is due to the value of the embedded call option. What is the result for Golddiggers from having issued this bond? If Golddiggers invests at 5% the $405 bond proceeds, then it will have $425.25cash in 1 year. Recall that costs are $380/oz.If the gold price in 1 year exceeds $420, Golddiggers will show profts of $420 + $9.21
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FI NANCIAL
ENGI N EERI NG AN D SECURITY
STRATEGI ES MOTIVATED
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October and November of 2000 (McMurray, 2001). Entering into a zero-cost collar reduces the risk of holding an asset, but it does not generate the cash that would be obtained from having sold the asset. However, once a
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stock position has been collared, the collared stock can serve as collateral for a lpan. For the example, suppose an executive owns stock and enters into a zero-cost collar where receive at least $90 put strike is $90. With this position, the executive is guaranteed to the stoek ptus put as using of value lend $90 bank present the mattllity. Thus, a can at collateral. lt is also possible to engineer a single instrument that hedges a stock position and 2003, Walt Disney Co vice-chainnn Roy Disney pays cash to the owner. In August sold a s-yearvariable prepaid forward (VPF) contract covering a large pereentage cif his Disney stock holdings. The contract called for Roy Disney to deliver to Credit Suisse First Boston a variable number of shares in 5 years. To quote from th Form 4 filed with
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The VPF Agreement provides that on Augst 18, 2008 (tettlement Disneyl will deliver a number of shares of Common Stock to Date''), glkoy the cash equivalent of such shares) as follows: (a) if the CSFB iLC (or WeightedAverage Price''l of the Common Stockfor average VWA.P the 20 trading days preceding and including the Settlement Date (&ISttle7,500,000 shares; (b)if the ment Plice''l is less than $21.751,a delivery of Settlelent Price is equal to or greater than $21.751per share CtDownside TllreshTlzreshold'') but less than or equal to $32.6265per share Threshold/settlement Downside old''), a delivery of shares equal to the Price x 7,500,000; and (c) if the Settlementpr e is greater than the Upside Threshold, a delivery of shares equal to (1 (10.8755/Settlement Pricel) x .
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United States.
Corporations may also wish to sell an asset without Tax deferral for torporations creating aconstructive sale. Acorporationhas an alternative not open to mostinvestorsnamely, the issuance of a note with apayofflirtked to theprice of the appreciated stock. A well-known example of this is the 1996 issuance by Times-Mirror Co of an equity-linked note, where the note was linked to the stock price of Netscape. In April 1995, Times Mirror had purchased 1.8 millio shares of Netscape in a private placement. The price (adjustedfor subsequent share splits) was $2/share. The
Disney Reaps $125 Million in a Novel Stock-sale Package,'' gRandall Smith and Bruce Orwall, I.I/// Sreet Jtpllr/lc/, p. el August 21 2003. ilkoy
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stock was restricted and could not be resold publicly for 2 years even if Ntscape were to go public. ln order to sell the stock, Times Minor would have had to 5nd a qualified buyer (a wealthy or professional investor) and sell the shares in a private placement. l () ln August 1995, Netscape issued shares publicly. In Mtrch 1996, Times Mirror had approximately $85 million in capital gains on the stock. tf the shares had been sold on the open market, the tax liability would have been approximately 0.35 x $85m $29.75 milliorf. Although tflx law did not at that time desne a constructive sale, a suciently wide collar seemd likely to avoid challenge by the tax authorities. Times Mirror's :
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A' stock, after the Securities Act section tbat defined it. 'The Netscape PEPS became an AOL PEPS when AOL acquired Netscape.
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ARATEGIES
DESIGN
AND SECURITY
ENGINEERING
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lzBanks worldwide are required to have capital to cover potential losses on Ioans and investments. The specic rules are in flux, but generally spealdng, one'' capital is equity, retained earnings, and preferred stock, while two'' capital s subordinated debt. Limited amounts of preferred tnlst securities, such as the M&1 convertible, can be counted as tier one capital. See Mishkin and Eakins (2003, ch. 18) for an overview of bank regulation. ''tier
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* Interest ?T the tntst: M&lissued $400million of subordinated debt (debtwith very 1owpliol'ity in the event of bankruptcy) maturing in 2038, paying a 3.9% coupon. bonds were placed into a trust.l3 Each unit of tlze Yhese convertible bond contains interest in the trust for $25 par value worth of these subordinated bonds. an After 3 years, the bond coupon will be reset so that the bond trades at pac The bonds are subordinated so they can count as regulatoly capital. * Stockpltrchase ctmlmcl; Each stock purchase contract pays a 2.6% coupon and requires the investor, after 3 yearsa to pay $25 for between 0.5402 and 0.6699
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Under U.S. tax law, interest payments on corporate debt are tax-deductible, whit' .. ( ( ividend payments on equity are not. The distinction between debt aljd equity itkt may t seerri clear-cut, but Fith inancial engineering it is easy to blur th distinction. Fof suppose a firv,l issues equity-linked notes that promise ctsfkt '('.,. P yments tlt ( q y debt) but have a payment at maturity contingent on the firm's stock price (like equityjt' i.)( i ls such a Enancial cla im debt or equity for tax purposes? t ;) rt. ln practice it is possible to design financing vehicles that have a sijlcant equity component, yetforwltichthepayments are atleastpartially tmvdeductiblfpythe E.ii, In fil'm. '' . this section wewill examinein more detailtheMarshall &llsley .( j . . . . j.. Col'p.(Mljonvertible bonddiscussedin Chapter3, withthepayoffdepicted inFigure3.l6. j j.j..j . As weditlssed, the ... bond payoff resembles a stock coupled with written and t purchased calls. Nekfthelesja the bond o n tax-deductible are partially payments for the issuer. Moreover, M&I bankholding compalm and the bond provided capital for regulatory purposes.lz ) is a With a structure like the M&I convertible, mytiad details hinge on complex ix, accounting, and regulatol'y considerations. Our purpose in discussing the bond ij .t . .( i( understand at a general level the kinds of issues that financial engineering can addres. We ignore specic details that aren't needed in conveying E a sense of the transaction. The M&1 convertible bond actually consisted of two components: an ownership stake in a tnlst containing M&I bonds, and stock a purchase contract requiting that the convertible bndholder make a future payment in exchange for slaares. These two components are separable, in that an investor could in theoly hold one without the other. Here are some of the details on these two pieces: . ..
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variable like that of Roy Disney selling the The effect o.f issuing the laEps was shares. deliver could received cash at issuance and plepa id forward. Times Minor hedgedposition. ln along with the payoffof the PEI'S seucture is graphed in Figure 15.2, issuanceleftTimesMila-oriluperfeetly aconstructivesale, tlae ordertoavoidchallengeas stockpl-ice was Iessthan $39.25,Times Min-orwould hedged. If at maturity the Netscape $45.14,Times payoffline is below zelo. Auove lose the interest payments; hence the net = 1.15 $45.14, that $39.25 x risk of holding 13% of the shares. Note
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The bonds hld in trust serve as collateral to ensure that the investor can pay the $2 to buy shares in 3 years, eliminating credit risk for M&1. Think of the investor as How can we understand the pricing of the convertible? paid for bonds and nothing the stock purchase the for having 3.9% contract. If you $25 0.6699 of M&I, the stock purchase with shares there contract are both costs and compare benefts of the stock purchase contract relative to the stock: I'he investor is obligated in 3 years t pgy the offering day plice for 0.6699 shares ($25)but could in three years receive as few as 0.5402 shares. The investor also does not receive the dividend on the undrlying shares (M&I had a 2% dividend yield). However, the investor can acquire f'uttlreshares for the offering day price. Taking all three considerations into account, the investor receives a 2.6% dividend in remrn for entering into the stock purchase contract at a zero initial cost.14 The structure is diagrammed in Figure 15.3. To summarize: M&I issues abond and stock purchase contract that together compse a coavertible bond. Investors receive a bonds and 2.6% from the stock purchase contract. For 3.9% the 6.5% fmm coupon: a M&I, the 3.9% coupon is tax-deductible because it is interest on the subordinated debt. The 2.6% payment on the stock purchase contract is not debt interest and hence is not deductible. At maturity of the stock purchase contract, the par value of the bond can be used to pay for the stok. At this point, it may be helpful to answer some questions that may occur to you: sintply stock, 1.W#7 didn .&J't:Q.J the a single f?/-/'?-l/l?/w/',convel-tible i'??/'t? * JF'/1.)) w??)epayoff? As we noted earlier, the bond payoff resembles stock plus options. ordinar/convertible bond, ' The payoffis unlike that for an ordinary bond oreven an ot both which proinise to pay at least par value at maturity. A single instrument with the structure of the M&lconvertible-with paymentno minimumpromisd would have been deemed too eqqity-like and the payment would not have been tax-dedtlctible. 't
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ltnit sltbordinated bollds) plus a stock (ct7nlt7f??f//,g @ The bondholder bottght the pw,l pllrchase colttract. Jfy/lf have to /75/# these (/. t ll?lfl, isn this //?d same J'/2f?lt as holdilq a silgle ?7./'?-lf??ldnr? The key to allowing tax-dedpctibility of the interest is that the bonds and stock purchase contract do not have to be held as a unit. The subordinated bonds expire 30 years afterthe stockpurchase contract mamfes. They are locumented as distinct entities. Moreover, the convertible-holder has the right ,/.
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rates ill 3 )'ct7?-' have risen tw7# the pt7/l/e oftlle .l/lvt/rtW?pt7rd# * What tfi/?ld?-d.l bolld /pe/t?I$7 hasfallen $252 The bonds are issued subject to a renlarketing agreemellt. Tis mans that in 3 years the interest rate on te bonds will be reset so that the bonds sell at par ($25).Thus, the bonds will be worth $25 at exactly the time when the shareholders need to pay $25 for the variable number of shares-'s did #7c stock ypl/rc/lt-lz contract /?&1?ca kink, instead ofjust beillg a ,W?/?p/c * 1,1//2,), fonvard c-lpwcr? The dividend on the forward ptlrchase contract compensates the investor fr the possibility of receiving fwer than 0.6699 shares at maturity and the loss of the dividend on the underlying shares, less the gain from defening the $25 share cost. ln exchange for giving up more appreciation, tlie investor receives a greater dividend. The kink is determined by the willingness o? seller anl buyers to trade appreciation for current income. '
Many financial institutions have used a trust structure like that in the M&I transaction. Relted strtlctures under different names (for example, DECS'') are used by companies wishing to obtain partially tax-deductible eqity-like snancing. tupper
l5we have not discussed the possibility that remarketing would fail or what would happen if h1&1 enters bankruptcy. The publicly available prospectus for the bond discusses these details.
498
% FINANCIAL
ENGI NEERI NG AN D SECURITY
DESIGN
%
PROBLEMS
Finally, note that the M&I convertible and the Netscape PEPS both have payoffs incorporating a collarlike structure, but the motivations are different. The Netscape PEPS was intended to avoid a constructive sale of stock, while the M&1 convertible incorporates a collar as a way for investors (and the company) to express a view.
Zero-coupon bonds, forwards, calls, and puts serve as building blocks that can be used - to engineer new financial products. Fair pricing of a product will depend upon volatility, the dividend or lease rate, and the cun-ency of denomination. Ordinary bonds that afe simply dnominated in something other than cash follow a simple pricing principl: The lease rate of the underlying asset becomes the coupon rate on the bond. The specifc charactelistics of a fnancial product can be varied, though when one characteristic is changed, another must be changed to keep the value the same. The dials ihat we can turn include the participation in the underlying asset (viaembedded calls and puts), the guaranteed minimum, and the coupon. Pricing theory tells us how to male these tradeoffs. Instruments can be designed specifcally to take advantage of tax rules and regulations. The Disney prepaid forward, Netscape PEPS, and M&1 convertible bond am examples of this. .
FURTHER
READING
In this ch>pter we focused on the creation of engineered instruments using bqsic building blocks such as assets, bonds, fonvard contracts, and options. However, using the BlackScholes technology based on delta-hedging (discussed in Chaptr 13), it is possible to will complicated instruments. We engineer more cover the more general approach in 22. 21 and applications in Chapter Chgpter see some Readings about structured products (icluding some not discussed in this chapter) include Baubonis et al. (1993),Mcconnell and Schwartz (1992),Arzac (1997),and Crabbe andogilagos (1994).Formore information aboutWestelwsouthern, a deal simhup://-.ltellojg.northwestern-edu/ Times-Mirror Netscape PEPS, see ilar to
facultpetersen/html.
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15.1. Consider a s-yearequity-linked note that pays one share of XYZ at maturity. The price of XYZ today is $100,and XYZ is expected to pay its annual dividend of $1 at the end of this year, increasing by $0.50 each year. The ffth dividend will be paid the day before the note matures. The appropriate discount rate for dividends is a continuously compounded risk-free rate of 6%. Suppose that the day after the note is issued, XYZ announces a permanent dividend increase of $0.25. What happens to the price of the equity-linked note? 15.2. Suppose the effective semiannual interest rate is 3%. a. What is the price of a bond that pays one unit of the S&P index in 3 years? b. What semiannual dllar coupon is required if the bond is to sell at par? c. What semiannual payment of fractional units of the S&pindex is required if the bond is to sell at par? 15.3. Use information from Table 15.5. a. What is the price of a bond that pays one unit of the S&pindex in 2 years? b. What quarterly dollar coupon is required if the bond is to sell at par? c. What qutrterly payment of fractional units of the S&P index is required if the bond is to sell at par? 15.4. Assume that the volatility of the S&P index is 30%. a. What is the price of a bond that after 2 years pays Sz + maxto, Sz S0)? b. Suppose the bond pays Sz + g x maxto, Sz For what will the bond sell at par? -
.%)1.
-
PROBLEMS Some of the problems that follow use Table 15.5 and the following assumptions: The spot price of oil is $20.90.Let St denote the time price of the S&P500 index, and assume that the price of the S&P 500 index is uV $1200and the continuous annual dividend yield on the S&P 500 index is 1.5%. =
15.5. Assume that the volatility of the S&P index is 30%. a. What is the pfice of a bond that after 2 years pays b. Suppose the bond pays & + ( x maxto, Sz will the bond sell at par? -
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ENGINEERING
AND SECURITY
($20.90)?
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b. Suppose that the oil payments are quarterly instead of annual. How large would they need to be for the bond to sell at par?
=
=
501
What payment would the bond have to make in order to sell for par
15.6. Assume that the volatility of the S&P index is 30% and consider a bond with the payoff Sz + x (max(0,Sz &) maxto, & ffll. -
%
PROBLEMS
DESIGN
=
c. lf
l for what K will the bond sell at par?
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The next six problems will deal with the equity-linked use the assumptions in that section.
CD in Section 15.3.4. If necessary,
15.15. Using the information in Table 15.5, stppose we have a bond that after 2 years pays one barrel of oil plus x maxto, Sz 20.90), where Sz is the year-z spot price of oil. If the bond is to sell for $20.90and oil volatility is 15%, what is ? =
15.16. Using the information in Table 15.5, assume that the volatility of oil is 15%. a. Show that a bond that pays one barrel of oil in 1 year sells today for $19.2454.
15.7. Explain how to synthetically create the equity-linked CD in Section 15.3 by using and a put option instead of a call option. a forward contract on the S&P index blillt: Use put-call parity. Remember that the S&P index pays dividends-)
b. Consider a bond that in 1 yea.r has the payoff 5'l + max(0, A-1 .S'1) maxto, 5'I -A). Findthe stlikeprices /71 and Kz suchthat ft':t-fl $2, and the price of the bond is $19.2454. HoW would you desribe this -
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=
CD in Section 15.3. Assuming that proht for the 15.8. Condider the equity-lilied bank is zero, draw a graph showing how the parttcipation rate, y, varies issuing with the coupon, c. Repeat assuming the issuing bank earns proEt of 5%.
15.9. Compute the required semiannual cash dividend if the expiration payoff to the CD is $ 1300 maxto, 1300 55.5) and tle initial price is to be $ 1300.
Payof-o
5'I $20.50+ max(0, A'j c. Now consider a claim that in 1 year pays where maxto, 5'1 Kz), /f1 and Kz x$'1) are from the previous answer. What is the value of this claim? What have you copstructed? -
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15.10. Compute if the dividend on the CD is 0 and the payoffis $l300-max(0, 1300S5.5) + x maxlo, 55.5 2600) and the initial price is to be $ l 300. -
if the dividend on the CD is 0, the initial price is $t300, and the 15.11. Compute 1200 is payoff $ + x maxto, 5'5.5 1300). -
15.12. Consider the
equity-linked
15.17. Swaps often contain caps or qoors. ln this problem, you are to construct an oil The initial cost is zero. Then in contract that has the following characteristics: each period, the buyer pays the market price of oi1 if it is between A-1and #2:, otherwise, if S < A-1 the buyer pays A-1 and if S > Kz, the buyer pays K, (there is a floor and a cap). Assume that Kz #1 $2 and that 4)il volatility is 15%. ,
b. For each paralneter change above, stlppose that we want the product to continue to earn a 4.3% commission. What price participation, y, would the CD need to have in each case to keep the same market value? 15.13. Use the information in Table 15.5. barrel of oil 2 years from now? a. What is the price of a bond that pays one b. What annual cash payment would the bond have to make in order to sell for $20.90?
15.14. Using the information in Table 15.5, svppose we have a bond that pays one baln'el of oil in 2 years. of oil as an interest payment a. Suppose the bond pays a fractional barrel the one barrel after 2 years. addition in after 2 and to 1 after year years,
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volatility, and a. What happens to the value of the CD as the interest rate, alternative volatilities of dividend yield change? In particular, consider yields of 0.5% 20% and 40+, interest rates of 0.5% and 7%, and dividend and 2.5%.
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15.18.. You have been asked to construct an oil contract that has the following characteristics: The initial cost is zero. Then in each peliod, the buyer pays S X, with a cap of $21.90 X and a ioor of $19.90 X. Assume oil volatility is 15%. What is X? -
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15.19. Using Figure 3.16 on page 84 as the basis for a discussion, explain under what circumstances an investor might prefer a PEPS to the stock or vice versa. 15.20. Consider again the Netscape PEPS discussed in this chapter and assume the following: the price of Netscape is $39.25.,Netscape is not expected to pay volatility of Netscape is 40%. dividends; the interest rate is 7%., and the s-yea.r What is the theoretical value of the PEPS? 15.21. A DECS contract pays two shares if Sv < 27.875, 1.667 shares if the price is above Sv > 33.45, and $27.875and $55.75otherwise. The quarterly dividend 35%, r 9%, and is $0.87kValue this DECS assuming that S $26.70,o' F 3.3 and that the underlying stock pays a quarterly dividend of $0.10. =
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The next two problems are based on the M&1 stock purchase contract.
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* lf the stock has a 2% dividend yield, what price in 3 years would you agree to pay for the stock?
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yirms often issue securities that have derivative components. For example, firms issue options to employees for inancing, and convertible debt is a bond coupled with a call option. However, even simple securities, such as ordinary debt and equity, can be viewed as derivatives. ln this section we examine both implicit and explicit options issued by .
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: ' ' :' .' . . -.:.
' ..
'
'
'
. . :
'
' :
'q
:
E.
rjyEyy..
:.
.-ttt;.
-
. ..
.
-y);.-.
.
.
.
- - . -..
.
:
:
.
.
.
-
.
.
.
-
.
-. - .-. . -. .
. . : - ... .
.
. ..
.
Valu b f e bt nd equity at maturi of the debt as a function of assets, for a firm that has a single issue of zero-coupon debt with . a maturty value of
.
. .
-
.
.--ii-,-;--'
.
.
.
(11:.11::
;!,1:. .,.:.k;l.. ll:E!,,-
d,::::),-
ordinazybond Convertible equity
14,000 12,000 10,000
slope
=
Default
slope =
-
,!
1
!i
Av + minto,
=
=
Av
-
W
=
W+
= W
minto, Av maxto,
-
W)
-
W -
Aw)
(16.4)
.
a ace value of = (16.1) and the ') maturity value of the debt is given by equation (16.4).The two payoffs are graphed in .(t..j . lf' Figure 16.1 as a function of corporate . assets at mamlity. Q .
! !J 4,000
.
-
y)
=
-
max(0, )'
.r)-
2A bond with a payoff specilied as in equation ( 16-2) is a debenture a bond for which payments are secured only by the general credit-worthiness of the company. Such a bond is said to be ltnsectlred. It is also possible for bonds to be secured by speci:c collateral. For example, lenders to airlines may have an airplane as collateral for their bond. .
' .
-
8,000
12,000
16,000
20,000
($)
1.fwe assume that the assets of the 1114.11 are lognormally distributed, then we can payoffs the Black-scholes model value the to to the frm's equity and debt, equations use (16.1) and (16.4).For purposes of option pricing, the frm's assets are the underlying asset, the strike price is th promised payment on debt, W,the volatility of the rm's assets?o', is volatility, and the payout rate from the firm becomes the dividend yield. Y rate is 1' and the debt matures at time F, we have the lisk-free Et
=
BSCall(A?
Bt
=
Af
-
,
W,o',
r, F
-
t,
(16.5)
t)
(16.6)
Et
Assuming that the debt is zero-couppn, we can compute the yield to maturity on debt, W-#(T-'). By dennition of the yield to maturity we have 1, = hence we can solve p. obtain for p to 3
p. r
'To follow these derivations, note that minto, a-
, '
Asset value at mattlrity
debt with a The interpretation of equation (16.4)is that the bondholders own lisk-free the with strike plice option W.2 assets on Pay 'off equal to W,but have written a put
)r Example 16.1 supposea firm has issued zero-coupon debt with ''). W $6000 The maturity value of the equity is given by equation
$6,000
I
0
W)
-
=
!
Awl
maxto, Aw
Assets
j
2,000 -
1
8,000
4,:00
W)
505
'
:
: . .
:LL. tyttt.jy
:).
jliiqillll:lt;,jjjjj. . -
6,000
Equation (16.3)says that the bondholders own the firm, but have written a call option to the equitpholders. This way of expressing the debt value explains where te call option in equation (16.1)comes from. Suinming equations (16.1)and (16.2)gives the te total plus debt-as Av. value of the srm--equity A different way to write equation (16.2)is the following: Sz
; -. -.
-
' ' . . .. .
' :
;
$6000.
This expression can be writtenl Bv
'
.
-
Ep
' E
$
AN D WARRANTS
1
jyjtwysy)
t This model of the 51-111 is vel'y simple, in that there are no coupons or dividends, no refnancings or subsequent debt issues, etc. lt is possible to create more complicated models of a srm'scapital structure; nevertheless, this model provides a starting point for understanding how leverage affects returns on debt and equit'y and determines the yield on lisky debt. -
% CORPORATE
506
EQu ITY, DEBT, AN D WARRANTS
APPLICATIONS
Viewing debt and equity as options also provides a f'ramework for thinking about credit risk. Equation (16.4)shows that defaultable debt is eqtlivalent to owning defaultfree debt plus a put option on the assets of the frm. An investor owning a cm-porate bond could buy such a put; the result would be economically equivalent to owning a defaultfree bond. Thus, the value of the put is the value of insurance to protect bondholders against default. We will examine credit risk more in Chapter 26. ' t@'(' $90, ?' Example 16.2 Suppose that W $100, zito qt!) 'il, have We 5 and F mtkes no payouts), years. =
=
tt arm )))
=
6%, c
25%,
=
=
0 (the
=
-
'1) t.it (::y
Eo
BSCall($90,
=
where rzt is the expected return on assets, 1' is the risk-free rate, and Qs is the elasticity of the equity. Using equation (16.5),elasticity is AtLE
Qs
=
(16.9)
Et
where As is the option delta. We can compute the expected return on debt using the debt elasticty, f2,:
l-e
1-
=
( 16.10)
r) x f)s
(rzl
+
-
The elasticity calculation is slightly more involved for debt than for equity. Since we compute debt value as Bt = A/ E:, the elasticity of debt is a weightd average of the asset and equity elasticities: -
$100,0.25, 0.06, 5, 0)
$gg Ljy
=
9!(
. C'p
..
f)s
$90 $27.07 $62.93
Bv =
;(.
i E'
-
=
(. .
qt The debt-to-value ratio of this lirm is therefore $62.93/$90 tt'. 1: IXZ turity on debt is :. :.
=
0.699. The yield to
l.E)
p
t@.
l =
-
5 0.09f6
.
)Et.
A?
-
Q,
Et
Et -
z4?
Et
-
late. .
Leverage and the Expected Return on Debt and Equity Example 16.2 shows that, because of the possibility of bankruptcy, the yield to matul-ity investor in the bond earns the yield on debt exceeds the risk-free rate. However, arl Accounting for the possibility of bankrupt. does if the not go firm to maturity only less than the yield to matulity and return will eal'n a bankruptcy, the investor on average default bears some of the risk of that efeet, In can debt risk-free rate. greater than the lisk with equity-holders. sharing the this the assets, We can compute the expected return on both debt and equit'y using the concept elasticity of an of optioll e/t'/l//cf/)/, vhich we discussed in Chapter l2. Recall that the tlnderlying expected asset and return the the between relationship the on option tells us expected equity the return 12.10), eqtlation compute on option. we can Using ( that on the
The elasticity of any asset with respect to itself is one, so we have Qzt = 1. Using equations (16.8)-(16.1l), you can verify that if you owned a proportional interest in the debt and eqpity of the firm, the qxpected rettll'n on your portfolio would be the expected return on the assets of the firm: rE4
+
(toebt
x rs)
=
'E
=
r +
(?u
-
r)
1 tzEquity
=
r +
t?u -
r) x Qs
rz
(16.12)
( 16.8)
(16.13)
This is the familiar Modigliani-Miller expression for the expected retuna on levered equity Equation (16.13)can be obtained from equation (16.8) bf assuming that the delta of the equity is one, which implies that the delta of the debt is zero. Viewing debt and equity as options, by contrast, allows us take into the account the effects of possible bqnkruptcy. Equation (16.8)assumes that debt- and equity-holders share the risk of the assets, so equation (16.3)will give a higher l'E than equation (16.8).
as rs
(16.11)
fzs
It befkrs emphasizing that this relationship requires that ?'s represent the expected ?'c?Ifr?? on debt, not the yield to maturity. Using the option model, we can also see how the dollar values of debt and equity change whn there is a change in the value of the assets. In other words, we can ask what the deltas of debt and equity are. From equation (16.5),if assets infease in value by $1, equity will incree by the delta of the all option. From equation (16.6),debt will increase by one less the equity delta. It is instructive to compare the expected ret'urn calculation for equity in equation (16.8) with a common alternative calculation. lf we asstlme the debt is risk-free, the; expected return on equity isa
ln(100/62.93)
t) lij 'j'(j The debt yield of 9.26% is 326 basis points gleater than the risk-free =
=
ttzEquity x
tE)
).)
A/
.
value of the debt is
'rhe
;E
l
507
'
jl::EEE
. ;j: y
%
sTh-ls expression is also sometimes wn-tten as rs
=
rz + (rz
-
?-)
x D/E.
508
%.CORPORATE
EQU ITY, DEBT,
Appl-lcvloxs
100, 0.25, 0.06, 5, 0)
0.735
=
The debt detta is 1 0.735 0.265. Thus, if te asset value increases by $1, the value of the debt increases by $0.735and the value of the debt increases by $0.265. Using equation (16.9),the equity elasticity is =
-
Q
509
At the same time, debt also becomes liskier as leverage increases and there is increased chance of default on the debt. Figure 16.2 graphs the debt-to-asset ratio (computedusing equation 16.5) and the expected tet'urn on equity (computedusing equation 16.8) as a function of the asset value (f the fit'm, using the assumptions in Example 1.3. For very low asset values, the debt-to-asset ratio is almost 1 and the expected return on equity is almost 40%. As the asset value exceeds $200,the expected return on equity is about 12%. For purposes of comparison, Figure 16.2 graphs the expected retarn on equity, computed assuming that the debt is risk-free. For asset values close to $200, the
Use the same assumptions as in Example 16.2, and suppose that the Example 16.3 expected return on assets, rA, is 10%. The equity delta is BSCallDelta(90,
AN D WARRANTS
90 x 0.735 = 2.443 27.07 The expected return on equity is therefore l
'
=
'
'!' pf .(('.y! i'lff ,'(' F::qrqrr!'rrf y'y r' t';'t' 1q'' 5* p' (lqzjif qs7i)i:f 1E1.!)4E)1* frpif 7* 1* F' ilf (' y'. y,jyyjgy(j'. )' q' @' ,'t-' j'y' j!f ;ff g'. yyyjf ''r' ;j))'. .y'. l'y t':1l'. jjtryjjy'll.ty 1111k* p' )' j')'y' y'. y' j)jj))..j' jy'j ('4111:44* iill i7. : ).!8! liftrii'(i ' !ii q( (;' 7i:',. r(y'tjjyjyyj.' )' )!)-),' ty'. til'tf tjf t'j jgj,yf t:;yf . . ii' ' j-''li'il.E.'i 't!.' .' (i'' ' : l;k'11ltli gg. ); ( tk;yy l.. tE ryjj jg y..jj.(j.y.. y jjy:. j ..li j j 44jjjjjjj,:4 .. ) y . j : l t;sy t j, .. j . :j..j.j j j . . j Iljj:iiilitl !1II:::::;k:ty. y. . . . !y. jj.g.j jyj.yyy yr . 'tlr));.tr--.:-.-t:...-..... jr . )-----jj).. ... ... ,. . . ,4* (IyI4'jjjjjj,,jjjj;E yq'jE F' '!t' !(' (' (',LLLLjj,L.-..L y' pf (k' 'h'f'lq'qq: f' (;E' j'gjyt'. ' .'..'
0 (% +
(0.1 0.06) -
.
.'
x 2.443
IL;'plE
E . .. ;
= 0.1577
:
:1E E
. E .
.
. .
,,-i
--..
-.-......
-
E ! ''.' . @
: j.jk!
.. - .
:
.
.
.
........jk.,-
-......,.
.
. ....,.
! : :: E EE
: : i 5* ! :q!'ltE ! : . . : .yjEfjz.( : ! jk.jjjjggjj;;;,.
: i
E E;.
.
......
E
:. ::. ..E
'
-
.
.
,.
.
. -.
-.
.
.
.
. -.
.
pq!E '
.
-
..,...,....- .-.-..,.--...
''' -
r
.;-jjjjj.
E- .
i
;.... iE... E.E.E E. E . E : . .. .!.. .jE. :yi.y ).L y y y jy y (,y. :.2:.,grj. . yq-...,- - -g -.. y. -.g.2-., , . . . -.
-
.
-
.... ,
.
.
,,
.
Debt-to-Asset Ratio 1.0 0.8
The expected return on debt is therefore
?-s
=
0.06 +
0.6
(0.1 0.06) -
x 0.3793
0.4
= 0.0752
0.2
Note that the 7.52% expected return on debt is greater than the risk-ft'ee rate (6%)and less than the yield to maturity on debt (9.26%). If we owned equity and debt in the same proportion in which they were issued by the firm, we would have a return of
90 27.07 27.07 0.1577 + x 0.0752 x 90 90
100
50
150 Asset Value
150
0.1000
Since 10% is the
expected return on assets, this illustrates equation (16.12). Finally, ifwe were to (erroneously)assume that debt is risk-free and use equation (16.13) to compute the expected return on equity, we would obtain
t
1O0
200
3O0
250
($)
(%)
Expected Return on Equity
-
=
1
.
0
-
'
-
Equity return, risk'y debt return, risk-free debt
t,qu j ty
h 'i
50
''.
x. .
'E
=
1 0.06 + 27.07/90
(0.1 -
0.06)
0
= 0.1929 This calculation
gives an expected retul'n on equity substantially greater than 15.77%.
%
This example computes expected returns for a particular leverage ratio. As the becomes tirm more levered, equity-holders bear more asset risk per dollar of equity. lf assets have a pos itive beta the expected return on equity 'will increase with leverage. ,
I '
0
50
N.
.
1
100
I '
150 Asset Value
I
200
'
l
250
'
I
300
(s)
The top panel graphs the debt-to-asset ratio as a function of the asset value of the firm, using the Black-scholes formula to compute the value of the dbt. The bottom panel graphs the expected return on equity as a function of the asset value of the firm, using equations (16.13) and (16.8). Both graphs assume that there is a single zero-coupon debt issue with maturity value $100 and 5 years to maturity, and also assume that r 6%, (y' 25% (forthe assets), and t 0. =
=
=
510
k. CORPORATE
ArapLlcvlorqs
EQU ITY, DEBT,
difference is less than 20 basis points. For a very highly levered (low asset however, the difference in Figure 16.2 is dramatic. Conflicts
between
debt and equity
value) 'lirm,
511
Multiple Debt Issues The option-based model of debt accommodates multiple issues of zero-coupon debt with different seniorities, as long as a1l debt expires on the same date. By de:nition, more senior debt has priority in the event of banlruptcy. Suppose that there are three debt issues, with mamrity values of $30, $30, and $40, ranked in seniority from highest to lowest. We will refer to each distinct level of seniority as a tralche. The value of equity will be the sarize as in Example 16.2, since it is still necessary for equity-holders to pay $100 t receive ownership of the assets. However, the option pricing.approach permits us to assign appropriate yields to each level of debt. Senior debt-holders are the first in line to be paid. They oFn the 'lirm and have written a call option permitting the next set of bondholders to buy the firm from them by paying the mattlrity value of the senior debt, $30. lntermediate debt-holders own a call option permitting them to buy the rm for $30, and have sld a call option permitting the junior bondholders to buy the firm for $60. Junior bondholders in turn own the call option to buy the 5rm for $60, and have written a call option permitting the equity-holders to buy the finzl for $100. The values of these ptions
The idea that equity is a call option on the 'lirm
and that corporate bonds are risky provides insights into relations between debt- and equity-holders. Since equitpholders control the firm, bondholders may be concerned that equity-holders will tke actions that would harm them, or may fail to take actions that would help them.
There are two decisions equity-holders make that affect the relative value of debt equity-holders can affect the volatility of assets. Equity-holders can - increase asset volatility either by increasing the operating risk of existing assets, or by xasset substitution.'' replacing existing assets with riskierassets. An increase in volatility, other things equal, increases the va fue o f the equity-holder's call option and therefore reduces the value of debt. In Example 16.2, the vega of the equity is 0.66, so an increase in asset volatility of 0.0 1 leads to an increase in the market value of quity of $0.66, which is 0.66/27.07 = 2.4% of equity value. Debt value woul decline by $0.66. A second decision that equity-holders can make is the size of payouts to shareholders, such as dividends and share repurchases. To see why payouts are a potential problem forbondholders, suppose that the 51711makes an unexpected one-time $1 payout to shareholders. This payout reduces assets by $1. The delta of the equity with respect to assets is less than one, so the value of equity declines by less than $1. Since the value of debt plus equity equals assets, the vtp/lfc ofthe debt inl/.? declilte !?,),olte less the delta oftlle Unanticipated payotgts to equity-holders therefore can hurt bondholders. Bondholders are well aware of the potentially harmf'ul effects of asset substimtion and dividends. Bond covenants (legalrestrictions on the frm) often limit the ability of the firm to change assets or pay dividends. Viewing debt and equit'y as options makes it clear why such restrictions exist. Bondholders also encounter problems from actions that shareholder fail to take. Suppose the has a project worth $2 that requires shareholde's to make a $1 investment. If shareholders make the investment, they pay $1, the valtle of the assets increases and the value of the shares rises by 2 x A. The gain to shareholders is less than by the increase in the value' of assets. The difference of 2 2 x 2. goes to the bondholders. In making a positive NPV investment, shareholders help bondholders. The shareholders in this example only will make the investment if the value of shares goes up by more than the $1 they invest, which will only occurif A > 0.5. ln order for shareholders to be willing to invest, the NPV must be great enough that shareholders gain after allowing for the value increase that is lost to debt-holders/ Thus, because of debt, the shareholders may fail to make positive NPV investrpents. A relatd problem is asset substitution: Shareholders might make negative NPV investmets ihat increase asset risk, thereby transfening value from bondholders to stockholders.
$
AN D WARRANTS
and equity. First,
re
$30, 0.25, 0.06, 5, 0) BSCa1l($90, $60, 0.25, 0.06, 5, 0) BSCa1l($90, $100,0.25, 0.06, 5, 0) BSCall($90,
$67.82 $47.25 $27.07
=
=
=
(16.14) (16.15) (16.16)
Table 16.1 summarizes the value, yield to maturity, and expected return of eacfl tranche of debt. The junior tranche has a yield to mattlrity of 13.69%, very close to the required ret'urn on equity. The senior tranche, according to the model, is almost risk-free. The expected returns in Table 16.1 are computed using option lasticities. To illustrate the calculation, consider thejuniorbond, which is created by buying a 6o-strike call on the assets of the tirm and selling a loo-strike call. The two option elasticities are and 2.4432 tloo-stlikel. Using the fact that the elasticity of aportfolio 1.7875 t6o-sttikel
'tzllf/y.
'lirm
J
'$2
-
.' ''.'
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.
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Equity
:
. ..
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bnds
'
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.
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lntermediate bonds
-.
.
-
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-
:
-y-r - -.
- -. . ..
-
' . . . . -- . -. . . . :. ' : ...
''
' . . . .. . . . .'
-
:
y
Junior bonds 'lrf'heidea that the debt may harm investment incentives is developed in Myers (.1977).
g
-. - -
-.-
2
. -...----
.......
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6.042
C(30)
22.18
C(30)
C(60)
20.57
7.54
7.03
C(60) C(100)
C(100)
20. 18
13.69
9.63
6.04
g. .j. . ..
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Assets
27.07
t.ls!il .... y
'' r::lii;i:E 11iEi!i: . 'E. E' .111;2:)1,.
-
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15.77
%
CORPORATE
EQU ITY,
Appl-lcA-rlolqs
is a weighted average of the junior bond is
elasticities
47.25 x 1.t875 47.25 27.07
of the portfolio components, the elasticity of the
-
47.25
x 2.4432
27.07
-
=
0.9077
The expected return on the junior debt is therefore
nunior 0.06 + (0.10 0.06) x 0.9077 =
-
0.0963
=
Table 16.1 makes it clear why debt cannot be treated as a single homogeneous when class sl'mswith complex capital stzucttzres enter bankruptcy. The interests of the most junior debt-holders may well resemble the interests of equity-holders more than those of senior debt-holders.
Warrants Firms sometimes issue options explicitly. If a 'Iirm issues a call option on its own stock, t'warrant'' is used here to denote options on a it is known as a warrant. (The term practice 111-111 the term includes traded options issued issued by the firm itself, though in the warrant-holder pays the firm the exercised, in lixed supply.) When a warrant is strike price, K, and receives a share wort.hmore than K (or else the holder would not have exercised the warrant). Thus, the act of exercise is dilutive to other shareholders in the sense that the 111411 has sold a share for less than it is worth. Of course, existing shareholders are aware of warrants outstanding and can anticipate this potential exercise. The problem is how to value the wrrant, and how to value the equity given the existence of warrants. This valuation problem does not arise with ordinal'y options, because they are traded by third parties and their exercise has no effect on the fin'n. To see how to value a warrant, suppose the firm has 11 shares outstanding, and that asst value the outstanding warrants are European, on nt shares, with stlike plice K. is A exercise the warrants, they pay K #er share and At expiration, if warrant-holders :4 ??l After the + ,?7#, shares. wvants are exercised, the firm has assets worth reeive worth exercised warrants are hence 11 A A + /71:7 K K = (16.17) rfhe
-t-
11
lll
-
-
ll
??l
The expression A/?2 is the value of a share of equity in the absence of warrants. I'hus, eQuation (16.17)suggests that we can value a warrant in two steps. First, we compute price, ignoling an option plice wit Ajn as the underlying asset and K as the strike/7/(?1 correction dilution factor, + ??/). This dilution. Second, we multiply te result by a exercise the changes second step accounts for the fact that warrant numer of sltar:s price of K. The watrant issued at a outstanding, with the new shares Black-scholes formula: can be valued by using the E
?? BSCall A K, -
?7+
,
??l
11
o- r, t, ,
.
(16.18)
$
513
srmscan
issue warrants embedded in bonds. A
convertible bond is a bond that, at the option of the bondholder, can be exchanged for shares in the issuing company. A simple convertible bond resembles the equity-linked notes we studied in Chapter 15, except that the bond is convertible into the company's own shares rather than the shares of a third party.vrfhe call option in the bond gives the bondholder the right to surrepder the bond's mpturity paypent, M, in exchapge for q sharesf The valuation of a convertible bond entails valuing both debt subject to default and a warrant. Suppose them are m bonds with maturity payment each of whch is copvertible into q shares. The value of the 'Iirm at time F is / w. If there are 11 origihal shares outstanding, then there will b 11 + nlq shrs if the bnd is converted. At exjiration, the bondholders Will convert if the value f jer share of th assets pfti- covrsion, Aw/(7? + nlql, exceeds the value per share of the matrity payment ihat bondholders would forgo: :
'
.
.
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.
.
.
.
.
.
,
Aw
M
-8-
/2
t?
??7tz
Or
11 n +
lnq
ztk
kslr11
?-
- q
71
lll t?
--
>:
(16.19)
/2
This expression is different from equation ( 16.12) for warrants, because rather tha injecting new cash into the 'Iirm when they convert, the bondholders instead avoid taking cash out of the 'lirm. Conversion occurs if the assets increase suciently in value. lf the assets decrease, the firm could default on the promised maturity payment. Assuming the convertible iq the only debt issue, bankruptcy occurs if assets are worth less the promised payment to all convertible holders, or Aw < I'tM. The payoff of the convertible at maturity. time F iS '
,
???#.- mllx (0,IIIM
-
-
-l-
p7
AN D WAR RANTS
Convertible Bonds In addition to issuing warrants directly,
27.07 -
D EBT,
Bond
'Written put
z4rl
+ nlq x
12 ??+
nlq
0,
x m:tx ,32:/
Aw
M -
pl
q
/1
+ nla ,?
(16.20)
Purchased warrants
Thus, owning ??7 convertibles can be valued as owning a risk-free bond with maturity paymept mM, selling a put pn the firm's assets, and buying nlq warrants with strike Mjq x n + ??2t?)/n. Equation (16.20)can be rewritten as
max min lllM, Awl
nlq
,
-1-
?2
lllq
Av
(16.21)
This version of the convertible payoff can be interpreted as follows: Shareholders give bondholders the least they can lrninlN, Aw/??2)); if it is optimal to do so, convertible
k
514
Co
Rpo RATE APPLI
EQU ITY,
cvlo xs
holders can then exchange this amount for the conversion value, which is their proportionate share of the assets lnqjfn + lnqj x Aw) .
$
AN D WARRANTS
D EBT,
515
Suppose a firm has assets of $10,000with a single debt issue consistingof six zero-coupon bonds, each with a maturity value of $1000and with F = 5 lisk-free = rate is r = 6% The yearsto maturity. The asset volat ility is o' 30% and the makes no payouts. 51-111 If the single debt issue is not convertible, the price is
Example
16.5
.
1* 11* )1 Example ''
Suppose a firm has issued ?n = 6 convertible bonds, each with 16.4 = mamrity value has ,7 400 M $1000and convertible into q = 50 shares. The 111-111 ttl ') common sh ares ou tstanding. Figure 16.3 shows the maturity payoff for the aggregate j 7,;) j',yvalue of the convertible bonds, comparing it with the matulity payoff of otherwise y', identical nonconvertible bonds issued by the same firm. The six bnds have a total ) promised mattu'ity value of $6000,so default occurs when assets re belw that level. Eqton (16.19)implies that conyeron occurs when assets exceed $1000x 700/50 = . I The slope of 4heconvertible payoff above $14,000is lnqln 3/7, + nlq) i itr .tijj . . because convertible investors share gains with existing )t) less than the slope . in default, . 1;) 't,')shareholders, but once in default, convertible bondholders bear additional losss alone ' )j (in default, shares are already worthless). %. 1j :
=
E
.
t14,000.
=
.
.
x $1000 .
d-0'06x5
i!!l..f .li..;.ltf )' (s'-li'':jq)f .--L'' ;'!!' ld t'r' 51:. :,ii;;' j't);f j(' )1* )' !i'1i:::;ki..' y' r' (' ).()' tf ..' !' y' jt'. (,111E!.;* )' 'Iii::-k.IIi'. j'j)' yyf tyf (' ki.'IEiit;C'. --':EiE!!!i-'..' ''ti!Ii'. '''jkd likEili:.f IEIt:'ili::'. 111:-* y' '1* ' (E E E' !E'' E E E' : t' !i '' i1: (r5r( ':..plE;ljq..(j'('E.E'.E(.(.'(..(.'E..(......k..'r(i./' g.iq.E.(j.. ..'(. .'...! .(('.('..' .yqq)''1:S'j(.(''':..'g'.(q.(g.yri'.('!' .(!( E. E; (!!jtE . j. E! E 'E E iI..(;p: E .q ;(. .EE ; E (.. j.r.E..... .q '.. E.( ; qE.. r. E ( E , . qq( . !i' @ . .tj. r.. (..j y..!. !...-';y j) ; ! . jy jy jj jj).yy..jEqyj.. y i!--'1IEI:Fp:. ;q(.. y..y.y. j -. j ..-y.r.j.q.....y.yy;.yy.-y.y E. . . .-. . . t-r:..-. .)......E.yy ..; ....'...;...--... E'..-;,...i r :(. . ..-. ..'... . .. . . . . . . . ;')' :j' -j' -1:::11.* .--)'' :r' E' j';'!' d.:ii!,f :'-'-k:;'. '.' j'-'''-.'F' !' l':d f' 'E' (' )'
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18,000
value of an aggregate bond and a ordinarybond, using
convertible
.
..
6 x $1000 x + 6 x 50 x
e-0'06x5
-
Bsput
400 zo()
-y.
y x 50
14,000
z
12,000
conversion (Slope
=
$1000,0.30, 0.06, 5, 0
6 x $1000,0.30, 0.06, 5, 0) ($10,000,
x Bscall
$10,000 $1000400 + 6 x 50 0.30, 0.06, 5, 0) zo() zo(j
(
,
=
3/7)
'
8,000
!
4,000 2,000 0
!
Default
(slope 5,000
=
1)
Assets
514,000
=
!1
10,000 15,000 20,000 25,000 30,000 35,000 40,000 Asset value at maturity ($) .
#
,
5:
$1000 x (400+ 6 x 50)/50
=
$14,000
Convertible bonds are typically issued at terms such that a signitieant increase in the stck price is required for conversion to be worthwhile. In Example 16.5, each bond gives tlle holderthe right to convert into 50 shares, so the jtrike price is $1000/50 $20. the ratio of te strike price to the stock price, which is Since the stock price is $11.809, 69.4%. In practice, conversion called the conversion prenzinm, is $20/$11.809 l premiums are most commonly between 20% and 40%. Why do hrms issue convertible bonds? One explanation is thit convertible bonds resolve one of the conoicts between equity- and debt-holders. Shareholders can tale value from holders of ordinary bonds by increasing volatility, even if this action has as a whole. However, convertno benefcial effect from the perspective of the 111311 ibles are harmed less than ordinary debt by an increase in volatility and may even be helped. Financing with convertibles instead of ordinary debt thus reduces the incentive of shareholders to raise volatility. In practice, valuation of convertible bonds is more complicated than in this example. First, convertible bonds are typically American options, convertible for much of the life of the bond. Second, convertible bonds typically are not zero-coupon. The p
=
-
6,000
$701.27
.
# z
=
= Each bond has a price of $879.39. The yield on a bond is 1n($1000/$879.39)/5 bond because otherwise equivalent nonconvertible is below the yield on an 0.0257. This of the conversion option: The bondholders have a call option for which they pay by t:: accepting a loweryield on the debt. The value of a share is ($10,000-$5276.35)/400 $11.809. Bondholders will convert if at mattll'it.y the assets are worth more than
M x n + mqllq
Ordinary bond Convertible equity
'
16 #ooo
the parameters in Example (16.5).
..
6
'
,
= 0.0710, greater than the The yield on the nonconvertible bond is ln(1000/701.27)/5 lisk-free rate because of the risk of bankruptcy. Suppose the debtisjue is instead the convertible bond described in Example (16.4). Using equation (16.20),the value of all convertible bonds is
= $5276.35 Just as we valued ordinary zero-coupon bonds with the Black-scholes formula, we can also use the Black-scholes formula to value a bond convertible at mattlrity.
$10,000
ssput
-
=
516
%.Co RPORATE
Appt.l CATI olqs
EqulTv,
payment of coupons complicates the analysis because banltruptcy becomes possible at times other than expiration and the reduction in assets stemming from the payment of the coupon (or any other cash payment to the 5rm s security-holdrs). This coujori feduces assets but is also paid to the bondholders. Third, many companies pay ciividnds. If the dividends that can be earned by converting the bond into stock exceed the bnd coupon, there can be a reason for bondholders to convert before maturity. Finally, intefest rates and volatility can change, and the circumstances of default may be more conzplicated than we have assumed (for example, when there are other debt issues, default could occur prior to maturity). The value of the bond can then change for reasons other than stock plice changes. It is possible to value convertible bonds incorporating early exercise, dividends, It is also possible to incorporate interest rate risk using techniques and callabiliys described later in the book. ''
'''
''
'
'
*
Callable Bonds Many bonds are callable, meaning that, prior to maturity, the company has the right to give bondholders a predetermined payment in exchange for the bonds. The idea underlying a callable bond is that the bond issuer can buy the bond back at a rlatively cheap price if itbecomes expnsive. Abond can become expensive becguse interest rates have fallen. in which case the issuer would like to xchange the existing bondj for new bonds carrying a lower coupon rate. A bond can also become expensive if the market percives that the company is-less likely to default. Again, in this case the company would like to exchange the existing bond for a newly issued bond with a lower default premium. A tirm can always buy bonds back at a market price, but there is generally no advantage to doing so. Callability permits the company to buy bonds back at a relatively low price. Of course the company pays for this light by receiving a lower price when it issues the bond. The predetermined price at which a company can call a bond is specifed by a call schedule. Bonds are typically noncallable for several years after issujmce, a period which the bond is said to have call protection. Callability is, in effect, an option ' that investors sell to the bond issuer. In return, the company issuing the bonds pays a higher yield on the bonds. 'duling
Convertible bonds are typically callable.
When the issuer calls a convertible
bond, the holder of the bond has the choice of surrendering the bond in exchange for the call price or converting. Callability is a way for issuers to shorten the life of the bond, potentially forcing holders to convert prior to maturity. The issuer--acting in the interests of shamholders-follows a call strategy that ntillintizes the value of the bond. Bondholders, by contrast,
bond.
follow a conversion strategy that nlaxinlizes the value of the
DEBT,
AXD WARRAXTS
Since bonds can be called prior to expiration, we cannot value the call provision
using Black-scholes. However, we can value a callable bond binomially. The call strategy for an issuer is like the exercise of an American option: The issuer calls if it is more valuable to do so than to leave the bond outstanding for another period. Figure 16.4 presents binomial valuation examples for four bonds: an ordinary bond, a convertible bond, a callable bond, and a callable convertible bond. We examine noncallable bonds as benchmarks in order to better understand the effect of callability. We value these bonds using the binomial pficing model outlined in Chapters 10 and 11. In each case, the assets of the firm are the underlying asset, pnd the value of the bonds is determined by the value of the assets. To perform the valuation, we move backward through the tree. as in Chapter 10. The assumptions in Figure 16.4 are the same as those in Example 16.5, so the results in the example and figure are comparable for the ordinary and convertible bonds. bonds We first consider the binomial valuation of an ordicallable nonconvertible bond (Panel B noncallable Figure 16.4) and its callable countelpart (Panel D in in nary, Figure 16.4). The yield on the callable bond (7.57%)exceeds that on the noncallable bond (7.29%)because of the option that bondholders have written to the f-m. How are bond prices in Figure 16.4 computed? For the ordina:y bond, the value in year 5 is Bs
Il'tintl/l
=
A'f, A5)
=
min($6000, A5)
There are six $1000bonds outstanding, and shareholders will pay to these bondholder's the value of the fil'm or $6000,whichever is less. Note that default occurs at the bottom node in year 5 since the value of assets is less than the required bond payment, $6000. Prior to mamlity, the value at each node is calculated as in a typical binomial valuation,
with
-r/;
p
?
=
e
f+?, Eps+
..y
(j
-
#)y-f+?, g
For a given node at time /, Bt*+ is the bond value if assets move up from that node, anct Bt-+iti's the bond value if assets move down from that node. The binomial valuation of the callable bond is straightforward. Let A7all de'note the call schedule. The call plice in this example is set to equal te price of a bond yielding 6.75%. At time 1.67, for example, the company could call by paying the bondholders $4791.10, which, for a bond maturing in 3.33 years, is a yield of 6.75%. We assume that rate is fixed; tus the only reason for the company to call the bond is if the te lisk-free company could issue replacement bonds at a lower coupon due to a decrease in default
risk.
Shareholders wish to minimize the value of the bonds, hence their value is Bt
=
min ( Leave the bond outstanding,
call j
-r/'t
'
(1977)and Brennan and Schsva.rlz(1977)discuss
the valuation of convertible bonds.
A'callj
p)B- J+/, ) t :+1, + (1 P B+ (j6 p,2) = min 11'you compare the binomial trees in Panels B and D, it is apparent why the callable bond has a lower price at issue than the ordinary bond. At the top node in year 1.67, -
Slngersoll
%.
'
k
518
Co
Rpo RATE AI7PLI CATI
y' t,' Ij.q))' )' 7* f' l'rr'!q'!7r77!'r7rq' 7'7!* :777* (' E l'lf l'1* 1* )' y' tttd ltlkl'-; t'(il':sjik ('. ;'7* ()' @1(75144*)):7* E''' ! j'jkjjjjj'. r' (' '''F' Cilllr.q; i!.'(''q( EE' ' 1. ; !!( E E'EE' !E i EEEEiE' EE7:;i:: (E'' ( E''' 'EE''EE'' ' E (k ' (!t.(tl. . (14': . ti .' ... . :(.;'(. ii.i .'ii:. ; .E . ..'...' .i . ; :.. i 2 qi ( (E; -'E E !! (t. ... . g jE! ... p...i.qy (y :r q j jyy.iy y j.!. y . j y y y ... j j y . . . .j.t!y. .jjjjjj;jjjjy . y E. . '. y l i . !! i 'll! ''lErilr ;. gjgjjy ; . . i(..... .; E.. .! jj. .y ( .. i . . . . .
yjjjjjkgjjjj'. j'j' jjiygjyy'jd t'r)' (' jjy'. jjtd j)f yy'r(' y' (j'.' (tj' ;'y'. yy'yy yfd t'r''f
.'''.
,.-.' .:' -'j'r'E)' --' '3* ?' y' '7* T' )' T' L' ,'j('.' ;r' E'
r yjjkjjjjjj(jj;' '( 111111::1111.*.* )( 'jjj. )yj ( r'''(j, y' ; (gg,., j.jjy. .y. ((.... jjjjjjjj(,. jjjrjyrgjjg;. ;yj. jyjy tjt.jj yj j j ty tjty. r . . ryj.. y (jkjgjjjjj,: . iiiii. j . j .. . ' ' i .. . ....j. j. '. . .. . .. . Panel
$
EQ.u lTy, D EBT, AN D WARRANTS
olqs
519
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:
Year:
A: Firm assets
0 10000.00
1.67 16279.12
7502.88
3.33
5 43141.27 19883.36
26500.98 12214.03
5629.32
9164.04 4223.61
the noncallable bonds are worth $4912.38,for a yield of 6%. (1fassets reach that node, default will not occun) The ;rm calls the callable bond at that node since it is now possible to issue default-free debt. The prospect of this call prevents the bondholders from receiving a capital gain. This in turn lowers the initial plice of the bond. Problem 16.15 asks you to compute share prices at each node so that you can see the effect on shareholders of the different bonds. Callable
4166.82
4912.38 4396.40
5429.02 5429.02 4471.64
6000.00 6000.00 6000.00 4223.61
N/A
4791.10
5361.58
6000.00
D: Callabl bond $684.86 Price Yield = 7.5t%
4109.14
4791.10 4371.73
5361.58 5361.58 +471.64
6000.00 6000.00 6000.00 4223.61
E: Convertible bond
5324.34
B: Ordinary bond Plice = $694.47 Yield = 7.29%
C: Call schdule
=
Price Yietd
=
$887.39
=
2.39%
11357.56
8521:44 6000.00 4223.61
4471.64
F: Callable convertible Price = $818.14 Yield = 4.01%
bond
4908.85
6976.77 4371.73
Binomal valuation of a callable nonconvrtible
11357.56 5361.58 4471.64
18489.12 8521.44 6000.00 4223.61
and a callable convertible bond. The
assumptions are the sam as thos in Example 16.5. The binomial tree for assets in d Parlel A is generated using equaton (10.1 0) (a forward tree) with u 1 0.7503, p* 0.4044, T= 5, and three binomil time steps (hencethe time between 1 In each case there are 6 bonds outstandinj with a binomial periods is h 5/3 total maturity value of $6004. Convertible bonds conveft inyo 50 shares. The yleld for each bond is coniputed aj In(600/&)/5, where Bois the time 0 value of the six bonds. The price is Bo16. The call lcheule in Panel C is the price of a zero-oupon bond maturing in year 5 and yielding 6.75%. Callable bonds are call-protected until year Prices in italics denote calls of the bond; prices in bold deriote conversions, and 1 prices in bold lt'tzllcx denote conversions in response to a call. .6279,
=
=
=
lt + nlq
A5)
In Panel E in Figure 16.3, bondholders convert at the top two nodes in year 5, receive the mamrity payment in cash at the next node, and the firm defaults at the bottom node. Prior to maturity, the convertible investor values the bond as the gfater value of letting the bond live one more period:
of its
conversion value and the =
18489.12
6351.59
lltq
maxgmintplf, A5),
B: 7578..78 4733.96
We now consider noncallable and callable convertible
bonds
convertible
bonds, Panels E and F in Figure 16.4. Note 'Iirst thar, as in Example 16.5, the yield on the convertible noncallable bond (2.39%)is lower than that on the ordinal'y bond (7.29%) because convertible bondholders receive a call option and pay for this wlth a lower yield. Using equation (16.21),the year 5 value for the convertible bond is
=
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Bt is the total value of the bonds. You may recognize this expression as almost identical to equation (10.10)for valuing an American call option on a stock. The difference is that the payoff is the conversion value instead of St K. -
When the bond is both convertible and callable, there is a tug-of-war between
the lirm and the bond investors. We can imagine the bond value being determined as follows: The bondholders decide whether to hold or convert (maximizingthe bon valuel.. Given this decision, the ;rm decides whetherto call (minimizingthe bond value).
If the 'Iirm calls, bondholders revisit their decision about whether or not to convert (again maximizing the bond value, conditional upon the behavior of the srm).This chain of reasoning implies the following valuation equation:
.671.
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% CORPORATE
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APPLICATIONS
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why this happens. In year 1.67 at the top node, it is optimal to wait to convert if the bond is noncallable. This gives a bond value in Panel E of $7578.78.However. if the bond is callable using the eall schedule in Panel C, the fil'mwill call at the top node in year 1.67. In response, the bondholders convert, giving them 50 shm'es worth $6976.66.The bond is worth less because shareholders cannot delay the conversion. The fil'm does not call at the lower node in year 1.67 because the credit quality of the bond deteriorates at that node.
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An alternative approach, often used in practice, is to base valuation of aconyertible bond on a binomial tree for the stock, rather than on assets. A standaydbinolial tree converttle bond for the siok, hwever, will never reach a zero stock price. and thus a This raises the default-free. question: How valued on this tree will be prlced as if it were j risk plicing procedure into the can we incorporate bond default Tsiveriotis and Fernandes (1998)suggest valuing separately the bond income and thestockincomefrom an optimally managedconvertible bond. Theirprocedure accounts for default by discoting bbnd income at a rate greater than the risk-free rate while the component of the bond income related to conversion into stock is discounted at the .
It is possible to issue bonds that are considerably more complex than those we have considered. Conversion and call schedules can vary with time in complicated ways. Bonds can be pltttable, meaning that the investor can sell them back to the 51711at a predetermined price. Bonds can pay colttillgellt interest, meaning that if a particular event occurs, the interest rate on the bond changes. As we saw in Chapter 15, particular structures are often a responje to tax and accounting considerations. Another example is the use of contillgelt ctulvcrrf/p/d bonds, alSO known as Firms report earnings on a per-share basis. For a til'mthat has issued only shares and ordinary debt, computing earnings per share (EPS) is straightforward since there is outstanding. However, if a :rm has issued no ambiguity about the number of shares should the 5rm use in computing EPS? shares eonvertible debt or warrants, how many the firm must compute the worstin the United States, Under linancial accounting rules issued a convertible seculity, the firm has When earnings share. per case fully diluted interest after-tax earnings back adding on the converible, and to this generally means outstanding. convertible shares shares to adding the number of ''
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6tlqew Bond Rules to Dent GM Earnings,'' by James Mackintosh and Jenny Wiggins, Fillancial Tible., London ed-, July 22, 2004. p. 26. 'F&GM acts on New Co-co Bond Rules'' by Jeremy Grant, Fillallcial 7'i'??lt?.tLondon ed-, August 6, 2004, p. 24.
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The box on page 521 discusses the reaction of one company to the FASB'S ew rule, and makes it clearthat the accounting treatment of a seculity is important for issuers.
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When shares are used to pay employees (as for example with compensation options), there is an increase in the number of shares outstanding. Companies making heavy use of share compensation frequently buy shares back from other shareholders (a share rcpl/?'c//t7lclso there is no netlncrease in the number of shares outstandinps Many companies that repurchased shares during the 1990s also sold put options on their own stock; a commonly tated rationale for issuing such put warrants (see, for example, Thatcher et al., 1994) is that the put sales are a hedge ajainsi tfi cost of repurclasing shares. lntel, Microsoft, and Dell, forexample, all sold signifcant numbers of puts, with Microsoft alone earning well over $ 1 billion in put premiups duripg the 1990s. Here is a quote from Microsoft's 1999 l0-K desclibing the put prtgram: Microsoft enhances its repurchase program by selling put warrants. On June 30, 1999, l63 million warrants were outstanding with strike prices rangingfromssg to $65 pershare. Theputwarrants expire between September 1999 and March 7002. The outstanding put warrants permit a net-share settlement atthe Company's option and do not result in aputwarrant liability on the balance sheet. .
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16.2 COMPENSATION
OPTIONS
Many firms compensate executives and other employees with call options on the company's own stock. The use of such compelsatiol options is common and signifcant in many companies, but has declined since 2002.
524
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Microsoft, for example, estimated in its 1999 armual report (10-K) that its grants that year of 78 million options were worth about $1.6 billion. This is approximately $52,000 per employee (Microsoft had 31,000 employees). Elsewhere in its 10-K, Microsoft reported that total outstanding options on 766 million shares (againsi5 billion shares outstanding) had a market value on June 30, 1999, of $69 billion, or $2 million
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Many other companies made signiscant use of compensation options. Eberhart (2005) found in a sample of 1800 sl'msusing compensation options in 1999 that options were on average l2% of shares outstanding. Moreover, the use o options has not been restricted to executives: Core and Guay (2001)found in a sample of 750 companies that two-thirds of option grants were to nonexecutive employees. The use of compensation options has declined in recent years. Several developments seem to be responsible for this change. First, the market decline in 2000 lef4 many employees with deep out-of-the-money options and created morale problems for companies heavily dependent upon options. The box on page 526 discusses Microsoft's to its overhang of out-of-the-money options. . response Second, both te Financial Accounting Standards Board (FASB) in the United States and the lnternational Accounting Standards Board IIASBIarmounced that tey would require companies to recognize employee option grants as a compensation expense. Throughout the 1990s most companies had treated compensation options as worthless when computing enrnings.g An attempt by the FASB to require option expensing in the early 1990s was defeated by companies opposed to expepsing. Mapy of these companies were concerned about the decline in reported enrnings that wou td r esult from expensing. The logic behind requiring companies to expense option grants is straightforward. If a company pays cash to an employee, the company deducts thepayment as an expense. If an otherwise identical company replaces some of the cash with an option grant, that ltnless the value of the company will report systematically higher earaings than the first also deducted is option grant as an expense. To tlzink about expensing, you can perform the following tought experiment: lmagine that a firm that issues options to employees were to hedge its obligations by buying a contract from a third party. Under the terms of this contract, the fhird pal-ty would pay the firm te value of the option when the employee exercises it. The f11711 could grant an option and buy the hedging contract, thereby insulating shareholders from the effect of te grant. The cost of this hedging contract is the cost to the firm of the
option grant. If a firm does not buy such a contract, it self-insures, meaning that shareholders the cost of the option grant. Either way, the cost of the insurance is the cost bear shareholders of the compensation option. The problem of valuing a compensation to option amounts to asking how much it would cost the tirm to hedge its compensation commitments.
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ln December 2004, the FASB issued Statement of Financial Accounting Standards (SFAS) 123R, which contained the final rules, effective in June 2005, for companies to follow in expensing options. The statement requires apublic entity to measure the costof employee services received in exchange for an award of equity instruments based on the grant-date fair value of the award (withlimited exceptions'). That cost will be recognized over the period during which an employee is required to provide service in exchange for the award-the requisite service period (usuallythe vesting Periodl.lo A company might grant options that vest after three years and expire after seven years. Under SFAS 123R, the company could value these options using the Black-scholes or the binomial formula, and then expense 1/3 of their value over each year of th vesting
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ose Valuation? Compensation options cannot be traded. An employee who cannot sell options will typically discount their value. As a result, you can expect that 'lirms and employees will value compensation options difkrently. Such a difference in valuation can occur for any compensation other than immediate cash.
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(Optionqvalue was estimated using an expected life of 5 years, no dividends, volatility of 0.32 in 1999 and 1998 apd 0.30 in 1997, and risk-free interest rates of 6.5%, 5.7%, and 4.9% in 1997, 1998, and 1999. Microsoft does not document how it chose volatilities, but these are close to hisyorical volatilities in each year. Using weelly data, historical volatilities for Microsoft for July to June were 32% for 1916-1997, 30% for 1997-1998, and 39% for 1998-1999. Why did Microsoft use a s-yearexpiration to value qptions expiling in 7 years? t ' We lenrned m Chapter 9 that it is never optimtl to eady-exercise a publicly traded call option on a nondividend-paying stock since it can be sold for more than its intrinsic value. However, compensation options cannot be sold. Thus, the value of the options to #?c holder may be less than intrinsic value. In this case, employees may exerciie th: options before expiration.lz This is in fact how employees behave: In ppctice, executives frequently exercise a large frction of their in-the-money options aj soon as they 13 vest. In addition to exercise by continuing employees, options are often caceled due to death, termination, or retirement of the employee. Taxes can also potentially affect te exercise decision, although a common tax-motivqjed argument for early exercise is incorrect.l4 Finally, compensation options canpotbe exercised until they vest. Arealistic f
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J'ylfrns/ i q whlii proclaimed: gplden age (?f syock optiphj g iu E y t y (Th buwtiji'fti kkntjl t g isoven'' (see '-Microsoft ushersoutThe kr: fri) tit. ilikizj ed t of Optionsr'' by Robert A. Guth an Joarm Sk tq pigt.rities wey ) y,'Ey Cyi Lublin, J;lF# Street JbII?77f'If, Juty 14, 2003.): tr tl+lt,' #ejtlt Aj. yqtt iplytjld rcentty dq Wji Microsoft stock had fallen and many yy V. ?.pj q nj uj g y y y ojjs j yjy s st yy y x s y y jy y o . Ln, employees had out-of-the-mone# options. signiicaptty less s tnap et i g : jjj r jkjj jujkitlnt tg Elzlployees expressed what CEO Stev fpr ( 1ptitjik X ( yyC(yy Ej hjjE !:qi.(jta 1@89 1:::11'..t LEdir (I. IE! <& 151t: 4::), 1t;1lcr?L 1('1::1. d:,r 1(-4:::1!t1l. IEJlr)k dpE) ltklk l fzr1:t1L ;EIL.II. (Iii!k 11 f.4::0/.. ih y y.y. t ..y(...y flii!: .q 2414; .; ... N.. . l::l s 1L::!lt1@:)/ (,.wtij gitld ..... . y )..4r#)(i,.. y, .. . !:7:1:!1711.. (t,kJhrit)I. ,:1... tE. ILi .1!j; qt:ri EI!'1:tjk.. .jI);p..!t. ,kjIL. ik.. 4::: 4:2, '::), 4::,1. IETIL d::rd:zr Jlk 4: ':,: llqqti'd:gr .... 1t:. .. 1ty. 15;... lpt. i@:i.. .i::ii. 1t) ' A.. 17I#. 1. IECI. .i:2:... .',#IL'. i!jr..).t.. d:jlL 1k71t :zr d:E)p jl;;p k::i'. .........., '.l(rik. '.'. 1i.. ii ... i. .. ..... i... (. '.'.. 4p. .... .. .,.. .. ....)........ t.. t .t . 5 . !((:,MiCr0SOft'S CO Waj r.i ltito' IEE ... . narj 1ft. Uftlf it ..ol. i'j jtrtlg lyr ygq tllotld t ayrtlcl tlt/dlt-hdmGetlle k k ( .;k,yt k. yu o y z r oy k ( ;.o y Optlo to q pgjltonjt 2)llgq as saying that employees culd no lpnerq vkf otktt expet fpi-, tlift hu/ /izu to becorne wealthy frorf stok oljtipp: tlj y E'j y y r ( (11#r::1L1 g j g'y E' y j y E y (ilqr':lt. .- Ey. ( ' '. .y( ''.. '.i .q'. 'r ' 'y jii). ( ' '. ' ( . q. yg j. y ;yy .'g.i ..j' ..q..';... ( y..'.'.j'...'.(. gy .. y.y ) (.y y..j y qj.y..y(.. .yyqy.gy.j ..y. .j. '.'.. .. yjgy,rjj;s . . :yjjyjj,s y . . . . . y y y y . . ; . . . . .. y ( rjjj, . . j '. jjjjyys .jg..yjjr, . . . (jjj,.rjj.s y.. g . . . .. ... . y.... . y.. . .. .... .. . .. . . tjjjy.jjj, . . . g y.... . . ... tjjjjjy:jj . j.j jy...jg ...g.j.. . .. y ..jyjgj . jjgggggjjy .kjj.s .jjj,... .. ', inctwft ! ... ,. . gjyjy,;y(,. yyjjjyg;jy ... j ... y.. y.j. . yjj uv ..(.....y.j...yy..y . ( .y q . .(..j.r...jy.jyj.. .......... then , bitit . tj hitlt itsfltitq Ftlr' O ( Vlijict E q Chlf it 's not, said Micfsoft y ki) jjj j ' E bilfkf fd. 1% jtj; k'lht' yjuj otjjj,j E r t cialOfcer John Connors. I rnt .E E tlgh /f.ttis btWyfi 20.01 d 2003,(j,q y E ) he descfib tj(Ithe p fiifktt intelwiew (ttsycik bdkt( AWIX ibtk ' ' t-t.':'jjk kyyjyyg ( n sz 12:11 - - 422: 1:211 1::1/ 42)I- EEI. 12214 1)/qErlL 1:. 1i1L 12:)8 1(r1CT1. IEt 15 IE1 jlglp !7'' 't 1(q1(qI. IIEIL 1EI. 1:. lElz '.. ' j.. jkr. .' .. .. ' ,, 1(71L 'i!, . . y.(..j..(.y...y. .i:;i..ki 1(:2.. i, E.. d@2f. 4tEI.. . ..,... jt:)l (gr'.:j,k IEr)g:I. )(r!:.,,... i.... i... irq j. jrir4..tk:)ir ...... ..t. .ii. ... ....q...y 1(11/:1:. i.ir?.,.(IE!:ik. i,.ttt... !:4 t.. t.. t tidl .. ... .f !j:j, jky t. . .
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Accounting standards require that companies deduct the cost to rc contpal,y. The is goal to measure cost to nonemployee shareholders, not value to employees. For example, suppose a company grants employees nontradable membership in a golf club costing $15,000per year. No one would value the membership at more than $15,000, because it can be purchased for that amount. However, someone who does not play golf might value the membership at zero. Thus, while for one executive the membership rnight displace $15,000of salary, for another, it might not displace any sally However, in either case, shareholders bear the $15,000cost. Thefact r//t71the c??lp/t/lc discottltts the /,/c??7/JcrJ/2f/J t vallte #t/?. llot l'edltce r/lc cost to #1d/?w?. For sharehollers, the key question is the cost to the company: How should shareholders value option grants, given the behavior of employees.
1lsince dividends reduce tlze value of an option, it is possible that widespread use pf compensation opuons has resulted in a reduction in corporate dividends. 12SeeKulatilaka and Marcus (1994)for a discussion of the employee's valuation of options. l3Hkddart (1998)shows that options are disproportionately exercised on the lirst through foulh anniversaries of the grant, in blocks of 25% of the grant. Since it is common for grants to vest 25% annually, this linding sgggests tat many options m'e being exercised as soon as possible. 14Anemployee is taxed at ordinary income rates on the exercise of a nonqualified option, wth subsequent gains on the stock being taxed at capital gains rate. Some have argued that employees optmistic about the share price should exercise compensation options early in order to mximize the percentage of income taxed at the favorable capital gains I'ate. This argument is incorrect. However. if the ordinary income rate is expected to increase, early exercise can be optimal. For a discussion see McDonald (2003).
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An Alternative Approach to Expensing Option Grants There are alternative ways to account for option expense. For example, companies could fully deductthe market value of the option at exercise; this is the approach taken in taxing option grants. A deduction at exercise gives an expense with the correct present value. However, by waiting to deduct the expense, the srm'sreported income in earlier years will not rellect the economic value of compensation in those yem's. Another alternatfve is suggested by Bulow and Shoven (2004). They propose option expense calculation that records an expense for each period that the option an and remains unexercised. Their approach neatly sidesteps many valuation diculties provides a correct present value of option deductions. Moreover, the method provides insight into why a long-term option grant is costly. We will illustrate the proposal in the specifc context of compensation options, but the methodology is generally pplicable to long-term contracts and contingent liabilities.
There are several points to make about this example. First, the full deductions with a present value of $28.15are reported by te company only if the option is held unexercised until year 3. Thus, the proposal solves the problem of early exercise and the employee leaving the company. Second, the act'ual deductions are substantially less than the muimum possible value of the option at exercise. Third, expenses behave in what might appear to be a counterinmitive fashion, with reported option expense g'reatest when the option is at-the-money. In year 2, for example, expense is greatest when the stock price is $100. reason is that the value of delaying exercise on a primarily option is deep-in-the-money the value of delaying the payment of te strike in Panel D, the price. $4.88is one-year present value of interest on $100. When 'l'he
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posal works for a 3-year option that vests irnmediately, and for which recipients can exercise the option in year 1, 2, or 3, or one year after resignation. The binomal value of the three-year loo-strike option is $28.15;you can verif'y this by applying the binomial plicing method to the stock price tree in Panel A.
In this expression, each option prie is weighted by the fraction of employees who historicily exercised at that time. A problem with this approach is that it does not recognize that employee behavior depends on the stock plice. If the option is deeply in early years, fewer employees are likely to resign before the options the money in .vest. If the option is out of the money in year 3, all employees who do not resign will wait before deciding whether to exercise, which lengthens option maturity. Thus, the assumptions about exercise behavior will generally be incorrect. Bodie et a1. (2003) point out that for these reasons, equation (16.25)will undervalue the optio. A binomial model or Monte Carlo valuation (whichwe will discuss in Chapter 19) permits a more flexible and realistic treatment of early exercise.
l5If a company alters its assumption about the exercise behavior of employees, the estimated value of newly issued options will change. Cisco, for example, changed its assumed option life from 5.6 to 3.3 years, reducing the estimated value of its option grants by 23%, from $1.3 to $ l billion. See t'Cisco May Profit on New Option Assumptions,'' by Scott Thunn, 1..Pt7#Street Joltlnal, December 7, 2004, P. C1
529
Bulow and Shoven make the observation that most option contracts are exercisable after a vesting period, and that employees or heirs are typically required to exercise their options within 90 days of resignation or death. Thus, in practice, a lo-yea.r option can be vieFed as a renewable 90-day option, extendible (by the employee's continuing to work) for 39 additional 90-day peliods. Bulow and Shoven propose taking the value of this extension as the deduction in each quartenl6 The value of the extension is the market value of a 90-day option. less intlinsic va'lue. The present value of expected option expense computed in this fashion is the fair market value for an option with the same time to expiration.
valuation would account for the likelihood of these various factors. The asstlmed s-yea.r life is intended to account for the expected life of an option.ls lt is possible to modify both the Black-scholes and binomial models to account for complications due to early exercise. For example, suppose that 4-yea.r options vest after 3 years. The company examines historical data and estimates that 5% of outstanding options will be forfeited each year during the vesting period. Furthermore, the company believes that half of the remaining options will be exercised at vesting, with the other half exercised at expiration. One could then value the option grant as being jliially a option: 3-year option and partially a zyear (1
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Reducing the exercise price of compensation options in response to a decline in price is called option repricing. The delta of a deep-out-ofz-the-money option stock the low, is so that subsequent stock price changes will not have much effect on the value of employee options. Companies in this case often feprice options.l? lfyou expect that options will be repriced if the price falls, how valuable is the option grant in the srstplace? We can answer this question using barrier options, discussed in Chapter 14. An option that is going to be repliced if the stock plice reaches a certain level can be modeled as alnock-out option ttheoriginally granted option vanishes), plus a knock-in option (a new option replaces it) with the same barrier. Specifcatly, suppose that the option strike is K, and that at the barrier, H, a new at-the-money option will be issued in place of the original option. A repriceable option is then worth Calmownoutts,
K,
tr, r,
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The second tel'm reiects the lnock-in call being at-the-money when it lnocks in.
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ln December 1997, the Company reduced the exercise price of approximately 20% of the outstanding common stock options held by the Company's employees to the fair market value per share as of the dte of the reduction in plic. The Company repriced these employee stock ptions in an effort to retain employees at a time When a significant percentage of employee tock options had exercise plices that were aboke fair market value. The Company believs that stock options are a valuable tool in compensating and rtaining employees. Executike oficers and directors were excluded from this replicing.
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Figure 16.7 shows the bilzomial valuation of an ordinary option and a reload option with a single reload. The binomial price for an option without a reload provision is price for this option is $36.76.) The reload price, by $38.28. (The Black and optimally occurs in the second binomial period. Let's reload is a contrast, $42.25, this works. how examine First, consider the valuation without a relad. When S $179.37in period l while value value left alive is exercised is $79.37.As we of the option the $94.153, the dividends, the option exercised early; since there is the value in would expect not are no 0 is $38.28. period When a reload is permitted, the one candidate node for a reload is when S $179.37. (A reload would have no value at S = $100 or in the snalperiod-) If a reload occurs, the option-holder receives $79.37 for exercising the option, and 100/179.37 options are issued with a strike plice of $ 179.37 and 2 years to maturity. Thus, we calculate the value of the option at this node as
0.4, 1' 0.06, t l0, J 0.01, and that Example 16.6 Suppose S $100,c options will be repriced if the stock prie hits $60. The value of an option that will llot be repriced is =
=
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$60 is
Ca1lDownOut($100, $100,0.4, 0.06, l0, 0.01, 60) + CallDown1n($100, $60, 0.4, 0.06, l0, 0.01, 60)
=
$41.1l + $20.30 $61.41 =
Thus, the possibility of repricing increases the value of the option by 13%.
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=
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Reload Options
+ ().605 x $o) $1:6.09 100e (16.2.7) (() a95x sl42.a6 $79.37179.37 From Figure l 6.7, the value of the reload option is $42.25, 10.5% greater than in the
A reload option gives the option-holder new call options when existing call options are exercised. The idea is that the option-holder uses shgres tp pay for exercise, andr new at-thermoney options ftre granted for each share given up in yhis fashipn. This type of optipn is best explained with an example. Assume that a lo-year option grant for 1000 shares with a strike price of $100 permits a single reload. Suppose the employee exerises the pption when the stock plice is $250, with 4 years of option life remaining. The exercise price requires a payment of $100 x 1000 $ 100,000. This amount can be = 400 shares. in cash by surrendering Ap exeutive paying the paid $100,000/$250 or at-th-money 400 price by surrendering shares receives options with 4 years jtrike new
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absence of the reload. In general, we can compute the value of the reload at every node by solving another binomial pricing problem valuing the appropriate number of newly issued options. The
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Arnason and Jagannathan (1994)pointed out that there m'e two important characteristics of reload options. First, the reload feattlre is valuable: A reload option can be worth 30% more than an otherwise equivalent option without the reload feature. Second, reload options cannot be valued using the Black-scholes formula because reload options may be early-exercised. However, they can be valued using the binomial option pricing
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Reload options migt seem esoteric, but Saly et al. (1999)show that 1135 reload options were granted, in 1997, out of a total of 9673 grants reported in the S&P Execucomp database. SFAS 123R accounts forreloads by ignoring the extra value of the reload feature when the option is grante yd and accounting for the additional expense when the option is exercised and reloaded. 1 This treatment is in the spirit of the Bulow-shoven expensing proposal, discussed above. Reload options can be valued binomially. This is accomplished by replacing the exercise amount at the time of exercise, S- A', with the value o t a new re load option. We will illustrate tlzis in the simplest possible fashion, with a two-peliod binomial example.
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of ordinary option (binomialtree on the Ieftl and reload option $100, #= $100, c 0.3, (binomial tree on the right). The calculations assume that and that there is a single reload. Stock prices and option J h 2, 0.08, 0. T= 4, r prices are shown at each node, with the reload value in italics. A reload occurs at the d 0.768, and p 0.395. boxed stock price. In this example, we have u 1 Binomial valuation
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1SSalyet aI. (1999)suggest that reload options may be a way to give management undisclosed compensation.
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option-holder reloads if doing so is more valuable than not doing so,just like the exercise decision for an American option.ig
S&P 500 index is at 1950. A a payoff of
$185
Level 3 Communications Level 3 Communications was one of the first companies to deduct the cost of compensation options in computing enrnings. However, Level 3 also granted unusually complex and valuable options and did not take this copplexity arld exea value into account when expensing. In a June 1998 proxy statement, Level 3 described its outperform stock options (OSO), granted to employees. This is how they are described ip the proxy: Participants in the OSO Program do not realize any value from awards unless the Level 3 Common Stock outperfonns the Standard & Poor's 500 Index. When the stock price gain is greater than the corresponding gain on the Standard & Poor's 500 Index, the value received for awars under the OSO Program is based on a formula involving a Multiplier related to how much the Common Stock outperfonns the Standard & Poor's 500 Index. The multiplier is then described as follows: 'rutperfonn
Percentage'' The Multiplier shall be based on the for the The Outperform Percentage Period, determined on the date of exercis. in the Fair shall be the excess of the annualized percentage change Market Value of the Common Stock over the Period over te annualized in the Standard & Poor's 500 Index over percentage increase or dcrease the Period. .
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Because of the multiplier, if Level 3 outperforms the S&P 500 index by at least an annual average of 11%, the option recipient will have the payoff of eight options. The options maturity and are exercisable and fully vested after 2 years. have a zyear
Suppose that at the Rrant of an option, the price of Level 3 is $100, q) Example 16.7 and the t)! and the S&P 500 index is at 1300. After 4 years, the plice of Level 3 is $185, jEqgj
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outperformance option would have had
$100 x
1950 1300
$35
=
on Level 3 and the S&P 500 index are 85% and 50%. The 1.
850.25 --
j 5:0.25 .
==
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0.05957 x
8
x 100
4.332
=
The payment on the option is
($185$1e0 x 1959) x 4.332 jyt;tl -
$151.64
=
Tl'tis option is worth between 0 and 8 times as much as an ordinary option. How can we get an inmitive sense for the value of the difference? We will irst examine the effect of the outperformane feature and then cosider the effect of the multiplier. First, what would be the value of an ordinat'y feature valuing the outperformance zyear-to-mattlrity at-te-money call? Using a volatility of 25% (whichLevel 3 says in volatility'' of its common stock), apd a risk-free its 1999 Annual Report is the rate of 6%, we obtain arl option price of 'Eexpected
$100,0.25, 0.06, 4, 0)
=
$30.24
The Level 3 1999 Annual Report discusses the valuat t'on o f the outperformance option as follows:
0
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returns (nonannualized)
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outperform percentage is
BSCal1($100,
Multiplier
Outperform Percentage
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The fair value of the options granted was calculated by applying te Black-scholes method with an S&P 500 expected dividend yield rate of 1.8% arld an expected life of 2.5 years. Th Company used a blendd volatility rate of 24% between the S&P 500 expected volatility ate of 16% and te Level 3 Common Stock expected volatility rate of 256. expected con-elation factor of 0.4 was used to measure the movement of Level 3 stock relative te S&P 500. 'l''he
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lgwhen 11 reloads are permitted, the problem can be solved by having the binomial pricing function call itself, along with the information tat one less reload remais. This is simple to program but computationally very slow because of the large number of binomial valuations. See Saly et al. (1999) for a discussion. '
1-
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+
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atysts:p zpo-uevel
where p is te correlation between Level 3 and S&P 500 returns and ?- is the risk-free rate. The net effect on value of granting an outperformance call depends upon the effect
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of these substitutions. The recent years, Jsa:p has been less than volatility is :blended''
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= . 0.252 + 0.162 = 0.2368
,
-
2 x 0.4 x 0.25 x 0.16
which is rounded to 24%. The price of the outperformance option is therefore BSCall($100, $100, 0.2368, 0.018, 4, 0)
=
$21.75
This is about 2 of the value of the ordinary option. This reduction in value is primarily 3 due to replacing the 6% interest rate with a 1.8% dividend yield. The volatility reduction by itself lowers the option plice only to $29.44. ;'
Adding th: multiplier Now consider the effect of the multiplier. Wecan approximate value of the multiplied option using gap options (describedin Section 14.5). The the
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Figure 16.8 shows the payoff of a single nonmultiplied option, plotted against the payoff and the payoff approximated by gap options. Note that the exact and gap exact ' . approximation are not identlcal, but they are quite close. Table 16.2 shows that, using the gap option approkimation, the value of the compensation option is about seven times the value of a single option. A more precise binomial valuation using 100 binolnial steps gives a value for the option of $ 156.25, so the gap approxilnation of $153 is quite close. Monte Carlo valuation, which we will discuss in Chapter 19, provides an alternative way to value the Level 3 option. Finally, it is interesting to note that it may be rational to exercise the Level 3 option early even in the absence of dividends. Suppose the option is clos to expiption and the outperformance percentage is slightly above 1l %. lf the holder exrcises, the lpultiplier is 8. If the share price rises further, the mltiplier remais 8. However, if the share price falls, the luultiplier may fall to 7-,by waiting to exercise, the option-holder can lose options. This extra loss from a share price decline can provide an incentive to exercise early. For very high prices, there is no incentive to exemise early since the multiplier remains constant. For low prices, the potential increase in the multiplier offsets the
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multiplier in effect provides additional options as outperformance increases. For every .1 1 1.% per year by which Level 3 outperforms the S&P 500, the multiplier increases 8 8 Thus, 1. by we can approximate the effect of the multiplier by valuing a strip of gap outperformance options.
For example, the multiplier is 2 if over 4 years, Level 3 outpelforms the S&P = by l l 146, nonannualized. To approximate the value of 500 a factor of l option, that if outperformance is between 0 and 1.375% per year, the we can assume ontion-holder receives nothina. Between 1.375% and 2.75%. the ootion-holder the option. At 2.75% receives one per year, the option-holder receiyes a second option, Above 11% the option holder receives eight options. Each additional etc. per year, option. For example, the option received if perforbe valued option can as a gap is above 2.75% would per year pay 5Lvel3 s'sapif 5kevel3 > 1.0275 xj x mance
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TH E USE OF COLLARS
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16.3 THE USE OF COLLARS IN ACQUISITIONS
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A common financial transaction is for one firm (theacqltil'el') to buy another (the taqe by buying its common stock. The acquirer can pay for these shares with cash or by exchanging its own shares for target firm shares. Collarlike structures are frequently used in these tl-ansactions. Suppose that underthe purchase agreement, each target share will be exchanged for shares of the acquirer @ is the exchallge ratio). Once the target agrees to the purchase. -'r acquisition will generally take jme to complete, often six months or more.20 Target the will be concerned that the acquirer's stock will drop before the merger is shareholders in wlzich completed, case the dollar value of acquirer hares will be lower. To protect prie it is possible to exchange whatever number of shares have a fixed drop, against a value. (For example, if the acquirer price is $ 100, exchange l share for each target dollar If the acquirer price is $50, exchange 2 shares for each target share.) However, share. also shareholders wish to participate in share price gains that the acquirer target may xing the exchange ratio rather than the dollar value. There experiences; this suggests offer four structures that address considerations such as tlrlese--zl common are
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Figure 16.9 illustl-atesthese four types of acquisition offers. As this list shows, it is possible to modify the extent to which the target bears the risk of a change in the stock plice of the acquirer. More complicated structures are also possible.
The Nortlzrop Grumman
TRW merger
Northrop Grumman's 2002 bid for TRW is an example of a merger offer with a collar. ln ' Grumman and TRW aglee d thatNorthrop would pay $7.8billton for July 2002, Northrop TRW. News headlines stated that Northrop offered per share,'' but the offer actually resembled a collar. The number of Northrop Grumman shares to be exchanged for each '
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TRW share would be determined by dividing $60 by the average Northrop Grumman plice over the 'live days preeding the close of the lnerger, with the exchange ratio to be no less than 0.4348 ($60/$138)and no more than 0.5357 ($60/$l 12). Thus, if the price of Northrop Grumman at the merger closing was below $ 112, TRW sllareholders would receive 0.5357 shares. If the price was above $ 138, TRW shareholders would receive 0.4348 shares. lf the price, S, was in between $ l 12 and $ 138, TRW shareholders would receive $60/5, wllich is $60 worth of shares.22 The deal closed on December 11, 2002, Gruluman was $96.50-,TRW shareholders therefore when the closing price of Morthrop
4:$60
20Inmany cases- forexample, regulatory agencies examine the acquisition to see ifit is anticompetitve. 21Fuller (2003)discusses the kinds of offers and the Inotives for using alternative kinds of collars.
539
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* Fixed collar offer: There is a range for A's share price within which the offer is a fxed stock offer. Outside this range the deal can become a floating stock offer or may be subject to calzcellation. * Floating collar offer: There is a range for A's share price within which the offer is a :oating stock offer. Outside this range the deal can become a fixed stock offer or may be subject to cancellation.
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I N ACQU ISITI ONS
22Tl)e acquisition terms changed between February and July. Intially, Northrop offered to exchange $47 worth of Nol-thropGrumman stock for each TRSV share: The number of shares to be exchanged was to be no more tllan 0.4563 ($47/$l 03) or less tllan 0.4159 ($47/$ l l 3). In May, tlle value of the bid was increased to $53 (no more than 0.4690 sllares ($53/$l 13) r Iess than 0.4309 ($53/$123)
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AppLlcvlotqs
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TH E UsE OF COLLARS
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How would TRW shareholders value the Northrop offer?
$
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Figure 16.11 depicts equation (16.28)using the lzistorical Northrop Gnlmman stock price from July to December 2002, assuming that the ofer would close on December 11.24 The theoretical value of a TRW share under the terms of the offer is consistently greater than the market plice of a TRW share. This is what we would expect to see, since in order to induce the target company to accept an offer, the acquirer generally has to offer a price greater than the perceived value of the target as a stand-alone coppany. Since there is some chance the merger might not ocur, the target share price is below the value of the offer. The difference between the vftlue of a TRW share and the theoretical value of the offer declined toward zero as December approached. Had the merger been cancelled for some reason, the value of aTRW share would have diverged from the value under the terms of the offer. tale positions in the two stocks in order to speculate on the Risk tzr/ppwtdl/r. merger.25 of the Equation (16.27)tells us that the offer is equivalent to failure success or portfolio. of and shares options. Thus, usipg the option replication technique Northrop a shares and borrowing or lending to synthetically of Chapter %10, we can hold No1 position equivalent the offer. Because the price of TRW is less than the to create a offer value, an arbitrageur speculating that the merger would succeed could then talce a long position in TRW shares and a short position in the offer, short-selling delta shares of
$51.69
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l N ACQU ISITIONS
Suppose that TRW
shareholders were certain the merger would occur at time r, but uncertain about the future Norththrop Grumman stock price, S. TRW shareholders could then value the offer by noting that the offer is equivalent to buying 0.5357 shares of Northrop Grumman, selling 0.5357 llz-stn'ke calls, and buying 0.4348 l38-strike calls. ln adition, the TRW shareholders would not receive Northrop dividends paid prior to closing and would continue to receive TRW dividends. The time t value of TRW shares would then be
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dividend yield of 1.2564, and volatility of 36+.
25Mitchell and Pulvino
(2001)examine the historical returns
earned by risk arbitrageurs.
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Northrop. If the offer succeeds, tle position earns the difference in plice depicted in Figure l 6. l l ; if the oflkr fails, the difference should diverge and the arbitragetlr would
lose money.
Three col-porate contexts in which options appear, either explicitly or implicitly, are capital strtlcture (debt,equity, and warrants), compensation, and acquisitions. lf we view the assets of the firm as being like a stock, then debt and equity can be valued as options with the assets of the 'lirm as the underlying asset. Viewing corporate securities as options provides a natpral way to measure bankruptcy risk, and illuminates conoicts between bondholders and stockholders. Compensation options are an explicit use of options by col-porations. They exhibit variety of oluplications, some naturally occun-ing (earlyexercise decisions by riska employees) reloads, outperformance, and some created by the issuer (repricing, . averse multipliers). this they provide interesting For and context in wlaich to use the reason an pricing fonnulas from Chapter l4. exotic Oflkrs by one firm to purchase another sometimes have embedded collars. The Grunzman offer to buy TRW was an example of this.
FURTHER
READING
The idea that debt and equity are options was lirst pointed out by Black and Scholes ( 1973). Merton ( 1974) and Mellon (1977)analyzed the pricing of pepetual debt and deluonstrated that the Modigliani-Miller theorem holds even with (costless)batzltruptcy. Two principal applications of this idea are the detel-mination of the fair yield on risky debt and the assessnent of bankruptcy probabilities. Galai and Masulis (1976)derivrd the link between the return on assets and the return on the tirm's stock. The discussion of wal-rants and convertible bonds in this chapter assumes that the Op tions are European. .With merican wal-rants the optimal exercise strategy can be more complicated than with European options. The reason is that exercise alters the assets of the tirm. The problem of optimal American wal-rant exercise is studied by Emanuel ( l 983), Constantinides (1984),and Spatt and Sterbenz ( 1988). McDonald (2004)exam.t ines the tax ilnplications of wan'ant lssues, including put warrants. Complications also arise with convertible bonds, which in practice are almost always callable. Thus, valuing a convertible bond requires understanding the call strategy. Classic papers studying the pricing of convertibles include Brennan and Schwartz (1977)and Ingersoll ( 1977). . Hanis and Raviv ( l 985) discuss how asynzmetric information affects the decision to call the bond, and Stein (1992)discusses the decision to issue convertibles in the srstplace. Finally, there is a large empirical literature ol1 the converyible call decision-, for example, see Asquith (1995). Giintay et al. (2004)examine the decision to issue callable bonds. Papers on colnpensation options include Saly et a1.. (1999), Johnson and Tian (2000a), and Johnson and Tian (2000b). Replicing is studied by Chance et al. (2000) and Acharya et a1. (2000). Petrie (2000)examines the use of collars in acquisitions.
%
PRoBuEus
Appl-lcvlorqs
543
PROBLEMS For all problems, unless otherwise stated assume that the 'lirm has asjets worth A $100, and that o' = 30%, 1- = 8%, and the firm makes no payouts prior to the maturity date of the debt. =
16.1. There is a single debt issue with a matulity value of $120. Comjute the yield on this debassuming that it matures in 1 yeai, 2 years, 5 years, or 10 years. What debt-to-equity ratio do you observe in each case? 16.2. There is a single debt issue. Compute the yield on this debt assuming that it matures in 1 year and has a matulity value of $127.42,2 yearsgwith a matul'ity value of $135.30,5 years with a maturity value of $161.98,or 10 years with a maturity value of $218.65.(The maturity value increases Fith paturity at a 6% rate.) What debt-to-equity ratio do you obsel've in each case? 16.3. There are four debt issues with different priolities, each promising $30 at mamrity. a. Compute the yield on each debt issue assuming that a1l four mature in 1 year, 2 years, 5 years, or 10 years. b. Assuming that each debt issue matures in 5 years, what happens to the yield on each when you vary o'? When you vary r? 16.4. Suppose there is a single s-yearzero-coupon debt issue with a matulity value of $120. The expected ret'urn on assets is 12%. What is the expected ret'ul'n on equity? The volatility of equity? What happens to the expected ret'urn on equity as you vary A, c apd r? ,
16.5. Repeat the previous problem for debt instead of equity. 16.6. tn this problem we examine the 16. 1.
effect of changing te
assumptions in Example -
'
a. Compute the yield on debt for asset values of $50,$100,$150,$200,and $500. How does the yield on debt change with the value of assets? b. Compute the yield on debt for asset volatilities of 10% through 100%, in increments of 5%. For the next three problems, assume that a firm has assets of $100and s-year-to-mattil'ity zero-coupon debt with a face value of $150. Assume that investment projects have the same volatility as existing assets.
16.7. The 5rm is considering an investment project costing $1. What is the amount by which the project's value must exceed its cost in order for shareholders to be willing to pay for it? Repeat for project values of $10 and $25. 16.8. Now suppose the 51-111 financs the project by issuing debt that has Iower pliority tan existing debt. How muh must a $1, $10, or $25 project be worth if the shareholders are willing to fund it?
544
k
CORPORATE
AppLlcvlorqs
P Ro
16.9. Now suppose the firm finances the project by issuing debt that has higller pl4ority than existing debt. How mud must a $t0 or $25 project be worth if the shareholders are willing to fund it? 16.10. Assume there are 20 shares outstanding. the'shm-eprice for each of the following situations.
Comptlte the value of the warrant and
a. Warrants for 2 shares expire in 5 years and have a strike price of $ l5. b. Warrants for 15 shares expire in 10 years and have a strike of $20. 16.11. A tirmhas outstanding a bond with a s-yea.r maturity and maturity value of $50, 10 shares. also 20 There are shares outstanding. What is the convertible into price of the warrant? The share plice? Suppose you Were to compute the value of the convertible as a risk-fre bond plus an option, valued usinjthe Black-scholes formula and the share price you computed. How acctlrate is this? 16 12 Supp'ose ahrm has 20 shares of equity, a lo-yearzero-coupon debtwith amaturity value of $200,and wan'ants for 8 shares with a strike price of $25. What is the value of the debt, the share price, and the price of the wan-ant? 16.13. Suppose a firm has 20 shares of equity and a lo-year zero-coupon convertible bond with a matulity value of $200,convertible into 8 shares. What is the value of the debt, the share price, and the price of the warrant? 16.14. Using the assulnptions of Example 16.3. suppose you were to perfonn a bond plus 50 call options on the stock. valuation of the convertlble as a lisk-free cinaive''
How does the plice you compute compare with that computed
in the example?
16.15. Consider Panels B and D in Figtlre 16.4. Using the information in compute the share price at each node for each bond issue.
each panel,
BLEMS
%.
545
a. What is the value at grant of an option tlat will not be repliced? b. What is the value at grant of an option that is repliced when the share P
Iice
reaches
$60?
c. What repricing trigger maximizes the initial value of the option? 16.18. Suppose that top executives of XYZ are told they will receive at-the-money call optons on 10,000 shares each year for te next 3 years. When granted, the .opt ions have 5 ears to maturity. XYZ'S stock plice is $ l00 volatility is 30% and 1- 8%. stilnate the value of this prolnise. Hl-llt: Se Problelu 14.2 l ,
,
.)
=
16.19. Suppose that S 1, and J $100, c' 30%, 1- 6%, t 0. XYZ European put option on one share with strike price K $90. =
=
=
=
=
writes a
=
a. Construct a two-period binoluial tre for the stock and price the put. Compute the replicating portfolio at each node. b. If the lirm were sylzthetically creating the put (i.e., trading to obtain the same cash flows as if it issued the put), what tl-ansactions would it undertake? c. Consider the bank that buys the put. Wllat transactions does it tlndertake to hedge the transaction? d. Why might a firm prefer to issue the put warrant instead of bon-owing and repurchasing shares? 16.20. Firm A has a stock plice of $40and has made an offer for lirm B where A prolnises to pay $60/share for B, as long as A's stock price remains between $35 and $45. If the price of A is below $35, A will pay l 14 shares, and if the price of A is shares. The deal is expected to close in 9 months. above $45,A will pay l 40%, 1' 6%, and Assume t:r 0. .7
.333
16.16. As discussed in the text, compensation options are prematurely exercised or canceled for a variety of reasons. Suppose that conpensation options both vest and expire in 3 yeary and that the probability is l0% that the executive will die in year l and. l 0% in year 2. Thus, the probability that the executive lives to expiration is 80%. Suppose that th stock price is $ l 00, the interest rate is 8%, the volatility is 30%, and te dividend yield is 0.
=
=
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a. How are the values 1 14 and l b. What is the value of the ofkr? .7
.333
an-ived at?
c. How sensitive is the value of the offer to the
volatility
of A's stock?
16.21. Firm A has a stock price of $40, and has made an offer for firm B where A promises to pay l shares for each share of B, as long as A's stock price remains between $35 and $45. If the price of A is below $35, A will pay $52.50/share, and if the price of A is above $45,A will pay $67.50/share.The deal is expected 0. to close in 9 months. Assume o' 40%, 1- 6%, and .5
a. Value the option by computing the expected time to exercise ging this into the Black-scholes fonnula as time to luaturity.
and plug-
b. Conpute the expected value of the option given the different possible times until exercise. c. Why are the
answers
for the two calctllations
different?
16.17. XYZ Cot-p.compensates executives with lo-year Etlropean call options, granted at the money. lf there is a signiscant drop in the share price, the company's board will reset the strike price of the options to equal the new share plice. The mattlrity of the repriced option will equal the remaining mattlrity of the original option. Suppose that
tr
=
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=
6%,
=
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.and
that the original share price is $ 100.
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a. Hosy are the valtles $52.50and $67.50arrived at? b. 'What is the value of the offer? c. How does the
value of this offer colupare with that in Problem 16.20?
16.22. The strike price of a compensation option is generally set on the day the option is issued. On November l0, 2000, the CEO of Analog Devices, Jerald Fishman, received 600,000 options. The stock price was $.44.50.Four days later, the price
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CORPORATE
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rose to $63.25after an earnings release: Maria Tagliaferro of Analog said the timing of the two events (option grant and earnings release) is irrelevant because company policy is that no option vests until at least three years from its granting date. ttWhat happens to the stock price in the day, the hour, the year the option is granted is not relevant to that option,'' she said. stock price only becomes relevant after that option has vested.''z6 ltrf'he
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a. Using the inputs from the annual report, and assuming no dividends, estimate the vlue to the CEO of an at-the-money option grant at a stock price of $44.50.
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why not?
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offers a 3-year compensation option that vests immediately. 16.24. Suppose that a 517.11 An employee who resigns has two years to decide whether to exercise the option. Compute annual compensation option expense using the stockprice treein Figure 16.6. Velify that the present value of the option deductions is $28.15.
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16.23. Four years after the option grant, the stock price for Analog Devices was about $40 Using the same input as in the previous problem, compute the market value of the options granted in 2000, assuming that they were issued at strikes of $44.50and $63.25.
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Estimate the value of an at-the-mony grant at a price of $63.25. . c. Estimate the value of a newly granted option at a strike of $44.50when the stock price is $63.25.
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!48
INVESTM
t. REAL op-rloxs
17.1 INVESTMENT
AND THE
NPV RULE
We first consider a simple investluent decision of the sort you wotlld enounter in a basic hnance course when studying net present value (NPV). Despite its simplicity, the example illustl-ates the issues that will arise again later in this chapter. Suppose we can invest in a maclzine, costing $ 10, that will produce one widget a year foreven ln addition, each widget costs $0.90 to produce. The price of widgets will be $0.55 next year and will increase at 4% per year. The effective annual risk-free rate is 5% per year. We can invest, at any tilne, in one such machine. There is no
Static NPV A natural first step is to compute the NPV if we invested in the project today. We obtain Npvlnvcst today .042
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Problem 17.4 asks you to verify this result. You will discover that 23.82 yers is not exactly optimal. Rather, waiting approximately 24.32 years-not 23.82 years--rmxirnizes NPV. At this point the widget price will be about $1.43. We will see the reason for this slight difference in Section 17.4. It occurs because effective annual interest and growth rates of 5% and 4% are not the relevant rates the decision the since to put off the investment is made on a day-to-day basis. It is instead continuoltsly ct??n//t/llnt-/ct rates that matter. equivalyt the This example demonstrates the important point that making an investment decision requ'es thinling careftllly about alternatives, even under certainty. We ate lef4 with (at least) two questions:
The Correct Use of NPV The NPV rule worked correctly in the above example. The NPV ment decisions entails two steps:
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Thus, it is better to wait 5 years than to invest today. 'What is the maximum NPV we can
all
When we computed the widget machine's NPV in equation (17.1),we neglected to into take account the NPV of alternative mumally exclusive projects, namely investing in the project tomorrow or at some other future date. Static NPV--'-NPV if we accept the project today-ignores project delay. Because static NPV measufes the value of an action we could take, namely investing today, it at least provides a lower bound on the value of the project.
$28
= $30.49 attain?
RULE
@ How do we approach this ldnd of problem in general? * Why didn't the NPV rule work? Or did it?
$10 $28
-
0.01
.
NPV
Common sense points to an approximate answer: We should notinvestuntil annual widget revenue covers marginal production cost ($0.90)plus the oppo>nity cost of the project ($0.50).,i.e., cost is at least $1.40. The widget price will be $1.40when ?? satisses
tlncertainty.
Before reading further, you should try to answer this qklestion: 'What is the most lights would to this project? pay to acquire the you
ENT AN D TI,I E
lerheprice must 23 82.
increase by a factor of 1.40/0.55 to reach $1.40, so we have ln(1.4O/-55)/ 1n(1.04) =
zlntroductory hnance textbooks state the NPV rule correctly. but in casual discussions it is sometimes stated incorrectly.
550
%.REAL
Op-rlolqs
IN V ESTM
In this example it would be correct to invest in the project today
ifllotclcrvt'lr?,jr
the
nautually project 1474/47.)'nleant #l7r It/d b,oltld Iose ?./b?-cT,t??: Under this assunption, exclusive alternative (nevertaking the project) has a value of 0, so taking it today would be correct. theE
To dvcide whether and when to invest in an arbitrary project, we need to be able to compute the value of delaying that investment. As stlggested at the start of the chapter, option pricing theory can help us to value delay.
'
Present value of costs
=
'
$10 +
0.05
=
$28
As we discussed earlier, we can view this present value as analogous to the exercise price in an option valuation. In returfl for paying $28, we receive a cash flow with a present value of Present-value of widget reventle
=
.+1 0.01
where .S'+1is the widget price the year after we make the investment. When $0.55, the present value of cash flow is $55. This present value of widger revenue is the price the revenue stream would have if it were traded separately. It is analogous to the stock price in an option valuation, and therefore it is sometimes called the the twin security or the t'raded present value of the project. Now recall the discussion in Sections 9.3 and l l about the three factors governing exercise of a call optio: the dividends forgone by not acqtliring the asst today; the arly interest saved by deferring the payment of the strike price; and the value of the insurance that is lost by exercising the option. It turns out that the same three considel-ations govern the decision to invest in the widget project. ,ol
=
.1
First, by delaying investment, we lose the cash flow from selling widgets. The cash
flow we do not receive is analogous to stock dividends we do not receive by holding an option rather than the underlying stock. The first period cash flow is $0.55. The present value of future cash Pows is $0.55/0.01 $55. Thus, the dividend yield is approximately l %. (We can also think of the dividend yield as the difference between the discount rate (5%Jand the growth rate of the caslz llows g4%j.) Second, once we begin widget production, sve are colnmitted to spending the present value of the marginal widget cost, $ l 8, along with the $ l 0 initial investment. =
The annual value of delaying investment is interest on the total ilwestment cost, or ' 0.05 x $28 $ 1 per year. .40
=
$
and therefore no
We can compute the value of the widget project option using the perpetual call calculation discussed in Section 12.6. The formula ajsumes continuously compounded 4.879%, and for the dividend yield we rates, o for the interest rate we use 111(1 difference compounded the continuously between the interest rate and growth rate, use 0.9569%. or 111(1.05) ln(1.04) .05)
=
'
=
-
=
$55 (thepresent value of revenue), K
=
$28 (the present value of
costs), '1- 0.04879,+ 0,a and t 0.009569, equation (12.19)gives an option plice of $35.03and investment when the widget price equals $ 1.4276. We will call this price the investment trigger price. We reach this price after about 24.32 years, which velifes the answer we discussed earlier. The example in this section illustrates the iluportance of thilzking dynalnically about a project and shows how this speci'lic problem can be modeled as an option. =
The decision to invest in the project involves a comparison of net present values. In what sense is this an option? We can interpret equation (17.1)so as to make the option analogy more apparent. 'When we take the project, we pay $ 10 and we commit to paying So.go/yearforever.The present value of this stream of costs is
UN C ERTA I NTY
Third and finally, in the widget project, there is no uncertainty
insurance value to delaying investment.
With S
The Proiect as an Option
ENT U N D ER
17.2 INVESTMENT
=
UNDER
=
UNCERTAINTY
In this section we discuss the valuation of rel investment projects when cash flows are uncertain. With the widget project in the previous section. waiting to invest was optimal because project dividnds were initially less than the interest gained from defelning the project. If we add uncertainty about project cash flows, the value of insurance (the implicit put option) also influences the decisiop to delay the project. ln such a case, waiting to invest provides infonnation about the value of the project. In this section we will use a binomial tree to value a project with uncertain cash flows. As before, the decision to invest in such a project is like exercising an American option: We pay the investluent cost (strikeprice) to receive the asset (presentvalue of -
future cash tlows).
A Simple DCF Problem We first examine a particularly simple valuation problem in order to better understand the link between discounted cash tlow (DCF), real options, and linancial options.
Suppose an analyst is evaluating a project that will generate a single cash flow, X, occulning at time r. As with many investment projects, it is not possible to observe market characteristics of the project. There is no way to directly estimate project returns, project volatility, or the covariances of the project wittl the stock market. Instead, suppose the analyst considers the economic fundamentals of the project and makes educated inferences about these characteristics. The ahalyst might also look for ptlblic firms with a business resembling the project. The nalyst could then use information about these public f'irmsto infer charactel-istics (suchas beia) of tlze project.
3lt is necessary to set o' to a small positive number stlch as 0.0000 1 to avoid a zero-divide en'or.
%.REAL
552
Opl-lolqs
INVESTM ENT UN DER UNCERTAINTY
After examining al1 available data, the analyst estimates that the cash flow will be Xu if the economy is doing well-an event with the probability p-and Xd if the economy is doing poorly. The project requires expenditures of % at time 0 and fw at time F. The analyst determines that projects with comparable risk have an effective annual expected cateof remrn of a. We can use this description to compute the value of the project, J/. The standard discounted cash flow methodology calls for computing the expected cash flow, and using as a discount rate the expected return on a project of comparable risk:
pxu + (1 pjxd (1 + J)F -
J/
=
Assuming that we either take the project now or never, we invest in the project if F 2: r)W. % + J/(1 + '
Example
17.1
Suppose that the risk-free rate is ?' 6%, the expected return on the )(l market is 0.60. 1.25, project is F = 1, Xu 10%, the beta rju $120, and' p p ).jjj )'t Xd gp $80. The expected return on an asset with the same risk as the project is =
=
E .E
hy (.j
a
.
.
t') The ('E)
('. q'ii :
q 'El'' E
=
0.06 + 1.25 x (0.10 0.06) -
EX)
j'.
=
(2.60 x
$120+ 0.40 x $80
=
p*Xu + Thus, we have
$104
t Using (17.2),the present value of the project cash flows is !( i j.l! $ 104 J/ p)( $93.694 ?).j
IZ
=
(1
involving options, suppose
.F
(1 + l )
p*)Xd
-
4 p
i
need to make f?2order
probability must therefore satisfy
0.11
=
expected cash flow is
!
.
lisk-neutral
=
EEtIIE
'
F0,T
=
=
. .
To see how to perform capital budgeting calculations
1p'
553
that if we invest Io to start the project, the subsequent investment of fl is optional'. We make this further investment only if the project at time 1 has suflicient value. Since the project has a good and bad outcome, it is namral to think about using binomial option valuation. ln order to do so, we need to lnow the risk-neutral probability of the high outcom. 5 Fortunately, we can easily compute risk-neutral probabilities for this project. Recall from Chapter 10 that the expected risk-neutral price is the fonvard price. We have computed U, which is the price of p.n asset paying a single cash flow.at time T. The forward price is The
j'('j'j' y' j '
valuing the project Jv' have already ppt-l#c alI r/?easswtlptions pt'I/l/c derivatives related to #?c plnject. to
$
*
Fo p
-
:
=
Xu
-
#(),w
=
Xd Xd
probability of a high This gives us the binomial tree xu and Xd) and the lisk-neutral using risk-neutral project distribution, then Notice that if value the the (p*). we outcome by c/n.lrldcrtwl we will obtain the original project value, F.
:
=
:.jyj
E ( 1.E
=
1 + 0. 11
i
t,) Suppose that Ifj $10 and f l $95 and that the manager commlts at time 0 to paying ti'r $95 at time 1. Then net present value is t.1 gqr .
).(the
=
=
Example 17.2 Consider the same parameters as in Example 17.1. The forward price for the project is Fo, l
E)
ijJ l..
The
Iisk-neutral
=
$93.694x (1.06)
=
$99.315
probability of the good outcome is
99.315 80 120 80 -
ps
=
=
0.4829
-
Valuing Derivatives on the Cash Flow
If we value the project using the risk-neutral probability, we obtain
0.4829 x $120 + (1 0.4829) x $80 = $93.694 1.06 Now male the same assumptions as in Example 17.1, except that we decide at -
The calculation in Example 17.1 is standard but it is nevertheless based on strong assumptions: We have specised the future cash flows in different states, the probabilities of those states, and the comparability of the project to a traded asset/ It t'urns wt that in
time 1 whether to
'lef'helast assumption in particulardeserves some additional comment. We are assuming that the returrls of the project are spalllled by existing traded assets-, in other words, th addition of the project to the universe of assets does not materially change the opportunities available to investors. If this were not true, we would have to know more about the preferences of investors in order to evaluate the project-
5In general we also need to know volatlity to value an option. As we discussed in Chapter 10, volatility detennines te vertical distallce between binomial nodes. Thus, in specifying the tree, we implicitly specined volatilitj?.
'incur
the
$95 cost. We will choose to produce output in time 1 only
)'554
% REAL )', when the
y(.
j
OpTloNs
INVESTMENT
cash flow is
$120, since
.. the output sells for $80. The value )t, ).). maxg () x ps) gI j ( j (,. q!)) p ?l + $i)) l.lytt :
E
)EE)
,y
.j.
-
,
y)j ljj
-
,,
.0.4829
$l5 by paying $95 to produce
we would lose
when
-
,
x
$25 + ( l j
Jj j -
=
10
0.4829)
.(j6
x 0
- $10
=
$1.389
. Given the risk-neutral probability and the cash flow distribtltion, we can value projects with options or other nonlinear cash fIows.6 You may be thinking that there appears to be little difference between standard discunted cash flow valuation and real options valuation. Recognize that ally snancial valuation entails assigning a dollar value today to a (possiblyuncertain) cash flow that occurs in the future. This descliption applies to the valuation of a project, as well as to valuing a bond, a stock, or an option. When we value an option on a stock, we rely on the market to have already pelormed part of the valuation naluely, valuing the stock. When we value an option on a project, we have to estimate the value of the project since we cannot observe it. This is p-llc vlletller or llot the p/r.#c/ colltaills optiolls. This discussion illustl-ates the point we made before in Section l l that riskneutral pricing and discounted cash flow are altelmative means of valuing a futtlre cash :ow. If done using the same assulnptions, the two methods give the same answeni ln practice, of course, it is common to make silnplifying asstlmptions for tractabilitjt Answers may differ because the simplifying assunptions for different valuation methods are inconsistent.
?-risk-free
-
0.07 + 1 0 15
.33(0.06)
PV
stcyj)
=
'-project glrwth rate $l 8 0.15 0.03 -
=
-
= $150 Static NPV is theyefore $150 $100 $50. Suppose we have 2 years in which to decide whether to accept the project; at the epd of that time, we either invest in the project or lose it. (Imagine, for exalpple, that the licensing rights for a technoloqc will revert at that time to the oliginal owner). The static NPV rqle will apply after two years because further deferral is not possible. However, at tilpe 0, we must evaluate the option to wait. The forgone initial cash flow (thedividend on the project) is $ 18 and the interest saving is $7 (7% x 100). Thus, considering only dividends and interest, it makes sense tl srart the project imrpediately. However, the project also has irnplicit insurance that we need to know the project we lose by investing in the jroject. To value the insrnce -
.2,
=
volatilityr
for project value Suppose that cash flows are lognormally distlibuted with a 50% volatility. Figure 17. l uses the Cox-lkoss-lkubinstein approach to construct a
A tre
,'?' .--.-' ' -' -' ,'--' !(j'
j'p' ,j,(' j''7,:'*(1* (' cd T' (jd ;'!' jfd t'.y' q-' )' iiii2-:-1I1E't,f j'ly-'t ((''l ()(-'-. ;(' tfd )' .-' !iIi:t:;;;k'. k' t.!)f r' rqd t)'. j'. y'l'y .y'.y' )' ttyf ))' tyy'.
.'iiiii:#!;. .' . .'p'jjjyj r: ; y ;. r .' . tjj ..lllii:tli!t. . yy . 1l1i. IIIEEEEE:. .yj.y.. jj(. ...-j.j j. j'.'j y yj --.. r, . .. ... .kj .,r. j t-.;.r..;...r. .(.. r.. E E EE( Ei ' EE:' ;' ! E' EtE : i; ' ' (l !E!( (... ..' (''.(. E E..i.. (.E'E!.iE (ll;Ej:EE'...'.''EE iq (E.'.yi'.: . jF . . .. !'(.EE. (..' (. ' : ;''7::.'(( . (' ((..!( q ' ()(j.. y.jjqjjj E .( . y... .y ;..qjgjy g j .. t.. yy g.yyyjy. ( .( q...q.yyy.j..qttjj .y..y y j y.yj.. y. yyy. yj. y.y..y.yy,, E yj y. . .y . i!. -. . - - j- - -.- . . -y . -. ( . :i -. - k;. . . . . .. . . . .. ''
.
Eyyjl -
E ::
..
. .
.
:IE! :
(iE
.
.
.
. .
:
.
E
'!
'
.
.
::. :
E . .. . . g
-.
:
' .
:'
...
E
: '
. .
: . :' .
-: .:.. :
.y.
:
.
.
- . ..
.... .. .
-
-
-- - -
-
.
.
-
.:
-.
y -. y y
r
-. . .
.
E'. . .
. .
'-;11111.
'
-
We now consider the problem of when to invst in a risky project. As before, the decision to invest in such a project is like exercising an American option: We pay the investment cost (strikeprice) to receive the asset (presentvalue of future cash flows). The widget project in the previous section had cash flows that were certain. Suppose a project costs $ 100 and begins prodtlcing an insnite stream of cash Cows l year after investment. Expected annual cash Pows for the srstyear are $ 18, and are expected to grow annually at a rate of 3%. Suppose further that the risk-free rate is 7%, the risk prelnium on the market is 6%, and the beta of the projet ls 1 Using the Capital Asset Pricing Model (CAPM), we compute the discount rate for the prject in
y
.
.
-
.
.
.
.
. y ., . -
. .
-. .
..
.. . . ....
Binomial tree for project cash flows, assuming binomial distribution with 50% volatility.
.31.
tiproblem 17.9 asks you to value a project paying the squared cash flow.
555
To value the project, we perform a standard discounted cash flow calculation. Since the project lives forever, we treat it as a pelmetual grwing annuity. The present value is
k
Evaluating a Project with a z-Year Investment Horizon
$
Jl(/-market'-risk-rree)
-1-
?-project=::: -
UNCERTAINTY
the usual way:
of the project is maxgo Xg
UNDER
56.62
5!6
k
INVESTM
op-rlolqs
REAL
=
The up and down moves can be modeled using any of the binomial
=
=
7'* (' ,',' )(j' ;(' j'jyy'tyyf t'ktf ''''7Tqq'':'q' y' q' )pf( (! IiIjkiI:@ )r' )' (' y' q' r' 'r(''' 'r:'r'* (' i ((( '' (' jjjL)jj,', y' )' yy'!. j').jf ' '( qy q;j '! . !(;j (( (g $)* .(. (' ( ..('('.' ( .rqr).
$100.,continuously = 0.50,* nd time to
t'';' t'';')':'F' t'y
-.'. '' -.,yjjjj'-'
!E.E'' .' ..
((j.i. .E;.. E! (. .' ' .'. '.'. i'..'.'.('.('. ('( r' ' '.'(:(''.'E.' E.' 'E..!'.'( (( . ; qq ( q.j ;;i;. q.'j.' ;E- y j' 'yE . . .q y. E ;E). . ; j;yjyj.(. q.... ;-jy q! r .. . jyy. . . yjyly y ..y yy jjyy y .y .y..g...yy.yyy. . !!!(E!ii:t::klj .y. ..y ..y.y.. .Ej..!-t-.-(,(.t-t)i..i-)..y.y.yjy.yyyy.yjyj. .y.y...y.yyjq,,,yy . . .j..yj., t.y --.!ky.q-ti.I12!q...... ! .....1....L.. ,-,..-.LLt. ykl. .j qq. .jtkyj, .. .... iE i E :
(.E --
ry. ttl llti ..
-
-..
.
-
-.
..-t)')'.'.:..-......
-. .-
'.
'
:
.
E'EE' ' : .E'i (. . . . E '(
E: : ..(.
'''
E
-.
E ' . ::
:
:
). .tj. ..('. it k, yy.yyy .
-
.
''
- .
. . . -
-
.
.
.
.
-
.
-.
.
-. .
-.
-. . .
--. .
-
-..
. --
-
.-...,-.-
-
-
.
-i#-.)...,.-
......-.-
. . . ...........
.-.. -....
. . ..
=
eees from
p*
=
e
0.0676-0.1 0.5
133 -
-0.5
-
e
=
/-0.5
0.335
Using #*, we work backward through the tree as in Chapter 10. The results are in Figure 17.3. Notice that the inithl value of the project option is $55.80,which is greater than the static NPV of $50. Problem 17.9 asks you to velify these calculations. ln practice. decision tzees are often used to analyze this kind of problem. Figure 17.2--11.1(:e a decision tree, albeit with probabilities and any binomial option problem-g nodes constructed in a very particular way. We saw in Section 11.2 that if the discount rate applicable to the underlying asset is constant, then when valuing an option using true probabilities, the correct discount rate varies across the nodes of the tree. Analysts using a decision tree often use true (notrisk-neutral) probabilities and a constant discount rate along the tree.? Binomial pricing per se does not imply that any particular true expected -' 'S' ;'L' -' jr;j'E pJ.r,'!)'(yyj;(yjyjyyj ;:' y)f )' :'7* '
k.--.;.-jjk'-' y' 1Iitk::iIIE-' j')' p' 1* (' ;ktjjjj'. !' '(k' ,qE7'77'(*'71'* (-'. )' '(' 'f t'' t'pJ' jtf jltf q;(7((j)();'(r':.:! .;.'. ,-L-', y' yyf r' ''F'j'q' rill)f r;'rg.. j-'.,)f .-' j'jy'jj i'-. (:!E!!!llr-'.--..,-. (' Iliii:qqqf )' r' r!sc' t;;f tll-'i rjf y' y'. '. i q (I'j(1)! j( iiE('. (.qE('.'(.( I. (j (i . (' ('E( (.( q'.'q('(.' (E'(' L--:'. t'l'l''if );j'. i: ('('E(')( '' j . ;')(:4(1. ti't-t . l r '.yy('j.(q;.j..j(:. t.(i.q....j ')i ..;!. :. (..: iirl.i((. . j.y ..gttjj(yyy-: .pjjy;q... (j :jjLi'''ibL. jjy;j..j.g.j.jj.y.y(jyy..y.(y.g...jyyy ..g..j j. .: E. . E ) .li lljlqiijli:. iiliiE!!:. '(IIt:t:I:;i.. . . i . . . i .-. ; 11111 j---.i r (. .... ..).. (..i... r. . .. . (. q-)!.)t''' . ..' .... . Value of the investment option for the project in Figure 17.2.
.'
.)tj)--'.',b,-''' ,IE!EE.-?-!!.,-.X
ll't'ff '' .
:
jE
iE
:
:
i
LE:. E .
,
.
E
'' '
'
:
'
. : :
.
.. .
'
' '' ' E ' E:' ' i E EE::. ' E' 'E frE E :E : : . E : E .. ' .
:
::: .: . . E'.7E: E E:: . ..E. ... . .. :.E : ! E q . -.. . ..E.y............:.. E . E
.
:E
.
. .
..
'!r'7q7r'7lq'77r7q
:
.'t!pE!.
.
..t
.......
. ....
...........
...
..
..
.. .....
.
.
,..
. . . .
''
'
.
: E . .
' :
.
:
: :. i .
:
.
.
---;(..
:
E:
:. .
IE :...
:. i . .
- . -. . --..- -.- - . .. ..
....
.
-
.
.
.
-
.
..
--
-.
.
ll:::,,. ll!!.,-?!!!.i-. IIF:i,:.
.-,.1-,-.
II:;I:. ...11
.--.1,,,.
j::;,. II;:;I:. ,1(!!!),-,,:,..-.
-
yoq
yojy y
()
Binomial tree for project
voR
If:
z
111/:
:407.74
ltdl
$150.00
value, assuming 50% volatility.
$247.31
$150
$0 J: $90.98
J(f: $55.17
557
Chapter 10. We can then solve for the value of the investment option just as we solve for the price of an American call option. The risk-neutral probability of the project value increasing in any period, p*, is given by:
The act of investing creates the project, therefore the value at each node in Figure 17.2 is the value of the project ifwelt/crd to ?lvl'r at l/?t7r?l(?#'. Figure 17.2 describes the evolution of the project's present value. The project does not exist prior to investment, but the tree provides the information we need in order to decide whether to invest. The tree in Figure 17.2 is exactly the same tree we would construct for the stock price of a company that had the project as its only asset and that paid dividends equal to the cash flow of the project. Such a stock would have an initial price of $150and a 50% volatility. being It ray trouble you that in valuing this project, option pricing formlas used in a context where literal replication of the option is not possible beause the twin lpnt/pptx/p/tpcLflf?'d also security does not exist. As we saw in Chapter l 1, however, the 1/t'l/i/t'#k't7?? Isl/?-/t'- f?? settiltg 'kW?c?-' ltsilg ntodel. J/' pelfonn #7' CAP.H or tprcrprfcn,g a notarbitrage-h'ee plices. Thus, we are using option plicing formulas to ceztefairprices,
The inputs are initial project value, S = $150.,investment cost, K = 6.766%) volatility, o' compounded risk-free rate, ?' = 1n(1.07)
$
=
=
We can use Figure 17.2 to solve the solvins for the optimal Investment decision would it in problem exactly investment use a binomial option pricing ljfoblem.' as we
NTY
=
=
-
U N CERTAI
expiration, t 2 years. Since the market value of the project today is $150and the cash flow in a year would be $18 if the project were developed, the dividend yield is l2% ($18/$150). Since project value is proportional to next year's cash flow, the dividend ln(1.12) yield never changes. The continuously compounded dividend yield is 0.1133.
binomial tree for the evolution of cash flows with a binomial period of 1 year. If we wait to take the project, initial cash flows in l year will be either $18c0.5 $29.677or $18e-0.5 $10.918. Since the project value is proportional to cash 'Ilows, the value of the project is also lognormally distributed with a 50% volatility. ln l year, project value will be either $29.677/(0.15 0.03) $247.31 or 0.03) $10.918/(0.15 $91. lf we will continue to learn about the project at the same rate over time, we can build a binomial tree with constant volatility that shows the evolution of project value. This tree, constructed by discounting at each node the cash flows in Figure 17.1, is in Figure 17.2. -
ENT U N D ER
7efhis is not necessarily incorrect; as a logical matter, a constant discount rate for the option could be correct #- the tree for the underlying asset had discount rates that varied across the nodes. However, in decision tree analysis in practice, the issue of discount rate determination is often glossed over.
558
% REAL
Op-rloxs
REAL OPTIONS
return is constant; instead it tells us how to perform valuation so that the assuluptions about the project and the asstlmptions about the tree are consistent with each other.
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The above example assumes that we must start the project by year 2 and that we evaluate it annually. Suppose instead that the project can be stafted at any time and then will live forever. The project is then a pel-petual call ojtion that we can evaluate using the perpetual option pricing formula. Using continuously compounded inputs, we
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When the project valu is $ 150, the option value is $63.396qnd tlie optimal investment trigger is $2.45.71. In other words, we invest whep the project is worth $245.71, more than twice the investment cost. If we invest immediately, the project ls wrth $50. The . ability to wait increases that value by $ l 3.396.
17.3 REAL OPTIONS
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In Chapter 6 we explained that electricity forward prices can vary over the course of day. They also varj? seasonally-. ln the United States, electricity forward prices are high in the summer and low in the winter. In addition to this pl-edictable variation, electricity prces can be volatile. On extremely hot days, for example, prices can spike to l00 times their average price. A peak-load plant, as the name suggests, produces only when it is protable to do spikes in the price of electricity. Stlch plants are designed so that they can exploiting so, when idled the price of electricity is less than the cost of fuel, but they can be qtlickly be online brought to produce power when the price of electricity is high or when thq price of declines. Because it is tulmed on only when prolitable, owning a peak-load plant is fuel
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investing in production capacityand facing uncertain demand the same peak-load production experience problem as electricity produces. Consider a manufacturer investing in productio capacity facing uncrtain dmand. How shouli the and manufacturel - c h Oose p lant capacity? Consider tlae plant's capacity to meet exkcted choosing demand. lf demand turns out to be less than the firm will either produce at a lods forecast, or have an idle plant. lf demand is gleatel, th (1E:,.4(::). ..:2),- dki:r:dzz:: 'kg:r II:E 4:::), 4k2:: 11:) ji(. :E1; ,58;: :rE1.. lIg'' llr'':!ii!;. 11.;1.. 11.... 1.,(.,. ii. qlr.' ..1L :Iq. ii:14;. 21. lll 11) n 'kzr:.::2:. !!i,;-. .. jlq:ii produce whether demand is high or low, then' capacity has no option value. However, extra it is when if possible to idle an unused plant capacity is low, thvn extra yp'-/t.lcf?p, demand Th extra capacityq is like a rw/c-/tmt-/-//c//fry. givesthe nrma call option. Intel in 1997 had to decide upn th ,, capacity pf a new plant. Seluiconductolr j 1(:)lE:-:rll. li-;EztL. IIE:)/ . 4::2:: 1!r.rii 1(::2,. dt::!r llg ft::)p 11:: .: . 4::22p ql.:ll. 1IE) dkiz:: 1221L. 11::1L d::z:' 16E; $!511. 118;. lllii. 11:2:, ii. qI..ii) iil IL::). t!;i (1,l:':!zlt. .
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Section 14.6). The difference This is the payoff to a European exchange option (see generation, H x Sgas,is called the between the price of electdcity and the cost of spread is the variable spark the spark spread. There are operating costs besides gas, but component of marginal profit. and volatilities for electticity. ln order to value the option Fe need forward prices ' ' Figure. 17.4 jhows . and gas and the correlation between the two. The top panel in has a shape representative folavard curves for electricity and gas. The price curve for gas electricity peaks. curve, by The winter familiar from Section 6.9, exhibitinc seasonal implied by spread spark shows the eontrast, exhibits summer peaks. The bottom panel the prices in the first panel. provides. Let Fz.ti and The value of a plant is the sum of the operating options it and electricity gas delivered at time t If Fs t; represent the time 0 forward prices for .j operating plant is z value the of the we ignore other marginal operating costs, then
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Research and Development Research and development is a capital expendimre like any other, in that it involves paying R&D costs today to receive cash flows later. lf R&D is successftll, a project using the new technology can be undertaken if its NPV is positive. This fnal option is a call option, just like the other projects we have analyzed. The R&D leading up to this project is therefore like an option premium: We pay R&D costs to acquire the project. R&D can be thought of as acquiring fumre investment options. Dnzg deveiopment by phannaceutical lirms provides a particularly clear example of the options in R&D since the drug development process has cleady delineated points at which there is a decision to abandon or continue development. Figure 17.5, based on a descliption in Schwartz and Moon#2000), summarizes the process, along with the probabilities of progressing from one stage to the next. In practice, stages sometimes l'un together, but Figure 17.5 re:ects a standard description of the process. As R&D costs are paid over time, phnrmaceutical frms are able to resolve uncertainties about their technical ability to produce and market the product. Specifcally, they answer the questions: Will the project work, and, if it works, will anyone want it? At a1l times, project managers have the option to continpe or stop the research. In effect, each ongoing investment purchases an option to continue development. Figure 17.5 shows that most potential drugs are abandoned before Phase 1 trials. As with pealoload electricity generation, value arises from what is not done. A pharmaceutical company that pursued a11potential drugs, nl matter how unpromising, would reap f'ull rewards from successful drugs but would be bankrupted by the unsuccessf'ul drugs. The put option to abandon a drug is what creates value for the finn.
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When we include other costs, the static NPV of a peak-load plant is typically negative. Adding the shutdown option, however, makes NPV positive. One implication is that, in equilibrium, after the optimal number of peak-load plants have been constructed, electricity prices will continue to be variable. Otherwise, the marginal peak-load plant would have negative NPV. Thus, the c-vu/wcc ofpeak-load l'c/??ltl/t7gl? will ?7t)/ eliminate prices. ct-/lfflibrlfp/lvariability in d/clrl'/y volatility curves in Figure 17.4 are declining oyer that the final point, note As a of Year standpoint From the 0, a z-yetrvolatility is less than a l-year volati tyk time. which for with stocks, in is contrast This we typically assume volatility is constant over time. To understand the behavior of volatility for electricity, recall the discussion of stock prices in Section 11.3. The assumption that a stock price follows a random wnlk implies that volatility increases with the square root of time. Thus, volatility enters the Black-scholes model as o' T r; this expression measures the volatility of the stock 'price over the horizon frortl t to T. By contrast, we do not expect electricity prices to follow a randc?m walk. When the electricity price is high, users consum less electricity and produeers increase production. 'When prices are low, usrs consume more and producers produce less. Thus, the price of electlicity reverts to a level re:ecting the cost with T at a rqte less of production. When prices revert in this fashion, volatility F r/?tw7 t. '
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To consider a specisc example, suppose it is January. From this perspective, the July price this year and the July price next yeltr have similar distributions; ske won't lenrn much about the July price this yar or next until we approach July. tTl'lis is not strictly true because economic activity and even weather can follow long-tenn cycles, but suppose that it is a good approximation.) To compute an option plice, we require annualized volatility, c', which the option pricing formula transforms into volatility over
l3lrlaug (1998,p. 59) discusses
approximations
that can be used to value spread options.
563
-
H x S :,s exceeds c. An option with this payoffcannotbe valued using the Black-scholes g formula because neiter Saec H x SgasnOr S x has+ c is lognormlly distributed. Equation (17.3)is rherefore an approximation once nonfuel costs are added to the strike
price.
$
-
Sgas+ c), 0J
X
CE
the life of the option, c' F t. If you believe the uncertainty this July and next July is the same, the annualized volatility will be lower for next July since a given amount of uncertainty, when annualized, is spread across a greater period of time. If o' T. t is the same for two different 6.s, the cs will be different and the volatility curve will declie with horizon.
This way of writing the plant's value makes apparent th difference between a static (NPV calculation and the real options valuation. The static ' alculation assumes operation at all times', we can value the plant by simply discounting the spark spread computed using pc/llc of a peak-load plant forward prices. Equation (17.3)also makes it clear that the Il/pt?/? prices 47?-: high-all ,1471 pDnloperating stenhntn operate Ip/ltv? prices t-frc does bvhe'l 7?'' Iov. shutting down prices high-but m//?d?'/r??7 ln reality, equation (17.4)is overly simplilied. There are marginal distribution, operation, and maintenance costs associated with an operating plant. Represent these costs as c. When we take these into account, marginal prolit is Profit
P RACTI
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process for a new drug. Probabilities are the percentage of Pharmaceutical drugs proceeding from one stage to the next. For example, 74% of drugs submitted for FDA approval receive it.
. -
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1% Preclinical Tesdng
7O% Phase I Trials
Phase 11Trials
47%
Phase III Trials
82%
FDA Approval
-r-lj-:--
74A
Marketing and Sales
Sottrce: Schwartz and Mpon
(2000).
k
564
CoM
op-rloxs
REAL
How do we evaluate pharmaceutical
investments?
The underlying asset is the
AS AN OPTI ON
$
565
P.eak-load pricing and research and development
are examples of how option
techniques are used in making investment decisions. In the next two sections we develop an extended example of commodity extraction, which is yet another area in which real option considerations are essential.
Assuming that all of these inputs are known, we can evaluate the sequential investment as in Figure l 7.6. The figure presents an example in which, in each period, row) to keep the it is necessary to pay an investment cost (shownin the Iinvestment''
.' .'i1iij:i:;;..
EXTRACTION
project alive for another period. The static NPV of the project is negative, since the initial value of the developed investment is $ 100, but the present value of the investment costs at a 10% rate of interest is $108.60. This static calculation ignores the staging of the investment, which pernzits maling later-year ipvestment costs only if the project shows prolnise. With staging, the value of the development option is $2.812. Schwartz nd Moon (2000),building on work by Pindyck (1993a),developed a general valuation model, with staging, which is applicable to phm-ntaceutical R&D.
value of the drug if brought to market. How do we find the value of this asset? With the gas) and peak-load electricity plant, we have forward prices for both the input tnatural the outptlt (electricity).We can estimate volatilities from market prices. However, in plaarmaceuticals, we must estimate development costs, potential revenues, volatilities, and correlations without the benetit of observing market prices. Project payoffs witl lisk, which must also be vary with the state of the economy and, hence, have systematic estimated.
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() $5
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Investment:
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$25
$100
EXTRACTION
AS AN OPTION
Natural resources investlpents are an important application of option techniques to inflom the ground exhibits many vestment decisions. 14 The extraction of a lesoulce siluilarities to the exercise of a Iinancial option. resotlrce has a value that can be realized by paying an extraction cost. The marketYorthe resotlrce is typically competitive so that the behavior of one producer does not affect the price. ln this section we will consider the problem of extracting oil frolu the ground. There is an initial cost to sink a well to columence prodtlction, after which we assume we keep producing forever. ln Section l 7.5 we introduce the possibility of shtltting down production when it is unprotitable. Our goal in studying the oil extraction problel'n will be to understand the ecoltonlics of this problem. The analysis is an exmuple illustl-ating the costs and beneEts of defen-ing investment and stopping and starting production. The specilic formulas do not pply in every sittlation. -
-
-
'he
5385.743 $285.743
$245.960 $96.216
$141.907
5156.831 $22.869
$41.907
$90.484
$100.000 $2.812
0.000
$52.205 50.000
557.695 $0.000
Single-Barrel Extraction under Certainty
$33.287 $0.000
Suppose there is a plot of land tlzat contains one ban-el of oil. The cun-ent price of a barrel of oil is $ l5, the oil forward curve is such tlzat the effective anpual lease rate, is 4% (constantover time and across matulities at a point in time), and the effective annual l'isk-free rate, r, is 5% (alsoconstant over time). There is no uncertainty about the future price of oil. The bal-rel can be extracted at any time by paying $13.60, which we denote ,
205 $0 000
$19
An example of staged investment. The value of the project, if developed, is in the top Iine at each node. The value of the option to develop the project is shown below the value of the project. In each year, it is necessary to pay the amount in the lnvestment row to keep the project alive in the next period. The tree is generated as a forward tree 0.15. ' 0.50, r 0.10, and f o' assuming S
E i
X. Finally, to make matters simple, suppose that the land is completely worthless once
the oil is extracted.
.$100,
=
=
=
=
l4see in particular Brennan and Schwartz ( 1985), McDonald and Siegel ( 1986), and Paddock et aI. ( l 988).
566
k: REAL
CoM
Op-rlolqs
If the price of oil at time 0 is
&, the time 0 forward price for delivery at time F is
given by
S 'T
=
1+
S0
1,
MODITY
$
AS AN OPTION
567
This expression shows that we defer investment as long as l't /
S
F
I
-<
wt-vt
(17.7)
Ja
X
(17.5)
1+ &
EXTRACTION
l+&
1.051/365 1 = 0.013368%, with the daily lease rate = In this case we have sd = Ij++l/ua is $16.918. The trigger price, at which s = xa,, essentially'th'e sa-nte that, since daily rates m.e as continuously compounded rates, we get the same answer by using continuous conpounding. We invest when -d
-
Since prices are certain, the future spot price will equal the forward price; hence, the spot 1 = 1.05/1.04- 1 = 0.9615% price of oil will grow frever at the rate (1+r)/(1 +J) -
PCI' XCILI'.
How much would you pay for this plot of land? The obvious answer a bid of $13.60) ignores the possibility of delaying investment. As with the Cllllnotjl6lhle Widget project, Xd Idllditllolftfl*stdecidillg lf?l#d?-11?/lJlcf?-cIf?)lJlJ?1&dS )?t?lf g?rIIn#. A bid of $1.40 is too low. The correct answer is to extract tp//-o/?l the yott 11IiW value of net extraction revenue, maximize the F present select to
$1.40 (= $15
( 1,
0.010746%. xote
ln(1.05) x $ 13.60 ln(l
-
Sv
.'r
-
(1 +
(17.6)
?')W
Using equatiun (17.3)to model the change in the oil price overtime, we can mechnnically find the F that maximizes expression (17.6).However, we want to discuss te reasons for delaying investment.
St
=
.04)
=
$16.91y
This shows why our back-bf-the-envelope answer of $17 is not exactly right. lnstead of the ratio of effective annual rates (5%/4%),we want to compui the ratio of computing compounded rates (ln(1.05)/ln(l.04)). continuously of the Iand We lnow that we will extract when oi1 reaches and appreciation How long will this take? The annual growth rate of the price price of $16.918+m4-e1. a .t094l5)J l = 0.9615%. We have to tindthe t such that $15 x (1 ofoil is 1.05/ l 12.575 years. At that point the value of extractin will $16.918. Solving gives us t NPV today is Hence, be $16.918 $13.60.
value
.0).z
=
-
=
-
The costs and benests of exaction are probably familiarby now. optlmal extrauion extraction, the barrel of oi1 in the ground appreciates at 0.9615% per year, lf we delay less than the risk-free rate. We-lose 4% per year-the lease rate--on the value of the barrel. However, exacting the barrel costs $13.60. By delaying exaction 1 yelr we earfl another year's interest on this amount.
Thinldng about costs and benests in this way suggests a simple decision rule, familiarfrom the widget project: Delay extraction as long as the cost exceeds the benest. The benet i tlziscase is constant from year to year since the exeaction cost is constant, with the but the cost of delaying extraction-the forgone dollar lease payment-grows ()il price. Tlzis line of thinking leads to a back-of-the-envelope extraction rule. Since the interest rate (5%)is 25% greater than the dividend yield (4%):,the dividend yield lost by not investing will equal the interest saved when S = 1.25 x $13.60 = $17. Thus, we should expect it to be optimal to extract the oil when S J7kJ $17. A more precise calculation is to compare the NPV of investing todqy with that of investing tomorrow. At a mirtimum, if we are to invest, we must decide that the NPV of investing today exceeds that of waiting until tomorrow to invest. If we let rd and d represent the daily interest rate and lease rate, then we defer investing as long as the present value of producing tomorrow exceeds the value of producing today. Sinee + #), we delay investing as tomorrow's oi1 price is today's oi1 price times (1 + rdlj long as
1 1 + rd
s
1 + rd 1+ d
-
X
>
s
-:
X
$16.918 $13.60 = $1 1 0512.575 -
.796
This is Iv/zclr J$?e bvottld ptzy-/r the /J??# today This substantially exceeds the value of $ 1 were we to extract the oil inzmediately. At whatrate does the land appreciate? The oil in the land is appreciating at0.9615% per year; nevertheless, the /tw?#itsefappreciates ('Ir 5%. If the land appreciated at less than 5%, no one would be willing to own it since bonds would earn a higher remrn. In fact, our valuation procedure ensures that the land earns 5% since that is the rate at which vhetherlqrodllcilzg or we dijcount the fumre payoff The p?rpc?-/l? opet-ated oil rdlE'?-p', ?-t?lld?7l ??,11-/ at alI tinles afaizthis the tnw?cr (in case, 5%). pt-l)' llot, .40
pricing formula
the option
Usins
This problelp is equivalent
to deciding when to
exercise a call option. By paying the extraction cost (the strike price), we can receive oil (the stock). As with a financial call, early exercise is a trade-off between interest saved by delaying exercise and dividends forgone. Olve we have possession of the oil, we can lease it; hence, oil's lease rate is the dividend yield. We can verify our answers by using the formula for a perpetual call option, Callpel-petual,discussed in Chapter l2. ln(1.05), and c.= 1n(1.04).l5 We get 0.0001, r Set S $15. K $13.60,tr =
=
=
=
$13.60,0.0001, callpel-pemaltsls,
ln(l
.05),
1n(l.04))
=
($1.796,$ 16.918)
(
l5To use equation rates. pounded
( I2.19),
ve must convert the interest rate and dividend yield to continuously
com-
%:REAL
568
CoM
Op'rlolqs
MOD ITY EXTRACTION
AS AN OPTION
$
569
oil in the ground is worth as much as oil out of the ground, so why pay the extraction cost? Equation (17.7) and the option pricing formula give the same answer, with the extraction barrier approaching insnity as the lease rate approaches zero. Thus, gold, t??' ally caowc/ve resolll-ce, u'f// llevel- be c-rl/-t-/c/'cfifthe lease rate is zeln.
The option price is $1.796 and the optimal decision is to exercise when the oil price reaches $16.918,exactly the answer we just obtained. The option formula implicitly makes the same calculations. Like the widget example, this situation illustrates the similarity between the exerof cise a nancial and a real option.
This discussion provides an answer to the question of why gold has a positive lease
rate. Investors hold a large stock of gold above ground despite the positive lease rate. The lease rate must therefore reqect a convenience yield earned by gold investors. This conveqience yield is reqected in the forward curve as a positive lease rate. The positive lease rate in turn makes producers willing to extract new gold.
...'
What if the cost of extraction, X, changes over time? costs ihanging extraction lnflation might cause X to grow, while technological progress might cause X to decline. lntuitively, real growth in the extraction cost will accelerate investment. The reason is that the benest from delaying investment is less: We earn interest on money set aside to fund extraction, but some of that money has to be reinvested to fund the growth in extraction cost. Thus, if g is the grosvth rate of the extraction cost, ourbeneft from elay is 1' g instead of r. lf we v iew the option to extract oil as a general optio to exchaj ne asset (cash) for anbther (oi1),our willingness to make the exchange depends on th relative dividend yields of the two assets. The option is equivglent to being long the ppderlying . asset without receiving its dividend, and short the strike asset without havipg to pay its dividend. A high dividend yield on the asset we are giving up (the strike asset) makes us less willing to make the exchange, other things equal. Positive growth in the extraction cost reduces the dividend yield on the asset we are giving up, maling us more anxious to give it up; hence, there is a lower trigger price. ln the example, if the growthrate of the extraction cost is 0.5% perannum (effective annual), then we would invest when, using continuously compounded rates,
Jmder
Single-Bnrrel Extraction
Uncertainty
-
'
.
0.04879 0.00498 $13.60 $15.19 0.03922 It will take 1.32 years to reach this price, and the land would therefore be worth 1.32)/1 051.32 $1 407. ($15.19 $13.60 x (1.005) 1. g X $
-
-
S
=
=
=
=
-
.
revijited
price.
-
.
'
In Section 6.7 we saw that the lease rate of gold is positive. We can now see that if the lease rate of gold were zero, it would never be optimal to mine go1d.l6 If a commodity has a zero lease rate, then the cost of delaying extraction is zero: It is always preferable to wait to extract. To see why a zero lease rate implies that we would never extract gold, think about the comparison wejust made between extracting oil today or tomorrow. If oi1 had a zero lease rate, then by defnition the forward curve would be growing at the rijk-fre rate. The present value of oi1 tomorrow would be the value of oil today; nothing is lost by leaving it in the ground. The gain to deferral, however, would be interest saved on the extraction cost. Thus there would have been no reason ever to extract the oil. In effect, Gold extrauion
l6-rhisalso
assumes that extraction cost grows less than the
fisk-free
rate.
X when S If we decide to extract the oil when the price is we will receive 12, know of barrier values in discussion how Chapter present we reaches From our Using the contingent the price is reached. X when payoff of presentz S S to value a valueformula dened by equation (12.17),the value of the extraction option is ,
.
Growth in the extraction cost hastens extraction and lowers the value of the property.
.' '
Now we consider the effects of uncertainty on the oi1 extraction decision. Before proceeding, tl.y to answer this question: lf we keep all variables unchanged (thelease rate? extraction cosq and so forth). except that the oil price is uncertain, how do the extraction trigger price and the value of the undeveloped land change? Option reasoning gives unambiguous answers to this question: The extraction trigger price goes up and the land becomes more valuable. The comparison of dividends (thelease rate) to interest savings in the previous example captures two of the three ,reasonsfor early exercise. The third is insurance that results from the ability to delay taking the project. With uncertainty the insurance has value, which increases the value of delay. The forgone dividend has to be greater before it is worth giving up the implicit insurance. Another way to think about the investment decision is that by deferring extraction of the oil, we have more time to see if the oi1 price will decline pr rise f'urther. This effect induces additional delay, in the sense that we will optimally invest at a higher
-
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('# $13.60)
=
-
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=
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-
-
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-
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This is the preseny value of our investment strategy. By varying S, we can see how the present value of the project is affected by different extraction trigger prices. Figure 17.7 compares the value of the land under different rules about when to pay $13.60 and extract the oil. When oil price volatility is 15%, the trigger price is higher and the lnd is more valuable. The trigger price that maximizes the value of the land is S = 25.3388. At this price, we have a project value
%.REAL
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Value of Undeveloped Land ($) 7 =.= 6
=
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(x
=
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=
fr
().000()1 0.10
Valuing
0.3
=
571
%
AS AN OPTION
invest 1 in order to turn the undeveloped reserve into a developed reserve, Exactly l year after that, the reserve will begin to produce one ban'el of oil a year forever at a cost c per baln-el. We solve this problem by working backward. We hrst compute the value of the firm supposing that it is already producing, and we then sttldy the decision abotlt when to invest.
.
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EXTRACTION
MODITY
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the producing
Once the lirm ha's invested, it will continue producing
firm
foreveltsince the price of oil is always rising. Recall from our discussion of commodity forwards in Chapter 6 that the lease rate is the discount rate linking the future commodity price with the current commodity price. Thus, the time ? valtle of a barrel received at J)F-J F),w/( 1,+ r)TStj l + time F is P P)(F;,w) The value the producing firm at time 0 is tlerefore
e'.v
#*
'of
=
a' P
V
=
X
J
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*;:.
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i= I
!,,,,,::,;,,,j,,,,,:,,,j,,,,.. .
.
>awoomx
/ t!t?7. hf
parea.
f
1
40 35 30 25 Extraction Trigger Price (5)
20
'
45
50
.796.
.$1
Valuing
-.5
=
$3.786.
the option
(1+ $3.7856
We can verify this calctllation by exploiting the insight that for option pricing purposes the lease rate is the dividend yield, and use the perpetual call formula:
Callpel-petualls l5, $13.60, 0. 15, 117(1.05), 111(1.04)1
=
The perpetual call calculation also gives s l3otlld occur.
=
($3.7856,$.25.3388)
$25.3388as the pcice
tt which exercise
'j'
Valuing an lnEnite Oil Reserve that can be Now suppose that the land contains an inhnite number of ban-els of oil firnz the that kYewill can at any time asstlme extracted at the rate of one ban-el per yeal-.
1-
sv (!
-
c
-,-
-
--i
-
1
=
1
l -j
(I +
,.)
v
(
Sw
-
g 1) c
-,.
+ 1
This is the value of the ltldeveloped oi1 reserve. Note the similarity with equation ( 17.6). If in equation (17.6) we replace Sv with the present value of oil extracted, Spj, and replace the extraction cost, X, with the present value of all extraction costs, cjl' + 1, then we have a problem that appears the same as in the single-ban'el case. We want to select F to maximize equation (17.9).The right-hand side of equation (17.9)expresses ban-els in the ground does the value on a per-barrel basis. times l /. Having luultiple not change anything fundamental about the problem if there is certainty and the oil price grows indeinitely.
ltI =
-
If the lirm invests at the price Sp, the value of the land at well less the investment cost, 1, or Sv/l cll' 1.
to invest
l r)T
$l 5 $13.60) 25.3388
c
-
that time is the valu of the producing The valge of the land today is
-
-
So
You might wonder why the present value of a ban'el of oil a year forever is So/. We know that a perpetual-coupon bond paying c/year is worth c/?' (the second term on the right in equation ( 17.8)). We saw in Chapter 15 that the lease rate on a colnmodity bond is analogous to the interest rate on a cash bond. The operating well is like a bond paying one unit of the comluodity forever. so the lease rate is the appropriate discount rate for a bond denominated in a commodity, and &)/ is the value of the well.
Value of Iand containing one barrel of oiI as a function of the trigger extraction price, = 0.00001, the maximum is -.S', for four different oil volatilities. For the curve where c of value For the curve where corresponding with 6.918 at 51 a attained value of corresponding = with attained $25.3388 at a c = 0.15, the maximum is
($25.3388
i= I
>==.=
( l + r4i
( l + &4
j
..f7
15
'-
=
()
10
?-)f
'>...
WMVQ
c
ua
=
xzyox
#:
..'
(l +
X
c
-
*>''>-.>>
*
1!
F(),i
.
Suppose So ti!)Example 17.3 )') ''jt the producing well is $15/0.04
i! .y'.
) l
$ l5, 1' $8/0.05
4%, c = $8, and the value of $215. lf the investment cost, 1, is $ l 80, then bleper-barrel extraction cost is cjr +. 14 = 0.04 x ($8/0.05+ $180) $ 13.60. The problem is the same as having 1/ options to extract at a cost of $ l 3.60; hence, the =
-
=
5%, J
=
=
=
572
k
REAL
Op-rlorqs
CoM
solution is exactly the same as in the single-barrel case. To appreciate the similarity, use the option pricing formula: Callperpet'ual =
jtl.t),yj $8
$l5
,
().:5
+ 180, 0.000001, ln(1.05),
ln(1.04)j
$422.956. Thus,
extraction
k
=
$15 $8 ,
+
$180,0.15, ln(1.05), ln(1.04)
=
($94.839,$633.4691
The well is wort.h $94.639and we invest when it is worth 5/0.04 = $633.469,or when S = $25.3388. On a per-barrel basis, the well is worth 0.04 x $94.639 = $3.7856. With these assumptions, the solution is the same as in the single-arrel case. k
ln the absence of any shutdown opyions, the single- and infrlite-barrel cases differ only in scale. The interesting difference arises when it is possible to avoid operating losses by shutting down, which matters pnly in the multiple-barrel case.
AN D RESTART OPTIONS
k
rjf F' j'(' y' p' t';')' EEL.)(.r.' .EE E'.'( 'y'yf yjf i'?'.'jE)'E(!E '''''E7'''7>/ES''':rqTE.i:' 'Eil''(.r''.'y ' .'.':.' .E):.E t'q'$'j'k' ).':;'' ...'E. '. . E .1.'L.. ..E.E:.. E' E. . . I !j.: . .. . .. .. l E : ;!.. !IEi. i..E!.E. . ... ... E. ;i... E.. E i.;E: : '. ..j.j(.y.kj..E.. E .. E .. . .E .E .. EE . . :..(; .... qq ..;. j(....g.q . . .. .j.. . . E.. .. .. jjjjjg,E) jq:.j..Ej @. E (y .E..g...j!(:2 .'. . .)gjy:'yIjy. g.yjE. .qF(!;EE..:.rjjj. . tjEl.iyyjj j q;i.j(...j.. .y.j . . .)j:.. . ;y . . ,j. .t..t.;.:t yy, .t . . . q. .j(j::ygjy......@ .;)Cgjjjj;i ..,...tk..q j..j .( jlj:,gjj .(. L,.LL ..L., . 1:::: jjE:;j:,. .EE.:. jg(:q;' :33433). 4(:(2:). ........,....-...'..3,,....-------.-,.1L-'.L.L.i.LL?L.-b'.,--'.3. .jgj .... ji.!I.1Iq2'' .....q......E)..... lrrb ..E.. ji. 4:5:,4:k11141, ;'!' L'
.' '''
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.
Investment and Operating decisions for oiI well. lnitial an investment
OCCLICS
:
2),
.5'
..--.--.-..---...-....--.-.-............--.-.
== ....-. .-.
A .
z
-.
,.z==..==
-
Oil Price . lnvestment Trigger Restart Trigger shutdownTrigger
...-..-....-.--,,-.--.,--..---..-..........-..-...
..
..
. ..
when the oil price crosses the investment trigger price, at point -5'
17.5 COMMODITY SHUT-DOWN
EXTRACTION AND RESTART
WITH OPTIONS
With production occurring over time and uncertainty about the price of oil, we face two new operating decisions'. 'Whether to keep the well operating, or, if it has been shut down, whether to reopen it. There are thus three stages of production'.
nitial investment in the well: We begin with an empty feld containing oil. This is an undeveloped well. At what point do we drill the well and begin extraction?
We answered this question in Section 17.4, assuming that the well operates until the resource is exhausted.
573
Thus, thewell can be in one of three states'. Undeveloped, producilg, and shutdown
Make the same assumptions as in Example 17.3, except suppose 17.4 the price of oil lognormally distributed with a constant lease rate and volatility is is that 0.15. The land value anct optimal extraction decision is given by
0.04 0.05
SHUT-DOWN
(developed but not producing). Figure 17.8 shows a hypothetical price path of oi1 over time and possible operating rules. Investment occurs the lirst time the oil price reaches the investment triggerprice, S. Prodution is shut down at the shutdown trijgerprice, S., and restarted at the restart trigger plice, S*. Thus, before pointAthe well is undeveloped. Between points A and B, and after C, the well is producing. Between B and C the well is shutdown. Key questions are: How do we determine the investment, shutdown, and restart triggers, S, S., and S*' and what is the value of the land on which the oil is located? Once again, we have to work backward, as in the binomial valuation of a stock option. Before we can decide the rule forinvesting (determining), we have to determine the value of a producing well (thisis the present value of future cash flows at the point w
Example
Callperpetual
WITH
Restarting an operating well: Having shut doFn the well, if the oil price rises agin, it may be pojlble to pay a cost and l'estat't the well, turning it back into a producing well.
occurs
With uncertainty, we could have the ability to shut a producing well. We will assun.e for the moment that production continues forever. ln tat case, the prolem is the same as in the single-barrel cse.
o'
EXTRACTION
Once we have made the investment in an oil well, we Contintling to produce: say that the property is developed. However, a developed well may or may not be producing. lf we are extracting oil from the ground, we have a developed twzt'/ producilq well. However, if the oil price drops below extraction cost, it may make sense to pay a cost in order to shut down the well and avoid future operating losses. Then it is a shlttdovll well.
$422.956) ($z1.4.914,
The value of the well at which extraction occurs is when S = 0.04 x $422.956= $16.918.
MODITY
A. Shutdown of production occurs
the shutdown trigger, & (point B). Production is restarted at the restar-t at point C. trigger,
c S*
.*,
n B Time
..
.
.
..
k. REAL op-rlorqs
CoM MODITY
invest). ln order to value a producing well, we need to understand operating decisions, specifcally how S,v,and S* are determined.
It is helpful to analyze shutting down and restarting by considering three separate
3. Production
once, this time permanently.
.%
..'.
Suppose that we are operating the well. If the current price is S and we ignore shutting value of the operating well is simply
down, the
P'operating, no
shutdown
.
=
S -
-
-
&
c
(17.10)
?-
Suppose that we can at any time pay a cost of kx, abandon the well, and never produce again. S,vis the price at which we shut down. What is the value of shutting dowh? There are three considerations:
Once we shut down, we no longer sell oil. Thus, we give with present value Sj8. 2. We no longer pay the exaction
up the revenue stream
cost, so we gain te present value c/?'.
-
c
-?
.
-
ks
=
c
( ) -p
.
-
ks
.% -
.,,..-.ty
'
,'
, .04)1
=
''''
''''
''''''
''''
-$53.
-
t) !
li.
E
ltt
When ks
) jy
PutPerpetual($10/0.04,
$8/0.05
) ltt The
-$69.863.
=
-
25, 0.15, 0.04879, 0.03922)
0.04 x
$1.8j
: .
..'
.'
q
.
'. .
E
( 17 11) .
This is #7' payoff to a pl// option wl/l strike price c/?kx and asset price ,S/1. If value this operating and the price is detenmine the value of th S, put to we can are we option to shut down, as well as the trigger price, %, for shutting down.
($5.778,$90.137)
=
$90.137 $3.605.We pay $25to avoid losses =
%.
=
=
'
-
The value of the producing
well Given that shutdown is possible, whatis the value of a producing well? The answer is that the value of the well is the value of the perpetually producing well plus te value of the shutdown option: optiont) Voperaungts')= Fkoshutdownts') + Vshutdown
l-/ef'hemodel with inhnite shutdown and restart was lirst analyzed in Brennan and Schwartz (1985)and subsequently by Dixit (1989).
. '
Ci
To interpret the shutdown resultsa there are two nattzral benchmark prices to consider. The flrst is tlze price at which te NPV of the operating will become zero, wlzich $6.40. The second is the price equal to the parginal ost occurs Fhen S 8 x c/r of production, c $8. lf S > $8, then we are mnking money by operating and we wilt not shut down. If $8 > S > $6.40,then we are losing money on an operating basis but the NPV of operating the well is positive. In this case we would consider shutting down if we could later restart production, but, because the NPV of operation is positive, we would not shut the well if the shutdown were pelanent. Finally, if S < $6.40,it may make sense to shut the well even if the shutdown were pennanent. Note that in a11cases, te initial investment is sunk and therefore irrelevant. In Example 17.5, shutdown is pelwanent so tll zero NPV price CS $6.40)is' natufal the benchmark. The usual option exercise logic applies: We won't shut down as soon as present value is negative, because the decision is Aeversible. In fact, we wait price might subsequently increaset by shutting down we until te plice is below $5. unable to benest from this reversal. This is the counterpat't to not investing as soon are as NPV is positive. =
.
.....
$53.17.
$25, the shutdown solution is
=
.
,..
($9.633,$10.301
'
' .'jji)By shutting down production, we avoid losses of (
. '').. '(t... i..; . .
.....'
'l'he
Thus, the value of shutting down at price Ss at a cost of ks is +
..
..
=
3. We give up k, the shutdown cost. sS.
$8/0.05, 0.15, ln(1.05), ln(l
shutdown trigger is then S lilt 1 with a present value of
Permaneqt Shutting Down
.'
shut down production when 5'/ = $106.83or when s = 0.04 x point, the present value of continuins to produce is $4 2-/3 $8.00 = 17 0 04 0 05
E.
j
..y
..
. Putperpetuallslo/o.ozl,
), ) nus, we lp $4.273,.At this i'
y tt' t)E)
. .
'
)
Each case layers a new option on the previous case. ln qddition to allowing additional shutting down and (estarting, we can impose costs of doing so. We focus in this section on the case where the well can be shut down never or once after the initial investment. Appendix l7.B adds restarting to the analysis.
..
Ejy
'i
an infnite number of times.l?
can be shut down and restarted
$
j')'pj Example 17.5 Suppose the oi1 well is operating and the oil price is =:u $10. We '(7't)i = = also ' have 0.15, and effective annual r gnd 8 are 5% and 4%, fjjeiikely. o' $8, c (? ))1If ks = $0, the value of the option to shut down is . .. )
.
2. Production can be shut down once, then restarted
AND RESTART Ol7Tlolqs
t'j' i'
tEj
1. Production can be shut down once permanently. After the well has been shut, the land has no additional value.
SHUT-DOWN
WITH
)'
) .ltr
C aS e S :
EXTRACTION
% REAL
576
Opn-lolqs
CoM M ODITY
E''' 7E jjjf j'. j.jf yrf )' r' *' t'y' j't;;;'. jIIjjj!(-'. 1* jlllrf . 'fL;;''q''. )' .' r:-' i. .. EE i fi. r ! E....k..jjjjy qk ; : ; .( r. ! . q.qy. q ....j( .... ;. qljljjk;pl!t. ).;jyyj. .... y.j... .qy..... g.y...jq..I. y.y t t Iillk:,,jjjl . jlj;::;ii::. . ig.)j.yyjlj;q:;:r .(il.. r . .'i(.t; j(t.-!. . ( .yj.jg .g...j j..jj.y.. ., j.y.j;y.j.y. . . yj lyj.yyy y (. . . . .... . . .. . . .'a,pikj-pip-.'-;j:jjs '''t' .F' s' ;';' !' 'Eyf :'r' -'tyyj -' :;' ''''' s' 7)* t'f.tf ;' jj:ygr:jjf ,j2jj,!' y' j'''y'' t)' t?f
..' .'
jsE' : : ' ' 'Ef't.rE ' j't' ''!' q' t';y' ('.' :E y' .y;..;jjy' rytf qty'i yf )' j'. f'. q' ' ;' i' .:''' . E '::' . '. ' : . E
.''.
'
.
' :
.
.
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.
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. ..
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.
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.
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-
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.... ..
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: E' ' E' ' .. . . E '....
'.
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'' 7* S''( C''' E'7'7*
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.
. -.
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,
-
Value of a producing well that can never be shut down (eqation 17.8), and that can be shut once wth no shutdown cost (equation 17.1 2). The shutdown triggers for 'ihedifferent volatilities
are $6.40 c 0.00001), $5.01 (c 0.1 0), $4.27 (tz 0.15), and $2.68 (c 0.30).
' ' : '' E' E .
-
-...
.. .
.
;.
'' '' ' ' ' .. . .. ' .
. .E. .y-...
E
.
kjjjy.
y
Developed A'Vell($) Weil withotlt Shtltdown Option 0.00001 tr
--
150
$
AN D RESTART OPTIONS
Value of well at time of investment
=
P'p
=
)
1
-
-' '
100
..*--.
o'
=
(r
=
0.15 0 3
..---c-:.r
.
.::.3
50
-!.-,
....-,,.
---1'%V.'''--''
o
=
:.
.
.
..
'
sr r
w
t
'
.:.z .r...*,--.
-ql,
To solve for we need to find the present pa/lfd of equation (17. l 3), and then choose maximize this egyation to present value. For a givn (l7.t3) tells us the value of lf the oil price today is S < S, we can compute the present value investing when S of equation ( 17.13) using equation (17.14).The value will depend upon the current oil price (thelower the price, the longel- it will take to hit S4,so we denote it as nvesttz; 'X). This present value is 'j
vp%
.
.
=
..e-
.....<*-
=
q w-we''x..-
=
slope
=
.w,'o/
=
-100
jj&
+=''
.
...ee
S 11l
+.e
.
-150
Jlnvesttk; -S)
,,.e','
'
-200
0
1
2
3
4 Current
5
6 Price oil
7
8
9
($)
-c/?.
How does the ability to shut the well affect the initial ilwestment decision? Once we drill the well, the maximum potential loss is less because of the shutdown option. The ability to shut down makes us willing to invest sooner.
-
=
=
S '.
-
-
c 1'
;
+ Putperpetual
-
-
c
,
1'
-
ks,
ln(1 + rJ, ln( 1 +
(y',
'
Equation ( 17.14) can be maximized with respect to m ri ca l p rogl-am.
S using
t5)
-
'
-
1
a spreadsheet or other nu-
1* J'
i'
Example 17.6 Suppose $ 0.04, ?' 05, c 0.05, o' $8, ks $0, and ('l I lf the oil value is S, $ 15, then the of- yhatmaximizes equation price, current $180. 1. g . . (17.14) is $25.12 an dthe value of the undeveloped well is l5, $25.12) $95.13. Wnvestts ':1' ' ' ' . is ' . l and ' Well . the udevelped . .the val . of tt If ks $25, then $25.2 $94.93.' Er'p
=
il7'!
-.-$160
en Shutdown ls Possible
-#
x
10
Figure 17.9 graphs equation (17.12)for a range of oilprices and four different volatilities, along with the value of the well without ashutdown option. Withoutthe shutdowp option, = the value of the well is like a stock and declines to wen S = 0. With (Recall that the option, the well is worth zero once it is shut. once te wll has been shut, the land has no additional valtle.) When the oil price is signifcantly above the shutdown price, the shutdown option worth little and. te value of the well changes by 1/ for each $1 change in the oil is price. (The A of the well is 1/.) Close to the shutdown price, however, the value of the well becomes less sensitike to the oi1 price, because the shutdown option is increasing in value to absorb the effect of declines in the oil price. ln each case, the value of the well smoothly approaches zero as we approach the shutdown price. This example illustrates how the shutdown option affects valuation of an operating well. The neyt question is how the shutdown option qffects the decision to invest in the well in the tirst place.
Investing
SH UT-DOWN
To account for the value of the shutdown option we work backward. Equation (17.12) gives the value of a producing well. Call this VpS). If we invest at the price J, paying an investment cost of 1, then the value at tll ?/p?' I/d inb'est is
.. .
.
-
,
EXTRACTI ON WITH
=
=
=
=
=
E! l'E
=
,
'
=
=
7!.(y ff we increase i:j
'
the current oil price to s = $20, the the value of the undeveloped . . k.v= $0 and $176.64when ks = $215.S is the same as
lt when ' ) well increases to $177.01 k1/,when $15. u
=
.
.
Q.
This example illustrates some key points. First, as discussed earlier, the ability
to shut down reduces the investment trigger, from $25.34 with no shutting down, to $25.12 with shutting down. Second, if there is a cost of shutting downa shutting down occurs at a lowerprice and provides less protection. This mitigates the benest of shutting down, raising the shutdown trigger to $25.21 Finally, a point that may be obvious but is important to understand: The investment trigger implied by maximizing equation (17.14) is independent of S, the current oil price. To see why, suppose that S $25. If .
=
S S
=
=
$ l5, it must pass $20 before reaching $25. Thus, if we evaluate the option when $20, we must obtain the same as when S $ l5. Thus, S is independent of S. =
% REAL
578
Op-rlolqs
579
Restarting Production The preceding example assumed that the lirm could never restart once it had shut dow. In this sectin We xamin te restrt stiatgy if the 5rm could restart after it jennanently had shut down. Suppose the firm can pay kr to restart production. The ability to restart is a call option in vlzich the 5rm receives Sj8 by paying c/r + kr, f'uture production costs plus
the restart cost. 1* t';':
q Example
t'E)
0.0
The value of a shutdown well is
17.7
)) c + kr, o', logs + r), log + 84 Callperpet'ual t.( ?'I' lr'i k 0.05, o' 0.15, c $8, and kr $0, the option 0.04, r t't Assuming that S $10, well as $94.46and $11.92as the price at which 1') of the h alue pricing gives formula us t e v . t) )) ), to restart. k. ,s
-
,
,
:
=
=
=
=
=
=
.
..
The ability to restart affects the decision to shut down. 'Whenwe shut down, we not only cut offfuture losses but u also acqltire a call optiol to ?'wMrl. ln equation (17.14) we acquired a put option when w ivested, stj in this case we acquire a call option when we exercise the put option! And when we invest in the srstplace, we acquire the put option to shut down, but the value of that put now implicitly contains the call option to restart. The solution for this problem appears in Appendix I7.B.
shutdownproductiopmany timej. Wecan determine sirategy in the previous following by value of th well and solve for the the triggers Appendix I-/.B. solution in Details of sections. the are Table 17. 1 summarizes theprice trigers forseveral differentcases. The qualitative results are intuitive. As with any American option, we require that the oi1 well have positive NPV before we invest-we are reluctant to ldll the put option implicit in the option to take the project. Ifshutting down in the futtzreis possible, there is an additional put option available besides tat f'rom deferring investment, and we are willing to invest at a lower price. The addition of an option to restart once we have shut down makes us more willing to shut down, and, hence, more willing to invest. Adding costs to restarting and shutting doFn makes us more relucta.nt to restart, to shuydown, and, hence, to invest initilly. More options generally mean more value and investment at a lower price; greater costs meap lower value and investment at a higher price. The results in Table 17.1 illustrate a phenomenon called hysteresis, which Dixit failure of an eflkct to reverse itself as the underlying cause (1989, p. 622) defnes as
'
abletorestartand srrnmightbe
4t-he
25
0.15
0
0
25.34 25.12 25.14 25.00
25 25
25.17 25.17
0
.--
4.27 3.60 6.03
11.92 13.79
4.33 4.37
is reversed.'' Suppose that a1l oil producers have a marginal extraction cost of $8. The current oi t prt ce i# $7 following a period in which it was $30 and there is a shutdown' cost. Uilprouction is currently unprofitable, and we Fould not invest,in new capacity at this price, but pmduction frop existing wellsis not unprohtable enough to shut down production. We are in a smp.tion where the cause (te oil plice) reversed itself, but the effect (thecreation of an oil well) did not. Oi1 producers lose money on an operating basts, but are not losing enough to shut down produciion. Real-life investment decisions exhibit hysteresis. To illustrate hysteresis in difa ferent context, Dixit (1989)considers investment decisions of a manufacttlrer with operations in a foreign countly Wchange rate iuctuations will change the prcstability of the foreign investment. Ho/ever, since investing and disinvesting afe costly, it will be optimal yo wAit until the investment is suffciently prohtable before investing, and suficiently unprtjstable before disinvesting. What appear to be sluggish investment decisions may s )pp ly result from costs of undoing what has been done.
Additional Options The
0.15 0.15 0.15
.
.
. ' . .
.
,
..
,
'
.
.
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Real options is tlil analysis of invstmnt decisions taking into account the ability to revise future opernting decisions. Examples of real options include timing options (the ability to choose when to make an investment), shutdown options (the ability to stop production in order to avoid losses), sequential investments where the decision to make later investments depends on the outcome of earlier investments (commonin R&D), and
580
k
REAL
op-rlorqs
natural resource extraction. lnvestment decisions in which such options are present can be analyzed using pricing tools from earlier chapters, such as the Black-scholes model, perpetual options, binomial trees, and barrier present value calculations. ln some cases the optimal decision is equivalent to the problem of when to exercise an American option. 'In general, however, as illustrated by the oil extraction problem, a simple option formulation is just a starting point for analysis. Even when standard option plicing models are not directly applicable, understandthe economics of investment
ing the econonzics of derivatives is helpful in understanding and operation decisions.
FURTHER
READING
In later chapters we will encounter more generi pricing techniques that expand our ability to solve real options problems. Early papers that used techniques from tinarlial options to analyze real assets include Brennan and Schwartz (1985),McDonald and Siegel (1985,1986). 'Fhese papers sttldy investment timing and the option to shut down and restart. Brennan (2000) insightfully summarizes the literature since then. There are several valuable books on real options, including Dixit and Pindyck (1994) and Trigeorgis (1996). A number of papers have applied real options to understanding the real estate market. These include Titman (1985),Grenadief (1996),and Grenadier (1999). Many sl'ms uje capital budgeting techniques more sopltistiated than simple discounted cash flow. Triantis and Borison (2001)survey managers on their use of real options, identifying three categories of real options usage: As an analytical tool, as a laguage and frnming device for ivestment problems, and as an orjnnizational process. McDonald (2000)argues that the use of high hurdle rates in capital budgeting could be an approximate way to account for real options.
PROBLEMS 17.1. Suppose you have a project that will produce a single widget. Widgets today lisk-free rate is 5%. Uder what cost $1 and the project costs $0.90. The circumstances would you invest immediately in the project? What conditions would lead you to delay the project? 17.2. You have a project costing $1.50 tat will produce /1ptp widgets, one each the srst apd second years aer project completion. Widgets today cojt $0.80each, with the price growing at 2% per year. The effective annual interest rate is 5%. When will you invest? What is the value today of the project? 17.3. Consider again the project in Problem 17.2, only suppose that the widget price is unchanging and the cost ofinvestment is declining at 2% per year. When will you invest? What is the value today of the project?
PROBLEMS
$
581
17.4. Consider the widget investment problem outlined in Section 17.1. Show the following in a spreadsheet. a. Compute annual widget prices for the next 50 years. b. For each year, compute the net present value of investing in that year. c. Discount the net present value for ach year back to the present. Verify that investing when the widget price reaches $1.43is optimal. 17.5. Again consider the widget investment problem in Section 17.1. Verify that with S $50, K $30, ?' 0.04879, (7' 0, and J 0.009569, the perpetual call price is $30.597and exercise optimally occtlrs when the present value of cash flows is $152.957.What happens to the value of the project and the investment trigger when you change S Why? What happens to the value of the jroject and the ilwestment trigger when you increase volatility? Why? =
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=
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=
17.6. The stock price of XYZ is $100. One million shares of XYZ (a negligible fraction of the shares outstanding) are buried on a tiny, othenvise worthless plot of land in a vault that would cost $50 million to excavate. lf XYZ pays a dividend, you will have to dig up the shares to collect the dividend. a. lf you believe that XYZ will never pay a dividend, what would you pay for the land? b. tf you believe that XYZ will pay a liquidating dividend in 10 years, and the continuously compounded lisk-free rate is 5%. what would you pay for the land? c. Suppose that XYZ has a 1% dividend yield and a volatility of 0.3. At what price would you excavate and what would you pay for the land? 17.7. Repeat Problem 17.6, only assume that after the stock is excavated, the land has an alternative use and can be sold for $30m. 17.8. Consider the widget investment problem of Section 17.1 with the following modiscation. The expected growt.h 4ate of the widget price is zero. (This means there is no reason to consider projet delay.) Each period, the widget price will be $0.25 with probability 0.5 or $2.25 with probability 0.5. Each widget costs $1 to produce. a. What is the expected widget price? b. If the lirm produces a widget each period, the NPV of the widget project?
regardless
of the price, what is
c. If the tirm can choose to produce widgets only when the widget price is greater than $l, what is the NPV? d. What happens to the NPV if widgets can cost $0.10or $2.40with equal probability?
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582
%.REAL op-rlolqs 17.9. To answerthis question, use the assumptions of Example 17.1 andtherisk-neutral valuation method (andrisk-neutral probability) described in Example 17.2.
In the following five problems, assume that the spot price of gold is $300/oz,the effective annual lease rate is 3%, and the effective annual risk-free raie is 5%.
value of a claim that pays the square root of the cash flow
17.17. A rnine costing $275 will produce 1 ounce of gold on the day the cost is paid. Gold volatility is zero. What is the value of ttae mine?
b. Compute the value of a claim that pays the square of the cash flow in period 1.
17.18. Amine costing $1000will produce 1 ounce of gold per yearforever at a marginal extraction cost of $250,with production commencing 1 yearafterthe rnineopens. Gold volatility is zero. 'What is the value of the mine? . 17.19. Repeat Problems 17.17 and 17.18 assuming that the annual volatility of gold is 20%.
a. Compute the in period 1.
lisk-neutral valuation, back c. Given your answers above computed using l?7Ic discount rate that would give the the you same value for each out claim. ln each case is this rate bigger or smaller than the 11% discount rate for the cash flow itself. Why?
17.10. Consider a project that'in one year pays $50 if the economy pedbrms well (the stock market goes up) and that pays $100 if the economy performs badly (the stock market goes down). The probability of the economy perfonuing well is 60%, the effective annual lisk-free rate is 6%, the expected return on the markry of the project is is 10% and and the beta '
-0.50.
a.
Compu te the present value of the project's cash flows using the true PrO
babilities
and expected return on the project.
b. Compute te risk-neutral probability of the economy performing well, valuation. then repeat the valuation of the project us ng lisk-neutral y
17.11. Verif'y the binomial calculations in Figure 17.3. 17.12. A project costing $100will produce perpetual net cash flows that have an annual volatility of 35% with no expected growth. lf the project existed, net cash Cows today would be $8. The project beta is 0.5, ihe effective annual risk-free rate is 5%, and the effective annual risk prernium on the market is 8%. What is the this static NPV of the project? What would you pay to acquire the rights project if investment lights lasted only 3 years? What would you pay to acquire perpemal investment rights?
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.
17.20. Repeat Problem 17.18 assuming that the volatility of gold is 20% and that once opened, the mine can be costlessly shut down forever. What is the value of the mine? What is the price at which the mine will be shut down? 17.21. Repeat Problem 17.lS assuming that the volatility of gold is 20% and that once opened, the mine can be costlessly shut down once, and then costlessly reopened once. What is the value of the mine? What are the prices at which the mine will be shut down and reopened? APPENDIX 17.A: CALCULATION OF OPTIMAL TIME TO DRILL AN OlL WELL Appendix available online at www.aw-bc.com/mcdonald. APPENDIX 17.B: THE SOLUTION DOWN AND RESTARTING
WITH
SHUTTING
'to
17.13. Aproject has certain cash flows today of $1, growing at 5% peryear for 10 years, tfter which the cash flow is constant. The risk-free rate is 5%. The project costs $20 and cash :ows begin 1 year after te project is started. When should you invest and what is the value of the option to invest? 17.14. Consider the oil project with a single barrel, in which S $15,?' 5%, & 4%, and X $13.60. Suppose tat, in addition, the land can be sold for the residual value of R $1 after the barrel of oi1is extracted. What is the value of the land? =
=
=
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17.15. Verify in Figure 17.2 that if volatility were 30% instead of 50%, immediate exercise would be optimal. 17.16. Consider the last row of Table 17.1. What is the solution for S. and S* when ks. kr 0:7 (This answer does not require caiculation.) =
=
Appendix available online at www.aw-bc.com/mcdonald.
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(r2)
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z-
(,
We can use the normal distribution to compute the probability of different events, but we have to be careful about what we mean by an event. Since the distribution is continuous, there are an infnite number of events that can occur when we randomly draw a number from the distribution. (This is unlike the binomial distribution, in which an event can have only one of two values.) 'T'he probability of any particular number being drawn from the normal distribution is zero. Thus. we use the normal distribution to describe the probability that a number randomly selected from the normal distlibution will be in a particular ralqe. ' . We could ask, for example, what is the probability that if we draw a number from the standard nol'mal distribution, it will be less than some number a The area under
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As an example, 1(0.3) is shown in Figure 18.2. In the top panel, 1(0.3) is the area the normal density curve between and 0.3. ln the bottom panel, M(0.3) is a point oyl'it(ki( i : . .( the cumulative distribution. The range all +x possible to . covers . outcomes for a ..tttt . j. (... singl draw from a normal distdbution. The probability that a randomly drawn number . E''''), will e tesstan cx) is 1,' ence, Nx) = 1. As you may already have surnsed, the C,)' Na) defined above is the same N ) used in computing the Black-scholes formula. ; y'E, There is no simple formula for the cumulative normal distribution function, equation (18.2),but as we mentioned in Chapter 12, it is a frequent-enough calculation that modern spreadsheets have it as a built-in function. (In Excel the function is called
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to the left of a, denoted Na), equals this probability, Probz < 7). We call X(fl) the cumlllative normal distribution function. The integral from to lj the area under the density over that range', it is cumultive in that it sums the probab from -cK) to a Mathematically, this is accomplished by integrating the standard nal density, equation (18.1) with Jt = 0 and c. = 1, from -G) to '.t(EE' a ' ..g. El(. y. . . yjyy
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NornlvsDist.l The area under the normal density from to 0.3 is 0.6179. Thus, if you draw a number from the standard normal distribution, 61.79% of the time the number you draw will be less than 0.3. Suppose that we wish to lnow the probability that a number drawn from the We have standard normal distribution will be between a and Probtz
Q
DISTRI BUTI ON
This result will be helpful in interpreting the Black-scholes formula. If we have a standard normal random variable z, we can gyerate G2), using the following: AJI/7,,
(18.3)
a varible
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x
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, Finally, if a variable obeys the standard normal distribution, it is extremely unlikely to take on large positive or negative values. The probability that a single draw will be below or above 3 is only 0.0027. If you drew from a standard normal distribution only about once a year. The probability every day, you would draw above 3 or below which, with daily draws, would occur on being below above 4 is 0.000063, of ol about 43.25 years. once evel.y average
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Converting a Normal Random to Standard Normal
If we have an arbitrary normal random variable, it is easy to convert it to standard normal. Suppose
Sums' of Normal Random Variables Suppose we have
variables i = 1, n, with mean and variance E (.vf) = a Ftzrt.pl and covariance Covxi xj) o'i o'ij. (The covariance between two Jzf random v ariables measures the ir tendency to move together. We can also wlite the covariance in terms of pq, the correlation between xi and xj'. o'fy = pijcicjj) Then the weighted sum of the 11 random valiables has mean ,
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592
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LOGNORMAL
TH E LOGNORMAL
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where i and j represent arbitrary weights. These formulas for the mean and are true for any distribution of the xi.
variance
In general, the distribution of a sum of random variables is different from the distribution of the individual random variables. However, the normal distribution is an example of a stable distribution. A distribution is stable if sums of random variables have the same distribution as the original random variables. ln this case, the sum of normally distributed random variables is normal. Thus, for normally distlibuted xi, 31
11
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i=1
f=I
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18.2 THE LOGNORMAL
ncluding the story of the normal distribution is entertainingly related in Bernstein (1996)3blost statistics books discuss one or more versions of the central limit theorem. See, for example, DeGroot (1975,pp. 227-231) or Mood et al. (1974,pp. 233-236).
593
DISTRIBUTION
A random variable, y, is said to be lognormally distributed if ln(y) is normally distributed. Put another way, if is normally distributed, y is lognormal if it can be written in either of two equivalent ways: .:
'
ln(y)
)?
The central
2The history of statistcs
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distributed. Since the central limit theorem is a theorem about what happens in the limit, sums of just a few random variables may not appear normal. But the normality of continuously compounded returns is a reasonable starting point for thinking about stock returns.
u/zj
Why does the normal distribution appear in option pricIimlt theorem ftequently in other contexts)? The normal distribution is important because it (and ing arises naturally when random valiables are added. The normal distribution was originally discovered by mathematicians studying series of random events. such as gamblinjz outcomes and observational errors.z Suppose, for example, that a surveyor is mnking obselwations to draft a map. The measurements will always have some enor, ynd the error will differ from measurement to measurement'. Errors can arise from observational or simply from recording the wrong number. error, imprecise use of the insrlpents, Whatever the reason, the errors will in general be accidental and, hence, uncorrelated. If you were using such error-prone data, you would like to lnow the statistical distribution of these errors in order to assess the reliability of your conclusions for a giken number of observations, and also to decide how many observations to make to achieve a given degree of reliability. It would sem that the nature of the errors would differ depending on who made them, thekind of equipment used, and so forth. The remarkable result is that sums of such errors are approximately normal. The normal distribution is therefpre not just a convenient, aesthetically pleasing distribution, but it arises in nature when outcomes can be characterized as sums of independent random variables with a snitevariance. The distribution of juch a sum approaches normality. This result is known as the central Iimit theorem-3 In the context of asset remrns, the continuously compounded stock ret'urn over a 'year is the sum of the daily continuously compounded returns. If news and other factors are the shocks thatcause assetprices to change. andif thesechanges are independent, then it is natural to think that longr-period continuously compounded rettlrns are normally
DISTRI BUTION
=
-v
=
e
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This last equation is the link between normally distlibuted continuously returns and lognormality of the stock price. By desnition,the continuously compounded return from 0 to t is S(0, t)
ln(&/&)
=
compotlnded
(18.11)
Suppose S(0, t) is normally distributed. By exponentiating both sides, we obtain St Equation
=
Ste R0,t,
(18.12) shows that if continuously
( j g j p,) .
compounded
retums
are normally dis-
tributed, then the stock price is lognorfnally distributed. Exponentiation converts the continuously compounded return, RQ, ?), into ne plus the effective total retur from 0 'A(0.f) Notice that because S: is created by exponentiation of A(0, r), a ft?,gnt/?-/?lcl/ to t, stockpric ct'I???7t# be ?scqt-l/vc. LWesaw that the sum of normal variables
is normal. For this reason, the prodttct and a are normal, then yj =
of lognormal random variables is lognormal. lf and y:t ex2 are logllormalk The product of yl and y2 is
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=
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=
e
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-3
x e
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=
e
l +.ra
is normal, cA'1+'Q is lognormal. Thus, because normality is preserved by Since + addition, lognormality is preserved by multiplication. However, just as tjye pro tjuet ojr normal random variables is not normal, the sum of lognormal tandom variables is not lognormal We saw in Srtion l 1.3 that the binomial model genetate: a stock price distribution that appears lgnormal; this was an example of the central limit theorem. In the binomial model, the continuously compounded stock return is binomially distributed. Sums of binomial random variables approach normality. Thus, in the binomial model, the contiuously compounded return approaches normality and the stock price distribution approaches lognormality. .:1
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example, E + implies that E (c') inequality. (&V)
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E ...
k.
i. EE E. :' . i E . ly . . .. .. .EE:!. . '
@(
..
:
...
: ''
'
': '
':' ' t( !' ' .. . . ... : : ..
.
. .
.
yjjyyj r t:ys .jtj. ( ( 'y( jy gjjjjjjyjy jllli:plli:. . . . .. j Graph of the Iognormal
.. -
-
. . .
-.
...
: -E .
-
-..
..
-
.
:
-
:. . ..
:.
-
-
-
.
j;,jjjj..
E' :. . . .
...
.
. ....
.
-..
.....,
.
. .
,,
: -.
,
:
-. .
-
g
,. .
yy
.
,
densityfor y, where In(y) A/-(O, 1 ) and In(F) .)V(0 1 'x'
.. . . g
yy. ......
. !
:. -
.
y
.
.
.
g
y.
.
;
0.9
eE
OF STOCK
MODEL
PRICES
%. 595
('K)
Since the exponential function is convex, Jensen's inequality c'@). Derivatives theol'y is replete with examples of Jensen's
>
The variance of a lognonnal random variable ij .jj
() 8
r
'
j)j 1n()z) () 7 j
.5).
-
,
=
-
zkr,
1)
While
(gr2
(jg.jy)
we can compute the variance of a lognormal variable, it is much more convenient only the variance of ln(y), which is normal. We will not use equation (18.14)in to use the rest of this book.
.
yls) 0.6 ). ) jt. j 0.5 r 4A j21* t't 0.4 y j
elnt-vl
val*(c.<)
:
'
1n()?)
.
pv-to,
-
18.3 A LOGNORMAL
1.s)
%
0.3
1.
'% 1k
How do we implement lognonnality as a model for the stock price? J.f the stock price St is lognormal, we can write
2
0.2
hx
0.1
N
0
'
1
3
2
4
5
6
7
8
9
-L = e,: Sfj
.
'
0
MODEL PRICES
OF STOCK
ontinously compounded return from 0 to t, is normally distributed. We want
where
10
wthe
x,
wr
plices.
If ln(y)
.x.
t)2),
.V(?n,
Letthe continuotsly compoundedreturnfromtime Suppose we have times to < /1 < tz. By the denition return, we have
the lognormal density function is given by
()'; ??l k%
,
1
'D) EB
yr
-
2=
e
(
jg
lnf.vh-?'l ''
)2
0, Figure 18.3 is a graph of the lognonual distribution as a function of y, assuming Jt = = nop-negative ty Notice that the lpgnormal is and 1.5. 1 istlibution and for both o' and skewed to the right. Figure 18.3 is based upon exponentiating the distlibutions in Figure 18.1. We can compute the ntean and variance of a lognormally diitributed random vtri-z ' If .J(/?1,tl i ), then the expected value of tr' is given by able. =
-7r
The stock price at tz caqtherefore
e ?,;+J :,2
=
'
'
=
Slo
SJ2
=
Slk e#(/l,zc) Rtk ,J2)
) Rt' /?(/(,,/1 e = Stoe
,Ja)
A(J(),fl )+A(Jl
,/2)
= Stoe Thus, the continuously
compounded ret'urn from tt to tz, S(/0, continuously compounded returns over the shorter peliods:
.
-0.5
=
Stz == St e
( j g jg;
We prove this in Appendix 18.A but it is inmitive that the mean of the exponentiated variable will be greater than the exponentiated mean of the underlying normal variable. Exponentiation iq asymmetric: Apositive random draw generates a bigger increase than an identical negative random draw does a decrease. To see this, consider a mean zero with probability 0.5. binomial random varigble that is 0.5 with probabilijy 0.5 and e0.5+e-0.o l 6487+0.6065 5 = = 1.128, which is 1.6487. Thus, You can verify that e 0 2 a = 1. obviously greater than e Appendix C at the end of This is a specifc example of Jensen's inequalit.y (see i this bookl: The expectation of a function of a random varlable is not generally equal to the function evaluated at the expectation of the random variable. In the context of this
eS(?t),?l)
StT
be ekpressed as
.x'
Ee' r)
t to somelatertime. be Rt, s). of the continuously compounded
Rtz,
/2)
=
Rtz,
/'1)
+ Rtk
,
r2)
la),
'
is the sum of the
(18.15)
('(l1Example 18.3 Spppose the stock prie is initially $100 and the continuously com)h pounded return o a stk is 15% one year and 3% the next yelm Ylieprice after 1 year (j . )) is $100,0.15 $116.1834. and after 2 years is $116.1834,0.03 $119.722.This equals '
=
'j
.
)/ l
.
-.
:
.
jgggo.
15+0.03
=
j()()e0.i8
.
.
'
'
'
.
=
$
As we saw in Section 11.3, equation (18.15),together with the assumption that returns are independent and identically distributed over time, implies that the mean and
596
k. THE
LOGNORMAL
DISTRIBUTION
A LOGNORMAL
variance of returns over different horizons are proportional to the length of the horizon. Take the period of time from 0 to F and carve it up into 11 intervals of length h, where /? = F/??. We can then write the continuously compounded return from 0 to F as the sum of the ,7 returns over the shorter peliods: R (0, F) =R
(0, /7) + R h 2/?) +
-
.
.
,
+ R ((:
l )/?
-
,
MODEL
OF STOCK
of which contains the random variable z and the other of which does 5';
@-Jc2)J
Sfje
=
e
Q
597
nqt:
ah
Next, evaluate the expectation of e o.xk using equation
r)
PRICES
(18.13).Since z
have
V(0, 1), we
x
11
:=
R ((f
-
,
ERLi
-
,
e
(r
Yz
=
1 2/ o'
ea
f= l
1)/2 ih) aIt and Var(#((i' period, the mean and variance are
Let
E
1lll ih
=
1)/1, ihjj
-
=
(7-/2, .
E (u%)= Sze
-E (S(0, F)) Varl R (0 F )) ,
=
llalt
=
n tr
(18.16) o L
(a-J-j.o'2)?
zbo.lt
(18.21)
e-
-
0r
(18 17) .
of Thus, if retu'rns Jtre independent and identically distributed, the ???ct'I?l and vtz/-?lc ctpnrnlftpll-/y compolulded ?-drll/w. are proportiollal to time. This result con-esponds the . with the intuition that both the mean and variance of the return should be greater over long horizons than over short horizons. Now we have enough background to present an explicit lognormal model of the stock plice. Generally we will let t be denominated in years and a and tr be the annual mean and standard deviation, with 8 the annual dividend yield on the stock. We witt assume that the continuously compounded capital gain from 0 to 1, nS:j%), is normally distributed with mean a & 0.5c2)2 and variance c2r: -
This gives us
Then over the entire
ES
l
)
Xet-ntt
=
(18.22)
The expression a is the expected continuously compounded rate of appreciation the stock. lf did not subtract 1tr2 in equation (18.20),then the expected rate of we on 2 This is fine (we can define things as we lilel, except appreciation would be a $ + 1c2. :.! that it renders a difcult to intelpret. Thus, the ijsue is purely one of creating an expression where it is easy to interpret the parameters. If we want a to have an interpretation as the expected continuously compounded capital gain on the stock, then because of equation (18.13),we need to SI.Ibtract 1c2 a The medign stock nrice-the value such that 50% of the time prices will be above or betow that value-ls obtalned by setting i = 0 in equation (18.20j. The median is -
;
-
-
-
.
lnts) /So)
ew
.Nja
J
-
-
0.50-2)/ o'2/.j
(18.18)
,
thus
This gives us two equivalent ways to write an expression for the stock price. First, recall from equation (18.7)that we can convert a standard nprmal random variable, z, into one with an arbitrary mean or variance by multiplying by the standard deviation and adding the mean. We can write
lnCS/ So) l
=
Second, we can exponentiate equation
&
=
a
-
J
-
-
l a t)- )t + 2
(7'
Vz
Sze @-&-Jo.2);+(rxfc
(jg po) .
2
To understand equation (18.20)it helps to compute the expected stock price. We can do this by breaking up the right-hand side of equation (18.20)into two terms, one
Estle-qo.
)1 =
2
t /
This equation demonstrates that the median is below the mean. More r/y(p750% ofthe distributed stock u?#/ earlt below its expected ?z/!/?w. Perhaps more r?nr, a D,g/lt//-?nt'/py surprisingly, if o' is large, a lognonnally distributed stock will lose money St < &) more than half the time!
.
(18.19)to obtain an expression for the stpkprice:
2
-o.
( 18 19)
We will use equation (18.20)often in what follows. You may e wondering how to itetpret equations (18.18),(18.19),and (18.20). The subtractlon of the dividend yield, $, is ncessary since, other things equal, a higher dividend yield means a lower futtlre stock price. But why d we subtract 1c2 in the
mean?
s 0e(a- 3
t' Suppose that the stockprice today is $100,the ekpected rate of return )) Example 18.4 lo%/year, and the standard deviation (vlatility) is o' 30%/year. ))) on the stock is a ,y..) ttj If the stock is lognormally distributed, the continuously compounded z-yearreturn is )) 20% and the volatility is 0.30 x 0.4243. Thus, we have ))Et)
.
y,)
.
=
,/-2
z-year
=
) q .(gThe expcted value of Sg is
xa-ytxx/i'u $100,(().j j()-aal
'z
-
=
.
:
jy .
i
'
.
E
'z)
=
(0 1 x 2)
$100c
.
=
'
$122.14
.
=
% THE
598
LOGNORMAL
LOGNORMAL
DlsTRlBuTloN
:
)g:E .
$ 100d j().
iE1
lti !) If the volatility were
l''!
' . .!. would be ,.
j
-g.5
x 0.32 ) x 2
Q
599
If the stock price today is &, what is the probability that St < K, where K is some arbitrary number? Note that & < K exactly when 1n(&) < ln(#). Since 1n(5') is normally distributed, we can just use the normal catculations we developed above. We
y j j j ygj
=
.
60%, the expected value would still be $122.14,but the median .
have
E).if
't) lt
(().l-().5x0.62)x2
1!j ?!Half k'
oNs
Probabilities
)i' The median stock price is yj .j
. jy.i E
l,) .j..j
PROBABI Ll'ry CALCULATI
$100c
$85 p,j
=
nstj'z)
.
tlze time, after 2 years the stock price would be below this value.
or
hf ((a
x
J
-
''su
lone
(7'2/j
.
,
eqkivalently, + a lnt5)l JV gln(&)
We can also define a standard deviation move'' in the stock price. Since distlibution, then if z has standard normal 1, the continuously compounded the z 1, the continuously stock remrn is the mean plus one standard deviation, and if z = compounded stock rturn is the mean minus one standard deviation.
0 5o'l)t
-
-
0.5t7'2)/, c'2/4
-
We can create a standard normal number random variable, z, by subtracting the mean and dividing by the standard deviation:
=
-
)'.r1 Example 18.5 Using the same assumptions as in Example 18.4, a one standard )y)deviation move up over 2 years is given by
C=
@;
s2
1E)
LILL
j
=
a)x2+(r yjx j
$jc(u(0.1-J0.3
C'iA one standard deviation move down is given by ).( a)x2-(r s/j S2 = $j00:(0.1-J0.3 tt
$jg() 6:),
=
Probts'?
'
Prob
K)
<
theseprices aslogarithmically
=
$e/g()g
centeredaround
Ckj,
.
the meanpriceof $122.14.
k
.
ln(&)
This discussion also shows us where te binomial models in Chapter 11 come from. In Section 11.3, we prvsented three different ways to construct a binomial model. All had up and down'stock price moves of the form Stt
Se a/;+o.W. ,
sd
=
sealt-o.l
where a differed for the three models. ln a11cases, we generated up and down moves :1. 0 the tllree mod:ls converge; the effects of the different c's As /1 by setting z probabilities. in the three cases are offset by the different lisk-neutral =
GVF
-
ln(u%)
-
(c
-
-
0.5&2)1
lnt#l
<
caf
Probts'; Since z
=
0.50-2)1
-
ln(&)
-
((z
-
-
-
Q.5o'1)t
cli
Since the left-hand side is a standard normal random variable, the probability that & < K is
-
x j
&
-
=
i.
t.l rt. Wecanthinkof 'j! =) j
a
-
.
iE(
. tt...): i
ln(,%)
-
We have Probts'; < K) = Pl'obllnt&) < ln(#)). Subtracting the mean from both ln(&) and lntffl and dividing by the standard deviation, we obtain
l)E)-
'i-t ))
ln(& )
ew
<
K)
)
Prob z
=
.V(0, 1), Probts) Probtu?
<
K)
<
<
K)
=
N
1n(ff)
ln(&)
-
(ct
-
-
0.5&24t
-
cv%
j lnt#l
-
lnt.%l
((y GVF -
-
-
0.5& lt
This can also be written Probts'?
<
K)
N-z)
=
-->
where 2 is the standard Black-scholes argument (seeequation (12.1))with the risk-free rate, r, replaced with the actual expected return on the stock, a. We can also perform the complementary calculation. We have Probtq; > K4 1 L- Probtuf < #), so =
18.4 LOGNORMAL
PROBABILITY
CALCULATIONS
If St is lognormally distributed, we can use tlzisfact to compute a number of probabilities and expectations. Forexample, we can compute the probability that an option will expire in the money, and, given that it expires in the money, the pxpected stock plie. ln this section we will present formulas for these calculations.
larobt-,> K)
=
x(Ja)
(18.24)
The expression Ni) contains the true expected return on the stock, e. If we replace l the risk-free rate in equations (18.23)and (18.24), with a we obtain the rik-neutral probabilities that St is above or below K. lt is exactly these risk-neutral versions of ,
% THE
600
LOGNORMAL
LoG N O RMAL PRO BAB I LITY CALCU LATI ONS
DISTRIBUTION
equations (18.23)and (18.24)that appear in the Black-scholes formulas.
call andput option plj ing
$
601
have
Lognormal Con6dence lntervals tlwhat is the range of prices We can now answer questions about future prices, such as will be in that range 1 year from such tat there is a 95% probability that the stock price today?'' To answer this question, we compute the 95% con:dence intervals for a numbr of different time horizons. StL < Stj and StU such that Pobvf Suppose we would like to lnow the prices Pobkhu > St4 //2. If the stock price is %, we can generate StL and StU as p/2 and follows. We llmw from equation (18.23)that
similarly,for
Stu we have su
s te
=
J
=
(ct-t-j(rz)?+rrx/j-1.96 -
$256.40
=
=
Probt.
Example
18.7 Suppose we have a lognonnally distributed $50 stock with a 15% compounded expected rate of return, a zero dividend yield, and a 30% continuously -L). The monthly continuously comVOlatility. Consider a horizon of 1 rponth t Ia pounded mean ret'urn is
5',f-)= N-z)
<
where
=
:;' f'
f
z
(ln(.%/5'))+ @ &
=
-
StL
Thpj, we want to find the
t.sclltj/o,.t
-
such that the probability that St is less than
StL
is //2,
or
(ty
'' .
p/2
=
-
&
l o'z.lt e,
-
N-z)
ln order to do this, we need to invert the cumulative standard normal distribution function i.e., ask what number dz corresponds to'a given probability. We can write this inverse cumulative normat probability function as N-t p). Then by de:nition,
NSolving explicity for
StL
p/
o
Similiarly, we solve for the S1 such that N-
l
'N
(1
-
p/2)
=
O'W =
h
=
eu--lo'lll+o'xfi'Nq
= and ,-1(0.975) 1.96. That 5%, N-k (0.025) = that a standard normal random variable will be outside the 1.96). Thus, if & $100, t = 2, a = 0.10. $ = 0, and o' 0.30, we
ilq range (-1.96, )
1
-.0788
ln
-.1644
ln
Sonemonti,
50
S 0.0954
Sonemonth
0.1819
50
ln
(
SO0e
monlh 50
j
'
=
month/so, s'one
prob
=
68.27%
prob
=
95.45%
(for example, $50,-0.0788 $46.22, =
etc.), we can express the confdence
interval in terms of
-1.96
=
! .
0.3
Eqllivalently, by exponentiating all of these terms
e prices
llpExample 1s.6 If p lt).! is, there is a 5% probability
'i'
-16.44%
dz
Thus, to generate a condence interval for a lognormal price, we need only :nd the values of z corresponding to the same consdence interval for a X0, 1) vadable, and then substitute those values into the expression for the lognotmal plice.
t'
1
.0866
(j-p/2)
0
:?
0.3
=
This gives us U
1 -
For the standard normal distlibution, there is a 68.27% probability of drawing a number in the interval (-1, +1), gnd a 95.45% probability of drawing a number in tlte interval (-2, +2). Thus, over a l-month horizon, there is a 68.27% chance that the continuouslyk compounded rettlrn on the stock will be 0.00875 ::Iu0.0866 (i.e.,the return is between -7.88% and 9.54%), and a 95.45% chance that the ret'urn will be 0.00875 ::lc 2 x 0.0866 and 18.19%): (the ret'urn will be between
s ()d (a-J-Jo.2)f+tr+iW-1 (,/2)
=
0
and the monthly standard deviation is
.-#z
gives us h
-
= 0.00875
N- (A'(wv))= x. Fortunately, this is a standard calculation, and Excel and other spreadsheets contain a built-in function that does this. (In Excel it is Nonnslnml We have l
0.15
=
=
=
$46.22 S $42.35 S
tone
month
uone
month
$55.06 S $62.09
S
prob
=
68.27%
prob
=
95.45%
k. THE
602
(,:::::,,f jj:,,f ds:::::,if 41:::2:11* C' ,'(-' t')' j);f pf '(#' :,j!;;' j,r,..f 7)r777,')** i'@' 16* (tf llr...:lf q' :')'i'r' (' 'i'' g:jijg;;f k:;;i!!:f g' llijllpf y' Ilq::::,,f y'yf jtr'. j',',*11r-.* jylf jql''l
LOGNORMAL
Dls'rRlBu'rlolq
LOGNORMAL
-''S' -' -j::2:r:' jj::y:yyf 7* -' r' -)'
PROBABI Ll7'y CAl-cu
%
LA-rlolqs
603
,:!j;;' ...11* ............' .--.' .-'.' .11:::*
)'. )'' 'yy',-jjkji,jjjkjyyy ttf t')'y' '' ( i.):((.ttiitrr1*(:qi '. (. i (j.'(.'i @.('' ('. (.' qf'T'?ltrl' . ( (.' q:.:(I ). t.r ;:qg.. .. (.'; ( ! yy q'yj. q. y.. ..y.r); j .,,rr..ti!,.,.ii..k.('.i,,, .;yg 44::4j rjy(.jjyy..y.g..y....yq.j-r..(.,,)y,,t)))k .jjj . .yyjr. ))iLq,L,'jjk, jjg:yy:,,.y..y. jj y .,jjgg, jy.-::y, .jy
.')
E: Ei- ?'' ::'.1; E -q -;; . -. --.
.....:
: tiE:EE '' . ... :
(
' ''
-.
..
,y;
.))q-.. .t .. ..-
'
:'
.
-
((-. .
: : E ' E : ' (')?'7q7'q;' ' E E'E.''..' 'E.'EE. '' E' !E.E5i.'! E.' E'. 'E EE.''i E EE''E : ( : '' .E'' . E.. . . .. . ... . - -.- -. -.- -. ---.- ..-'t,.,l)..q),.,yr,,,rjkttyy)j)y,,;,,.g,.g,, -. .- .. - ( i , -. . - - - .y -y ' : .
:
:
.
-
. .
.
-
..
....
. ....
........
.
....
.-..-....
.
...
.
;:
-
.
jj::yy:,,,
standard
-l:::::),l,.... ll-... e ::!!;;. l,'...,l,ii'I,''..,., Inital deviations from
j,::ryyy
-y'. :ir' (i';.. ;')' (' -' j')t'. )' y' ,lljr-!il,ko :,!!;i;o jto (' ()' .j)))-;', ,iIj,j1;kr'. :'-'-iiijj:j'. )'. j'yyo ;-111j2,1;,. ;'. y.jo .y'/'q'. y'-y'. y' y'. -. . i. !. E. (E.EEj.E. iE.: (Ei . .
y' y')' yyy.', kllir.-,ii.' y'. yy'q .y' )(-' 2f'. yy'k-y.k'. j,''i ,j' y'. ;';'( (y,':iiif' y.'. ,')' k)' ji.,.--;' )'. jliliriji,;yo y'. t't(' tittyyt)o y:i;k;j;'. kliijkqy'. .r'. 'yjy',rl'. 111* .'t-1k:j:,--'. r' k rj,i,,;;j.. lillj. .:jI1i!--
.';r ..
(2:),-
the
-
.k. ir..
. .
-
-
. . ...
.
.
.
,
'
'!
'
:
-
-' -;kk-k;,,-'. -'. --'. ,';';.,' '','-,' :'yf --j;::z!,jo -' --,ik:,!,;'.
-'ti ,'.'-'.y' .j;;i' .y'r,'. --'
. . .. -.. ...
-1:::::111
.2:;;4E!2. a stock price of 50. .-'.
- .- - . .
t'X
r. E.
-
- . . -.-. . .. . . . --. i - . . - -.. ... .. - . -.-;. .. EE E -- . - -. E - .. .. - E ..:-E..E . . . . . . . .. . ... . . . . . . ... L. .
. -. . .
..
,;k:-!jj,-. -kI,:tj:. .:
tkkjj,,;-. ....kjliljiq. .. r . :;;ij;ijk-. . .. . .... .qiijj .......:r iilr--ii,li .
-kkkj:il;,- -k1i::i,,-. .
.
..
-
::.!.-. ) . .
-
.
-
- .. .
-
-
-
.
-
.. .
. -. - . .. . . . . .. . . . . . . ... . .. . . . .. .
.
. .
.
-. -. . -- -. .-. :,. y.- - .y-qy :q - -. yr-, .. . ..-.. rlliE:;ll,:r :j1It--k-j. .-1IIj!,IIjj. rqlg:rr;, ji !i'j,.: i,. y ..j-.. yj-.. -j..--jyy-y.r.i,,kjj,L;.. r..-..y..-;r--y;yy-...j.y...)-r. ,,,12.,,11r .-ty.. jyj ..E........... . !.;.(.,:.Et,!jj:tijjr:. !!gt;.. ...y k-(j.y-... .i.y.,.. ..-...y...y-.-.....y....y. ... y....-..... .--..... yj..- ...,,-(:.y,y.y-..j... .y..y-;.y,...g.yy .i ,E j.yy y.j.. y.. t. ;. ... . t.., .q ..- q....j...... ..r.tj.i .q . .. . . . . .. y. . tlll-!-. . . . . . ..
,!!jggk,jj,,tj,Ii..,I.... -jkiri:Il-k. --. lqlii,:iE .
-
-
y. E- --. . E . . ..
.
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ondEkpij. bok on page 299, %ve discussct th ndflerfork ( iro'lcjit E result that the ost of insuring a stock l sikj, on the size of ihe riskpieiiur t weli kh it that performs 1: t at least as so as 4 r, flatik t the portfolio 7 a yy cz r If the rik ptenrtitll is hikh, pugy Nkiujj xj,.y zero-coupon bond is increasing With th r E time to maturity of the insurgnce. Th incresinalv lss llkely,t 'pp.# ff ifi the'lj k .( . . this in Chaptr . 9 relies. on . . . . tta ftpfil demonstration of .i . lfli: . ev ..n thugtt . . runk y ' 1i.'71L dt:i:l- (.E dl:z:: 4:j:1. dl:::: !lE 4:!2:: ;i!ri d::: 1E1k. IIFTIL IIEL iIr.'ii.. lk::i,(::)4d:i,:: '. 1!;: rlq'' CrC3l.!l:'' 1!'' llr')t IEEIL. 1i::l, lil:lp iil 1EE;k iil ...i.. llr:lt. i 21114;q (Iqil. IIF:II. 1IE)I'1l.. ...q..... ;.y....... j.. ...'j.., .t. ....... . .. . g ! ... ( . .... .).y..: .-:.........y ;. .!.(....(. .. .; ..'.j:. .... ...... .: .. g .. . y(t .j. .(. .... .t-.. . . . incontrovertible but does not alkaj?s proyide q j .j. y . . .k- u . .. . .. .' .. .E intuition about th: result. Usirt the resltg i ) ( 1 tybgbiliWti , th jtk.- Wltl (( 'lE ( unlttjdbrrri this section, we can reconcile the historical Pfb*tir' (Q k)L 'tij bd, 1''9 t/lr'ktiij probability of stoeks undfpedbfhg ct Q d W tlki' 1' ' ' . . . ' . ' E E E E bond s w ith the increasing cost of insranc as q ; ; y,j y y,r y j q 41k:)) y ( y .'. g E.. l:ij:l jlr)ls 4(t4:). 4k1,:: .' .- .( t .:. E'y 4:E2)1 jlg..dkzz:l ..l 4k:2:: 4::2)) jlr:lt :;p!r; j!r:)L..,-..'. '(:)::):)'.....' .ji. :jp;. .:.:')... 4(4). :jk'.' 11.g1r glylk :r(,I. (...i,LLL. qlrr, 217- (Irr.jil (14 .y'y: gjjjj;' '..! ...(. 'I::), 'jk,;lk,' ;'-)t.) y. .' .. t y...' '..t..' (ltqtl''ljgr.. y.:Lf';. ) .! y. . .... (2)1 .... q( . . Lib'ii;t'. (EEI l,yyyyyl.q 'iir: 11:::!5/8, k ' 144;'11. 2) qy,q, llr:llg'!l. 1lJ')k 4(1:: iElt.ll:::,. g E i. .E ..'.iE.. j j, j ( .E. irEl.qltz ',i.F'' ll/l l::ik r .J!ii7' 11::), 11....(:E:1. :1L ii). 11. .iiL '. ... ts!;''*11(7,. .. qllrll t,g.y...yq..;.. ......y... L?jt''L. t' . ..... . .'. .. . (. (. r. ;;.. q j.-.....y.. r;!r.. ..... t. y . .t.)yq... $kltty.E ilbiluitit i Tl (18.23).By sttiflg the strike price t #fobittWt,t ihjtit ' t'' pay' ' ittfatl = wltl.i equal the fonvard pr ce, i e K r c'z s() er s.e ttf ttttt' r y qj yq q tqy can use equation (18.23)to calculate t j y j rlih that stocks bought at tiriz 0 Wiltq ifeji,j ti) l t ttli ljfi ikij y r probability y'' ( ltpld i: oditial kljdtttt ? haveunderperfonned bonds at time :/'. Aftf k)..) ;, stljkc pri S . equation (18.23)cafl be written fftjr ih tli fii ik ttk g ) 55 l E I'lojkk j tjyjy; kliij' 1' y .1./2 duij tjliji fitijkj y.) ( wxnatkjj:oE txo yzt ij j # y: w..Q (# x x ltztszztu x 2 u u u hijttzutz Iz lte: I!E;I tzu :::::::7 il!:)' ltr't:E)p ( zttgtgut 1E:), :?r. . tzuuuuluqc;a I:T:hr'' .J!it:7' ()k .. . . .. li: .. . q (. r . . .'. ltllLqi .. . . .. d::)r7 4:1:' d::)k il. .E'.. 4:21 ltllL 4:29 2(, ttl ..L..LL :54; ilr:iL .tj(y..'yy....., .t'... (Iihlir)z tjt.lirit '..'q..y. t ...''.... .. .)'.y ...'. .
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=
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0.0849
48.47
49.24
50.81
51.61
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The calculation in Exnmfle 18.7 is often used to compute loss probabilities and risk exposure. We will see in Chapter 25 that this is how value at risk (VaR) is calculated. The idea behind V:R is to assess the magnitude of a possible loss on a position that an occur wit.h a given probability oyer a given horizon. So, for example, if we examine the l-day horizon in Table 18.1, there is a 2.275% prbability that over a l-day hol-izon the stock price will droj below $48.47.4In practice, it is common to evaluate the magnittlde in 20 dayf') probbility of of moves of 1.96c since this corresponds to a 5% CGonce
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Chapter 9.
The Conditional Expected Price Given that an option expires in the money, what is the expected stock price? The answer expected stock price. For a put with strike price K, to this question i the collditital 15'J < f('), the expected stock price conditional on & < K. To we want to calculate E St expectation. compute this we need to talce into account only the portion of the probability density representing stock prices below K.
4You can verify the 2.275% probability by computing N(-w2).
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To tlnderstand the calculations we are going to perform in this section, consider a binomial model in whicjthe strike price is $50, and the stock plice at expiration can be $20, $40, $60, or $S0, with probabilities 1/B,3/8, 3/8, and 1/8. lf a ut is in the money at expiration, the stock price is either $20 or $40. Suppose that for these two values we sum the stock price times the probability. We obtain Probts'i) x St St <50
.
=
1 x $20 + 8
-
3 -
8
x
$40
=
$ 17.50
(18.25)
The value $ l 7.50 is clearly not an expected stock price since it is below the lowest possible price ($20). We call $ l 7.50 the partial expectation of the stock price conditional upon Sl < $50. When we compute a conditional expectation, we are conditioning upon the event S < $50, which occurs with probability 0.5. We can convert a partial expectation into a conditional expectation by dividing by the probability of the conditioning
y
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The box on page 603 illustrates how the probability calculations in this section be can used to analyze the cost of portfolio insurance over time, previously discussed in
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lt U sing this s'ame logic, we can compute one standard deviation and two standard deviation intervals over different holizons. This will give us 68.27% and 95.45% confdence (j')i .intervals over uwsehorizons, which are displayed in Table 18.1. yor example, there tii)is a 95.45% chance . over ..... a l-day horizon that a $50 stck will be between $48.47and )..)). Over . a s-yearhorizon, there is a 95.45% chance that the stock plice will be ) $51.61. 1tt! )': .
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event & < $50). Thus, the conditional 1 Probtu
<
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50)
TH E PARAM ETERS OF A LOGNORMAL
EsTl MATING
x
St <50
s:
=
is
1
1
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x
$20 +
those probabilities, the price of a European call option on a will be
3
j. x $40
DISTRI BUTION
$
nondividend-paying
605 stock
(18.26) = $35 The calculations for a lognormally distributed price are analogous, using integrals rather than summations. The partial expectation of St, conditional on St < K, is
j
K
&, (e-)'x
.S'/:(.%; szjdst= =0
s
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-
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r)
x(-l) sea-&b'
=
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The Black-scholes Using equations
(18.23)and (18.30),with
>
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=
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stock
S4
We can rewrite this as
and using (18.24)and (18.28),with P ( S K .?..- r t J)
0.5G2)/
,
,
,
,
a
r, this becomes
=
=
Ke-rtN-dz)
e-bv%N-dkl
-
(18.29)
18.5 ESTIMATING
THE PARAMETERS DISTRIBUTION OF A LOGNORMAL
ln this' section we will see how to estimate the mean and variance of lognormally distributed price data. When stocks are lognormally distributed, a price St evolves from the previous observed price at time t h, according to -
(18 30)
uf
=
Stne
Suppose we have daily observatjons. deviaion? We have
Formula
(18.23),(18.24),(18.28),and (18.30),we
K) x Prob*tk
which is the Black-scholes formula. Similarly, the fonnula for a European put option on a nondividend-paying is derived by computing
.
Nz)
15'>
r, this becomes
=
.)
K)
-
As before, except for the fact that it contains the expected rate of return on the stock, a, ins'teadof the risk-free rate, th second term isjust the Black-scholes expression, Ndk ). The conditional expectation is
=
e-rE*K
Using
>
(18 28)
N-,)
11(S0)
=
E (s, I.St
e-rt /(7*(u$'15* > K) x Prob*tu
=
..1
For a call, we are interested in the expected price conditional on S partialexpectation of St conditional on & > K is K
t)
C(S K c
E (s? I-C t <
*
CS, K, c, r, t,
.(a-,),x(-tt)
is the probability density of St conditional on %, and l is the Black'Scholes :,/1 (equation(12.1))with a replacing ?'. Thus, the expectation of St conditional The probability that & < K is N-thl. < K is St on
where gst
We can rewrite this as
can now heuristically delive
the Black-scholes formula. Recall that the Black-scholes formula can be derived by J.n this case, the expected return on stocks, a, will equal r, assuming lisk-neueality. Iisk-free lognolpal probability density E* the rate. lf we let g. denote the lisk-neutral denote the expectation taken with respect to risk-neutral probabilities, and Prob* denote
(e-J-o.2/2)+o.x/jfc
How would we estimate the mean and standard
Thus E
(ln &/,%-/,)) (a =
Var (ln
vstjvt-l,lj
-
-
=
c2/2)/7
&2/7
k. THE
606
LOGNORMAL
DISTRIBUTION
EsTl MATI N G TH E PARAM ETERS O F A LOGN O RMAL DISTRI BUTI O N
By using tle log ratio of prices at adjacent points in time, we can compute the continuously compounded mean and variance. Note that to estimate a, we have to add one-half the estimate of the variance to the estimate of the mean.
$
607
illustrates, mean returns are hard to estimate precisely because the mean is determined by the difference between where you start and where you end. lf you start at a price of $100 and end at a price of $104, the in-between prices are irrelevant: If you had a big it must have been offset by a big posltive rettll'n negatiye weekly rettlrn (say +20%), (on the order of or you would not have ended up at 104! Having many freqtlent observations is not helpful in estimating mean remrns. What is helpful is having a long time interval, and seven weeks is not long. -20%),
)'
jq Example
Table 18.2 contains seven weekly stock price observations along compounded returns computed from those observations. You can l'Eil )t Compute the mean and standard deviation of the values in the third column (forexample, jq 'lt 1't1using thezdbltdmgeand in Excel). Since these are weelcly observations, we tlt weekly -)j)) of log plice ratio and the veekly standard deviation. estimating the the mean are j j( . in Table 18.2 is 0.006745 and the standard deviof the second column The )'y mean 1,y ) ation is 0.038208. The annualized standard deviation is
18.8
.t(j)
with continuously
krtvfunctions
j .
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.
t
Annualized standard deviation '
j
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i Two adiustments k(t, !(
0.038208 x
=
52
0.2755
=
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.
are needed to in terpret the mean. First, we have to annualize it. Second, E(' since we computedthe mean of the log returns, we have to add backone-half thevariance. ii'i' Thus, t't we obtain ''.'''
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f)
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Annualized expected return
0.006745 x 52 + 0.5 x 0.27552
=
'li 't))The prices were generated
=
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0.3887 .
assuming using a standard dviation of 30% and a . . observations,'the standm'd deviation estimate is O having only six loo. ) ii'!mean oespite . tl) value l't quite close to the true of 30%. The estimated mean, however, is quite far off. % randomly
We used hypothetical data in this example in order to compare the estimates to the true underlying parameters, something we cannot do with real data. As this example ;r' jk.,,' y' jjy,..gjjf ,''2* (y' kkjjyjjyf ;'j'g ktf .jjyyy' 'q'(' r' jgytf y'. ''J'(';'J((' jr:::::,,f j')')'y' jjyrggrj,f tj'. jjryjjjjjf jiiE!ijE,f jjf jyf jjj,,,yjjf .,kIijI'. .g:jj,,j,,' y' 'j'tgf (' jqf '''y )'. yjf )@').@' rt-'iijjrir''y )y' )' ttf .' j;(!' (!('(:j(;'j';((gjr!pj.(k.)('( .(k'.j'...('.(..'.( (.('.''. yj.'(j ( j.(! (..(' 'y ('y' ''*1'157: (. ;.'.(' (('r t'tt'; ljf r:'' ' 'j (yj qy q. (. 'i(iEi)l j.,yl g y. .yq q. yy .t-ryjkr )( . jq ;y.(tj k. jk' q. jy.i q'. ),y jiiiii!,,r;. y. yyj. j .. r.g.yy.jjj y . g k j j . j . . yyy j y yj. . j . tyyy.y..,y-y.g.r.yyy)y. .r. ..y.yg '.11j... )y jq . .. ..r.y. . . k- ..jjy-. ;yyj. , (r. y. ..'.. .,. ... ... ..
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means.
In this discussion we have assumed that the variance is not changing oyer time. There is good evidence, however, that the variance does change over time, and sophisticated statistical methods can be used to estimate changing variances. You should also be aware that, in practice, using data from vel'y tiny intervals hourly prices) may not increase precision. Over short time periods, factoj s suc jj (e.g., spreadsEdu bid-ask bounce-the movement of te price between the bid and ask to as some orders being sells and others being buys--can introduce into prices noise that is not related to tfle values we are trying to measure.
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Statistical theory tells us the precision of ourestimate of the mean. With a normally distributed random variable, the standard deviation of the estimated mean is the standard deviation of the variable divided by the square root of the number of observations. The data in this example were generated using an actual weekly c of 0.3/ 52 = 0.0416. Divide this by (sincethere are six return observations) to get 0.017. Thus, one deviation for our estimate of the mean is 1.7% on a weekly basis, pf 12.25% standard There is a 68% probability that the annualized continuously plpounded annualized. :1c 12.25%! A 95% confidence intelwal is 38% ::1u 24.5%. the range 38% mean flls in This is a wide range. Even with 10 years of weekly data, one standard deviation for our estimated annualized mean would be 30%/ 520 = 1.3%. bf When we estimate a standard deviation. we are interested in the mokeft observations price. The have, the precisely stimate the we more more we can movement. Wit.h six observations, an approximate 95% confdence interval for the standard deviation is approximately :18 percentage points.s With only 26 weelcly observations, the 95% consdence interval shrinks to :178 percentage points. Moreover, unlike the mean, we can increase the precision of our estimate of the standard deviation by making more frequent observations. In general, standard deviations are easier to estimate than
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The lognol'mal model assume: that Stock returns are independent over time (today's rettlrll does nOt affect futllre retllrnsl, that mean and volatility are constant ovr time, and tat the distribution of continuously oppounded returns is nonnal. I.towever, we saw i n Chapter 12 that implied volatities differ for options with different strikes. One possible explanation is that stockprices are not lognorrpally distributed. How can we tll whether lognormality (or some other pmicular distribution) is a reasonable approximation for acmal stock prices?
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peak around zero; the presence or absence of this peakedness is refen'ed to as kltrtosis the peak of the distl-ibution is), and the graph displays lep(a measure of how thin, delicate''l.? For a normally distlibuted random tokttt'tosis (lepto meaning variabl. ktlrtosis is 3. For the data plotted in Figure 18.4, kurtosis for the S&P and IBM are 8.03 and 9.54 for daily rettlrns, and 4.68 and 5.21 for weekly returns. Accompanying the peaks are fat lt7/., large returns that occur more often than would be predicted by the lognormal model. These shapes are typical for stock returns. There are several possible explanations for retunas appearing nonnormal. One is that stock prices can jump discretely from time to time. We will discuss jumps in subsequent chapters. Another explanation is that returns are normally distributed, but with a variance that changes over time. lf actual daily returns are drawn from a distlibtltion that has a 1% volatility half the time and a 2% volatility half the time, the stock price histogram will appearfat-tailed. This blend of two distributions is commonly referred to as a mixture of normals model. Long-horizon returns, which result from summing short-horizon returns, will still appear normal. ttsmall,
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None of the histograms appears exactly normal. All of the histograms exhibit a
Histograms
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500 index and for IBM. The bottom row is histograms for weeldy returns. Also plotted on each graph is a normal distl-ibution, computed using the histolical mean and standard deviation for each return series-6 Several observations are pertinent.
18.6 HOW ARE ASSET PRICES DISTRIBUTED?
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ARE ASSET PRICES
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6An equivalent approach would be to nonnalize returns by subtrating the estimated mean and dividing by the estimated standard deviation. The resulting series should then be standard normal if returns are truly lognormal. A 7rf'hekurtosis of a distribution is the fourth central moment (i-e.,E ((.v Jz)4 1,where t is the mean) tr4. divided by -
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k. THE
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y'r Example 18.9 Suppose we draw from a distribution :ve times and obtain te values i .. tk 41 The order statistics are (3,4, 5, 7, 111. 3 11, 5, l.( i (7 '''''
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Suppose z is standard normal. The 10% quantile is the value such Example 18.10 .q ).:1 ))t that there is a 10% chance that a draw from the standard nonnal distribtion is less than l that number. Using the iverse cumulative dis tribution N-1 (0 l0) 1.282. Thus, r tjt) #-l The 10% is the 30% quantile is quantile l % (0.3) ,
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The idea of the normal probability plot (whichcan be done for any distribution, not just normal) is to compare the distance between the quantiles of the data with the distance between the quantiles of the nonnal distribution. If they are the same, the normal probability plot is a saight line. To see how this Works,suppose we have the five values in Example 18.9. We want assign quantiles to the data points, so with five data points we need 5ve quantiles. to 40%, and so forth. Assign the order Divide the range 0 100% into 0 20%, 20% statistics lthe ordered data points) to the midpoints of these ranges, so that 3 is assigned a quantile value of 10%, 4 a quantile value of 30%, 5 to 50%, 7 to 70%, and 11 to 90%.8The normal probability plot then graphs these pints against the pointg from the corresponding quantiles of the standard normal distribution. 'rhe top lef4 panel of Figure 18.6 presents the nonnal plot for the data in Example 18.9 with the data points plotted against the corresponding z-values of the siandard normal distribution. Appendix 18.B explains the construction of tls plot. The top right panel is exactly the same, except that the y-axis is labeled with probabilities corresponding to the z-values. The data do not appear nonnal, though with only five points there is a large possibility for error. 'I'he bottomrow of Figure 18.6 presents nonnal probability plots with two different y-axes for 1000 randomly generated points from a M(0, 1) distribution. ln this case the data lie along the line and, hence, appear normal. J.n Itll of the normal probability plots, the saight line is drawn connectink the 25% ad 75% quantiles of the dta. q l essence, the normal probability plot changes #?d scale on th y-trW- so /'/2d cltlnktlative ?/t/rn/tW If?3/dIille ?-t7l/7/r //7:0/ L1ll S-*@/tf distribution is fl Strtgllt -
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and qttalltiles. Suppose thatwe rapdomly we have to defne two concpts, ordrsiatistics 11 1, draw random variables xi i n, from some distribution with the cumulative #@)). lf we sort the distribution function F@). (For th normal distribution, F(-v) in ascending sorted data called order statistics. order, the data are ,
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9The straight line can be htted in numerous ways. For example, Matlab connects the quartiles. ln the case of the sample data, the 10% and 30% quantiles are 3 and 4, so by interpolation the 25% quantile is 3.75. Similarly, tlze 70% and 90% quantles are 7 and l l , so by interpolation the 75% quantile is 8.
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The norm?l distzibution has these characteristics: and left sides are mirror images of each other. possible (albeitperhaps unlikely) * ltruns to plus and minus infinity, which means it is the distribution. from draw when could occur you that alty number the mean. * lt is unimodal', i.e., it has a single hump, which occurs at
* It is symmeic;
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The lognormi distribution arises from assuming that continuougly compounded characteristics: remrns are normally distributed. The lognormal distribution has these * It is skewed to the right. that negative outcomes are impos@lt runs from zero to plus insnity, which means sible. single hump), which occurs to the left of the mean. * It is unimodal (i.e.,it has a Products of lognormal random variables are lognormal.
o
The Black-scholes fonnula arises from a straightforward lognormal probability calculation using risk-neutral probabilities. The contribution of Black and Scholes was in th formula. not the particular formula but rather the appearance of the risk-free rate and weekly daily probability plots for From examining hijtogratns and normal large returns continuously compounded returns, we can see that there are too many compounded continupusly returns relative to normally distributed returns. Although do not appear to be exactly normal, the Black-scholes model and the accompanying assumption of lognormality is used frequently and we will continue to use ard develop this model in the rest of the book. We will also explore extensions that are consistent with departures from normality we have seen in this chapter.
FURTHER
READING
ln Chapter 19 we will use simulation to price options assuming lognormal stock prices. We will also extend lognormality by allowing stock prices tojump discretely. In Chapter 20 we will introduce the continuous time model of stock retrns used by Black and Scholes, which is the basis for modern option pricing and which, with their assumptions, generates loglrmal stock prices.
Both the hlstogram and normal probability plot verify that continuously compounded retulms in practice are not normally distributed. The question is whether this and pricing matters for Xi'cing, and if so, how to modify theassumed price distributions fonnulas to obtain more accurate derivative prices. Two modifications we will examine in later chapters are to allow tlae stock to jpmp discretely and to permit volatility to be
stochastic.
614
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An excellent discussion of the basic characteristics of stock remrns is Campbell (1997, chs. l and 2). The history of the normal distribution is entertainingly recounted in Bernstein (1996). (See in particular the accounts pf DeMoivfe, Gauss, and
et a1.
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1. 'What is Est 1,%< $98)? What is E St 1u/< $120)? How do both 18.11. Let t expectations change when you vary t frorp 0.05 to 5? Let o' 0.1. Does either answer change? How?
18.2. You draw these tive numbers from a standard normal distlibution: (-1.7, 0.55, What are the equivalent clraws from a normal distribution -0.3, variance 25? 0.8 and with pean
Sverb. Compute Probts'w < Kv) and Probtur > Kv) for a variety 18.12. Let Kp of Fs from 0.25 to 25 years. How do the probabilities behave? How do you reconcile your answer wit.h the fact that both call and put prices increase with time?
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and 18.4. Suppose Jrl .V(2, 0.5), and x?. X'(8, 14). The correlation between distribution of is What is the What is the distribution of + .::2 ? l -'t ew
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18.7. Suppose you observe the following month-end stock prices for stocks A and B: Day StqckA Stock B
()
1
a
a
4
100 100
105 105
102 150
97 97
100 100
For each stock:
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18.14. Select astockorindex and obtain atleastsyears of daily orweekly data. Estimat the annualized mean and volatility, using all data and 1 year at a time. Compare the behavior of your estimatej of the mean with those of the standard deviation. 18.15. Select a stock that has at least 5 years of daily data. Create data sets consisting of daily data and weekly data, Wednesday to Wednesday. (The veekday fgnction in Excel will tell you the day of the week cormsponding to a date. Wednesday is 4.) For both data sets, create a histogram of remrns and a normal plot. Are the stock prices lognormal?
18.A: THE EXPECTATION VARIABLE OF A LOGNORMAL
APPENDIX
a. Compute the mean monthly continuously compounded return. What is the annual return? b. Compute the mean monthly standard deviation. What is the annual standard deviation? estimate of the mean depends only on the c. Evaluate the statement'. beginning and ending stock prices; the estimate of the standard deviation depends on all prics.''
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In this appendix we verify equation (18.13). Suppose that y lognormally distributed.The nprmal distribution is given by
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probabilities of reaching each of the 10.5, with the addition of the total lisk-neutral terminal nodes. Figure 19.1 demonstrates that the option can be priced by cpmputing the expected payoff at expiration using the probability of reaching ech 'Iinal node, and then discounting at the lisk-free Figure 19.1 rat. You an verify that the option price is the same as that in Figure 10.5. We can also use the tree in Figure 19.1 to illustrate Monte Carlo valugtion. Imagine gambling wheel divided into four unequal section, where each section has a probaa bility corresponding to one of the option payoffs in Figure 19.1: 9.5% ($34.678),34% ($12.8 14), 40.4% (0), and 16% (0). Each spin of the wheel therefore selects one of the linal stock price nodes and option payoffs in Figure 19.1. lf we spin the Fheel numerous times and then average the resulting option vlues, we will have an estimate of the expected payoff. Discounting this expected payoff at the risklffee fate prvides an estimate of the option value. It is easy to compute the actual expected payoff for the option in Figure 19.1 without using a gambling wheel. However, the example ill.ustrates how random trials can be used to perform valuation.
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t using true probabilities, we need to compute the discount rate-/r eh #Ir/3: 'Thf . eight possible paths for the stock price, four of which result in a positive .optio pyfft annualized continuously compounded discount rate Au of these paths have a srst-period discount The subsequent 35.7%. rates depend on the path the stock takes. Table o 19.1 verifes that discounting payoffs at path- (je p endent discount rates gives the correct osyou jsyx. so tjjja just tjw jjrst rowy the disctmnte (j expected option payoff for that 14::2j11.. :lijg;:j;:. ()1(, .1j,,)!It. ::!j:Il., (jlqr-. (ii. (41(,:!EE;:. (jjgl::j)-
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CARLO
VALUATION
S l M U LATI N G
Using Sums of Uniformly Distributed Random Variables
LOG N O RMAL STOCK
PRI CES
$
623
distributed: I.f you draw from a distlibution, by desnition any quantile is equally likely to be drawn. The algorithm is therefore as follows:
One standard technique to comput normally distribpted random variablej is to sum 12 unifonn (0,1)Tandom variables and subtract 6. Thus, we compute the X(Q, 1) random variable 2 as
1. Generate a unifonnly distributed random number between 0 and 1. Say this is 0.7. 2. Ask: What is the value of z such that #(z) 0.7? The answer to this question is computed using the iltverse cll?nlf/lrvc dtriblltiol htllction. ln this case we have #-1 (0.7) 0.5244. This value is a single draw of a standard normal random variable (0.52+M. =
=
lli are distributed uniformly on (0,1). This technique works because the variance of a variable that is uniformly distributed between 0 and 1 is 1/12 and the mean is 1/2. Thus, if ytj sum 12 uniformly distributed random vaziables and sqbtract 6, you get a random variabl Fith a variance . of 1 and a mean of 0. The sum of 12 Jmiform variables is not precisely nonnal, ut it is close.This technique is an apptication of the cential limit theorep.
where the
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Uw) X()')
=
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=
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.
This procedure simulates draws from a normal distribution.
To Simulate a log-normal
random variable, simulate a nol'mal random variable and exponentiate the clraws. This procedure of using the inverse cumulative probability distribution is valuable because it works for any distribution for which you can compute the inverse cumulative distribution.
.
Using the Inverse Cumulative Normal Distrlbutioq .. .
3 Repeat. .)
x.
S wj
As discussed in Chapter 18, w is the Uw) quantile and y is iheX(y) quantile of the two distributions. If we randomly draw a uniform number tt, how can we use 11 to construct a corresponding nonnal random numbe', z? lt mrns put that the same idea we used to conjtnlct nonnal plpts in Sction 18.6 i ts perm' us to geperate a pormal tandomnumberom a unifonp rapdom npmber. Instead of interpreting a random draw from the uniform distribution as a nltlnben we intqrpret it as a qltantile. So, for example, if we draw 0.7 from a N(t, 1) distribution, we interpret tllis as a draw corresponding to the 70% quantile. We then use the inverse distribution function, #-l (If), to find the value from te normal distribution corresponding to that quantile.3 This technique works because, for any distribution, quantiles are uniformly
19.3 SIMULATING
LOGNORMAL
Recall from Chapter 18 that if Z
Sje (a-&-)o.2)w+gxfz
=
written
( jq a) .
Suppose we wish to draw random stock plices for 2 years from today. From equation (19.3),. the stock price is driven by the normally distributed random variable Z. Set 2, a T and o' 0.30. tf we then randomly draw a set of standard 0.10, normal Z's and substitute# the results into equation (19.3),the result is a random set of lognormally distributed Sz's. The continuously compounded mean ret'urn will be 20% (10% per year) and the continuously compounded standard deviation of 1n(u%) will be 0,3 X 'Ui 42.43%. .0
=
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Simulating a Sequence of Stock Prices 'I'IJere is another way to create a random set of prices 2 years from.now. We can also price. This will give generate tw??7ll/f random prices and compound these to get a z-yea.r exactly the distribution for plices. Here how do is it: to same us z-year .
* Compute the l-year plice, 3The Excel function Nornlslnbt computes the inverse cumulative nonnal distribution. Unfortunately, there is a serious bug in ths function in Oflice 97 and Office 2000. ln both versions of Excel, Nol'nlSInj'Q.999%9%6j= 5.066, and NonnSIn3,Q.99%9991) = 5,000,000. Because of this, Excel will on occasion produce a randomly drawn nonnal value of 5,000,000, whch ruins a Monte Carlo valuation. I tank Mark Broadie for pointing out this problem with using Excel to produce random normal numbers.
PRICES
.V(0, 1), a lognormal stock price can be
x
Sp
STOCK
SL
.
as
S3
l+o../fz(l)
x Sve (0.1 J0.32) -
=
* Using this 5'1 as the starting plice, compute Sz.. Sz
x 1+0.3xf(2) 5'l e (0.1 $0.32) -
=
-
ln these expressions, Z(1) and Z(2) are two draws from the standard
normal distribution.
'
624
k
MONTE
CARLO
VALUATION
MONTE
If we substititute the expression for 5'1 into Sz, we get Sz
(0. l Soe -
=
x2+0.3Yi1Z(
J0.32)
The difference between this expression and equation ./iZ we have (Z(1) + Z(2)). Note that
l )+Z(2)1
as Vv.,
(19.3)is that instead of the term
V
and
.
.
.
2
Therefore, equations (19.3)and (19.4)generate with the same distribution. If we really want to simulate a random stock prie after 2 yearsj there is no reason to draw two random variables instead of one. But if we want to simulate the pathof the stock price over 2 years (forexample, to price a path-dependent option), then %ve can do so by splitting p the 2 years into multiple periods. ln general, if we wish to split up a period of length F into intervals of length /7, . the number of such intervals will be 11 = r/ /7. We have qz's
S
Szl, = She @-J-Jo.2)+o.Xz(2)
F, zzz: -!7,,
-
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a-t-qcrljlt-o.xfltj
l stock prices can be intepreted as equally spaced points on the stock price path between times 0 and F. Note that if we substitute % into the expression for %h the expression for Szh into that for Ss, and so on, we get
maxto,
,
.usxge(a-:-Jo.2)F+gx/igj7.j (a-&-)o.2)w+(rx/Tg.J.;w
= Ste
=
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-
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=
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.
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This expression gives us an estimate of the expected optiopayoffat time /. We discount the average payoffback at the risk-free rate in orderto get an estimate of the option value:
'' p' $'
=
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1
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N
= l
i
sv
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Suppose we wish to value a 3-month European call option where p Example 19.1 )1 the stock price is $40, the stnke price is $40, the risk-free rate is 8%, the dividend yield ltl lt1is ljj? zero, and the volatility is 30%. We draw random 3-month stock prices by using the t.yexpress kon l' )). 53 months Ste (0 os-o.32/2)x0.25+0.3x/(EB'z .
.
In Monte Carlo valuation, we perform a calculation similar yo that in equation (19.1). The option payoff at time F is a function of the stock price, Sp. Represent this payoff
r. We generate random
N
1
.
VALUATION
=
Average the resulting values:
ztflj
Since x/v E''i=z l Zi) -wA'(0,1), we get the same distribution at time F wit.h equation (19.5) as if we had drawn a single X0, 1) random variable, as in equation (19.3). The important difference is that by splitting up te problem into 11 draws, we simulate yhe path taken by S. The simulation of a pat id useful in computing the value of pathdpendent delivatives, such as Asian and barrier options, the value of which depend on ie path by which the pce arrives at Sv
CARLO
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uaetr-d-o.so.zlrio'xffzi
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19.4 MONTE
(19.6)
standard normal variables, Z, substitute them into equation (19.3),and generate many random futtlre stock prices. Each Z creates one trial. Suppose we compute N trials. For each trial, i, we compute the value of a call as
These
sr
w)
Monte Carlo Valuation of a European Call We assume that the stockplice follows equation (19.3),with a
and so on, up to -i2-(?,
vs
i=1
rl'he
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,
,
=
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=
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z(2)1
625
31
1
z(,%,0)
=
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626
CARLO
VALUATION
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N'rE CARLO
VALUATI
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To assess accuracy, we need to ltnow the standard deviation of the estimate. Let Cvki) be the call price generated from the randornly drawn v lf there are 11 trials, the Monte Carlp estimate is
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.
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of European call wth 8%, t 91 days, and & 0. The Black-scholes price is $2.78. Each trial uses 500 random draws. =
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2. 2.75 2.63 2.75 5
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Average
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=
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-
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Option payoff
t.)
=
maXto,
c(l) iz=l
Let c'c denote the standard deviation of one draw and o'tt the standard deviation of draws. The variance of a meh, given independent and identically distributed Si is
. .
1,
's,
2
Gtt
=
1 a
-Gc lt
''''''
c/)
=
1
W
cc
Thus, the standard deviation of the Monte Carlo estimate is inversely proportional to the square root of the pumber o? draws. ln the Monte Carlo results reported in Table 19.2, the standard deviation of a draw iS a Ot1 t $4 05 tTllis value is computed by taldng the standard deviation of the 2500 Price estimates used to comput the average.) For 500 draws, te standar deviation is Ss mnths
-
.
$4.05
$40)
500
.i...
' tjt We repeat this procedure many times, average the mjulting option payoffs, and discount ' IJ ' ' tjl the average back 3 months at the lisk-free using 2500 draws, rate. With @.single estimate jl' t') we get an answer of $2.804(seeTable 19.2), close to the true value of $2.78. % In this example
11
.
it' For each stock price we compute i l..
11
-
..
.
j'''j'
j
'&?,
we pliced a Europearsstyle
option. We will discuss in Section
19.6 the use of Monte Cado simulation to value American-style options.
Accuray of Monte Cado There is no need to value aEuropean call usingMonte Carlo methods, butdoing so allows us to assess the accuracy of Monte Carlo valuation for a given number of simulated stock price paths. The key question is how many simulated stock prices sufce to value an option to a desired degree of accuracy. Monte Carlo valuation is simple but relatively inecient. There are methods that improve the efciency of Monte Carlo; we discuss several of these in Section 19.5. To assess the ccuracy of a Monte Carlo estimate, we can run the simulation different times and see how much variability there is in the results. Of course in this case, we also lnow that the Black-scholes solution is $2.78. Table 19.2 shows the results from running 5ve Monte Carlo valuations, each containing 500 random stock price draws. The result of 2500 simulations is close to the correct answer ($2.804is close to $2.78). However, there is considerable variation among the individual trials of 500 simulations.
= $0.18
Given thatthe correctprice is $2.78,a $0.18standard deviation is a substantialpercentage of the optionprice (6.5%).Wit 1500observations, the standard deviatin is ctt $0.08, suggesting that th $2.80estimate from akeraging the five answers was only actidentally close to te correct answer. ln order to have a 1% ($0.028)standard deviation, we would need to have 21,000 trials. .
.
.
'
.
.
'
.
.
.
Arithmetic Asian Option ln the previous example of Monte Carlo valuation we valued an option that we already could value with the Blk-scholes formula. In practice, Monte Carlo valuation is usef'ul under these copditioj: * Where th nmber of random elements in the option valuation problem is too great to permit direct numrical solution. o Where underlying yariables are distributed in such a way that direct solutions (lifitl tllt. @Where options are path-dependent, i.e., the payoff at expiration depends path of the underlyinj asset price.
re
upon the
For the case of a path-dependent option, the use of Monte Carlo estimation is straightforwrd. As discussed above, we can simulate the path of the stock as well as its terminal vlue. For example, consider the valuation of a security that at the end of 3 months makes a payment based on the arithmetic average of the stock price at the
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M o NTE
end of months 1, 2, and 3. As discussed in Chapter 14, this is an arithmetic average plice Asian option: tlAsian'' because the payoff is based on an average, and average price'' because the arithmetic average stock price replaces the actual stock price at expiration.
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19
.
E
Tgble 19.3 lists plices of Asian options computed using 10,000 Monte Carlo trials each/ The first row (wherea single terminal plice is averaged) represents the price of an ordinary call option with 3 months to expiration. The others represent more frequent averaging. The Asian plice declines as the averaging frequency increases, with the largest plice decline obtained by moving from no averaging (thefirst row in Table 19.3) to monthly averaging (the secod row of Table 19.3). Note also in Table 19.3 that, in any row, the arithmetic average price is always above the geometric average price. 'Ihis is Jensen's inequality at work: Geometric
g(r-J-o'2/2)F/3+o.x/r/Jz(2) .(r-J-(r2/2)zy3+o.x/r/Jz(3)
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where Z(1), Z(2), and Z(3) are independent draws from a standard normal distribution. We repeat the trial many times and draw many Z, The value of the security is then computed as 's.
CAsian
Example
=
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-
( 19.7)
19,2 Let r 8%, o' 0.3, and suppose that the initial stock price is F'igure19.2 ompares histograms for the actual lisk-neutral stock plice distri-
$40. tyd rtyd tt'. $111/(*, t'ttd tgptd .-'yy' .'
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HoF will the value of an option on the average compare with an option that settles based on the actual expiration-day stock price? Intuitively, averaging should reduce the likelihood of large gains and losses. Any time the stock ends up high (in which case the call will have a high value at expiration), it will hav traversed integnediate stock prices in the process of reaching a high value. The payoff to the Asian option will reflect these lower intermediate pcices, and, hence, large payoffs will be much less likely. We compute the l-month, z-month, and 3-month stock prices as follows: =
VALUATI
)t bution after 3 months and that for the average stock price created by averaging the !@ l') As expected, the nonaveraged distribution has signifcantly .: three month-end prices. j(' . higher tail probabilities and a lower probability of being close to the initial stock price tjk ).1of t.j $40. k.
tialithmetic
s'j
CARLO
i' ..
'.. ..g
4A trial in this case means the computation of a single option price at expration. When 40 prices are averaged over 3 months, each tlial consists of drawing 40 random numbers; hence, 400,000 random numbers are drawn in order to compute the price.
'. . . .
.-.
(
.. ...
630
k
MONTE
CARLO
EFFICI ENT MONTE
VALUATION
CARLO
$
VALUATION
631
This is a control variate estimate.u Since Monte Carlo provides an unbiased estimate, E'-) = G. Hence, S(A*) = '(A) = z4. Moreover, the variance of A* is
averaging produces a lower average stock price than arithmetic averaging, and hence a lower option price.
*) vartW) + var ('&) zcovtW vartz't
'&)
=
19.5 EFFICIENT
MONTE
CARLO
,
As long as the estimate 'J is highly cormlated with the estimate W,the variance of the :stimate A* can be less than the variance of W. method can be dramatic. hpractice, the variance reduction from the cntrolvariate Figure graphs the results from the frst 200 simulations in pricing an arithmetic Asian option. The control variate estimate converges in just a few trials to the correct value of about $1.98. For example, the very first draw in the graphed simulation gave an arithmetic option plice of $0.80and a geometric price-using the same random pricesof $0.75. The correct geometric price is $1.94. Correcting the estimate gives a price
VALUATION
Monte Cado, making no attempt We have been describing what might be called for givn nulpber of trials. There are a the variance of the simulated reduce answer a to valuations.D . number of methods to achieve faster Monte Carlo
'naive''
.19.3
Control Variate Method We have seen that naive Monte Carlo estimation of an arithmetic Asia option requires many simulations. ln Table 19.3, even with 10,000 simulations, there is still a standard deviation of several percent in the option price. In each row of Table 19.3, the same random numbers are used to estimate the ' option price. As a result, the errors in the estimated arithmetic and geometric prices are cofrelated: Whe te estimated price for the geometdc option is high, this occurs because we have had high returns in the stock plice simulation. This should result in a high arithmetic price as well. Tlzis observation suggests the control variate.method to increase Monte Carlo estimate underlying this method is the The idea to error on each trial by using accuracy. the price of a related option that des have apricing formula. 'I'he error estimate obtnined f'romthis conol price can be used to improve the accuracy of the Monte Carlo plice on each trial. Asian options provide arl effective illustration of this idea.6 Because we have a formula for the price of a geometric Asian option (seeSection 14.2), we know whether the geometric price from a Monte Carlo valuation is too high or too low. For a given set of random stock prices, the arithmetic and geometric prices will typically be too high or too 1ow in tandem, so we an use information on the error in the geometric price to adjust our estimate of the adthmetic price, for which there is no formula. To be specisc, we use simulation to stimate the arithmetic plice, W, and the Let G and A represent the tnle geometric and arithmetic prices. The geometric price, error for the Monte Carlo estimate of the geometric price is (G V). We want to use this enor to improve our estimate of the arithmetic plice. Consider calculating
-
of
Control variate plice
=
$0.80+ ($1.94 $0.75) -
This example illustrates that if the con-elation control variate method works very well.
=
$1.99
between the two estimates is high, the
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Price of Arithmetic Average Price Call 40 #
3.5
-
-
Naive Monte Carlo Cont-rol Variate
3.0 ;
2.5
'-.
1.5
-
*
:4
=
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(G
-
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'-)G
sExcellent overviews are Boyle et al. (1997)and Glassennan (2004). See also Judd (1998,ch. 8), which in turn contans other references, and Campbell et al. (1997,ch. 9).. ti-fhisexamplefollows Kemna mzdvorst (1990),whoused thecontrol variate method topricearithmetic Asian options. '
0.5
0
20
40
60
80 1O0 120 Number of Trials
140
160
180
200
''naive'' Comparison of Monte Carlo estimate of arithmetic average option price with method. Graph depicts first 200 simulations for an option with variate control $40, (7' 0.3, r 0.08, T= 0.25, f 0, and the final price computed with three # .
=
,$40,
=
averages.
=
=
=
---.*,'k1-.
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--
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% Molq'rE
CARLO
Boyle et al.
VALUATION
VALUATI
(1997) point out that equation (19.8)does not in general provide
*
=
W+ p (G
-
'-)G
( 19 10) .
The variance of this estimate is Var(A*)
=
'J) var(W)+ p2var('J) 2pCov(W,
(19.11)
-
Var(X*) is minimized by Setting p = COVIX, V)/Var(V). One Way to of trials, perform small nunber Monte Cado obtain p is to nm a regression of equation a remaining trials. The optimal value of p will obtain and then (19.10) to use p for the p, depending the application. on vary
The
Valiance
19.6 VALUATION
Other Monte Carlo Methods
-0.5.
Tere canbe an efciency gain because the two estimates are negatively correlated; adding them rdces the variance of the estimate. ln practical tenns, this means that if you draw an exeme estimate from one tail of the distribution, you will also draw an extreme estimate from the other tail, balancing the effect of the srstdraw. Boyle et a1. (1997) nd modest benefits from using the antithetic variate method.
,
.
.
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OF AMERICAN
OPTIONS
American options. We will discuss pricing a 3-year put optin with a stlike of $1.10, the example used in Longstaff and Schwartz (2001). In order to analyze early exercise we need to consider times before maturity, so we must simulate stock pjcepaths. Figure 19.4, taken from Longstaffap.dShwartz (2001),illustrates eight hypothetical simulation paths, with intermediate stock prices generated annually. The in-the-money nodes (thosefor which
calculation.
=
O PTI ONs
It is generally more dicult to value American-style options than to value Europeanstyle options, and this remains true when using Monte Ctrlo valuation. Standard Monte Cado entails simulating stock price paths fonvayd,then averaging and discounting the matulity payoffs. ln American option valuation, te difliculty is knowing when to exercise the option', this requires worlcing backvard to determine the times at wltich the option shouldbe exercised. Recently, Broadie and Glasserman (1997)andtaongstaffapd Schwartz (2001)have demonstrated feasible methods for using Monte Carlo to value
The control variate example is just one method for improving the efficiecy of Monte Carlo valuation. The antithetic variate method uses the insight thatfor every simulated realization, there is an opppsite apd equally likely reylization. For eyampl, if we drpw By ujing a random normal number of 0.5, we could just as well have drawn the opposite of each normal draw we can get two sipulated outcomes for each random path we draw. This seems as if it would help, since it doubles the number of draws. But drawing a random number-is often not the time-consuming part of a Monte Carlo
Another important class of methods controls the region ln which random numbers ar'e generated. Stratitied snmpling is an example of this kind of method. Suppose 100. With naive Monte Carlo you have 100 unifonn random numbers, IIf i = 1, (IIf). #-1 This calculation each random number as would treats compute you zi representing a random draw from the cumulative distributon. However, because of random variation, 100 uniformrandom numbers will notbe exactly uniformly distributed and therefore the zi will not be exactly normal. We can improve the distribution of the Iff and therefore of the zi if we treat each number as a ?w??#(??n drawh'om wc/? percentile of f/?c ttniform distribtttion. Thus, tale te first draw, If 1 apd divide it by 100. 'I'he Take the second draw, divide it resulting I11 is now uniformly distribtuted over (0,0.011. by 100, and add 0.01. The resulting I12 is uniformly distributed over (0.01,0.02).For the This value is unifonnly distributed over the fth draw, compute i = i 1 + IIf)/100. th percentile. Proceeding in this way we are guaranteed to enerate a random number for each percentile of the normal distribution. You can select a number of intervals
AM ERI CAN
different from 100, and you can repeat the simulation multiple times. A generalization of this technique when the payoff depends on more than one random variable is Latil santpling, discussed by Boyle et al. /?lwd?'cal?c (1997). There are other techniques for improving the efciency of Monte Carlo. The approach called importaltce samplillg concentrates the generation of random numbers where they have tle most value for pricing a particular elaim. futnp discrepancy seqltences use carefully selected deterministic points to create more uniform coverage of the distribution. Boyle et al. (1997)provide an excellent summary and comparison of the different methods. lf you are performing a one-time calculation, the simplicity of naive Monte Carlo is appealing. However, if you are pedbrming a Monte Carlo valuatin repeatedly, you may achieve large efliciency gains by analyzing the problem and using one or more variance reduction techniques to increase efciency.
the minimum variance Monte Carlo estimate, and in some cases can even increase the variance of the estimate. They suggest that instead of estimating equation (19.8),you estimate A
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Soltrce: Longstaff and Schwartz (2001).
634
%.Morq'rE
CARLO
VALUATION
VALUATION
Continuation value at time 2
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where S is the time 2 stockprice. Now foreach node where exercise could be optimal, we insert the stock price at that node into the regression equation and obtain an estimate of continuation value. By comparing this to intrinsic value, we decide whether to exercise 1.08 in row 1, the esimated value of waiting to at that node. For example, when S exercise is
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early exercise is optimal?
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The idea underlying any method of American option valuation is to compare the value of immediate exercise to the continuation value of the option-i.e., the value of keeping the option alive.? The problem is therefore to estimate the continuation value at each point in time. It is wol'th noting one potential problem in estimating continuation value, which stems from the use of future stock prices on a given path to decide whether to exercise pn that path. Consider path 1 in Figure 19.4. The option is out-of-the-money, and therefore worthless, at t = 3. Therefore, on this path we would be better of'f at t = 2 exercising rather than waiting. However, in d.eciding to exercise by looking ahead on the path, we are using knowledge of the future stock price, which is infonnation we will not have in real life. Valuing the option assuming we lnow the future price will give us too high ttlookahead bias'' is to base an exrcise decisipn on a value. The way to mitigate such pointfomard. There agiven are at leasttwo ways to characterize average outcomes from point. is from given One to use a regression to characterize the the average outcome a analysis of multiple paths. This is the method proposed by based value continuation on Longstaff and Schwartz (2001).Another is create additional branches from each node, providing multiple outcomes that we can average to charactelize continuation value at that node. This is the basis for the Broadie and Glasserman (1997)procedure. To price the option using regression analysi, we work backward through the nmning regression each time of Figure 19.4, at to estimate continuation value columns a backward because the continuation value at of stock price. We work the as a function = exercise is optimal given path at t = 2. At t = 2, will depend whether 1 on a upon t where the option is in-the-money and exercise ;ve paths 3, 4, and 7) 6, there are (l, value exercising of of paths, know immediately be optimal. For each these the could we regression of Longstaffand Schwartz the the value of waiting. and present value of run a against the price continuation value) stock and stock price exercise the waiting to (i.e., result: time obtain the following At 2, squared. we
OPTIONS
Since the immediate exercise value is l l 0.02, which is less the 0.037, we wait at that node. Table 19.4 summarizes the results. We then repeat the analysis at t = 1, using the results at t = 2. The final decision about where to exercise the option is summarized in Figure 19.5. We can value the option by computing the present value of cash flows based on exercising at the nodes Where doing so is optimal. The fnal American put value is $0.1144, compared with $0.0564 for a European value computed using th same simulated paths. A problem with the regression approach is that it is not obvious how to select an appropriate functional form for the continuation regression. Longstaff and Schwartz (2001) report obtaining similar results for a variety of functional forms, but for each new problem it will be desirable to experiment with different functions. Broadie and Glasserman (1997)adopt a different approach, pointing outthatAmerican option valuations are subject to different ldnds of biases. As we discussed above, an estimator will give too high a valuation to the extent it uses information about the future to decide whether to exercise at a given time. Estimators will be biased low to the extent that early exercise is subopfmal (sinceoptimal exercise maximizes the value
$1 are candidate nodes for exercise of the option; in Figure 19.4 they are in bold (this ignores exercise at time 0). How do we determine at which of these nodes S
OF AM ERICAN
Path 1 2 g 4
5 6 7 8
t
=
1.00 1.00 ) j (s 1.00 1.00 1.00 1 .(m
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636
%.MONTE
CARLO
TH E PolssoN
VALUATION
of the option). To address the two errors, Broadie and Glasserman use two estimators, one with high bias and one with low bias. In constructing these estimators, they create sample paths in which there are multiple branches from each node. The resulting set of paths resembles a nonrecombining binomial tree with more than two branches from each node. The high bias estimatorassesses thecontinuation value by averaging thediscounted values on branches emanating from a point and exercising if the value of doing so is greater than the value of continuing. Because the subsequent branches are constructed by simulation, there will be sampling error. To see the effects of such error, suppose exercise is optimal at a node. lf the subsequent branches are too high due to sampling error, we will not exercise and assign an even higher value to the node than would be obtained by (optimally)exercising. Now suppose that exercise is not optimal at a node but subsequent branches are too low due to sampling error. We will then exercise and again assign a higher value to the node than we should, given the subsequent branch
values-s
.
The 1ow bias estimator is obtained by splitting the branches from each node into Using the frst set, we estimate the value of continuation and decide whether sets. two to exercise. lf it is optimal to contintle, we use the second set of nodes to estimgte the continuation value. By using separate sets of nodes to make the exercise decision and to bias'' discussed above. But estimate continuation value, this estimator avoids the to the extent the exercise decision is suboptimal, the inferred option value will be too low. Both estimators are biased, but both also converge to the trtle option value as the number of paths increases. The Broadie and Glasserman approach is computationally involved, but provides general method for accomodating early exercise in a simtllation model. a 'lzigh
DISTRI Bu'rl
olq
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637
unlikely yo occur more than once over a sufciently short intelwal. Thus, is like an annualized probability of the event occurring over a short interval. i Over a longer period of time, t, the probability that the event occurs exactly ??? tils is given by ./7(/1 ,
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The cumulative Poisson distribution is then the probability that there are events'from 0 to r.l0 l1t
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Of
the Poisson distribution is kt.
Suppose th ptobability of a market crash is l = 2% per year. Exqmple 19.5 Then the probability of seeing no market crashes in any given year can be computed as p(0; 0.02 x 1) 0.9802. The probability of seeing no crashes over a lo-year priod would be /(0, 0.02 x 10) = 0.8 187. The probability of seeing exactly two crashes over a lo-year period would be /(2, 0.02 x 10) = 0.0164. % =
-
19.7 THE POISSON
Figure 19.6 graphs the Poisson distribution for three values of the Poisson paramSuppose we are interested in the number of times an event will occur over a kt. eter,
DISTRIBUTION
We have seen that the lognormal distribution assigns a 1ow probability to large stock price moves. One approach to generating a rore realistic stock plice distribution is to permit large stock price moves to occur randomly. Occasional large price moves can generate the fat tails observed in the data in Section 18.6. The Poisson distribution is a discrete probability distlibution that counts the number of events-such as large stock plice moves-that occur over a peliod of time. where The Poisson distribution is sununarized by the parameter is the probability that one event occurs over the short interval h. A Poisson-distributed event is very ,
9By desniton, the number of occurrences of an event is Poisson-distributed if four assumptions are 'V satished: 1 1. The probability that one evelitcwill occur in a small nterval 11is proportional to the length of the interval-
2.
The probabilit'y that more than one event wll occur in a small nterval 11is substantially smaller than the probablitythat a single event will occur.
3. The number of events in nonoverlapping time intervals is independent. 4. The expected number of events betakeentime t and time t + The Poisson distribtion
-
ca be deiiked from these four assumptins.
lDln Excel,
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9e Casella and Berger (2002). and' the cumulative distlibution,
T(?? 1l'rhe probability that no event occurs between time 0 and time t is p(0, )vJ) C-ZJ The probability tliat one or more events occurs between 0 and ? is therefore l e-kt. This expression is also the cumulative dstribution of the exponential distribution, which models the time until the hrst event. The density function of the exponential distribution is f (?,2.) ke-kt. ,
SNote that the other two kinds of sampling errors do not matterfor assessing the value of early exercise. lf it is not optimal to exercise and subsequent branches are too high, we wll not exercise and therefore not erroneously attribute value to exercising. Similarly, if it is optmal to exercise and subsequent values are too low, we will exercise, giving the correct value to early exercise.
you can compute p(??), kt) as Poissontpl, kt,fae), ?), as Poissontn, kt, ?rI/c).
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640
CARLO
51 MULATING
VALUATION
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where a is the
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DISTRI BUTION
jump return is 10%, then over time the stock price will drift down on average due to jumps. ln equilibrium, the stock must appreciate when not jumpingin order to give the owner. a fair return unconditionally. If ag we would need to raise the stock in expected order i the for return to on earn a fair rate of return on average
lf there are ??7 jumps, we must then pick ??l additional random variables to determine the magnimdes of thejumps. Each jump has a multiplicative effect on the stpck price. Speciically, suppose the stock price is St If a stock cannotjump, 12is t+
JUM PS WITH TH E POISSON
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where the expected stock return, condional on nojump, is We will see in a moment why we use a different notation for the expected retprn in this expression. If the stock jumps ??; times b.etween t and t + l1neach jump changes the plice by a factpr of .
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........
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from the standard normal distribution.
a aczao,,l j 30 0/ o, z. g c' 2% an d a)
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.
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;....?...b....
Simulated stock price paths over 10 years (3650 days). One stock cannot jump;the other is the same except that 1'umps can occur. The simulation assumes that
from the Ppisson distribution. =
.E.t ....
E .. '' :
-
&+ using this expression. There are three steps:
3. Select m draws, J7(), i
-0.5(r+o'./
e
:'. :.'' '1*:
'''''''''
m
(e-5-k-0.5(r2)+o.Jjfz
.':j'j;rj'. .,i'jjjjjjjj. ''
1. Select a standard nonnal Z. 2. 'Select
e
where a.l and O'J are the mean and standard deviation of thejump magnimde, Z and Bs are random jtandard nonnal variables and ?n is Poisson-distributed. A similarexpression appears in Verton(1976). Figure 19.7 displays two simulated stock price series, one for whichjumps do not occur, and one generatAusing equation (19.16).In the absence of jumps, the stock
-0.5cJj+o'J EJLl 1P(f)
Notice that the cumulativejump is lognormal, since itis the product of lognonpal random variables. The stock price at time t + h, taking account of both the normal lognormal retul'n andjumps, is then
ul
''* (a-J-k-0.5tr2)+o'x/J'z = Ste
llt
Yi = e ??l al
'v
=
f=0
B?(f) e a-l-0.5o'J+o.g
=
??l are standld Where Z and F(f), i = 1, jump is the product of the l$.'s, or .
ll1
'' St+ll
'
,:
'
: ..
k.
'1:: (14(:2. (IEF)' i. t!iii d:::)h 4::22
,(:2: d:i!r (1:7
900 y(o
with Jump
xo pmp
....
700 600
.()
By inserting these values into equation (19.14),we generate St+It, which is lognonnal since it is a product of lognormal expressions. We have not nswered the question: What is & There is a subtlety associated with modeling jumps. When a jump occurs, the expected percentage change in the 1. If a.i # 0, jumps will induce average up or down movement stock plice is e'l in the stock, depending upon whether ag > 0 or a.l < 0. Recall, however, that we the Ifp?ctpntfrtNltv tl/at I$pd do assumed jumps are idiosyncratic. Therefore, (lneallin.ft ( . ' . . . . ?-e/lfr?i expected vhether vill occltr) tliat does ??t# not /c?zt?w for a evpected rdflfrpl f0l-f!p7 Othel-se idelltictll should be Xd Salne Ja tlle Ifnctmtf/fNlf7/ When jumps have no systematic risk, the jump does not affect stock r/ltzrdoes the stock's expected return. However, we have to adjust the nonjump expected return, in order for jumps not to affect the expected return. For example, if the average ,
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2500
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30%. For the 8% and tr plice is assumed to follow a lognormal process with a 3 (an average of three jumps per year), al jump component, we assume 5%. ln the fgure, we can detect jumps because the no-jump series is drawn and O'J using the same random Z's. Some of the disparity, for example between days 1000 and 1500, is due to the approximate extra 6% return (k) that is added to the stock when it does notjump. What happens if we apply the normality tests from Chapter 18 to the stock price series in Figure 19.7? Figure 19.8 displays histograms and normal probability plots for the two series. Without jumps, continuously compounded rettlrns look normal. With jumps, the data look nonnormal and resemble Figures 18.4 and 18.5. The jump results =
=
-2%,
=
=
=
N G CORRELATED
STOCK
PRl CES
$
643
in data that do not look normal. The kurtosis of the continuously compounded returns without jumps is 2.93, very close to the value of 3 expected for a normal distribution. With jumps, kurtosis is 7.40.
Multiple
Jumps
Wheqwe assume lognormal moves of the stock conditional on a singlejump event, we can only get large up alld down moves by assuming a large standard deviatipn of the jump move. The reason is that we are drawing from a single lognormal distribution, conditional on the Poisson event. An alternative is to assume there are tvo (or more) Poisson variables, one controlling up jumps and one controlling down jumps. The lognormal movej associated with each can have different means and stanard deviations. This obviously provides for a richer and potentially more realistic set of outcomes.
19.9 SIMULATING
CORRELATED
Suppose that S and
STOCK
Q are both lognonnally
distlibuted stock pries such that
ln(u%) lntuol + as =
1n(P?)
ln(:0)
=
PRICES
+ aQ
0.5c2.:)/ +
-
0.5G2
-
l'sx'jl
lt + o' CXLZ
lf S and
Q are uncorrelated, then we can simulate both prices by drawing independent I'I/ and Z. However, suppose that the correlation between S and Q is p. Here is how to simulate these two randop variables tnking account of their correlation. Let I and z be independent and distributed as X'(0, 1). Let
Then
4
orrtz,
Z
=
p:l +
l
62
(19.17)
p2
-
<) = p, and Z is listriuted pV0, 1). TO See this note frst tat Z and JF both have zero mean. Compute the covariance between Z and J&'and the vadance of Z: 't ,
(.q.
E (
=
=
E (eI
(p61+
E:2
Sg(p61 + ca 1
-
1
-
p2)j
p2)2j
=
=
PE (c2) 1 p =
/)2 + )
-
/.)2 =
j
Thus, F and Z are both X0, 1) and have a correlation coefticient of p. Now we will check tat the continuously compounded returns of S and correlation p. The covariance between lntxf ) and ln(:?) is E
0n(q%) '(ln(&)))(ln(Q?) f'(ln(Q,)))) -
-
=
=
E
Q have
ICSWVV()ZV1
csccpt
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CARLO
VALUATION
FO RTH ER READ I N G
is
The correlation coefcient
Conrelation
o'sccpt o's/to'cqt
=
,-
-
=
llot an arbitrary valid.12
p
Thus, if J;7 and Z have correlation p, so will the continuously compounded retzlrrls of S '
and
Q.
Generatina n Correlated Lognormal Random Variables .
N .
Suppose we have ,1 correlated lognormal valiables. The question we address here is how to generalize the previous analysis. The first of the 12 random variables will have n 1 pairwise con-elations with the others. The second will have lt 2 (not counting its co/elation with the lirst, which We have alredy counted). Continuing in this way, we will have 1 1) + 1 n n l + n 2+
lf this condition is not satised,
condition'.
$
645
the set of correlations is not
The point and the reason for mentioning this is that correlations and covariances cannot be arbitrary. In practice, depending upon how a covariance matrix is estimated, this can be an important concern. The true covariances among hundreds of bonds, stocks, currencies, and commodities ntttst create a positive-definite covmiance matrix. However, estimated covariances might ot be positive-defnite. lf there are m assets. and 11 > ??7 observations. the covaziance matrix estimated from theje data will be positive-desnite. However, if different covariances are estimated from different data sets, positive-desniteness is not assured. The results of a simulation based on such covariances may produce nonsensical results.
-
-
'
-
-
.
.
-?!
=
-
-
pairwise correlatios we have to take into account. between kariables i and j as pf,y. .
'
. .
.
We denote the original uncorrelated random The correlated random variables tre Z(1), Z(2), .
F(Z(f)Z(.j)) We can generate the Zi)
We will denote the correlation ..
.
=
.
. .
Xo, 1) variables as
.
.
,
.Z(?l), with
E: l
,
c2,
.
.
.
6,,.
,
pf,y
as i
Zi)
aitjej
=
;=l
FURTHER
selected to make sure the pairwise correlations lre correct. where the ai,j are coefcients Creating the coefficients i,j has a recursive solution. That is, we construct Z(1), the'n Z(2) using the solgtion to Z(1), and so n. The formula for the ai,.j is Cli'j
j-t
1
= a
9i
j.j
'
j
Monte Carlo methods entail simulating asset rettlrns in order to obtain a futtlre distribution for prices. This distribution can then be used to price claims on the asset (for example, Asian options) or to assess the risk of the asset (we will focus on suh uses in Chapter 25). Performing simulations requires that we draw random numbers from an appropriate distribution (for example, the nonnal) in order to generate future asset prices. There are adjustments, such as the control variate method, which can dramatically increase the speed with which Monte Carlo estimates converge to the correct price. It is possible to incorporate jumps in te price by mixing Poisson and lognol'mal random variables. Simulated correlated random vatiables can be created using the Cholesky decomposition.
Lvkcli'k
-
READING
The rst use of Monte Carlo meths to price options was Boyle (1977)and the technique is now' quite widespyead. An excellent survey of the use of Monte Cado in valuing derivatives is Boyle et al. (1997).We will see how Monte Carlo is used in value-atrisk calculations in Chapter 25. Bodie and Crane (1999) use Monte Carlo to analyze
k.1
( 19 18) .
(' i .-. l
ai
9
;
1
=
*
-
a;2k I .,
lzsuppose there are three random variables, a, b, and c, each with a variance of 1. Suppose that a is perfectly correlated wit.h b pa /) l ) and b is perfectly correlated with c pb.c 1). It must then be the case that c is perfectly correlated with a. lf pa c z 1, the matrix of correlations is not positive-definite. To see this, suppos that pa.c 0, then compute Vartc b + c). You will find that the variance is -1, which is impossible. To take a different example, suppose ps.c. 0.9. You will then hnd that Vartc which is again impossible. 2b + c) If a matrix of correlations is not positive-definite, it means that there is some combination of the random variables forwhichyou will compute anegative variance. (Formanycombinations the variance will still be positive.) The ntemretation of a negative variance is that you had an invalid correlation matrix to start with. =
=
=
'
For the case of two random variables, this reduces to equatlon (19.17). The matrix of azj's is called the Cholesky decomposition f the original corgive coefcients, matrix. order for equation the set of In relation correct (19.18) to which positive-definite, that the correlations be correlations must must be such means a' variables and negative variance. is random This is that'there no way to sum compute
-
=
-0.2.
-
=
646
% MONTE
CARLO
VALUATION
retirement investment products. Schwartz and Moon (2000)use Monte Cado to value a by simulating f'uture cash flows. 1117.n The papers by Broadie and Glasserman (1997)and Longstaffand Schwartz (2001), which present techniques for using Monte Carlo to value American-style options, have clear discussions of their respective methodologies.
1 e. slok
19.8. Assume that the market index is 100. Show that if the expected return on the market is 15%, the dividend yield is zero, and volatility is 20%, the probability of the index falling below 95 over a l-day horizon is approximately 0.0000004.
jumps. Naik and Lee (1990)illustrate option plicing in the presence of systematicjumps.
19.9. Suppose that on any given day the annualized continuously compounded stock ' return has a volatility of either 15%, with a probability of 80%, or 30%, with a probability of 20+. This is a mixture of normals mdel. Simulate the daily stock returfl and constnlct a histogram and nonal plot. What happens to the normal plot as you vary the probability of the high volatility distribution?
Risk aversion affects the option price in such cases.
PROBLEMS 19.1. Let
N(0, 1). Draw 1000 random Iff and construct a histogram of the results. What are te mean and standard deviation? 'w
Iff
l Ef=I 2
19.10. For stocks 1 and 2,
19.3. Suppose that of c'' and e'''2
ew
.x1
.h (0 1) and
.h:2
c'K2?
19.4. The Black-scholes price for a European put option with S 0.08. J 0, and t 0.25 is $1.99. Use Monte o' 0.30, r price. this Compute the standm'd deviation of your estimates. need do you to achieve a standard deviation of $0.01for your
=
=
=
19.5. Let 1' 0.08, S $100, 0, and tion, simulate 1/u$'1 What is EjzLl =
=
=
.
Paying 1/,1 ?
tr
=
.
0.30. Using the risk-neutral distribuWhat is the fonvard price for a contract
=
=
$40, K = $40, Carlo to compute How many trials estimates?
=
100, r 1l 6 Supp ose Sj 0.4 and 0.06, o's prices for claims that pay the following: .
=
0. Use Monte Carlo to compute
a. 5'l2 b. 5'1 SI c. -2 -0.3
=
'
,
a. 5'l :1 b. VSL/QL c.
SL
:c2
=
=
19.11. Assume
$100, r = 0.05, o' = 0.25, = 0, and r = 1. Use Monte Carlo valuation to compute the plice of a claim that pays $1 if Sv > $100,and 0 othenvise. ('rhis is called a cash-ormothing call and will be further discussed in Chapter 22. The actual price of this claim is $0.5040.) us'n
=
a. Running 1000 simulations, what is the How close is it to $0.50402
:1
=
.
,
=
.
=
estimated
price of the contract?
lour
Monte Carlo estimate? What is b. What is the standard deviation of the 95% consdence interval for yot!r estimate?
c. Use a l-year at-the-money call as a control using equation (19.8).
variate and compute
a orice -
'
d. Again use a l-year at-the-money cgll as a control variate, only this time use equation (19.10).'What is the standard deviation of your estimate? 'Forthefollowingteeeproblems, assumethat Sz $100,r 0.08, c4 = 0.20, c = 0.30, and J = 0. Perfonn 2000 simulations. Note that most spreadsheets have built-in functions to compute skewness and lturtosis. (1nExcel, the functions are Skew and KulKl For the normal distribution, skewness, which measures asymmetly is zero. Kurtosis, discussed in Chapter 18, equals 3. =
and that s'a $100, 19.7. Suppose that ln(5') and 1n(:) have correlation p 0 06 o' s 0 4 and o' c 0.2. Neither stock pays dividends. po $10t l Use Monte Carlo to find the price today of claims that pay =
$40, Sz
,
Why? and a. What are the means of b. Cfeate a graph that displays a frequenc'y distribution in each case =
?'
=
correlation.
M(0.7 3). Compute 2000 random draws E'
c'VI
=
Let
-
x
,
SI
$100, and the return coryelation is 0.45. 0.08, (7' l = 0.30, (7'2 = 0.50, and J1 = (z 0. Generate 1000 l-month prices for the two stocks. For each stock, compute the mean and standard deviation of the continuously compounded return. Also compute the remrn
lti 19.2. Let lti. 6, 1000 times. (This will use 12,000 N(0, 1). Compute random numbers.) Construct ahistogram and compar it to a theoretical standard normal density. What are the mean and standard deviation? 'x'
647
d. 1/(S'1 Q l )
(1976)derived an option pricing formula in the presence of idiosyncratic
Merton
k
PROBLEMS
=
=
1/52. Simulate both the continuously compounded actual return and 19.12. Let /? the acmal stock plice, St-v What are the mean, standard deviation. skewness, and kurtosis of both the continuously compounded return on the stock and the stock plice? Use the same random normal numbers and repeat for /2 l Do any of your answers change? Why? =
.
=
.
j'
@ 'j ) 'j :
t
rilltsr''k'-jjd 'j' p' ;yy' j'jj'. yjjjjj'..., ,'y' i'. t'j))d (' j'. j'k' r'ld j)yd :j' t'i't' 'jjd r' (' j';'j' pjt'jjy. ty'. ,j'.' 'i--t'y; .;'..6b'. ..F' r',;'. ''i f'....k,,lE .q'. .fi.' ?' .L,' .@' .f' .tjy'. ()' )' y'. k' j';j;('.' rd tr'jyd y' ).)tjE;t';)-' ktjt-'prd (t' ,''j'j' :j' yjjjk..' iyd t'. jqd j'j' r'r' 't'j pryd )' jjjd j.t)')E)' ji;d ;LL,'' :!' kjyr'jtd /'?' j)' y' y;' !j' i;d '') .(!.' r:.' .)!.' r' .:J'. j;t'. :'(' y'tj(. )'. )'))j yyyjtj'jytyy. :4)j'-,jyy' gjd yyj;tjjyd jjjj'. yyjr'jjjj:. jy'j(y-(t' jj'yj yjd jyd :y' 48 jjyjd kjjr-.d (' (gjjjyy,'j;:yyy;jjd kjjyg-,d jyy'. lp'j j'yd jjjd l/'yd jj'. rr'yj r' )' ;'@' .jy', t'. jjj'. jjtj' .111::24(((((;111.* .111::((((((5;11,.* 1;* !' 1* t'1..;*. ji','.,.-' l';'.y'j;'' )jj' jtd jllk:;kd j;' kliq,' t'j'r')'1* jyjy'ld jydly'j.jyd (j.,' jjgy'yjjd #' jy('jyj'j-. y)'. jtr'r' tjd tt'. yy,jy'yr'. j;'yyyjj-,' yy'. jty'j'. ty')' j'; )t'. jjd yl'tjj ;j-,'LL?(', $' jjsd jt'jyj.j j.yj'; y' /':2* jkdjyjjj,d j'y 'risjd )' Jtjjdl'jt.g r'j (:()))'. (((((i-' r,-,td jj!jd (jy' jj'--yj'j jjy'. jy',(j'-j'lqi:jd j'jd p' )).yyj'jy $yjj444,,* t'..jjr' )y' j'. j'j.jd j'j)j';))jj', jyyd y'. l'r j-'jyy jyd jy;lr')t'. jy'. 'jyd jytld yt'. .j'i kyy'y' j)d )'pj j'i y)d y('. yjyj(j.' jj)', j)'jjj(j k' jj'. yjd j'jjj.' ''; jg'' jjd js'r ('.' '':(' j-!,' r'' @' )))j!.tjjyj't(' $;)f'. j(' y)jd .y'. )' /))))q.pj;j)Ij(jjkpy)j'.jytjjjj j)jt(jj;(yk)'(' jtjd lr'yi !q' (jr'. )'kq;-r' y')' jjdj.jl'y)r jjl'py jrj(';
;)'yr!jy' 11*. ijdlttjlt.ltill-'!id jrjt'ty'jjy j'jg )('jj#',).;y'jj) (' ;j'kt)p((j' j'lyyjjjl';jd y' pj'jjd gjyyj'yg y'! $)k' q' ))'. ,'yyyyj'. )'(' jgd jjd q'. lkdjtiytljjyjr't,d yj'jtd ;'it;rd )'. yytyj'.d y'q jy', r)', j': j'! y'. (jd lt'j .y' jyd jjtjjtd j.jyr((tt'., ,j;LL)b.' ()jj,'. ''p' (t' y,'t)' jjjy:,' qqdtj l'jd )*?1 j'. (j'. y'qd y'td yyy'yy ;y' yj'j'. j'q l'i yjd yjj;',, jjjj-'., t'jjjj.d ).jyd j-td yryjd Sl'y yjjyt'j y;;rkd yyd jjjjjjj,,,-,d jgjjjjj'-j;s;rjjjs jjjjjjj,,,tjjjjjgd tyjjjjj;;d .)' y,-jd (j' jryjj'.' j'jy'. y'j.yyj', yyy'yyd ('jd y'g y'! yt'yg )p' ;'y' )' pld ):kL', )-1*, 1* yy'yj j'jd .y' y'j. ((',j)tj' )'. k' lt'j jt'jj p'y yq'. (' t-,' ytd (jy'ki lt't' yg'. y'j )'-('. jyd jjd yy;'')ktd ' hilrh')d ,t'-!-! jyjy'y'yjy'' ('y )-'.. 711)4;;*,* j.j-rr!,d p'rygr)z .)j'. g'. ptd jtd .LLblLfk'jl.j' .t' t'y.('. j.!d q' )'-jy j).yjjjjjyd j'ydtly'tr'qj)d )(y'. jt'. y': y)' J)' lt'jjj jLjLLL')L))jj3jL.)jLqqjjLLLjLL))'jqL..LLLLjI('. jp'yk qtjrtl')td jqd ))'(! )y' jytjj)'. )t'tjt:-' p' jj'y jjld )' 1*. yp;'-' tkr'yy ;;'yjj. 1* yjjj;d yjy)', yjyd y' )'. )'!' (' q' q!y'. ?' jj)jkd rjjrtd yyjjjjjjjl:,d jyjjjgyd rk.jt';yd jjjjjjjjjjg,d jjjjjgjgjjjd g' yjd y'. y';rd (j'-. l'i t': q'. 111*). )'t.)iyr' 1*, )' y' tid ygtd ('. )t' (!y'r! yty'ptlr'yy tij'j ri:';l'@.'j-'.pr'j'! ..1j-'. (t'i gyjj'jy$)))'. y'ry ryd )y;)'(' yryd ;'));'. ))' #' .(t'. t')1* (iq'tjk)j(i. tyjjyry'. :!!44(;;1,,* yjyyjj'tr'. tr';l tjdyyyy, t;d jyjjjjjgy:jj,d yjjjgjy'-j;s:zyyyd t't'. jjjjjjjjjjgygjjd tj'trd t't$' t'yj ty'. rjd t't' tlrh:yd t'j yll' t))..'-jy'. ttd 'ttd tyj'. tgd ttj'k ty'. r' ry'jt'! t)' ))y', .-t'. rj'yjy )j)'. r' r'jdyyij .)'!-
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'''(iIIl:---'-''--'' '''jiI!!r...,' '''iiI1!r,--,' 1(*
648
% MONTE
CARLO
VALUATION
19.13. An options trader purchases 1000 l-year at-the-money calls on a nondividend0.20, and ty 0.25. Assume the options are paying stock with Sv $100,a according 0.05. priced to the Black-scholes formula and r =
=
=
=
-
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FOR GEOMETRIC
;
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APPENDIX 19.A: FORMULAS AVERAGE OPTIONS
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b. Repeat for an at-the-money put. 19.14. Repeat the previous problem, only l-year at-the-money straddles.
-. . . ..
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.
a. Use Monte Carlo (with1000 simulations) to estimate the expected return, standard devtation, skewness, and kurtosis of the remrn on the call when it is held until expiration. Interpret your answers.
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k.
651
These properties imply that Z(l) is a martingale: a stochastic process for which
ELzt
ln(5'(r)) ''w.Y1n(kS'(0)j+ (ct 0.5c.2)7* o'3T) -
,
The assumption that the stock follows geometric Brownian motion thus provides a foundatin for our assumption that the stock price is lognormally distlibuted.
Lognormality tells us about the distribution of the stock price at a point in time. For many purposes, however, we are interested notjust in the distlibution at a terminal point but also the path the stock price takes in getting to that terminal point. With barrier options, for example, the price of the option depends upon the probability that the asset price reaches the barrier. Geometlic Brownian motion allows us to desclibe this path. Our goal is to provide a heuristic, rather than technical, understanding of equations like (20.1).
IZ(l)) = Z(f). The process Zt) is also called a diffusion process. To represent this process mathematically, we can focus on the change in .Z(l), which we model as binomial, times a scale factor that rriakes the change in Zt) small over a small period of time. Brownian motion is then the limit of a sum of insnitesimal increments over a period of time. Denote the short period of time as h, and let F41) be a random draw from a binomial distribution, where i'(/) is +1 with probability 0.5. Note that F(i'(r)1 0 and Var(F(/)) = 1. We can write .)
+
=
Z(l + hj
Z(l)
-
ROAVNIAN Mo-rloN
?1
Z(F) =
-
Z (0)
-
* Zt +
,)
Zt)
is normally distributed with mean 0 and variance s. Z(!) is independent of Z(f) Zt * Zt + sz), where 'I n > 0. In other words, nonoverlapping increments are independently distlibuted. * Zt) is continuous (youcan draw a picture of Brownian motion without lifting your Penc il) -
F i ll)
/?
11
1 Yih) W J-l ,-I
./j7
z(r) z(0)
=
To understand the properties of Z(r),
rst understand the properties of
we must
the term in square brackets in equation (20.3). Since E (l'(f/7)) 0, we have =
?1
1
E
n.r
,*G
Also, since Var(1'(/7))
=
Yih)
0
=
i= l
1, and the l''s are independent, we have Va.r
0.
=
.
1)/?q) =
zg(f
-
i= l
-
?1
1 11
Lt
* Z(0)
11
.
(Z(f /7)
=
F/?7, we can also write this as
DeOition of Brownian Motion Z. t) represent the value of a Brownian motion at time r. Brownian motion is a continuous stochastic process. Z(l), with the following charactedstics:
(20.2)
=
Since 11
A stochastic process is a rndom process that is a function of time. Brownian motion is a stochastic process that is a random walk occurring in continuous time, with movements that are continuot!s rather than discrete. It is basic building block for standard derivatives plicing models. A rapdom walk can be generated by flippig a coin each period and moving one step, with the direction determiiled by whether the coin is heads or tails. To generate Brownian moiion, we would flip the coins infinitely fast and take innitesimally small steps at each point. Since all steps are inlinitely small, movements are continuous. As you might guess, a careful mathematical formulation of this concept requires mathematics beyond the scope and purpose of this book.l
Ftf + hj.vl'
Over any period of time longer than h, Z will be the sum of the binomial increments specifed in equation (20.2).Let 11 Tj /?be the number of intervals of length /7between ( 0 and F. We have
i= 1
20.2
=
i' ih)
1
=
11
1=
-
11
i=u I
1
i=c I
Thus, the term in square brackets has mean 0 and variance 1, since it is the sum of independent random variables with mean 0 and variance 1, divided by V-/l. By the Central LimitTheorem, the distribution of the sum of independentbinomial random variables approaches normality. We have
11
'1)
-
-
-
,
.
l'l'lle development in this section draws heavily on Cox and Miller (1965,ch. 5) and Merton ( 1990). For a more abstract approach to this matelial, see Dufie (1996)and Karatzas and Shreve
(1991).
lim ?/-.s co
11
l
g-. py
l'(f)
./'(0,
''x-
1)
y.j
./'7
The division by in this expression prevents the variance f'rom going to infnity as lt infnity. to goes Remrning to equation (20.3), the multiplication by F on the right-hand side multiplies the variance by F. Thus, in the limit we have Z(F)
-
Z(0)
-->
X'(0, r)
Q BROWNIAN
AND IT?s
MOTION
LEMMA
BRowlq
To summarize, we have verised that the Z(F) we have constructed has some of the characteristics of Brownian motion: it is normally distlibuted with mean zero and variance F, and increments to Z(F) are independent. We have not verified that the Z(F) dened in equation (20.3) is a continuous process, hence we have not demonstrated that it is tnle Brownian motion. However, it is plausible 't hat Z(F) is continuous because the magnimde of the increments is X-h F/?), and 13 0 as ?1 x. It is common to write down expressions denoting the Brownian increments. As 11 becomes small, rename /? as dt and the change in Z as dzt). We then have
hence
11
1lm
?1..-.:(13
=
F(r)
dt
i
Z(F)
Z(0) + lirpuVF lle
!j
J-lYih) ym j
Z(0) + u
.
j(j
The integral in expression (20.5).iscalled a stochastic integral.3
11
l
(z(i'/?.) zgt -
JI'-FIXI
i= l
-
1)/2.1)2 =
1im
/?Iy,
lt''-?
i= l
2
17 =
/7F2f/,
lim
/1-:X
l
lim
i=l
1
F
a''+(X)
i= I
1, qnd hence
=
11
X-11 IYilll =
/7IYiltl
X
11H
c-
F lim
=
Qll
f=I
N-'YX
V'
=
(x)
Arithmetic Brownian Motion o
The Brownian motion process described above is a building block for more elaborate and realistic processes. With pure Brownian motion, the expected change in Z is 0, and te variance per unit time is 1. We can generalize this to allow an arbitrary vtriance and a nopzero mean. To make this generalization, we can write
i= 1
X t + /7) This equation implies that XT)
zshould we think of F(1) as being binomially or normally distributed? For any t and E:, Zt +c) Zt) is te sum of innitely many dztlks. Therefore. we can think of i'(?) as binomial or normal; either way, Z(/) is nonnal for any finite interval. 3Because dztj is a random variable, considerable care is required in dehning equation (20.5). See Neftci (2000,ch. 9) for a discussion of stochastic integration. Karatzas and Shreve (1991)provide a more advanced treatment.
lirn
=c
This means that the absolute length of a Brownian path is inlinite over any finiteinterval. I order for a path to hate innite length over a firtite interval of time, it must move up and down rapidly. This behavior implis the ilfinite crossing propel-ty, wich states that a Brownian payhwill cross its starting point an in:nite number of tipes in an interval of any finite lengt.
Properties of Brownian Motion
lim
1)/11)2 = F
-
i= l
11
?l'-+(x7
is
h l'f/, l
Again, treating l$. as binomial, we have IFf/;l
1im
We now use equation (20.3) to understand some of additional properties of Brownian motion. The following derivations will be informal, intended to provide inmition rather than actual proofs. In particular, we continue to use the binomial approximation to the Brownian process. The quadratic variation of a process is defined as the sum of the squared increments to the process. Thus, the quadratic variation of the Brownian process Z(/)
Z((f
-
11
lt-
(20.5)
#Z(l)
F
=
l
=
12
lirn(X)
2*
/%
1 and
Surprisingly, the quadratic valiation of a Brownian process from time 0 to time F is not a random vatiable, but it is inite and equal to F. An important implication of the fact that quadratic variation is finite, is that higher-order variations are zero. Thus, for example, the sum of the cubed inrements is zero. The snitequadratic variation of a Brownian exiemely important fesult that we will encounter again. Process t'ul'ns out to be art The total variatlon of the Brownian process is
itover
-->
w 11 '
I
=
(Z()
lim
(20.4)
11
-
31-yco
i= I
JI'-y())
-
ar'
'''
=
l
This representation of the Brownian process is mathematically informftl but surplisingly useful. Equation (20.4)is just like'equation (20.2),except that Zt + h) Zt) is now called dzt), and X-his now dt. 2 Equation (20.4)is a mathematical way to say: small periods of time, changes in the value of the process are normally distributed with a valiance that is proportional to the length of the time period.'' Although expressions like equation (20.4)appear in the derivatives literattlre, it is ' mathematically more convenient to deal with sums of increments rather than increments. These sums are written as integrals, for example: =
.
11m
42,
In other words,
-->.
dzt)
/1
2 = /?F;?,
.
%. 653
Mo-rlolq
Since we are treating Yi as binomial, taking on the values +1, we have
=
-.+
IAN
-
X t)
=
ah + o' F t + 11) X-h
is normally distributed. Since /7
'
11
-
x(r)
-
x(0)
=
J'q a- r izcj
l
+ cvih) ll.
-
r 11
i' ih)
-ar+. zf;)'-lW i= l
=
F/?7, we have
654
k
BRowrq
IAN
AND I'r's
Mo-rlolq
LEM MA
GEOM ETRIC BROWNIAN
We have seen that as ?) x, the term in parentheses on the right-hand side has the distlibution .V(0, F). We can write -->
X(F)
X(0)
-
=
aT + O'ZCT)
The differential form of this expression is
dxt)
(20.8)
This process is called arithmetic Browninn motion. We say that a is the instantaneous (7'2 is the instantaneous variance per unit time. The variable Xt) mean per unit time and is the sum of the individual changes dX. An implication of equation (20.8)is that XT) - is normally distributed, or
x')
=
x(O)
x
M(aF,
cr2r)
As before, there is an integral representation of equation '
F
-Y(r)
=
7(0) +
drif't term: dxtj
+ o'dzt)
gc Xtldt
=
-
(20.9)
When a = 0, equation (20.9)is called an Ornstein-uhlenbeck process. Equation (20.9)has the implication that if X rises above a, th dlift, l(y X(?)), will become negative. lf X falls below a, the drif4 becomes positive. The parameter measures the speed of the reversion: If is large, reversion happens more quickly. In the lon nm, we expect X to revert toward a. As with arithmetic Brownian motion, X can still become negative.
20.3 GEOMETRIC
ln general we can write both the dlif't and volatility as functions of X (or other variables):
(20.t):
dxt)
o'dzt)
and volatility, o', depend on the stock price, is called
Suppose we modify arithmetic Brownian motion to make the instantaneous mean
(20.8):
and standard deviation proportional to X(r): dxt)
drkft into the process. Adding
adt
has the
e
Being able to adjust the drift and variance is a big step toward a more useful model, arithmetic Brownian motion has several drawbacks: but
o There is nothing to prevent X from becoming negative, so it is a poor model for stock prices. @The mean and variance of changes in dolla.r tenns are independent of the level of the stockprice. ln practice if a stock doubls, we would expectboth the dolla.r expected ret'urn and the dollar standard deviation of remrns to approximately double. We will eliminate both of these criticisms with geometlic Brownian motion, which in Section 20.3.
we consider
Process
Another modiEcation of the alithmetic Brownian process permits mean reversion. It is natural to consider mean reversion when modeling commodity prices or interest rates. For example, if the interest rate becomes sufciently high, it is likely to fall, and if the low, it is likely to rise. Commodity p ces may also exhibit this value is suciently tendency to revert to the mean. We can incorporate mean reversion by modifying the
=
+ o'xtldkt)
axtldt
This is an 1t process that can also be written
dxt)
-
.1(0).
(y,
(20.10)
an It process.
* Xt) is normally distributed because it is a scaled Brownian process. * The random tenn has been multiplied by a scale factor that enables us to change variance. Since dztj has a variance of 1 perunit time, o'dzt) will have a variance of tr :t per unit time. The adt term introduces a nonandom ffectof adding a per unit time t
ce(X(r))#l + (rEX(l))#Z(l)
=
This equation, in which the drift,
0
This expression is equivalent to equation (20.7). Here are some of the properties of the process in equation
MOTION
BROWNIAN
T
dt +
0
The Ornstein-uhlenbeck
655
-
adt + c#Z(r)
=
$
MOTION
=
Xtt
adt
(20.11)
+ o'dzt)
'Ihis equation says that the dollar mean and standard deviation of the stoct price change are aX t4 and O'X (r), and they are proportional to the level of the stock price. Thus, the Malue is /l&rDlflPlldistlbtlted 'kWl/iilstcllltclneolls Nldflpl t:t pel-centage ct-lnqd ill tlle JIJJV/' c2. equation The in variance instantaneous and process (20.11)is lnown as geometric will of the book, motion. the For frequently rest Bro&vnian assume that prices of we equation 1). follow and other stocks' assets (20.1 The integral representation for equation (20.11)is XT)
-
X(0)
r =
'
axtjdt
o'xtjdzt)
+
0
0
Lognormality We now circle back to our discussion of lognprmality because of this fact: A variable that follows geometric Brownian is lognormally distributed. Suppose we strt a process at xY(0) and it follows geometric Brownian motion. Because the mean and variance at time t are proportional to Xt), the evolution of X implied by equaton (20.11)generates compounding (the change in X is proportional tp X) and, hence, nonnormality. However, while X is not normal, ln(X(/)) is normally distributed:
lngA-(l)J M(ln(X(0)j x
+
(a
-
0.5(r2)/, o'ltj
(20.12)
656
%
BROWNIAN
MoTlohl
AND I'r's
:'. ''s' r'. y' jjjjjy,;jjjd ,'j,,rgr:,jf j,:yyyryf j,::ry;,,f j,::yyyyf j;:ggg,,f jjyyjg,,f jj:ryy;,,f j,:yyrr,,f ,'''';' :'71:,7*,,)'7:'*755q* j.j..,,'. F' 'j' : iE .jj:g!jjj' jlE j'$' @;' ''' 1* p' 4jj:jjjjjj.'. (E . E (iE ( i E E
:,jjj;' j,,.,,,,y,,.,,f :';'t''. ,!jg;;' q' i'tlf .':' (1): (' -jjjIr'j;j!r. j'jjyE,,f .y' ,;,,,,,j' .jjjyy' .jjyyy' )' 'jjjjj,.' q' y' (.' ('''i' q'l'' '!'l' (' ' (.!'qq' !( i i r! (' ((4( (lqjjjtj t'r' ggjjjjjj;yljjjjryd q: .'j.'.i('. (.'j.''(.' q( ;.ii i('' ( /;j jy'. jjf jyf jjj,,,!,;f j'jjj.,,r,jf )' y' j'-jk,j (' (jljj.jjjj.d (qjjjljqy,,jj'. rjjijd .. . )'(. (). )' 1* y'. .' yt('; t'ilq''jy ttf ; .' ;yj (j;' !ti .yyy.y, '( (lj ' ry .''tlitlffk yq .y,t.-,. '...'.. ... gy'- ;. jrj;,jjjjy. (yjyy ..y'y..y ygjty jj(.:(y ... yy ; y . . y . . y . lit-.it.t.i . jjrjjgjjjky. . .j.. . gyjyyjyyyyg .. .t.yy (j j j ---j.;:;;Lj;;j.j,:j. .,,,jjj,,,.' .;jj::' ,4,::)* ,,:,' '''q' ;' .',,,,,' '....,)4,6''.jjj4qjjjj..'.
As a result, we can write X(0)d (e-0.5(r2)?+(rs'z
=
(atl .
jg;
where Z X'(0, 1). This is the link between Brownian motion and lognormality. lf a variable is disibuted in such a way that instantaneous percentage changes follow geometric Brownian motion, then over discrete periods of time, the variable is lognormally distributed. Given that X follows (20.11),we can compute the expected value of X at a point in the f'uttlre. lt follows from the discussion in Section 18.2 that (a-0.5(r2)f
,E'(.Y(?)q= Xje
= Xle =' X (0)eJ
(20.11)is the expected,
Relative Importance
(e-0.5o'2)J
g.()(go'Wz) e
0.512/
continuously
(20.14)
compounded ret'urn on X.
-. !E E.--..)y.
. .
:!.-...
'x.
Thus, a in equation
.' E.
Xt)
.
:
-
......
'
:
:
-
.
q
...,.
. . ...
.j..) :-
......
..
l
..( ' ' ' . .
E ..
E '' ' ('
EEE E:E' '' EE.Ei :'' '':!! E' '' ' EE '' ' ( .. . i(' j E(g . i . . . ' E . . ' . .. -.;.. -. -
:
- . - - .- .E.. .E - . .. .-
t..-....,..,i-.....-,J!. 1?-.!-:-.
i:' : .
:
.
!E -.
.. ........ . . ..
'
( .
( r.
.
...
.
..
-.
(..--
.. .
( '
:
'.
.
.
'.
...
'
.
.
.
g
.
'. .
.
.
' ' ' '' : . . . .. .. .
. .
' '..(.' . . E.EE...'....'... .'. .... . ......
J
.
. . .
.
(' . . r . .. .
.
.'.
Flve years One year
.
.
.
. .
'
..
.
g .
.
.. . .
j(
.
.
..'.
' .. .....
X(r)
=
+ o'xtlytl.-
axtlh
Overashortinterval of time, there are two components to thechange in X: adetrministic An important fact is that and a randop component, o'xtlYtlx-h. component, axtlh, J/zt/?-? oftime, the c/zt-f?wcldr ofthe S?-(n$/?)n process is #'/'!?p??k7W allnost periods over entirely by the ?-t'I/7#tp/n component. The drift can be undetectable amid all te up and down movement generated by the random term. To understand why the random term is important over short horizons, consider the ratio of the standard deviation to the drift: tr X (t )XLl
ax
tlh
tr
=
5
1
This ratio becomes innite as h approaches dt. A numerical example shows this more concretely. Suppose a 10% and tr = ' 10%. Over a year, the mean and standard deviations are the same. Table 20.1 shows that the ratio increases as the time interval becomes smaller. Over a period of 1 day, the standard deviation is 19 times larger than the mean. This is important in practice since it means that when you look at daily returns, you are primarily seeing the movement of a random variable following pure Brownian motion/ The deterministic drift tthe expected remrn) ig virtually undetectable. =
are other considerations when you look at prices over short periods of time, including the bouncing of prices between the bid and the ask, and the effects of trades such as large bloks tbat may temporarily depress prices. Brownian motion implies that even in the absence of these effects,' prices would still bounce around significantly. .
. .
. .
.
:
.
(
:
.
..
. .
.
.
( j( . yy. (. .j y.g. i ..... . .. . . . .. . .
y
.
j. .
.
:j.
. .
.
.
,6'..
-
-. .
-
.
0.0833 0.0027
One minute
0.000002
jj:yyy:jjj
j,y,., ,,y,., jj, jjryry:,,. jjyyyyr,,.
,jj:yy.
(ll
(
.
.
j
gAj
g xa ( r .j q'.. . . . .. . . . . . . .. . ( 'r . .. . ...,j,,,.jj,' j. . j (. . .('j ('.. j IjLLLLjjLLL,L j. j (.j.g.j j .. j y . . j . . . . g . j . j . 'k:vvbb,j j . y . .... .g. y j j . .... ... y y.... . y. j. . . ..yy..y ..j. .. .. . yg..y . y. y ..y y. ( j. j . j g g.. .. . .. g y.............. y.......... y. ... y .j .. . . '... . .. . . ... . . .. . . ...
.. .
.. .
. y.
: g
......'
..'.. ..
-
-
.
:
.
. . .
:
. .
. ..
.
.
.. . .
. .
'.'.jj.. g... . .. :
.
. y . ..
.
.
.
.
.
. . . .
.
g .. .
l(
' y
..
:
.
..
.
.
..,...:...
E
'.. j.( j. .. .
.
,
-
y.
.
.. . ,,,,4,j,,,,.,,j1,,.j,r..-.
. .
.
.
.
.
.. g. .
g
g
y
g
..... .
,
y
..
.
g
. y :
..
.
.
.
. . .
.
g y
. . ..
. .
. .
.
..
..
.
.
.
..
.
:y g
k j .jj
. -
:
. . . :. . . . .. . . . . . . . . .. . . . . . . .
0.5 0.10
0.2236 0.10
1.00
0.0083
0.0289
3.464
0.00027 0.0000002
0.0052
19.105
0.00014
.
.
.
0.447
724.98
As the time interval becomes longer than a year, the reverse happens: The mean becomes more important than the standard deviation. Since the mean is proportional to /? while the standard deviation is proportional to VV,the mean comes to dominate over longer horizons. Since we take the ratio of the instantaneous stfldard deviation to the instantaneous mean, Table 20.1 also holds for arithmti Brownian motion.
Correlated lt Processes Suppose that
we have the 1t process =
(20.15)'
ct(2E:(r)1#r+ o-pE:(r)1#Z'(r)
where Z'(/) is a Brownian motion. The Brownian motion Z'(/) in equation (20.15) can be correlated with Z(l) in equation (20.10)kFor example, if X and Q represent stock plices, X and Q will typically be correlated. Let F1 t) and Wi(r) be independent Brownian motions. We can then write Z(r) Z't)
=
Fl
=
pFl
(r) (r) +
1
(20.16)
p2W:?(r)
-
You may recognize this is as the Cholesky decomposition, which we discussed in Section 19.9: Using equation (20.16),the correlation between Ztj and Z't) is '(z(l)Z'(?)1
Verhere
..
One day
One month
dotq
,/-#
a
. .. . .
jy .. ..
.
Consider the discrete counterpart for geometric Brownian motion: -
jjyy!g,,r jjyyyy,,r jj.,,,jjy kyjg,j, jjj j,,,,,,,j ,,.,,,,,,,,,,,, ,,,,.. jjyryyy,j-
jjyyg,,r
:. '. ( .. g ' : . . . . . . . .. ( ( ( . :. . ' .( ( ' '. .jjj; y j. . . . y. . .. ... . . . .: . .. . ... . g. E. .. . .. . j jjiii!!:. . .. .g . . :. . E . . . . . gj . . .. .q q.. g. y;. :. ... ..g . . q.y.. q y: y .E..(. E. j jjt:::ljt!:. .y.yg y. iyq(. q.. .y. ., . y j.. :.j....g .... y. .g. . .. . . y. . . . . . . . . .y . . .. . y y . . . . .. . y . . . . . . . . . y. . y.;y : '' ' E ' ' ' ' E ' '' ' : : .E .. . . E EE.'. (. ... .. . . .. . ... . .. .. .. . ..' .. .' . '. ..... '. . ... .. E . E.. ... .... ... .. . '.....'. E . .y . . . . . ........... . ....... ........... . . . . ....... .. . '
-
.jjjyy, k;jjg,,i jj,,,,,jj
standard deviatin to yhe per-period mqan for different time intervals. The ratio becomes infinite as the time interval goes to zero.
. (.j rjjijr.,. tjjjjkrjlky -: (jjjitgg:llll .. . . ((.. ! .. (' ' ll ( ! '
'' y.... ''.
,jjjj!i:
657
' .
:
-
.
'
'. .
.. .
' :
-y.. -y. -- E.- - .. -.yE -. -! . y- - .g y y .. . yy,yyl
.....--try
) '
' . . . . ' g. . :
'
'(' E' . - .
(
'
qy. iiyjjjj-
. .
of the Drift and Noise Terms
Xt + /7)
$
GEOM ETRIC BROWN IAN MOTION
LEMMA
=
=
pF(F1
(/)2) +
1
-
p2F(FI
(r)W(/))
pt + 0
The second term on the right-hand side is zero because FI We then say the correlation between JZ and #Z' is pdt.
t4 and lP2(1) are independent.
658
k
BRowlqlAx
MoTlox
Axo
I'r's
LEMMA
THE SHARPE
Multiplication Rules
(#z) 2
The dominance of the noise tel'm over short intervals has another implication. Since the ehavior of dx is dominated by the noise term, the squared return, dX4l, re:ects primarily the noise term. We have + hj
(Xt
-
.Y(/)j2
Laxtlh + cxt)
=
1:(/)./jfj2
Expanding this expression and simplifying, we have
Lxt + h)
2
.;t-(/')1
-
=
a 2X(/)2/?2 +
dz
Suppose that h is 1 day. Then /7 0.00274, /1 = 0.000143, and /l2 = 0.0000075. lf 11is 1 hour, then /? = 0.000114, /11.5= 0.0000012, and /?2 = 0.00000001. Clearly, the relative magnitude of the term mltiplied by /1is much greater than the other terms aj It becomes vel'y small. ln addition, if we tllink of (' as binomial, ten i'(r)2 = 1. Tis leads us to write .Y(/)) 2 ;kJ o'lxtllh
-
or E#X(/)1 2 = c zxtjzdt
,
We can also consider terms like
Rewriting this expression
jx(r)/?
-
Xtjh
gives us
+ axtljrtl/i
F,
ax(?)2
-
+ o-xtjvtlh
''5
Since the smallest power of /7 is 1.5, this entire term vanishes relative to 12as /7 goes to ZeFO.
dt
(20 17c)
pdt
(20.l7d)
.
J.f asset i has expected return ai the
lisk
premium is desned as
Risk premiumf
=
ai
-
1.
where r is the risk-free rate. Abasic idea in nance is that th return on an asset is lirtked to its risk, where risk is measured as the covariance between the ret'urn on the asset and the risk that matters is the investor utiliys In the Capital Asset Pricing Model ICAPMI, covariance between a stock and the market ret'ul'n since investor utility depends on the market rettll'n. There are other models of lisk, but for our purposes we need not take a stand on a particular model. The Sharpe ratlo for asset i is the risk premium, ai r, per unit of volatility, cf : Sharpe ratioy
=
i
1*
-
i
(20.18)
The Sharpe ratio is commonly used to compare well-diversilkd portfolios and is not intended to compare individual assets. In particular, if diversisable risk is different, two assets with the same c can have different Iisk premiums (and hence different Sharpe ratios) if they have diffejnt covariances with the market. However, we can use the Sharpe ratio to compare two perfectly correlated claims, such as a derivativ and its underlying asset. The main point f this section is that two assets that are perfectly correlated will have the same Sharpe ratio. To seethattwopefectlycoaelated assets musthave the same Sharpelptio, consider the processes for two nondividend paying stocks:
.
Suppose we have two differeptlt processes such as equations (20.10)and (20.15),. We can write dz ' t) Y'XX where E (1'(r)1''(r)) p. Thus, p is the correlation between Yt) and Y't). We have =
f (E.X(/ + = E
=
.Y(r)1(P(r+ ) P(l))) LLaxtlll + o'xtlyt + hlxlLac :(?)/7+ )
-
-
o'c
Qtlkr't
+
hj.Xj
:(r)/7 +
dt
x dz
(#/) 2
=
0
(20.17a)
=
()
(20.17b)
dS3
=
cl Stdt + cr I S3dZ
d%
=
azbdt
+ o'zhdz
(20.19) (20.20)
Because the two stock prices are driven by the same dZ, it must be the case that ,-)/c2, (al ,')/c l = ag oy else tere will be an arbitrage opportunity. Before we examine the arbitrage, let's explore the intuition. For example, in the CAPM, the lisk premium of asset i, ctf r, is -
(termswith power A: 1.) = a One way to make these calculations mechanical is to use the following so-called rules'' for tenus containing dt and #Z: o'o'cp Xt)
'Rmultiplication
=
=
-
essentiily ignoling all terms that are higher powers of h. This equation tells us that if we lpok at the squared stock price change over a small interval, gl1 w are seeing is the effct of the variance. (X(/' + /2)
659
The reasoning behind these multiplication rules is that the multiplications resulting in powers of dt greater than l vanish in the limit.
,
l'5
Lxt + /7)
%
20.4 THE' SHARPE RATIO
gagxLtjlyLtjh.s + o'lyLtjlyyjzjt
=
x
t'/z'
RATlo
-
-
ai
-
r
=
piaju
-
r)
(20.21)
660
k
BnowxlAx
Mo-rlox
AND
I'r's
LEMMA
TH E RISK-NEUTRAL
where au is the expected return on the market portfolio. The beta of asset i is
//
=
Pf, A:rGi
(20.22)
G' A;
where py,,v is the correlation of asset i with the market, o-i is the volatility of the asset, and 0-51 is the market volatility. Using equation (20.22),we can rewrite equation (20-21) aS i
1-
-
J
k,/
= p ; wj
1*
-
(20.23)
'
c Thus, if two assets have the same correlation with the market pi,5g), they will have the same Sharpe ratios. In equations (20.19)and (20.20),the fundamental uncertinty driving theprocesses forboth 5'1and Sz is #Z. Thus, both assets have the same correlation with the market, and in the CAPM would have equal Sharpe ratios. We will now demonstrate an arbitrage if the Sharpe ratios in equtins (20.19) and (20.20)ure diffrent. Suppose that the Sharpe ratio for asset 1 is greater than that ) for asset 2. We then buy 1/(c' 1 S1) shares of asset l and short l/to-iual shares of asset 2. These two positions will generally have different costs, so we inkest (orborrow) the bond, which has cost difference, l/c' 1 1/c2, by buying (orborrowing) the lisk-free the rate of return rdt. The return on the two assets and the risk-free bond is 11 ?1 1 1 dSL cej az rdt d% + dt (20.24) (7' l c'zxs'a Sj I o'z o- 1 o-z o-
t)-i
yv
-
-
-
-
=
-
-
.
This demonstrates that if the Shyrpe ratio of asset 1 is greater than that of asset 2, we lisk-free return. Therefore, to can construct a zero-investment portfolio with a positive preclude arbitrage, assets 1 and 2 must have the same Sharpe ratio.6 This link between volatility and lisk premiums forperfectly con'elated assets arose in Chapter 12 when we discussed option elasticity. There we saw that the Shrpe ratio for a stck and an option on the stock are the same. The reason is that the stock and option have the same underlying source of risk-the same #Z. They do not have the volatility of a call option is greater than that of the stock-and, same volatility-the hence, they do not have the same risk premium. Nevertheless, they do have the same Sh e ratio.
PROCESS
$
661
lisk-neutral pricearn therisk-freerate and discounting expected payoffs at that rate. The ing method alises from no-arbitrage pricing, but there is underlying economic intuition that we discuss in this section.
The notion of a
lisk-neutral
process for the stock arises with 1t processes,
just
as in te binomial tree. We saw in Section 20.2 that Brownian motion is a process for
+ J)) = Z(/). In other words, Brownian motion is a martingale. ln order male such to a statement, we must specify a probability distribution, or nleasltre, for Z(/ + s), conditional on Z(r). This point requires emphasis: in order to say whether a stochastic process is a martingale, we need to specify a probability distribution for the Which Efzt
PFOCeSS.
Suppose the tnle price process is dst)
Stt
=
(a
-
sldt +
(20.25)
o'dzt)
where is the dividend yield on th stock. This process represents the stock plice we observe in the world, with #Z(r) the unexpected portion of the stock return. On average J, and the deviations from this ret'urn we expect the stock to appreciate at the rate a words. should be a martingale under te true Z(l) should have mean zero. In other probability distribution. -
investor assesses the stock price process. Now let's consider how a lisk-averse defnition risk-averse investor by values $1 of gain less than $1 of ln utility telnns, a receives such investor the loss. When return on an asset, the portion of the return an negative expected value in utility terms: The gain from an martingale will have that is a worth less than the loss from an equal decrease in Z(l + /?). in /7) will be increase Zt + offset expected utility is by the asset paying a risk premium, equilibrium, this lower In (Z
--
1-.
This description suggests that there might be another way to write the rettirp on the asset. Suppose we create a new process ktj such that (/) does generate a martingale investor. (Note that this implies that Qt) is not a in utility terms for a lisk-averse the with mmingale respect to true probability distribution.) In order to offset riski modi the distribution for Zt + ll) so that positive outcomes are aversin, we must negative However, than if we want to substitute dkt) for dztl outcomes. more likely transformed will the equation not give us the correct stock in (20.25), process drift for the reduce the tenn to compensate greater expected value of #Z(/). unless we It turns out that if 2 (r) is a martingale when evaluated in terms of the investor's utility, the appropriate offsetting change in the drift is to remove the lisk pmmium. This gives lisk-neutral plice process us the ,
arice
20.5
THE
RISK-NEUTRAL
PROCESS
We saw in Chapters 10 and 11 that we can interpret the binomial model either as representing a risk-neutral stock plice process, in which case we stprobabilities so that assets ).., earn the risk-free rate and we discount cash flows at the risk-free rate, or as rejresenting the true stock price process, in which cas we use true probabilities and discount at an interest rate. In valuing options using Monte Carlo simulation appropriate (notIisk-free) in Chapter 19, we again used the risk-neutral approach, assuming that stocks on average
dst) S(r)
= (?'
-
jdt +
o'dkt)
(20.26)
In this equation, we subtract the lisk premium, a r, from the dlift a &,and we replace dzt) with dkt). As in the binomial model, when we replace a with ?', we must also modify the probabilities associated with stock price movements. Also here and in the binomial model, when we switch to the risk-neutral process, the volatility remains the -
Same.
-
662
k
BRowrqlAx
MoTlorq
Alqo
lT's
I'r's
LEMMA
The probability distribution associated with the lisk-neutral process, d2(l), is pricing, we will said to be the risk-neutral measure. When we perform lisk-neutral implicitly assume that we are using the risk-neutral measure. With this revised price lisk-neutral, because we have process, we can perform valuation as (/' the investor were transformed Zt) to (l) in order to make the investor behave risk-neutrally with resgect to the revised process. Thus, using equation (20.26),we can perform valuation as if Zt) were a martingale, because a risk-averse investor will treat it as such in valuation. It is important to emphasize that in the applications we have discussed to this point, we need not do any extra work in order to use equation (20.26)for pricing.? In P articular, we do not need to worry about how to construct (/) from Z(l). For example, when we perform Monte Carlo valuation of a derivative with a stock as the underlying asset, we draw mean zero random variables to simulate kt) and we use those random variables for pricing--discounting e'xpected payoffs at the risk-free rate-without any further adjustment. The technical result underlying the transformation from equation (20.25)to equation (20.26)is called Girsanov's theorem-s Girsanov's theorem states that a Brownian ' process, Z(/), can be transformed into a new process, d 2 t) = dzt) + qdt, that is a martingale under a transformed probability distribution.g Girsanov's theorem explains how to create the transformed probability distlibution. ln the case we are discssing, the transformation is based on the Sharpe ratio: set ?? = a ?')/tr. Thus, we have
((z
ldt +
-
o'dzt)
(r
=
-
l
=
ldt +
-
. -
gjvjt
dzt)
o'
+
---
/e
G
20:6
IT'S
dividend yield of Suppose a stock with an expected instantaneous return o f follows geometric Brownian motion: ,
'
on Girsanov's theorem include Karatzas and Shreve (1991,sec. 3.5), Baxter and Rennie
(1996) and Neftci (2000.ch. 14).
gAsimplihed version of Girsanov's theorem states that if Z(l) aBrownian motion underthe probabils = Z(J) + qt, is a Brownan motion under the probability density ity density ft), then (tlf (J), 0.5p2/). The probability density f'unction for a nonnal variable with where ( (l) = exp(-r/Z(?) .2(?)
-
variance t is
dvtj
=
l lt
CXP --Z
l
::!
2;.r1
Writing out (tj and
a
l lz.t
=
exp
l --
21
(Z + qt) a
l
zz.t
The Iast expression is the probability density for a nonnally distributed vmiable with mean
-.r)t.
f)
-
(S(l),
J1)
dt +
(20.27)
(S(r), r1#Z(r)
=
ztj,
(
Recall that d. is a geometric random walk with drif't. Suppose for a moment that the drift is zero, in which case S obeys a geometric random walk with equal probabilities of uP and down moves. Now look at Figure 20.1. Notice that equal changes up and down in S do not give liseto equal changes in Vs, t). Since 7 is an increasing convex function of S, a change to S + 6 increases JZ by more than a change to S 6 decreases 7. Thus, if the expected change in S is zero, the expected change in J/ will not be zero. The actual expected change will depend on the curvature of J/ and the probability distribution of S, which tells us the expected size of the up and down moves. -
.' ..' .'.';
)' t'kjj:j;ij:'. :''','(r',jjjk,,jjjjj :5:7:7* ;'p:' r' ,',qjL..jlk..' if @' :'r' ;'$' (' ()' k' y' i7'#E-' 'F''E 'T'ql'7qlf q'E'' '' ! '''q' )' llllitlji:)f (' k)' ''( ; ( :' ( ' ('' E E((' (! 7* j'ij 'j,;yj;jj'. ;'y' )'. rrqf ''1*(';(71jk*)(41* ('' t',))',' /'. yyf '#'t')if' 'g' ; (ii'(.(i'1* '(.'E('q k' )' ' 'tEi. E;(.!.yq r' t'q)' ttf .('( E.' (';'' (.(i!(i..; (k(ii'r! .(:i' (@ .!':@l'(' (q1));( ..'y. ' ..T''' :.E .i!iI !j.! : ')' ' .q.,y .rr; ...,.r!;. 'j.q .. ; i!ir )!,jjj;:jjj;j;:. r..y . ... i. .;y.(lyy q . lki )E. ; y .. y yy ( : y r tr j . '.r . . . .. g y!rjLj;. . yq....jy(.yj.yrr . ikllr .. . ykk-, 2.; ).j. ;....,... r.,...rgy ) L.L...L..L .y tjg y):-. 1.L.....:...2,.,,,,.L',j...... . .... ..... . ..
ilii!i!lf (1)*. ';jj'tti ilkl:;q ...r. 5* ;--'' '';'-' ,.jjjj::jjj,jF' 57* qE:f 7* )' E '' (E: E jE
:
'
.
-
. ---
-
.
-
-
-
E
! :
E' . -E
-
.
-
-
'' :' i ..
E.-... -..
.
.
CE ' E'.
:'
'
!
''
-
'.
E-
E :.E ' '
-.
-
...';:
...........
.
.......
......
........:...E.............
. . ....... .
' ' '' ;!E EE.. E '
'
.-
-
- -
-
-..)-y);
.
y-. .
'
:
-
.l1EE..
'
''
E E'' E E ' '
- . -.-. -E -
.
.
-
-
.
-
. . .
E.-
-
-.
.t)jk... .
-kyj.y-yj;. -
.
.
.
.,. .
.
.
------
-
.
-
Illustration of Jensen's
inequalty. The function 7(5)) is convex.
Equally spaced changes give rise to in unequally spaced changes ln /(.(02. In particular, 7(5'(t) + E1 yg5'(t)j> 7g5r)) 7(5'() E') because 7 is an increasing convex function of S. -
-
1 It) exp --Z(/) 2/
t3ES(l),
Functions of an lt Process
-
You can see how the Girsanov transformation works by examining Itlftj. combining terms, we obtain
and
ES(/), J) = Gstt. Now suppose that we have a derivative clqim that is a functio of the stpck price. Express the value of this claim as JZ(5'(l), /). Givtn that we know hw the jtockbehaves, how dos the claim behave? ln particular, i'tow can we describe the behayior of the claim in tenns of the behkior of S
.(t)
.(/)
=
=
The economic intuition behind the change of measure is that investors in equilibrium are indifferent to a small change in the allocation of their portfolio between risky
Slkeferences
,
ln this equation, tr, J, and c can be functions of the stock plice. 'When Stj follows ast), and eometric Brownian motion, we have (5'(1),1) g5'(/),/'1
#r
7We will see in later chapters. particularly when discussing hxed income, that in some contexts we need to take account of the risk premum on the underlying llsset when moving from equation (20.25) to equation (20.26).
LEMMA
instantaneous volatility
o'cjkqtj
..y
663
and risk-free assets. The Sharpe ratio appears in the transformation because it measures the equilibrium remrn investors require to absorb additional lisk.
-
(
k
LEMMA
-
-
-
. .
.
.. .
-
.
.
664
k
BRowxlAhl
MoTlorq
I'r's
Alqo
LEMMA
l'r's
In Figure 20.1, the second derivative is positive', that is, the slope of 7 becomes greater as S increases. As is evident in te sgure,the expected change in 1/ will then be E (1/(u)) if J/ is convex positive. The sgureillustrates Jensen's inequality: F(' (')) Appendix C). (see Using a.rfaylor series expansion (seeAppendix 13.A), we can see how JZ depends S. on We have Lvssd'jl
+
+2
CLst + h) t + hj
CLSt)
-
,
/.(1
,
+ ll) (u$'(/
+ terms
The multiplication rules aheady discussed in Section 20.3 tell us that since S is an 1t process, the tenns dtll apdtfu xdt vnish, alongwith all higher-orderterms. lntuitively, since the interval of time is shott, the noise tenn dominates, and the squared noise term This result stemj from the quadratic is the same ordr of magnimti as the drift. vadatio p'ropertywe dijussed in Sction 20.2. If we integrate the Taylor expansion w'ith respect to time, then the term containing squared changes will be pffrtional to time. Higher-order terms will sum to zero. This calculation is the basis forlt's Lemma.
r
Prosltion
Lemma)
20.1 (It's
(20.27).If CLSt), in C, dCLSt4, ?), is 6ICLS' t)
=
=
1)
Let the chage i the stock price be given by is a twice-differentiable function of Stj, then the change
Example 20.1
ast) + Ctdt
t5',tjjcs
-
(20.28)
J(5',tllcss
+
)/
+
c,
)
dt +
t5',tlcsdz
lt's
=
a
=
8
-
-'.5
1
o
2
=
r,
A, and 0
Ct
(20.29) Lemma to
=
p5'trl avst) dzt) dt + pg(?) nt
-= (a(alstjdt -
J
gt(v -
8)SCs +
jclslcss cij dt +
+ O'SCSCIZ
If there is no uncertainty-that is, if o' O-then It's Lemma reduces to the calculation of a total delivative familiar from ordinary calculus: =
dCCS, t)
=
Csds +
o'st);
=
p2u(r)
.
-
=
)Z(?)2
o'=St)
is given as
.)
,
t)
+ 0LSt), tjh
5'(0)c*--lo.2)'+':rZ(')
avt) oztl
,(/);
-
-
-
dCS,
J'j
dSs and recall that
-.>
The expression for a lognormal stock price is
Lemma states that dst)
dstj
=
=
5'(l))2Pg5'(/'),
-
0
Css ;
/.)
The stock plice is a function of the Brownian process Z(l). We can use It's characterize the behavior of the stock as a functipn of Z(r). We have
lCj&S1, and Ct = pC/j/.) The tenns in We use the notation Cs = DCIOS, Css braces are the expected change in the option price. ' In the case where St) follows geometric Brownian motion, we have &LSt), r) = ast), (St) r) = stj and (u(/), rj = o'stl, hence ,
-
,
-
Csds + lcssdl 2
(E(5', t)
665
The delta-gamma approximation over a vel'y short period of time is lt's Lemma. We can use lt's Lemma to verify that the exprssion for a lpgnormal stock plice satisfies the equation for geometric Brownian motion, equation (20.27).
st) equation
>
) St)
-->.
in dt43ll and higher
5'(r)) Ag5'trl
-
+ J(5'(/
Male the substimtions /? dt and St + /7) partial derivatives of the option price: are just
+ Vstdsdt
2
%
The extra term involving the variance arises from dS)l and is the Jensen's inequality correctin due to the uncertainty of the stochastic process. naming it-when we discussed deltaWe encountered lt's Lemma-without approximations in 13. Equation Chapter (13.6)stated gnmma
+ 1V)J(#l)2
LEM MA
tr2j
+
1 alStl
i
pz(?) a
g#z(r))z .j1
tldt + cutjdztj
+
o'lsstjdt
+ o'vtldzt)
In going from the second line to the third we have used the fact that dztjl calculation demonstrates tat by using Ito's Lemma to differentiate equation recover equation (20.27).
=
dt. This
(20.29),we
%
Cdt
Multivariate It's Lemma ';
losee Karatzas and Shreve (1998)or Merton (1990)for more details.
.
So far we have considered the case where the value of an option depends on a single It process. A delivative may have a value depending on more than one price, in which case we can use a multivariate generalization of It's Lemma.
666
%.BRowlqlAx
MoTlohl
Proposition It processes'.
29.2 (Multivariate
LEMMA
l'r's
Axo
vALul
lt's
dSi t) aidt + St) =
Lemma)
o'idzi
Suppose wehave
i
,
Denote the pairwise correlations as Edzi x dzj) twice-differentiable function of the Si we have
1,
=
.
.
FP
(5,(F):'1
.
.
&, tj is a
.,
The forward price for
s'la
,-rFJ(0)r',Er'(r-)+l,(c-1),2)T
=
0,z' ,
dc (k1 ,
.
.
Sn t)
,
.
,
Cs d uf +
=
i= 1
)
The lease rate for a claim paying
1
11
E
Ldc .
,
.
.
.
,
& r)1 ,
/1
ai
=
&Cs +
i= I
qi1Example 20.2
j..j jjj
Suppose
CCSL
,
Sz)
:.
=
J
it
o-i c'yyfy
zh
Csih + Ct
uy
i = l juzzI
5'1Sz. Then by It's
4
!( @
7 This
E E LLLLL ..
t5'jXj
=
uat/xj
+
s'jdy +
dSL
j
Edcj,::zz
py
,
(al +
(r
-
)
-
1c(7
1)c'2.
-
2
%.
az + pc
equation (20.30).
jtycluj
dS'
sg =
=&
%.
0.
=
m,
aS'-$dS
)N(t7
+
#u s'l s%+
1t7(/
L)9-lLo'vS)1dt
-
1)5'f'o' 2 dt
-
2
we get
dsa sa =
(tz
)+
-
JJ(7
-
1)o'2j dt + aadz -
aa
-
?')
+
Suppose we have a. claim with a payoff depending on S raised to some power. For example, we may have a claim that pays ST)l at time F. In this section we examine this claim with two goals. First, we want to compute the price of such a claim. Second, we want to understand the different ways to approach the problem of valuing ihe claim. The following proposition gives us the forward and prepaid fonvard prices for this claim.
(20.33)
.
'.
.
..
.
-
ST) Using equation
a
-
1-
r). premium is aa Thus, the There is another way to obtain the drift term in equation require the use of lt's Lemma. We can wlite
Sd
(20.32)
1)c2 and Thus, S'' follows geoletlic Brownian motion with drift aa ) + 1:1(/ 2 14ska dZ. Hence, if a is the expected return for S, the expected retul'n of a claim with price S6lwill be
lisk
ON
a
-
We obtain
Sz) does not depend explicitly on time, Ct
A CLAIM
?-
=
=
Example 20k2 is interesting because we know that the product of lognormal variables is lognonnal. Hence, we might expectthatthe driftforthe productof two lognormal variables would just be the sum of the drifts. Howeyer, Example 20.2 shows that the drift has an extra term, stemming from the term dSL dsz, due to the covariation between the two variables. The intuition for this result will be explored further in the discussion of quantos in Chapter 22.
VALUING
*
is
SCl
To prove this proposition, we will first use lt's Lemma to determine the process followed by S'. We will then use three different arguments to obtain the pricing formula,
Dividing by
20.7
.
ST)a. If S follows Consider a claim maturing at time F that pays C(5'(F), F) equation (20.27),then we can use lto's Lemma to determine the process followed by S'.
lu'a
implies that
) i). )y )1 .t) )) Note that since C'j
(20 31)
The Process Followed by S'
Lemma we have
j. 1E(
d
S(0)fldL&trYJ)+f'(J-1)G21F
=
i = I jcuzl
The expected change in C per unit time is
dt
(20 30)
lt
dSi dSj Cs h + Ct #/'
667
is
F 0,T. (5'(F)''1
lt
%.
-
's,
Jl
.5/
ox
20.3 Propositlon Suppose S follows the process given by equation (20.27).The the prepaid forward plice is value at time 0 of a claim paying ST)'l
11
.,
pi,-idt. If Csj
=
11 correlated
A CLAI M
xc
'
(20.32)that does
not
y(g)cgtI(a-(-0.5t7'2)W+co'Z(r)
=
(18.13)to compute the expectation of a lognormal variable, we have
'. ...
ELSLT) a)
=
S(0) t7 c
() a = 5'( ) e
c(tz-J-0.5o'2)F+0.5f2&2T (J(a-J)+0.5:(f'J-
1)c'21F
668
% BRowrqlAx
MoTlox
I'r's
AND
LEMMA
vALul
Thus, the expected continuously compounded as in equation (20.32).
return on
is aa
SLl
-
t)
+ Q.5ap
1)tr2,
-
Proving the Proposition Given equations (20.32)and (20.33),there are three arguments we can use to compute the time-o value of a claim that pays ST)t at time F. All three methods will confinn Proposition 20.3. Risk-neutral
aLa
-
?'),
for #k&:
First we use lisk-neutral pricing. Subtract the lisk premium, prlcing from the drift, aa obtain the following ), to as the risk-neutral process -
dsa g
a;
=
-
1
J) +
-
j.aa
-
1)&2 dt + aadz.
E* (5'(F)r'4
()
5'(0)&c''tr-J)+lt'(,-1)o.21T'
=
(20 35)
We can- also value the claim on S1 by discounting the true value. To do this we must compute te expected value of expected (nonrisk-neutral) claim and discount expected payoff appropriately. From equation (20.14),the the the ST)Ct value of is expected cash flow
,j.a
) z::z =d =
e-(r+tl(a-r))T' -(r+t'
-
'
-
.
=
=1
-
t7(?-
-
ta
-
-
&)
-
-
1(7(t7
-
-
1)4y2
-
2
The value J* is the lease rate of the claim paying S'. The prepaid forward price is then FP
&
0,F
CS )
=
SQ)'e-&'T u$(O)flc(-r+fl(r-t)+JJ(fl-
l)(r2)F
=
This is the same as equation
(c-r)1F
g
ST)Cl
at
We now examine four special cases of equations and 2. Claims
on J'
First, suppose a
=
ty(rl(ct-()+.).c
(t'J,
l )o'2jF
(20.30)then gives
1. Equation =
(20.30)and (20.31):
Se
us
-:z'
This equation is just the prepaid forward plice on a stock. Claims
(u(y.)c) (())t?
,
(20.30).
Examples
Specc
5'0
=
If a = 0, the claim does not depend on the stock price; rather since on J0 1, it is a bond. Setting a c.=: 0 gives us V(0)
y(0)Jc-rTe(J(r-J)+rl(c-l)c2)F
=
e-rT
-
Note that the risk premium on the stock, a r, drops out. The forward price for 5l, which we will denote F9,w S'), is just the future value of the prepaid forward: -
F0.r( S'' )
-
J/(0)
The discount rate is expression (20.33).The price at time 0 of a claim paying time F is the prepaid forward price, which we will denote Foppm4l w(
%. 669
Finally, we value the claim by nding its lease rate. We ask the lease rate the claim cash would have to make in order for us to willingly hold it, or what payment what equivalently, payment we would have to make to short-sell it. We can then treat lease dividend yield and compute the forward price. the the rate as Fromequation (20.32),the claim has risk ao'dz and so must be expected to earn r) + r. Equation (20.32)also tells us that the acmal expected a rate of remrn of aa 1)(r 2 capital gain on this security is aa ) + 1zaa In order to hold the security, we would need to eana the difference btween the expected return and expected capital gain as a cash payment. Thus, the payment would have to be . ?-) 1)g2j &) + lzutt,I + r (* aa
J(0)r'cIr'(e-J)+),(,-1)G2JF
El (5'4F)''(1 =
FoP
.c
ox
.
We saw in Section 10.1 that the expected price under the lisk-neutral measure is the forward price. Thus, equation (20.35)gives the fonvard price. Discounting this expression at the risk-free rate gives us the prepaid forward price, equation (20.30). Discounted
A CLAI M
Finding
(20.34)
Using the drift term in equation (20.34),the expected value of the claim at time F under *, 'the risk-neutral measure, which we denote E is
xc
=
=
e
r?- FP 0,r(
5'r')
which is te price of a F-period pure discount bond. Claims
price is
.$2
on
When a
=
2 the claim pays 5'(F)2. From equation
(20.31),the forward
y(0)rle(fl(r-J)+JJ(&-1)G21F
The use of a single discount rate works in this case because the payoff to the claim is simple. In general, computing a plice as a nonrisk-neutral discounted expected value is more difctllt than this. -
(20.36)
670
k
BRowtklAlq
MoTlox
I'r's
Arqo
LEMMA
vAuulxc
Thus, the forward price on the squared stock plice is the squared forward price times a variance term. The squared forward price is intuitive, but the variance term requires some discussion. One way to think about equation (20.36) is to perform the following thought experiment. Suppose that we have an ordinary stock with a price denominated in dollars. Now imagine that we have a second stock that is identical to the first except that instead of receiving dollars when we sell the stock, we receive one share of ordinary stock for each dollar in the quoted price of the second stock. This conversion from dollars to shares is what it means to have a squared security. With the squared stock, when the stock plice goes up, we not only receive the extra - dollars a share of stock is worth, but we also receive the appreciated value of each share we receive in lieu of dollars. We therefore receive an extra gain when the stock price goes up. The effect works in reverse when the plice goes down. ln tat case, we receive fewer dollars per share, and each share received in lieu of dollars is wfth less as well. However, the lower price per share hurts us less because we receive fewer shares! Thus, on average, the extra we receive when the plice goes up exceeds the loss when the price goes down. This effect becomes more important as the variance is greater, since large losses and large gains become more likely. The result is that we will pay extra for the security, and the extra amount we pay is positively related to the variance. This example provides anoter illustration of Jensen's
Claims
with a
on 1/J -1,
=
Finally, 1et a we get
=
-
F0,r(1/&
1, so the claim pays I/S.
(l/u(0)J
=
-1 = F 0,T e
Using equation
dQ
W
Now we generalize the previous example by having two prices. Consider a claim paying ST)t' QTjl' where S follows -
tsldt
cldt +
-
o'adzc
+ o'sdzs
(20.37)
671
(20.38)
hlt
=
Proposition 20.4 Suppose that S and Q follow the processes given by equations (20.37) and (20.38). The forward ptices for S6land Qbare given by Proposition 20.3. The forward price for S' Qbis the product of those two folavard prices times a covariance correction factor: Ft,Ts
a Qb )
:::
(sa) s f,F
F),F
Lcbleabpo.so.cr-Ij
'l'htr variance of S Qb is ooiven by 2 2 a 2o'. + yl c c + gaypcscc
%
.2,
Note tat te squared security, is a specitl case of Proposition 20.4. When S = Q, a = b = 1, and p = 1 (sincea vadable is perfectly correlated with ittelt), th covariance term becomes
abpcso'c same result as equation
=
tr. 2
(20.36)for
the forward price for a squared
Using multivariate Ito's Lemma, the process for St
valuing a Claini on Saob
as S =
ac
dzsdzc
stock.
czp
sc k
where
This gives us te
As with the squared security, the forward price is increasing in volatility. The payoffs for bot,h the Sl and 1/S securities are convex; hence, Jensen's inequaltells ity us that the price is higher when the asset price is risky than when it is certain. In both cases the forward price contains a volatility term, and in both cases the price is increasing in volatility. lf we considered a concave claim, for example X-S,the effect of increased volatility would be to lower the value of the claim. End-of-chapter problems 20.5-20.8 provide examples.
61S
=
(20.31)
etJ-r)Fecr2F
ox
and Q follows
'
inequality.
A CLAIM
d (5'cp') sa :d? =
tay -
$s) + bac
Qbis
1)c2y + kbb 8c) + 147(47 2 2
-
-
+ abpaso.c
dt + acsdzs
-
1)0,2
(2
+ bccdzc
The expected ret'urn on this claim depends on the risk premiums for'bot S and
r + aas
-
r)
+ bac
-
(20.39) ::11
r)
As before, there are tltree ways to find the price of a prepaid forward on tllis claim. Here we use risk-neu.al pricigk Problem 20.13 asks you to use the discounting and lease-rate methods to find th'e answer.
llproblem 20.11 aslcs you to velify that this expression gives the expected return.
672
%.BROwNIAN aas
-
MoTloN
I'r's
AND
LEMMA
JuM ps llq 'I'H E s'rocK
'Fhe risk-neutral process for d5l Qb is obtained by subtracting the risk premium, r), from the drift in equation (20.39).This gives r) + bac -
saQbunder
The expected time-r value o
:(r)bq E *(5'(r)c
dt + aasdz)
abpaso'c
+ bccdz*c
the risk-neutral measure is
Wecan write a stockpriceprocess
(20.31),this expressiop can be rewritten
Using Prpposition 20.3, in particular equation ZS
F0,v CSa
/)
Q)
(z/pyltryo'tj
=
F0,w(S a ) Fz,v (Qb )c
T
(gtj y(;; .
The expression on the light is the product of the forward prices times a fpctor that accounts for the covariance between the two assets. This is an important result: The price that results when we multiply two prices togetherrequires a correctionfor the covariance. We will see this result again in Chapters 21 and 22. Proposition 20.4 can be generalized. Suppose there are 11 stocks, each of which ' follows the process
(20.41) where dzidzj
=
(1976)proposed modeling the stock price as lognormal
=
=
-
Probuump) Probtno
dutljst)
(20.42)
The forward price for 7 is then F0 z.(F)
1-+' (F0,r(&)1f''c E;'-i (2'! -'
i
20.8 JUMPS IN THE STOCK
=
1
-
kdt
@
=
hkldt + o'dz
-
(20.44)
+ dq
=
0 1'
-
if there is no jump if there is a jump
1
drift term contains for the reason discussed in Chapter kkdt. and Edq) dq tel'm has a nonzero expectation, so we subtract kkdt in order to preserve the 19: interprtation of a as the expected return on the stock. We have 'l'he
-kkdt
=
E dSjS)
11 =
=
rl''he
i= l
'
kdt
where
11 SC
jump)
=
We can then write the stock price process as
dq =
withjumps as follows. With the Poisson process,
the probability of ajump event is proportional to the length of time. Furihennore, for an insnitesimal interval dt, the probability of more than a sipgle jump is zero (this is part of the desnition of the Poisson process). Let qt) represent the cumulative jump and 0. dq the change in the cumulative jump. Most of the time, there is no jump and dq When there is ajump, we let the random variable (' denote the magnimde of the jump, 1 is then the expected percentage change in the stock price. If is the and k E (F) expected number of jumps per unit time over an interval #r, then
pijdt. Let V (l)
with an occasional dis-
crete jump. One way to model such jumps is by using the Poisson distribution mixed with a standard Brownian process, as we did in Chapter 19. As discussed in Chapter 19, the Poisson distribution counts the number of jumps that occpr in any interval. Conditional on ajump ocurring, we assign some distribution to the change in the stock price. lt is convenient to use the lognormal density to compute the price change if the jump occurs.
s(0),:(0):eEr'(r-Js'+(r-:c)+l,(c-1)&+?,(-1)&$+a:'c,c:1r
=
%, 673
in the lognormal model. On a smaller scale, consider the stock price of a company that reports unexpectedly favorable earnings. To account for such nonlognormal behavior,
Merton
+
PRICE
/% o.o.iaajl-
(20.43)
l
PRICEIZ
=
(a
-
kkldt + E (tr#Z) + E dq)
,
adt
=
Thus, for example, if there is on average a downward jump, then k < 0, and, when no > 0 to compensate for the occasional jump is occurring, we need extra drift of bad times due to the jump. The upshot of this model is that when no jump is occurring, the stock plice S evolves as geometric Brownian motion. When the jump occurs, the new stock price is fact that it is straightforward to modeljumps does not necessarily mean that it is YS. easy to plice options when there arejumps. We will discuss this further in Chapter 21. -kkdt
'l'he
A practical objectio to the Brownian process as a model of the stock price is that Brownian paths are continuous-there are no discrete jumps in the stock price. In practice, asset prices occasionally do seem tojump; a famous example is October 19, 1987, when the Dow Jones index fell 22% in one day. A move of this size is exceedingly unlikely
Proposition 20.5 Suppose an asset follows equation (20.44).lf CCS, t) is a twice continuously differentiable function of the stock price, the process followed by C is dCCS ' t)
=
Csds + lcssclfdt 2
+ Ctdt + XEyLCLSY, t)
-
CLS,
/)j
(20.45)
674
% BRowlqlAlq
Mo-rlolq
AlqD
I'r's
LEMMA
The last tenn in equation (20.45)is the expected change in the option price conditional on the jump times the probability of the jump. %.
'I'he last term in equation (20.45)accounts for the jump. That term is not present version of It's Lemma for a stock that cannotjump, equation (20.28). the in
PROBLEMS
%
675
PROBLEMS For the following four problems, use It's Lemma to determine the process followed by the specised equation, assuming that St) follows (a)arithmetic Brownia motion, equation (20.8).,(b) a mean reverting process, equation (20.9)',and (c)geometric Brownian motion, equation (20.27).
20.1. Use It's Lemma to
evaluate
#(ln(&).
20.2. Use lt's Lemma to evalugte dSl. 20.3. Use It's Lemma to evaluate dS-6 HAPTER
U
ARY
A stochastic process Z(l) is a Brownian motion if it is normally distributed, changes independently over time, has variance proportional to time, and is continuous. The .change in Browpian motion is denoted #Z(r). 'I'he process Z(r) and its change dzt) provide the foundgtiop for J'nodel.n derivatives pricing models. The Brownian process Zt) by itself would be a poor model of an asset price, but its change, dzt), provides a model for asset risk. By multiplying dzt) by a scale factor and adding a drif't term, we can control the variance and mean, and thereby construct more realistic processes. Such processes are called It processes or diffusion processes. Black and Schols used just such a process in their original delivation of the option pricing model. Given that a stock follows-a particular It process, lt's Lemma permits us to compute the process followed by an option or other claim on the stock. The pricing of claims with payoffs St' and St3:, where S and Q follow geometric Brownian motion, illusates the use of It's Lemma. An important objection to Brownian motion as a driving process for a stock is the continuity of its path. It is possible to add jumps to a Brownian process, and tere is a version of It's Lemma for such cases.
READIXG
We will use the concepts in this chapter throughout the rest of the book. In the next chapter we will directly ajply the concepts of this chpter, in particular lt's Lemma, showing that prices f delivatives must satisfy a particular partial differential equation. ln later chapters we will use these concepts to discuss the plicing of ekotic options (Chapter 22), options bqsed on interest rates (Chapter 24), and risk assessment (Chapter 25). Many books cover the material in this chapter at a more advanced level. Merton (1990) in particular is an outstanding introduction. Other good sources include Neftci (2000), Dufie (1996), Wilmott (1998), Karatzas and Shreye (1991), and Buter and Rennie (1996).
t(,/J1.
20.4. Use It's Lemma to evaluate
20.5. Suppose that S follows equation (20.37)and Q follows equation (20.38). Use 1t ' s L emma to 'lind the process followed by Sl :0.5 .
20.6. Suppose that S follows equation (20.37)and Q follows equation (20.38). Use It's Lemma to 'lind the process followed by lntkol. 20.7. Suppose 0.4 apd $100, r 0.06, o's compute prices for claims that jy the follWing: u&(0)
=
=
=
0. Use eqation
=
(20.30)to
a. Sl b. S ,s'-2
c Compare your answers to the answers ypu obtained to Prpblem 19.6:
20.8. Suppose that 1n(5') and ln(:) have correlation p and that $100, :(0) $100,?' 0.06, o's 0.4 and (y c 0..2. Neither stock pays dividends. Use equation (20.40)to find the price today of claims that pay -0.3
u(0)
=
=
a. b. FURTHER
.
c. d.
=
=
=
=
SQ S!Q SQ
j.(SQ4 s1Q e.
Compare your answers to the answers you obtained to Problem 19.7.
20.9. Suppose that Xt) follows equation (20.9). Use lt's solution to this differential equation is
.;' l
=
xe-k
+ a
(1 -
e-')
Lemma to verify that a t
+ c.
eks-tsdz
.
0
(Hint: Note that when t increases by a small amount, the integral term changes by dZt).)
676
%
BROWNIAN
AND IT'S
MOTION
LEMMA
PRoBLEus
20.10. The formula for arl infinitely lived call is given in equation (12.19).Suppose that rVdt. Use S follows equation (20.27),with a replaced by ?- and tlat f'*(tf7) lt's Lemma to verify that the value of the call, J/(5'), satises this equation: =
1atr25'27
(r
ss +
20.11. Suppose that the processes for
l.V
-
=
0
=
ajzkdt
=
azubhdt + o'zszdzz
+ o'js'l#zl
=
(2
I#Zl Qdt + Q (?7
+ ?7a#Z2)
Show that, to avoid arbitrage, ac
-
r
=
Pl
cl
(a j
.
-
?
)+
P2 0-2
(tz2 ? ) .
-
(Hint: Consider the strategy of buying one unit of Q and shorting Qlj /,5*1& 1 units units of %. Finance any net cost using risk-free bnds.) of St and :722/5'2c.2
20.12. Suppose that S follows equation (20.27) with follows the process
dQ/Q
=
What is ac, expressed in terms of a S and Q that eliminates risk.)
acdt
-
=
0. Consider an asset that
ldz
llint'. Find a zero-investment position in
20.13. Suppose that S and Q follow equations (20.37)and (20.38).Derive the a claim paying u(F)& Q'lb by each of the following methods:
value of
a. Computing the expected value of the claim and discounting at an appropriate rate. llint: The expected ret'urn on the claim can be delived using the result of Problem 20.11.) b. Computing the lease rate and substituting this into the fonuula for the forward price. 20.14. Assume that one stock follows the process dS/S
=
adt + o'dz
(20.46)
Another stock follows the process
(IQ/Q
=
acdt
+ c#Z + dq3 + dqz
(20.47)
(Note that the o'dz terms for S and Q are identical.) Neitherstockpays dividends. dq3 and dqz are both Poisson jump processes with Poisson parameters l anct 2. Conditional on eiterjump occurring the percentage.change in the stock price is F1 1 or F2 1 -
.
Consider the two stock price processes, equations (20.46)and (20.47). 0), what would be the 2 a. lf there were no jump terms (i.e., l and between lation aq a re b. Suppose there is just onejump term (2 0) and that 1'I > 1. In words, what does it mean to have F1 > 1:/ What can you say about the relation between a and ac =
=
'
c. Write an expression for ac when both jump terms are intuitively why ac nzight be greater or less than a.
Note that the diffusions #Zl and dZz are different. In this problem we want to lind the expected ret'urn on Q, ac, where Q follows the process
do
677
=
and Sz are given by these two equations:
5'I
dSf d%
jl5'ys
-
k
nonzero.
Explain
y' ('.;';' j'kt(y'.' t'(jp((,j' f' j.-pd (jd '('jkd jlr.d 4(:2j1(* jjd lr' j'yyd :jryg' !j' '1111:44))(4(::* f'f' y' f' 41222:::* pjytjyjj.,!d ' )' !(' '(' t.,d (' jjjj:yjjjjjyyrq'd ;'(jtlp'fjjq)))d 1* $' E2' j';.j;' yjd jid )g' k' gtd ((' )y*.* 'j.j);' )' yyjjjjjj'.' j';'j' yjd p' yj..-jd .j' yjj'jjjjjjjgyjyyyd ld sfiyjtd 411:qjjj1;;* (' '/' (tEj,j()4yj..' (jj' (jk' )' 2pj' $jk:* kd 'jd itd jsd lj' !:' j'jkjd ;y' )t' )' (IIE.)j)X 'jj))jj'.' .(t':)' @)' j'jd (jd 'j' (()j' k' t)' )y' y' i)jd jjjjd jyt'jj;d ;jy' jyy..' iyt!rd jy';tjjy'.' tppird j'1('1(' jyqd ;r:t' jjd j;';' yj,yr,,d jjjyjd jqpd jypd (4!g.:),.' y-yd rrd p)' J' j'.' jjd ?p' tjd gyk)y)j(jjj()k'.' kjd j)j)j'.' jqd y' '''(' ltd t)y.' @' r' jt' (jj' .(j)' j.jjd j',' jyyjjjjj';j))d ;')' glli'ilkht'.rrqqd y)j'..jyjjj$yj'.' ():id (' q' y' .jyjk.' jyyyytjy'yd j'jd yjjjd jyjd jy.,;ry)'(yjj);)',' gt.t.jyjjjjj'.' jyjgj,..' jjpljd !' (1)* rpjjjy'j)jtd g' 1@* jjjrr'jtd jtyjjy!-' yytyytd jy'rjy.y)d yrd ).j!t' jjd y')d (ppd )k' jjyj)yd p' l'jlt'jd pd j.jd l'1* lyttd jgk.' jj'trrd (1* 14* j'jtj'gyjd $44j* j.;d )' @' jktjd y;hd ))'yy' ''Eyj'IX y' .));' j.yjd j)' jy'jjyjyyy.,' r' j'jd yyd (' jjj;.' j,' ;;' jjd j(' ;'j'1* g' k' jjjd !' yrd j;yd jjjjjjyjjjqd )' p' /jjj((ytyj.' j'yd J' y.yjd )y' 'i)d (jy('.' )jkjjjq;;y,;' kjjjjjj((.' 'y'
(' yy'jjj' p)' jjy,'.j(.tk'jj(tjyy'.' $' yyy'yrttjjj!y;d )' t'jjjjjyyyjjtd y' (rd g' 'jjj)j,',' t'yjd lq'yd yjd 1* tyjyttj..sp/;d ))' J' )!trd )(' ()'tr' q' ;;' (;tjd qj)d yy-t!yyd y;jd 7)* p' y.jd jyjjtd (' f' )' 4* )jj' l'y' h' yyd 1* )'jjj' $*j44y* y')d (' ;'t'?' qjd @' k' tjd ''y' jj'yjjjy'tjd yjjjjyidjtijkyjjjjjz'--d t't' tyl' ytjjyjy'.' rjd t,kjrd t'((' tyd jljjjqyd jqjr:!',,..kkjj''d jljjljj;;l::,..d jjjjljjjjq:,d t.tjd trj)d ttj)jd ryd ttd t.)' yqjl:!''''!kkj!''d (ljjjjljqlg:jj)d (jjjjjg;;l::...d t.jg)))d t?d t't'yyd 1*1* t--' r'tyd kyjggjj(jjj(jjjj(tj'...' r' tyj...gd 'ts.s.-yjgd yjjy;d rjd rr-.)'
'j' -' .?' -,td -',')' -'.',.' -.' .-' .,' ..' ...' .'' --' -'.y;';' ''''!!jll(t!'''' ..'y;yy' ..jjjt!!!!jjk..' ''11jjj.,,jkjj''' 'j' ')' '.'.,,' 'j' -i.))' -''jjjjj!jjj(r'' '''jjjj!y,,,,' ''.''j' :';'...!' ..9* ;',-',r.-jyyry,j(yyjjjgyy'.ygjyyyyjjjyjyjyjjjjyyy.(kyyy;..' -y' -'-.',';' 't!kj' il'ss;tpj;d (t'jyjgd r.)qd !g' Eyd yyd -()' -t' (' ,.jjkt' E(' -j' j',';?' y.y.d '?' 5,* :.' y' E;' :r' C.;' Ejd r' :'' k' --;it.' .r' ijd t'jd j'-'''' ;'yytEj.ld ':j'jjzrzr!:d j'!':,,' ,'y' yjd iy'jd ?' g' y' jkd j'Ejd y'sd .;' j)jjy)y)jy(E.' 5* y;.t(yyjjjjy;'
.'.' .'.'
.'' ''.' ......' ' '', . . .. . . . . . . . ' ' . ')tr'(.1E.E.'
i 'E E. ; ! ii E E E j(' :((.j ( !.jE j j:j' .E.y'i .yj.E jj'' i.g': i'i i' E! 41 ' E. ( '!j ' EE'' j(y ' 'E rj' qq ' Eljjyj. ( q;'. '.j;. jyg(j@g'' j.' !'jj(;jjj' '.yj 'jjj tjE jp' '. E.L/LLLjI. !q !Ej;ytt j(ijjl (j;jrj jii i jyEtkjyj;jjjjjjjk)t'.. j jy ;jg. ( ;).gLLLLLLLL. (('j ;!1!:. ; ' )jj(LL:LL jy'.j,'' g'g ......jLLjL(jLLLL))(j?. LLLLLLLLLLL.. kg .('.'. .jjjjjjjjLjj. j((y. g . 2 ...j.(.j. 'LL.L ;rjrjj'.j. ', . (.. .. (t(. yk(. (.' t.)(:. ) .;jj. yjj)yj(j'. j)j). jjk. jkkljj. (.j.j..j)y': . (((j.'(y, .(i. kj. j.i jjjjjjtydyjjjjjjjjjtjgdzrkjjr.:). yj'. ..(... kj)jj'. .. jj)L''jj()L)L'. .... yj.. .(j'i..)'' )jky. . tj . (yjjkjlljj. . r', jk)rk@'. ....ry:.g. (j(jjj('. . . . . (jj'. ..... . tyt)gyy((.. . jj))j.(... .y( )jj'. . .))jj'. . jyjjkt())g'. ..j,.'.,,: .' k:... j j)j)t)))(. .r! k...j'. ('. . jjyjtjjj)(. ())j'. .... ()gjj)'. y(yy);jjj!j'. .. .. tjyj(. . r: .LLLLjjLLLLL)LLj(.. . tjl. LLj;L(,. )j()j))'. r j)( .tyj;. . !.y t)j.tjj)y'. ).yj )'.((y..t.. . ti .. .. . ...... j)j))j .. '.jjLLjj'. ...yjjj)j(t;'. 1111. ij j l (.l p)'tlqj j( ygy ' '.. j('( jjjj!jjjjjjjjj!rjjjj:yjjjjjygqr' ;jy'.j'. jjjj!jjjjjjjj!!,jjjr:yjjyyjj;;jj)l jjjr:yjyjjjjj;jj)l i!jy litrji ' ...'. .r( 'kji;()'.tq. !jjj jyyrjj .jj.j.yj't'y . !j t'-y'. jj(.... '. '... ; .ir'.;;.p: t'; y( '.';t;y.g r ....'. ! ;kyii;;.i;'.y't('l ji . yjt(. ..'.. j j.qjp..!.)r'g j:j jkj'. j . r j '. . .. y y i g .piq!( !. j;.yj kjyj .jyj'. jj'yyj jjj (r ;:.jy ry y.4*jyj!(t't'. jyy .. ;.y.y . y'yy'(y ,; ..... ll!ly!;).!.jyg : l i1yy p)k ;rj(jy.jjj jjljjtyt!ll j. !y(.j. y j . . j r ty t.. . . . ) ; . . gy (rj ((('. ..' ''. pj(j !j'l t) j (.. jj f:..(; j(.. . yy. j .. j (. .Ij()'.. y . ..;. jjyyy'. y.r) j . '...i.) ;.)'.2lkjjj.gj-y);)y.''. j ..j..( jy .. k k j ... . r y. j. y..;(t.krjk.y. . l ;j.l j!)(. faced by tjjytty the option pricing formula Black and Scholes studied the problem '. '. lIply;.(g. . j . jr.pJ; 71j t! . .'r;r ;.l (t.. t-')): tr j(;( j 13 the market-maker who sells delta-hedging market-maker. As in Chapter we saw a a i )k.. jj y;.jr)jjj r jg ;). . . jj option buys shares to offset the lisk of the wlitten call. To analyze this situation it . . 2yr'. . ( y j ( ; .. . t.j . js necessary to characterize the lisk of the position as a function of the share price. It s ', pi ltdlj i k (pjl t'i.y j; ; . j jj . E. ..''.E.EE..'-!EE';Ei.(-. E.! ; ijiy,'y.. Lemma, discussed in Chapter 20, provides a tol that permits us to see how the option gyjy j; y. yg.; g'.qrr!; y.g.yyy.yE. .-'. y.yjjjyyyyj. yjl.rjtitjlty rtlyj-, j.,j(!.();y...-.. . yy.yttijj gjgy.jjg gjy.yjjs jtyy:r jyggri jyggry jtyg;j j:yrr jtyg;j jkk,rigjry jrrg;jyjy, jkj:rr j;::.jkygry j;yy:y$,yj(. jkygri gjyy j..:4;; gyy.jj :jg; :jj;i . E!(E',E,;qq.E.E,',iskiEllsjyji.i.gj.yjj.y.iijj.gljjt :jj;ijjjs gjrr;js gjy., gjrr)js gjyr;js gjyr)js (jjky;, ) (tjr iI tjjjj-.-,.. E E gjjg;j, gjjr;j p t!.j. .q.... y,jojyrttftiE .. . EE;;.(.r.iI . iji. )'(()r';(j. .. ..EEi'E.l',lE!';'E''Eiirrj.yp'tt)'y'j'r''(E'yjjyf'jj)) 'EE'q'.!'.ElEq.l(y;yq;.kij iIj;!. and Scholes assumed that the stock follows geometric Brownian motion t;j(EE..(. EE. jytjjtqyE .j jjjy ; ry)j'ty)t':it ly . 'E.E jyy ..'..EiE..;.ii''ii'E( !'. ...;q.. ()).(j .11,,(,1:. (j It,s Lemma l!..I ; )r.. EE g (.y.)i;l:rt Ili!!:k E;q iE.j': ;pEIl,. ! E ....EE! i to describe the behavior of the option price. Their. analysis yields i y q . . ; ' ' ( ); . ;. r.. . . . ryl .y..... ..'...E.....''EEk'iEE''l'ii!'',ti'tiII)q''ii!!jijt'jt.$t.)i .ttrtlryijjljy,y . . yr?jjjjyjyjy, .....ygl,qqr,yEEl.gij,jijijij.,ylj.jjjlljiitttgjkjyjjrj (jifferential equation, which the conect option pricing formula must satisfy. araj r(' 'j (; liE j!i i @ ap t;'. E (.E pjjI.iI. ! jjj'. ' .' ...EiE;.iEii,.!i'!;Ep'j!jq';'j(jly!j!jlj!ji)'.tjt;@.l,!tIyt)yt . (.(.;j.( ((. yjlilii1 . ryqy.i . jtilltyj.i t4'. . i.Ei ,I ?'. smdy the Blackcholes approach to plicing options. j,s This this chapter . In we (ILLLLL .q.E(E'.;;(;)f;'i'.'l.';Ei);'.ii)j)!! ( ! (;.jEq j p i.jq ( y q j ; ( jy j ; ! j j ). . . jj y . jjjyjyiy, )';('. jjj ..E'.'..' iE. ! :)l)'. y (tljj methodology is important not only for pricing European call options- it provides the ;. !.ili).Ei ri!;!.(i. ;jq y q yyjj ( ;iEy;j '.''.E.f.j:'.E(iE Iig ; j! ,E!'r!'.l)E.7;lEli.!,!j1(t)l.yit#jk!qg4j;t .y .k.. j yjyjj'. ry(. jjj! . .. . tg yyjjilyyy .. . . yt'yjy'. E ' E i furtdation ft . priciir virtally all dedvatives, and also. undepins . the risk' . ...! !E!7 intellecmal i i ,;..'.'E'.'...'...i...i....!.i.l.ji.j.j.;,,y.y..(..,,,pj). . . . i g i .. r.i' .lr.r! j!!l!!iiI;rjjyjy ;.kyE.jy.Eyy .)jy jy. '.'.( q,.E..(,.)i..;i jj). j:jjj (.tgj'. j .y.,; : ylt.j jz j:g;j j:q;j yj $:jj'. jz gjyij j. j jy jz jrrj. 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j. .. .-( ;)-, j r). ty'. 3L3',;.-r.(... ;y'! lj)ki. t$t'. j.(jj')! (S?r'!j ''.' l r(y; jgyjj'r )r,. (. j jy y'.y jyjyjl';j. )),,)(.jyy,.., Ey'( ('( (Ej!.iq1 ' . ....(..'irq'iiijEil'Eiljkjll'illpjyjjjillll'tsll/ljtrj jj'jjksjyjdgjjdtjtldtyyyjyjjtdy.yjjkl'yy. y tj)j j'. jy j. y.. . y..jy',. y yj'. j rytjtjtjyy i.. ; f:, l:i ! : , : i i )); jt. jj q j y ttqqdtrdttjqph,'p;jj),jyyr E' .j E.(i .-i;r. j!; jjt)'. y, . !I gjg . i(.....! ri (tly)jj ))jk', ( ty'. jyjq;jyq... )4*. rj.. yjt'! ........'E. qy.E..E;.E(;y!.kjjr.y;yj'.;j)'yj; ji; yyjj jj). jjyjjjjjjjyjj... .. ..j j.: jjjr'. . r, ((jjjjjjq. . .kq ry(!j'1!l lllllr ' qy (.y rjljjtrj,jy . jj r ' ' i . (j;.)(j--')')'fI'I')j'LL 1. jrjjjj.. .. ' ' ....y...( j.j(;!rj.;jj..jyjjjj.. j;'.j;'. !(t'. j j(jj..j. jj': j'yyjj'y(j'. ;i.jjj(;. jyy-rrj', . .. . .EE.(q;!yqq. j((j;.(.(i@ j.(g (tjjlyjt'jr i ( g q j @ ( .LLLLL'f.LjL. q(.: yj'. g ; y ; . g. ! i ..''...' ; .E.;..'.... .i.'.r;;E(rjj'.tjj. ! iq j('.LL;L'. jp)jltk r..;.(( jp)) j j ''.'.''.E.E.E.. j.i jy'-.ij.jjilj(!!jr!jtE!t!t;! ! li j j.. /8*, 'jjjl',jj'l.jjtt,r '' ' E..yk.yEj'q j;!q( jjy jjtj. y)jtjq rjq jjj!. . ,'.ry.ygj.q i.q.; y! j t('. j! . j))j)),,.')jLL1LL'.. , q;LLLjjLL),'. yj. pjjryy, j-. , .y jjjrg;))'y......' ...y....... ..(.;q;y.(p.qyjj;(..qjgjy(i!jjyjj!;jy.ljtij'kj')fyy'. . j yr..gryg ))j)j)j);. '.444j2 yt('y)'gy)'. rjjj!ty jy yj jtk'. .. i.r....;.j.Ey.ry(ikkijr;y r j. y.r j,yy qjylplrtr y-yyyjjy-,.j;jj;yj-. ! ; . . ( yy.yyt. yjyjy y'jg,syy'g, yy.. .... jjjyj'rj--jgy.y))j', yjjj,, ........y.... . rr.jjg.y j.jj jj.y.jy'. jgjtjr jg yj. ('.jjj((yj.(jyj.. ' ' ' (.. ..y).E..E!'qijIjyjj .
.
.
-
.
-
.
.
.
-
'.
-j
-
.
.
.
. :
'
:
:
.
' : E. E : i ii q. . ( y
j g.r
:
.j
j
: (.yE. y. .. y
:- . . - i. . . - -. .- . . y . ,tjjjyj'... yy . . . .. - E: E E. . - . E . jEtr !! E E q. E: - . E .. :. : E .. . . E . . : . .-. y , E : . i E - Ei . i E--. : . E . . . : EE : . : .E . . . . q -, , E . . .- . : i EgEE :g , qq ..jj)q), y.: .. :.! . . q ' .. . . : E . :.: . ii E. . . . :q . .: ..')3 g q -'iE Lf,;3:):L'. , ,,'- '')i '' E- E.: : ' . . i iE ''jt'p: .... .-:.. q';i .E . . E : E. - . . g . E. ' : .. ' -.-..'kjkLi-....tf:i,', )r(t@14!l?'f'.':'!j$j).jj i :. - g ' ' : '!. ::..E'j ! tE; :jgt'' i'.tr-''5'''jpjtjt; . E 'kll'. !y- E . E . . : k t)Et.! E E: g ; E y ' ' :: E . EE. : : -i . : . - . E. !g. -. E.. ..: - - -.- i- - --j.yy . -. -- -y:. -. . E-. ; . . - i y -. y-( -- . . ... . . . . . -i --tj'y E . --E. . . i --tjg. - . -. . i . . -. - E . - - -E. - .. -r', . -. - -. q E - E yjjj,yyjyyy y --. . . .. q q y . . . y yjjjj'yyjjjjjyjjjs : . . q::. . . - , gk ,, y. .. . . :. E i E y : : r. : . y q . gg r. '
...
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.-.
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.
l Robe,.t Merton also contributed
fundamentally to understanding tlae option pficing problem. As a ' result, the analysis in this chapter can also appropriately be referred to as the tBlack-scholes-Meron jkjgr,. ,,22r),.,,22::,:j(..,,y.:l; (jj.,,j. jj,(,222),, g:jj,;,. -pjjjI;k ,j,
-'.q4;;;p,,.
:!!4(((g,1,-
680
% THE
BLACK-SCHOLES
TH E BLACK-SCHOLES
Equvlorq
shows that ST) can equal $1 only if A
The Valuation Equation
S(J)
A farniliar equation from introductory 'linance is the following:
St)
Dt + 11411 + St + /?) . 1L (q ? 2)
=
--h-
-/,
This equation says that the stock price today, St), is the discounted value of the future stock price, St + /1), plus dividends paid over the period of length /7, Dt + hlll. The discount rate over a period of length h is ?v, We can also interpret St) as the price bf a bond and D(/) as the coupon payment. Whatever the interpretation, we can rewlite equation (21.1)as .
St + /7)
-
+ Dt + Illh
S
?v
Change in Stock Ice
St)
= Retum
Cash Payout
(21
on Stock
-.>
e-rT-'t
The cpndition ST) = $1 is called a terminal boundary condition because it sets the u$'(0), bond price at its maturity date. If instead we lnew the bond price today, we could set z4 so that the equation gave the correct value for 5'(0). That value would be an initial ' boundary condition. Tfle solution consrms what you already lnow: The price of the bond is the present value of $1.
Dividend-paying Stocks We can interpret St) as the price of arisk-free stockthatpays acontinuous lixed dividend of D and has a pdce of J at time F. Equation (21.3) then says that at evel'y time, t, dividends plus capital gains on the stock provide the risk-free rate of return. Since we know the value at time r will be we also have the boundary condition .,
sT) Equation
(21.3)with
,j
=
this boundary condition haj the solution F
+ Dtj
=
l'st)
St)
(21.3)
Equation (21.3)is a differential equation stating the condition that the stock must appreciate to earn an appropliate rate bf return. The transformation from equation (21.1)to equation (21.3)illustrates the sense in wlzich an equation describing the evolution of the price is linked to valuation.
De
=
-r(.-/)
t'j s
+
#g-r(F-l)
l
The stock price today is the discounted value of dividends to be paid between nw and time F, plus the present value of the stock at time F. Again, the discrete time version of this equation is the standard present value formula taught in every introductory finance C aSS.
The General Structure
Bonds Let Stj represent the price of a zero-coupon bond that pays $1 at time F. Since the bond makes po payouts, the evolution of the bond price jtisfes equtipn (21.3)with D = 0. The interetation is that at every time, t, the percentage change in the price of equals the interest rate. This is a familiar condition tht the bond the bond (dst dt jstl satisfy is if it fairly priced. The general solution to this equation is j should
st)
=
Ae-rT-'s
(21 4) .
where A can be any number. You can check that this is in fact a solution by differentiating it to be sure that it satisses the differential equation.
The differential equation describes the bond's behavior over time but does not tell what A is. In order to price the bond we also need tp ltnow the bond price at some us particular point in time. This price is called a boundary condition. Ifthe bond is worth $1 at maturity, we have the boundary condition ST) $1. Examining equation (21.4) =
Undercertainty, abond orstock will bepriced so that the ownerreceives aliskrfreereturn. The differential equation in these examples desclibes how the secul'it.y change from a given p'oint. The boundary condition desclibes the price at some point in the security's life (suchas at a bond's maturity date). By combining the differential equation and the boundary condition, we can detenuine the plice o t t e on d at any point in time. By analogy, if at every point you ltnow an automobile's speed and direction, and if you know where it stops, you can work backward to gure out where it started. Essentially the same idea is used to price options: We lnow the price of the option at maturity (for a call it is maxgo, S #1), and we then need to ltnow how the opti (jn p ri ce changes over time. -
21.2 THE BLACK-SCHOLES
EQUATION
Consider the problem of owning an option and buying or selling enough shares to create a rislless position. Assume that the stock price follows geometric Brownian motion: ds &jt'lt + o'dz T = tce -
2You might wish to verify that St)
681
.2)
Written in this form, the equation says that the change in the stock price plus cash payouts (such as dividends) equals the return on the stock. Equation (21.2) is written to emphasize how the stock price should evolve over time, rather thap the value of the stock at a point in time. Dividing by /2 and letting /? 0 in equation (21.2),we obtain
dst) dt
$
$1. Thus, the bond price is
=
$1 X
=
EQUATION
Ae-rl--tt
+ a satisis
the differential equation only if a
-
0.
682
%.THE
BLAcK-5cHoLEs
TH E BLACK-SCHOLES
EquATloN
where a is the expected return on the stock, c is the stock's volatility, and is the continuous dividend yield on the stock. The option value depends on the stock price,
and time, t, so we write it as 7(5'(f), l). Also suppose there are risk-free bonds that pay the return r. tf we invest 1,1/in these bonds, the change in the value of the bond St),
position is
#W'
rWdt
=
(21.6)
in the option, stocks, and the risk-free
bond. Let 1 denote the total investment Suppose that we buy N shares of stock to hedge the option and invest F in risk-free bonds so that our total investment is zero. Then we have
1
J/(5', t4 + NS + F
=
=
0
The zero-investment condition ensures that we keep track of snancing costs. It imposes the requirement tat in order to buy more of on asset we have to sell somethipg else. To buy stock, for example, we can short-sell bonds. Applying It's Lemm' a to equation (21.7),we have
dl
=
dv + Nds
+
sdtj +
#JP
10'2,:27 dt + NLCIS+ ss = 1/1dt + J/, dv + a
+ rwdt s.%dtl
(21.8)
If we own the physical stock, we receive dividends; this accounts for the N&sdt term.3 As in Chapter 13, we delta-hedge the position to eliminate risk. The option's delta (A) is F-. We delta-hedge by setting N
=
i
-
Holding this number of shares has two results. First, the ds and, hence, dZ terms in equation (21.8) vanish, so the portfolio is no longer affected by changes in the stock price-the portfolio is risk-free. Second, because we are also maintaining zero invest21.7) our holding of bonds is whatever is necessary to fnance the net ment (equation sale of the option and the hedging position in stock: Purchase or F
Substimting N
=
dl
-Ji =
=
I/,s.
-
(21.9)
J/'
and this expression for J/I/into equation
Vdt + lclfvssdt 2
Vs8vdt +
-
l-vss
(21.8)gives -
Vjdt
(21.10)
With a zero-investment, zero-risk portfolio, we should expect to earn a zero ret'urn or else there is arbitrage, so that #f = 0. lmposing this condition in equation (21.10)and dividing by dt gives y'ss + (?. Jl5'J& X + 1c25'2 2 -
-
rv
=
0
EQUATION
$
683
This is the famous Black-scholes partial differential equation (PDE), which we will call (We will refer to the formula giving us the price of a the Black-scholes eqltatiol? European call as the Black-sclaolesfo?'p??lf/rl.) Appendix 21.A derives the generalization of equation (21.11) when the value of &' depends on more than one underlyig asset. Yhesignicance of eqtlation (21.11) is that the price of an option must satisfy this equation, or else there is an arbitrage opportunity. In fact you may recall this equation from Chapter 13. There we examined delta-hedgin'g and saw that the delta, gamma, and theta ofa fairly priced option had to be related in a certain way. Since IZy.C is the option's gamma, JZ. the option's delta, and P) the option's theta, equation (21.11) describes the same relationship among the Greeks. We started this discussion by supposing that we owned an option that we wished to delta-hedge. Nothing in the derivation uses the fact that 1/ is the price of a call option or indeed any particular kind of option at all. Thus, equation (21.11) descl.ibes #?' challge p?p/'.'5 7J-ll???/.#kb??To be ill vt-l/lfc ofally contillgellt c/clkn./br vhich J'c ltllderlyillg are assumed deal: That underlying follows geometric have the great asset (a) a sure, we Brownian motion with constant volatility, (b) the underlying asset pays a continuous proportional dividend at the rate J (thiscan be zero), (c)the contingent claim itself pays no dividend and has a payoff depending on S, (d) the interest rate is fxed, with equal borrowing and lending rates, and (e) there are no transaction costs. These assumptions are unquestionably violated in practice. There are transaction costs, volatility and interest rates change over time, asset prices canjump, etc. However, our goal is to have a thorough understanding of how derivatives plicing and hedging works in this basic setting. This is a starting point for developing more realistic models.'
Verifying the Formula for a Derivative We can now answer the main question of option pricing: Given that asset prices follow geometric Brownian motion (equation21.5) what is the correct fonnula for the price of an option? As discussed in Section 21.1, there are two conditions. The pricing formula must satisfy the Black-scholes equation, (21.1l), and it must also satisfy the appropriate boundary conditions for the option. ((f we satisfy both conditions, we have the conect option plice. Almost all of the nonstandard option formulas we looked at in Chapter 14 solve the Black-scholes equation.6 The pricing formulas seem different, but they differ only
4When the den-vative claim makes a payout, Dtjd,
then equation
(21 1l ) becomes .
(21.11) sEquation (2l 1) holds for unexercised Amercan optons as well as for European options. 6The exception is Asian options. Since the Asian option payof is based on the average stock price, prices of those options solve a different partial differential equation, in which there is a term reflecting the evolution of the average. -1
t
684
%.TH E BLACK-SCHOLES
TH E BLACK-SCHOLES
EQUATION
in the boundary conditions. Appendix 21.C discusses a general set of solutions. Here, we discuss severtl particular solutions in order to convey the basic idea of how the Blackcholes
equation
works.
Let's begin by considering
two familiar calculasimple present value tions: the price of a zero-coupon bond and the prepaid forward contract for a stock. Suppose the bond matures at tim F and pays $1. The boundly condition is that it must be worth $1 at time r. ln addition it must satisfy the Black-scholes eqpation, equation (21.11).Consider this formula for the price of the bond: ce-rr-t)
J/ 1 t, y,)
(p,j
First, tis satis:es the boundary c6ndition since &'l (F, F) = $1. Second, the price of the bond does not depend on the price of a stock. Thuq, 7.9 = 0 and J/.- = 0. Equation (21.11) then becomes J$l
=
l
.
l
,
f.j
=
) #(:,/1)
#l
lntu$'/ffl
=
o' T
dz
t./1
=
-
-
o'
t
stle-r-'t
We will verify that equation
(21.13)solves 2
J1
=
2 JG
=
e
rj
=
$1. the
ST)
-
F
+ 1.o. a1 '
(?' -
'
are both equal
-
t
c'
t
-
-
-
-->
=:> 1n(5'/ff)
>
knS!K4
<
equation.
1n(5'/#)
0
(7' F
the Black-scholes
We have
-J(F-J)
t
lntuv/ffl
0
q
J/(5'(F), F)
#(4f1)
-> +tx)
T
-
Ndz)
=
=
l
->
-X
t
maxgo, K
5'(r))
-
) expressions at
'
-
l-stle-&T-t
=
0
Zero.
/1-1
It turns out that both terms in the Black-scholes formula indiW#l/(7#ysatisfy the Black-scholes equation. Consequently, each of the two rxpressions
All-or-nothing
(21. 14)
options
/'3(5'(f),?q F4g5'(/.),
A European call option has the boundary condition -
=
The put formula contains N-dj ) and #(-#2); as a result, the N maturity equal 1 when S < K, and 0 when S > K.
Equation (21.13)thus satisfes the Black-scholes equation and the boundary condition. Notice that for both claims, /&. = 0', their gamma is zero. We already saw in Chapter 5 that we can replicate a prepaid forward by buying a tailed position in the stock. No further trading is necessary. This static hedging strategy works because gamma is
maxgo, S'j
-
K if S > K, Thus, at expiration the Black-scholes formula for a call evaluates to S and 0 if S < K, so it satisfes the boundary condition, equation (21.14).The call formula also satisses equation (21.11), but we will not verify that here. Puts can be analyzed just like calls. European puts have the boundary condition
&stte-'1TMl
=
and e-rr-t
-
-
Fg5'(F), F)
Ndzj
.
substitutingthese into' the Black-scholes equation gives $stle-bTMt jst) x e-t--tb + 12 G lSt)l x 0 + r
Call option
-r(F-J)
As t approaches F, the difference between #1 and dz goes to zero, since the term F t goes to zero. Moreover, t he term (r 8 + 1&2) r t also goes to zero. -c c F Thus, both #1 and dz are governed by the term lntu$'/ffl/o' t. If S > K, then the option is in-the-money and LnSjK) > 0. lf S < K, the option is out-of-the-money and ln(5'/ff) < 0. Thus, as t r, we have
(21 13)
O
5=2
+
F
-
,
Ke
-
'
Since this contract pays a share at maturity, the boundary condition is that it is worth a share at matulity: J/ 2(ST)
685
-
v.
Equation (21.12)satisfies this eqtlation, with the boundary condition Fl (F, F) = Now cosider the prepaid frward contract for a share of stock. W lnw value is
7 2(st)
-J(W-f
For an option at expiration, since t = r, the terms e-'T-'s N(#I ) and A'(#2):? to 1. What happens to the definitions of tlt and dzL slightly will rewrite We
ja)
.
$
We Let's velify that the Black-scholes formula does satisfy the boundary condition. approaches l option expiration F, the date. formula as can examine the behavior of the From equation (12.1),the value of the call is Se
talculations
EQUATION
/)
e
=
=
.---tts
e-r
x N
w-,)x x
1nES(r)/A-1 +
-
F
c
1nES(r)/ff1 +
8 + 0.5o-2)(F
-
-
c
on its own is a legitinate price of a delivative. What
0.5c2)(r
-
F
-
r)
t
-
Er
-
-
?q
t
are they the prices of?
(21.15) (21.16)
686
%
THE
BLACK-SCHOLES
TH E BLACK-SCHOLES
EQUATION
#F
A cash or nothing option pays $1 at time F if ST) > K, and nothing otherwise.s has the same value at mattlrity as a cash-or-nothing option and satisifes Equation (2l the Black-scholes equation. Thus, equation (21.16)gives us the time-l value of a cashor-nothing option. Both asset-or-nothing and cash-or-nothing options are examples of all-or-nothing options, which pay a discrete amount or nothing. A European call option is eguivalent to buying one asset-or-nothing option and selling K cash-or-nothing options, both maturing at time r. The price of a European call is the cost of this strategy:
-o'
=
-
-
2
S a J/&&+
(J
In the foregoing derivation of the option pricing formula we required that a delta-hedged position earn the risk-free rate of return. A different approach to pricing an option is to impose the condition that the acmal expected retul'n on the option must equal the
IZJ
'
dt +
S FJITT/Z Unexpected
retum
Thus, the instantaneous expected return on the option is Edvj
1
1
7 v
-
H
)F
F,g. + a g1:0'25*2
l5'li + Jjj dt
-
(2 1.17)
v
Jopton
The unexpected portion of the ret'urn is E (# JZ)
# J/
-
J/'
S J/y
=
J/
J/
ln interpreting this expression, have
recall that
SVs(Jf is the option's
o'option =
'
(21.18)
o'dz
(21.19)
c'option#z
H
elasticity,
f2. Thus, we
(21.20/
S-ZG
This is a result we presented in Chapter 12. We know from Chapter 20 that two assets with returns generated by the same #Z must have the same Sharpe ratio. Thus, we have (Z
tz option
1*
---
1*
*--
=
coption
G
(2l
Using equation
.20),
we can rewrite equation Joption
The Black-scholes Equation and Equilibrium Returns
l5'k%+
-
Expected return
K x J/4g5'(/'), lj
You should verify that this is in fact the Black-scholes formula. (See Problem 21.7.) The fact that 73 and J/' solve theBlack-scholes equation gives us pricing formulas for two new derivatives, asset-or-nothing and cash-or-nothing options. Also, however, because74 by itself solves the Black-scholes equation, we could have sold any number of cash-or-nothing options and still had a valid price for a derivative claim. ln order to create a standard call, we buy one asset-or-nothing option and sell K cash-or-nothing options. The options. However, suppose we had instead sold Q.5K cash-or-nothing 0.5# if ST4 > K and 0 otherwise. This is a resulting claim would have paid 5'(F) gap option, discusqed in Chapter l4. This analysis verifies that equation (14.15)gives the correct pric for a European gap call.9 The boundary conditions we have considered thus far are all tenninal boundary conditions, meaning that they are satisied by an option at expiration. American options and some nonstandard options have a boundary condition that must be satisfed prior to expiration. For example, barrier options have boundary conditions prior to expiration rlated to lnocking ip or out. Nevertheless, their price still solves equation (21.11).
l
2
.16)
lj
-
1-
=
S 7. F
(21.21)to give (J
-
r)
lisk
elastictty.
in a particular way.
is also called digital cash.
9In practice, all-or-nothing Chapter 22.
and gap options are difficult to delta-hedge. We will discuss this further in
(21.22)
premium on the option is the risk premium on the Stock times the We can interpret equation (21.22)as stating an eqltilibriunl cpdition option ln other words, if we view the option as just another asset, it option obey. the must that its expected d that return is related to the expected return on the stock be must p rice so
ln words, the
7-rhis claim is also called a digital share. S-rhis claim
687
We can decompose the return on the option into xpected and unexpected components. UsingIt's Lemma, we have
option.
,
k
equilibrium expected return.lo As we saw in Section 11.2 in the context of the binomial model, the expected return on the option changes as the option moves into or out of the money.
option pays one share of stock if ST) > K, and nothAn asset-or-nothing K, and ing othelavise.? Examine 73 closely. We have F3(5'(F), F) = 0 if ST) 73g5'(F), Fj = ST) if 5'trl > K. Thus, at time F, 73 has the same value as an assetor-nothing option. Moreover, because 73 satises the Black-scholes equation, it gives the correct value at time t for this payoff. Thus, 73 is the value of an asset-or-nothing
F 3ESt)
EQUATION
'oBlack and Scholes also used this method to solve for the option price in their original paper.
688
k
THE
Bl-Acx-scuot-Es
Using equation
TH E BLACK-SCH
Equvlorq
(21.17),substitute for Joption in equation (21.22).This gives
If you compare equation
O LES EQUATI ON
$
689
(21.26)with (21.1l), the dividend yield, $, has been replaced
with I1, the difference between the equiliblium expected return and the actual l expected return on noninvestment widyets.l e Let y.. We can interpret J as follows: Ix is the return you get from holding and is the return you must expect if you Jtre to voluntalily hold a widget. id t w g a -
1
gy
+ (a Jl5'k', g147'25*2/.$, JZ
+ y;j dt
-
2
us'gy
l zz:
(a
.
-
JZ
r)
-
(21.23)
When we multiply both sides by J/ and rearrange terms, the expected return on the stock, a, vanishes: We once again obtain the Black-scholes PDE, equation (21.11). Thus, an intelpretation of the Black-scholes equation is that the option is priced so as to earn its equiliblium expected rettlrn. When we equate expected and actual returns, we can interpret the result as giving gfair price for the option, as opposed to a no-arbitrage price. This is eqttilibriltln us j'ricing. The no-arbitrage and equilibrium prices are the same. The equilibrium approach makes clear that determining a fairprice for the option using the Black-scholes equation does not depend upon the assumption tliat hedging is actually possible.
What If the Underlying Asset ls Not an lnvestment Asset? So far we have been discussing option plicing when the underlying asset is an investment asset, meaning that the asset is pliced so as to be held by investors. Stocks and bonds are investment assets. Many commodities are not (seeChapter 6, especially Sections 6.3 and 6.4). Suppose that widgets generate no dividends; and that the price of widgets, S, follows the process dk lxdt + c#Z (21.24) S = From this equation, widget price risk is generated by the term #Z. Let $ represent the Sharpe ratio associated with JZ and let represent the expected return for an asset with ?')/ty, adz risk. Since the Sharpe ratio is $ ( we have =
&
-
-
Thus, in order foryou to hold a widget you wotlld need an additional return of = Given the expected widget price change, y,, the only way to receive the extra return is through a dividend. This is the reason that Jz replaces the dividend yield in the Black-scholes equation. is the Iease rate for the widget, or We have encountered this concept before: more generally the lease rate for an asset with expected capital gain g, and risk c#Z. When you lend a commodity, you receive its capital gains. The lease rate is the extra income you need to make you willing to buy and lend the asset. In the same way, J is the extra income you need to make you willing to hold a widget as an investment asset. In practice, a widget-linked bond could be used to hedge the l4sk of a widget option. If the widget bond were constructed so that its price equalled the widget price today and at maturity, we saw in Chapter 15 that the bond would pay the widget lease rate as a coupon. This coupon, being a cash payment on the underlying asset, would play the role in the option pricing formula of a dividend on the underlying asset. This idea of a hypothetical lease-rate-paying, widget-linked security is also like the tbvin Jcclfr/y mentioned in Chapter 17. It provides an investment vehicle for owning the risk #Z. If such a twin security existed, we could use it to hedge the risk of the option, and its dividend yield, would appear in the option price. An equivalent way to write equation (21.26)is to replace with 1' + $)'. We then -y..
-
,
obtain
2 2 W+ 1a tr S liy + (/2
In this version,
the coefcient
-
$G
)S V
(aj L.;)
.()
.v
-
l
.
on the 5'Fs term is th e d r ift on the widget less the risk
premium appropriate for widgets.
r + o'#
=
=
'
= Note that when the asset is an investment asset, a and /2 = a and reduce equation Both equations to (21.26) (21.27) (21.11). $c To summarize, the Black-scholes PDE, equation (2l 1), also characterizes derivative prices for assets that are not investment assets. ln the case of an asset that is not an investment asset, the dividend yield, f, is replaced with the lease rate of the asset, =
The important characteristic of an investment asset is that Jt = What happens if an < is and investment ? not asset asset an /2 which says that the expected return on the option Consider again equation (2l equals te actual rettlrn on the option. When we delive this equation again using as the equilibrium expected return for an asset with risk #Z and Jz as the acmal expected remrn for widgets, we obtain .
.20),
$+
1G25'27
z
ss +
gSVS
q/y
-
J/'
?-
=
(& -
1/
r)
(21.25)
r+
-
-
.
.1
.
To see how to use equation (21.26),suppose we have an option for (') Example 21.1 ))i whih the maturity payoff is based a upon the stock price raised to a power, S vus type ! .
Rearranging this equation, we obtain J/t + lczvvss 2
+
y
-
(& yljvs -
-
rv
=
()
(21.26)
ll'rhis modification andSiegel ( 1984).
to the Black-scholes
equation is discussed in Constantinides
(1978)and
McDonald
%
690 j'y'
TH E BLACK-SCHOLES
EQUATION
RISK-NEUTRAL
For example, we could have a call option with a
a power option.
) of option is called !i payoff of
.j'. (! .y .
The Black-scholes
equation,
(21.11), can therefore be 1 dt
:
!((i
mftxt
v)a
-E*dV)
Ka g;
t: tj pitwe have already seen in Proposition 20.3 that the lease rate on an asset paying 51 is t1i 1zaa 1)(r 2 From It's Lemma the volatility is ao-. Thus, using J) r ar t) lti equation (21.26),we can plice the option by using St, as tjje stock price, K as the strike jlt ''ilprice, as the dividend yield, and ao' as the volatility. Q j
. ' j...
.
,
.
*
=
-
-
Under the risk-neutral
rewritten
PRIcI lqc
691
as
1. J/
=
%.
(21.31)
process, the option appreciates on average at the risk-free rate.
-
-
.
*
21.j Rlslt-xsuvRac
The Backward Equation Closely related to equation (21.31)are the following equations, which charactelize both the actual and risk-neutral probability distributions:
PRlclxc
does not appear in the Black-scholes equation, return on the stock, when delivatives on investment assets, only the riskpricing equation (21.11). Thus, expected free rate matters', the acttlal return on a stock is irrelevant for pricing an option bipomial pricing formula (see Chapter l0) also dpends only on the on the stock. The
The expected
,
risk-free rate.
This observation led Cox and Ross (1976)to the following important observation: Since only the lisk-free rate appears in the Black-scholes PDE, it must be consistnt with possible world in which there is no arbitrage. lf we are tlying to value an option, we any world in which it is easiest to value the option. Valuation that in the we are can assume risk-neutral world, in which (if it actually existed) all assets would easiest in will be a lisk-free of and the return rate we would discount expected future cqjh flows at the earn options and other derivative claims by asstunillg that value Thus, risk-free rate. we can risk-free of stock the the rate return and calculate values based on that premise. earns We assume that the stock in this world follows the process
ds
fldt
= (?-
+ o-dk
(21.28)
As we keep emphasizing, the risk-neutral distribution is not an assumption about investor risk preferences. It is a device that can be used when pricing by arbitrage is possible (see Appendix l 1.B fol: a discussion).
lnterpretingthe
Black-scholes
The actual expected
change in the option plice is given by
1 Edv)
J?
/)
=
Equation
+
y c a u a J ss +
.
@
(ylqyu
-
(21.29)
Let E* represent the expectation with respect to the risk-neutral distribution. Under the risk-neutral distribution, the expected change in the stock price is E*dS) (?' ldt. The drift in the option price can thus be written =
1
,y
L-.E (Jz) dt
j (#7) 0 #r l -f*(#F) 0 dt quation (21.33)is -E
=
v',+
=
c a s a p'ss +
(? .
-
j;
s j/j
(21.32)
=
(2l
=
For the risk-neutral process,
F) + 1 zo' lSl
/
ss +
(?. Jju/j -
=
0
(21.34)
E quation (2134) is called th Klmogorov backward equation fpr the geometric Brownian motion process given by equation (21.28).Whraj the Black-schols PDE characterizes prices, the backward equation characterizes probabilities. The backward 12 equation is just like the Black-scholes PDE except that there is no ? term The Black-scholes equation can be interpreted as saying that the expected return on the option must equal the risk-free rate. The backward equation pertains to probabilities of events, such as the probability that an option will expire in-the-money. To understand how such probabilities should behave, suppose we decide that the probability is 0.65 that the stock price 1 year from today will be greater than $100. We lnow today that if the stock price goes up tomorrow, we will then assign a greater probability to this event. If the stock price goes down tomoirow, or estimate of the probability will go down. However, we should not expect our estimate of the probability to change on average: Our expectation today, of toltlolanwl probability, must also be 0.65. lf today's estimate of tomorrow's probability were not 0.65, then 0.65 could not have been the correct probability today. Thus, whereas the price of a financial asset is expected to change over time, the expected change in the probability of an event is zero. This is why the backward equation does not have the l'V term. If fsp; St4 is the probability density for Sv given that the price today is St, both of these expressions would satisfy the backwtrd equation: .
-v'
.
'
-
(21.30)
.33)
l2The backward equation is covered in detail in standard texts (see,forexample, Cox and Miller (1965), and Karlin and Taylor (1981)). Wilmott (1998,ch. l0) contains a particularly clear heuristic derivation of equation (2l .34).
f
692
% THE
Bl-Acx-sclqouEs
C HAN
EcuvloN
K
Sv ; 5'?)#q%
Sr
f
Sr ; St l#kr
,
The Erst is th probability a call is in-the-money at time F. The second is the partial The e-kpectatiollof the stock price, conditional on Sv > K. Both are undiscounted. equation holds for both the and risk-neutral distributions generated by It backward true PFOCCSSCS.
21.4 CHANGING
The solution to equation (21.31)is equivalent to computi. ng an expected value of the delivative paypff under the risk-neutral probability distlibution and discounting at the risk-free rate. The specisc form of the integral depends upon boundary conditions and payouts. We can see how this works with our ssumptions (in particular a constant lisk-free interest rate) by considering a simple European call option on a stock that pays continuous dividends at the rate In thatcase, equation (21.11),along with the boundary A-), is equivalent to the condition that the option at expiration is worth maxlo, ST4 expectation discounted .
-
,
l-,
T
-
t,
(5)
=
e
-r(F-J
X
) K
(SLy')
zy -
ju (w;
g
,
,
y.,
j.
ggqtjjcjy ,
(y,;
where f*LST), c', ?-, J; S(1)) is the risk-llettral probability density for STj, conditional In general it is possible to write the solution to equation on the time-l price being (21.11), with appropriate boundary conditionj, as an explicit integral-l3 lf a probability 1'P(5', t4 satisises the backward equation under the lisk-neutral distlibution, expression (21.33),then J/'(5', 1) e '-rl--tt Ft5' t) the present value of u$'(r).
=
F(5',
will satisfy the Black-scholes
/'),
,
equation, equation
that W#(56,t) satisfes the backward equation, i.e., 1 l)) .
E (#1'P(5',
since 1/
=
e-rT-tlw
=
,
(21.31).To see this, suppose
0
we have j dt
Es
(#JZ (S,
l))
j =
dt
=
This is the Black-scholes
?'
Es
(#
.mqv..t
; j,jy
qs
,
j;y
y
J/
PDE, equation
(21.11).
13seefor example Cox et aI. (1985a,Lemma 4).The integral fonn of the Black-scholes equation is alsocalled the Feynman-lac solution. See Karlin and Taylor (1981 pp. 222-224) and Dufhe (1996, ,
Chapter 5).
.
THE NUMERAIRE
derivative conNow we consider what happens when the number of options (or other adset price. This determined by random, expiration is some tracts) that we receive at example. the following Consider payoff is cornmon. odd-sounding
' Derivative Prices as Discounted Expected Cash Flows
CLSt), K, c.
693
plnbabilities t7?l# partial expectarisk-neutral probability or partial expectation Thus, any tions are prices of derivatives. of this, we saw in Chapter exaymple price. derivative As corresponding an also has a that an option is probability risk-neutral N(#2) is the 18 that the Black-scholes term -r(r-l)x(#a) is therefore the probability, discounted The e in-the-loney at expiration. expiratton. option at if the in-the-money is plice of a derivative that pays $1
X
K
$
This result means that discoltltted riskmeutral
X
.f
G l N G TH E lN1U M ERAI R E
j' ')' y-'-' )' The priee today of a nondividend-paying stock is $100, and the ll Example 21.2 j .tjj forward price is $106.184.Joe bets Sarah that in 1 year the stock price will be greater loser . than $106.184.Joe wants the loser to pay one shm'e to the winner. Sarah wants the qj) q . to pay $106.184to the winner. The share received by Joe would be worth more than $106.184if he wins. Sim.., she wins. ti ilarly, Sarah's desired payoff of $106.184is worth more than one share if i bet valuable side of the if it is ltt)Are either of these fair bets? tf not, who has the more ') denominated in shares? Who has the more valuable side of the bet if it is denominated ll.
-'.
!
;g Ei# .
.E
lj.. . EE .
E!).
k
k..in cash?
will If Strah wins (i.e.,the share price is below $106.184),a payment of $106.184 thap $106.184), shae is the greater price exceed the value of one share. lf Joe wins (i.e., worth more than $106.184. However, it ip not obvious a payment of one shltre will be whiehbet has a greaterfair value given a current stock price of $100.Assuming no inside bet information about the stock, would an investor pay a greater price for Joe's desired ,
or Sarah's desired bet? nllmeraire or We can describe the two fonns of the bet as each having a different desired whereas Sarah's denominatedin shares, desiredbetis unitofdellomilation. Joe's bet share-denoininpted paying the interpret as You can bet is denominated in dollars. of price dollar dollars of vatiable number (the either a lixed number of shares (one)or a of number a ($106.184) dollars or bet fxed dollar-denominated pays a one share). The general The value with shares the number of $106.184). variable number of shares (the denomination) for question we want to answer is how a change in the numeraire (unitof derivative. of plice the a derivative changes the Here are some other examples where a change of denomination is relvant: oliginating in yen (forexample) can be valued o Currency translation: Acash flow in some other currency. We will discuss this example in depth in Chapin yen, or
ter 22
694
k. THE *
BLAcu-scHoLEs
clqAxcllqc
EQUATION
An agricultural producerwho wants to insure production of total revenue-the product of price and quantity-rather hedge an entire Eeld must alone. than quantity
Quantityuncertainty:
* All-or-nothing options: All-or-nothingoptions, which webrieqy discussed earlier, can be stiucttzred either to pay cash if a certain event occurs (suchas the stock price exceeding the stlike) or shares. The payoffs to the stock price bets above are in fact all-or-nothing payoffs; thus, the bets can be valued as all-or-nothing options. To see what happens when we change the denomination of an option, suppose
is the price of an asset that follows #Q = (ac
cjdt +
-
-
Q
ty cdzc
Q
=
Jzg5'(F) K c., r,
Q(F)
,
,
r, J)
(21.36)
Qb,of claims, 7. The value of this payoff
Equation (21.36)represents arandom number, is given in the following proposition.
k
695
(21.16)we have that
equation
IZ -
I()
u%(0) ,
K c ,
,
T J)
1',
,
=
e-rT
F+g5'(0) K
-
,
=e
,
S(F), Fq
RE
This expression is the discounted risk-neutral prob.ability that the bet pays off; it is also the second term in the Black-scholes formula. If you hold both a bet that pays $1 when ST) % K and a bet that pays $ 1 when ST) < K then for certain you will receive $1 e-rT and we have at time r. Therefore, F+ + J/- =
(21.35)
Let F(5', t) represent the price of an option denominated in cash, where S follows the process in equation (21.5). The correlation between dZc and dz is p. Suppose we receive the time-r payoff 1'(:(F).
therefore from
THE NUMERAI
-rT
(1
,
tr
s
1.,
T Eq ,
Ndzlj
-
Now consider 4he bets denominated in shares. Let F+ denote valpe a bet that pays one shar when 5'(F) > K and Y- the value of a bet paying one share when ST) < K. Holding a share-denominated bet is like having a random number, ST), of cash bets. By Proposition 21.1, the value of the share bet is obtained by multiplying J/ by the &2 l and p = 1 since S forward price for S, and replacing $ with (we have b multiplies a claim basd on S). Maling these substitutions, the value of the share bet is 5'(0)e(r-J)TF(5'(0) K o', ?', r, J tr2j, or =
-
-
Suppose the process for S is given by equation (21.5) and the 21.1 for by equation (21.35),with p the correlation between d. and dQ. Let Q process 7(u$',K, tr-, r, F-r, 8s) represent the price of aEuropean derivative claim on S expiring at time F. The price of a claim paying Qb7 is given by
Proposition
QU)ber-&*j--lj
yg,$t/.l K as, ,
,
$c) br r here,7 8 bpo'o'c and valueQbclaims, each with value 7, we replace the Q(/)'dtr-*)(F-r) mu ltiplythe resulting price by W
=
-
*
=
-
-
-
.
t.,
lbb
r
t,
-
(p,j
;yj
.gey)
1)(r2
In other words, to dividend yield on S, (!, by ?7, and 2
-
q
.
k.
The proof is in Appendix ZI.B. Equation (21.37)is quite important and deserves further comment. We encountered J* in Section 20.7, in Proposition 20.3., it is the lease :(?)@'dtr-J*)(T-J) is the forward price for a claim paying Qb. The rate for Qb. Thus value of a claim paying QbJ/ is thus the forward plice for Qbtimes 7 evaluated at a modifed dividend yield. We know from Section 20.7 that if Q and S are correlated (in which case Q and F are correlated), there must also be a covariance term. Ttle term ?? replaces the dividend yield 8 to account for this covariance. We will now value the share-plice bets described in Example 21.2. t! Example 21.3 liqLet J/ + denote the value of bet that pays $ j at time r if vST) > K, and 7- the value a y)) il', of a bet 7,7,,;
that pays
$1 at time r when ST4
<
K. Both bets are. cash-or-nothing options,
and it is the frst tenn This is the value of an asset-or-nothing option, equation (2l formulmlzl view the hrst Black-scholes Thus, we can term as a in the Black-scholes discounted risk-neutral probability with a change of numeraire. If you hold both a share-denominated bet paying one share when ST) > K, and also a bet that pays one share when ST4 < A'. then you will for certain receive a share e-Tvo, and we therefore have at time r. Thus, 1'+ + Y- = .15),
- (.S'(0) 1-,
K c .
,
1',
T
,
t1
=
SQleMT
= Sloje
-
FV(5'(0)
-
y(;;e-JF
,
-&T
K c', r, ,
r, Jq
,y(#j)
In the case of Joe and Sarah's bet, suppose that the share-price volatility is 30%, compounded lisk-free rate is 6%, the iime to expiration is 1 year, and the t.he share price will be above $106.184.The dividen Joe bets that ls. the share pays no cpntinuously
This argument link-ingthe two terms in the Black-scholes 14' inatonis due to Geman et a1. (1995).
equation by changing the units of denom-
696
%.TH E BLAcx-scHoLEs
EquATloN
an empil-ical Ultimately, theimportance ofjumps andtheirsystematic of jumps. effects the into insights useful provide Merton's formulas issue. Nevertheless, componentis
value of this bet is Value of Joe's Bet
=
l'+E100,106.184,
The opposite side of Joe's bet-receiving
0.30, 0.08, 1, 0q =
one share when ST)
value$100 $55.962 $.44.038.
<
$l06.184-has the
=
-
sarah'sbet
$55.962
pays $106.184 if the price is below
valueof Sarah's Bet
=
e-0.08x
l
-
$106.184;this bet has the value
v-hglto106.184
0.30, 0.08, 1, Oj
=
$55.962
-
=
Problem 21.8 asks you to 5nd the strike prices such that the cash and sharedenominated bets have equal value. We will return to changes in the unit of denomination in Capter 22 when we discuss more nonstandard options.
OPTION PRICING CAN JUMP
WHEN
THE STOCK
Suppose that the stock follows the process dutljut)
PRICE
+ O'6'IZ+
(21.38)
dq
ln(y) -
where (8' 1 Thus, if the stock price is S before a jump, it is I'.Nfollowing the jump, 1) is the EY is the percentage change i the stock price due to th jump, and k expected percentage jump. Asume that the occurrence and magnitude of the jump are uncocelated with the stock return. and with Merton (1976) shows that with the stock following equation (21.38), the Black-scholes PDE becomes jumps diverable, 0'2.2 /(5', /)) l.V (21.40) J kklvs + fygyts'y, tj + /ytr k' + 1U 2 SS When jumps are lognormal, as in equatin (21.39),Merton shows that the price of a European call isl6 =
-
=
-
-
(
oo
-;.?
d
/=0
r Cyl y ;f i
!
Bscall
an example of equilibrium
option pricing when there are .
jumps that
s, K
0-2 ,
+ io'llT, ,
?-
-
/f + iaJ!T,
T,
t
(21.41)
ke'l The price of a call is obtained by put-call parity. provides the option value as ap expectation of European option Equation (2l occurring. Conditional prices with respect to the probability of a given number ofjumps j(y2 o'IT o-lT, with + variance, a quantity that reoects J cm i jumps, we replace the price lognormal discrete moves. We also replace the the added variance from having i 1k, is increased by dri, instantaneous The risk-free rate, r, with (?' kklln + ial. iaJ the cumulative rpean of i jumps, is An interesting special case occurs when the only possible jump that can occur option call becomes stock If the to jumps zero, a a jump of the stock price to zero. Hence, with ajump to zefo, the PDE for a call worthless'. k'(5'1', t) = 0, and kk =
where
'
=
.
.41)
,
-
-
.1?
-.
obtained as l6when jumps need not be lognormal, Merton ( 1976) shows that the solution for calls is formula: an expected value of the Black-scholes
f=0
cannot
kkldt
-
x (1990)for
-
0 if there is no where, over an interval dt, ajump occurs with probability #/, and dq distributed lognormally magnitude, l', is jump and J' 1 if there is ajump. The jump such that (21.39) Aal, c.2,)
-
We discussedjumps in the stock price in Chapters 19 and 20. Jurpps pose a problem for the Black-scholes option pricing methdology. When the stockprice canjump discretely as well as move continuously, a position that hedges against small moves will not also hedge against big moves. As we saw in Chapter 13, large moves in the stock typically cannot be hedged. The fact thatjumps cannot be hedged does not mean that option pricing is impossirather, it means that riskmeutral option plicing may be imposslble. When moves in ble', the option plice cannot be hedged, we can still price the option by computing discounted probaexpectd payoffs using the act'ual probability density rather than the lisk-neutral bility density. The problem.is that the option has the lisk of a leveraged position in the stock, and we do not lnow what discount rate is appropriate. Some assumption about appropriate discount rates (wlch is really an assumption about investor preferences) option.lD will then be necessary to price an Merton (1976)derived an option pricing formula when the stock price can jump assuming by tat the jump lisk is diversifiable. Tlzis assumptin neatly sidesteps the discounting issue since diversifiable lisk does not affect expected rettlrns. White jump risk for a broad index is not diversisable, arguably many of the discret moves for individual stocks are. In that case, by holding a portfolio of delta-hedged positions, the market-maker can diversify the effects of jump lisk.
l5See Naik and Lee be hedged.
(J
=
Jumps. =
The opposite side of Sarah's bet pays e-'08 x $106.184 $55.962 $44.038. Thus, bothSarah and Joe wish to denominate the bet in their favor. Moreover, Sarah and Joe's k desiredbets have the same value!
213
Merton's Solution for Diversifable
e-;.w
(z.,z,)f il
(5'l',,c-Z2F, Ei gBSCall
conditional on jumps. whereEi denotes the expectation discussed in Chapter 19 the reason for subtracting k.
l7we
K, c,
?-,
F,
())
;
% THE
698
BLAcK-scHoLEs
EQUATION
PROBLEMS
becomes O'ZSI &)S (?.+ ) / + y.(?. + 7, + lFj a s Every occurrence of ?' is replaced by 1' + ; hence, when the stock canjump to zero with instantaneoqs probability #/, the value of a European call, BSCaII). is =
-
BSCall(5', K,
c, r, F
r, &)
-
BSCa1l(5', K, c.,
=
1'
+
F
,
(21.42)
)
t,
-
The formula for a put, BSPut;., is then obtained by put-call parity:'s Bsputzt5',
K, o-, r, T
t,
-
)
= Bscalltu,
K,
-+
t7', ?
,
F
t J
-
,
)
,
uSe-tT-'s
-
+ Ke-r--tb
jump formula further in Chapter 23.
We will discuss the Merton i';'t)tjtitid h!'d @)('((E j'1* S'... :'55* $' '()' jltk))jjd r' E'. 777q*. !!'!' 8q15*. )' jld j';' t'E)' il!zd r' jqr)d ld ?' (IE' jfd jll)d j:jd (TX !J()' ('rrd (' jld ))' ()))))' j;j)d (k*84 .)' qd jqd q' ((' j'j)l;))d '(#' ijd qqd ()' ld lj' JEjj)X lil')ri t;d q;' r'j!I(I(' q)' (p' ;:;r;q)' J?'j::(r?((T' (q' ;(' 5r4)* (' )' (q()l' Tjqzrllrll.glld ;)d (5.;)* @)' (td q' 1:j;*%s:1:). (j' ))' fjtjd r(' r!' t').14;1j11;)4r* r'
%
699
jumps are systematic. The Heston model is described in Heston (1993). ln addition to Bakshi et al. (1997)and Bates (2000),recent empirical studies of volatility skew include Benzoni (2001),Andersen et al. (2002),Eraker (2001),and Pan (2002). Cox and Miller (1965)and Wilmott (1998, h. l0) discuss the backward equation and its counterparq the fonvard equation, which characterizes the probability density for St, conditional on Sv. Geman et a1. (1995)studied the role of changing the numeraire as a pricing technique. Schroder (1999)extends iheirresults, inclding examples with stochastic volatility and jump-diffusionmodels. Ingersoll (2000)provides some additional exnmples of the use of this technique. Marcus and Modest (1984, 1986) examine quantity uncertainty in aglicultural production. PROBLEMS
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The Black-scholes equation, equation (21.11),charactedzes the behavior of a derivative s a function of the price of one or more underlying assets. (The Black-scholes equation also appeared in Chapter 13 as a break-even condition for delta-hedging marketumakers.) We can interpret the Black-scholes equation as requiring tat a derivative earn an appropriate rate of return, which occurs when the delta, gamma, and theta of an asset satisfy a particular relationship. The Black-scholes equation is ths a generalization of the idea, familiar from introductory fnance, that zero-coupon bonds apprecite at the risk-free rate. Probabilities and partial expectations satisfy a related condition known as the backward equation. Along with the Black-scholes equation, a derivative must satisf'y an appropriate boundary condition. A change of the units of an option payoff is called a change of numeraire. Proposition 21.1 shows that the price effect of a change of numeraire is accounted for with a simple transformation of the pricing formula.
FURTHER
READING
21.1. Verify that equation (21.12) satisfies the Black-scholes boundary condition for which this is a solution? 21.2. Verify that
satisies the Black-scholes
ASL'e?t
a
:zz
l
y.-
1-
1-
-
:j:
-
-
1 2
J
analysis to exotic options and in Chapter
Two classic papers on option pricing are Black and Scholes (1973) >nd Merton (1973b). Merton (1976)extends the Black-scholes model to allow diversi/able jumps in the stock price, and Naik and Lee (1990)develop a model to price options when
There is a different boundary condition when the stockjumps
to zero:
y)
-
cra
2
21.3. Use the Black-scholes equation to verify the solution in Chapter 20, given by Proposition 20.3, for the value of a claim paying S'l. 21.4. Assuming that the stock price satisses equation (20.27),verify that Ke-rT-'b + vtle-nT-t' satises the Black-scholes equation, where K is a copstant. What is the boundary condition for which this is a solution? 21.5. Verify that Stje-&l--sNdk 21.6. Verify that e-rT-tsNdz)
) satisfits the Black-scholes equation. satisses the Blck-scholes
equation.
21.7. .'Use the answers to the previous two problems to verify that the Black-scholes formula, equation (12.1), satisses the Blak-scholes equation. Verify that the boundary condition V(5'(F), T) maxlo, ST) A') is satished. -
21.8. Consider Joe and Sarah's bet in Examples 21.2 and 21
.3.
a. In this bet, note that $106.184is the forward price. A betpaying $1 if the share price is above the fomard price is worth less than a bet paying $1 if the share price is below the forward price. Why? b. Suppose the bet were to be denominated in cash. If we want the bet to pay if S > Jr, what would have to be in order to male the bet fair? .v
ISFor a put option, the solution does not entail replacing eve:'y occurrence of r with r + 2.. The reason is that the PDE for the put option is different from the PDE for the call option in the case of ajump.
2(?
.
.j.
-
(7 z
=
ln Chapter 22, we extend the Black-scholes 24 to studying interest rates.
-47
c. Now suppose that we pay one share if S in this case to make the bet fair?
>
x. What would x have to be
21.9. Consider again the bet in Example 21.3. Suppose the bet is S $106.184if the price is above $106.184,and $106.184 S if the price is below $ 106.184. What is the value of this bet to each party? Why? -
/7(0, tj rather than 0 in the case of a call.
=
Ke -rF
What is the
PDE for -
tr a
equation.
-
700
%.THE
EQUATION
BLACK-SCHOLES
21.10. Suppose that a derivative claim makes continuous that the Black-scholes equation becomes 1 2 2 kJZ. P) +
(r
c S V.s. +
)
-
payments at the rate
r.
Show
/1
lM
+
?
-
to
11.11. What is the value of a claim paying QTj1ST4 Proposition 20.4.
Check your answer using
21.12. What is the value of a claim paying Q(F)-l ST) Propositipn 20.4.
Check your answer using
(P(F)
21.14. An agrieulmral producer wishes to insure the value of a crop. Let Q represent the quantity of production in bushels and S the price of a bushel. The insurance payoff $50. is therefore QT) x J/g5'(F), FJ, where F is th price of a put with K Wh at is the cost of insurance? =
1 2
?1
11
o-ic'ypf,y Si Sj Vsh i = l j= l
Consider a elaim for which the payoff depends on the where ai
ijdt
-
11
asset prices,
u1
,
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,
Sn,
,
.
.
??
.
% t, F) be the assets, and bonds, ,
,
OF PROPOSITION
21.1
-
=
-
t
2
By the multivariate Black-scholes equation described in Appendix 21.A the claim Yz, Q, tj must satisfy ry (2) Pl% + 1 + o'l :2):: + gpcccvoyscj jv''s + +
(r
(r
-
-
2
Vlslyss
=
Q
Ac-(r-J-)l QbF where A is determined by boundary conditions, Guess the solution 1' J* is to be determined, and lJ; satisfes the same bounda.ry condition as F. Compute the derivatives of this guess and substitme them into equation (21.44).After simplification (in particular, the (h'multiplying evel'y tenn divides out), this yields
(*
-
r + la(?' -
+ 1ty2Qbb
t5(2)
+
-
1
1)
-
2
(J'JS+
p'
-
(J
2 bpo'c (2))5'W- + 1aca S J,I/'.u) = r
-
Using the multivariate version of lt's Lernma (Proposition 20.2, Multivariate lt's Lemma), the change in the value of the portfolio is dl
=
$dt +
1 Z-sdSi + 2
?1
11
11
dSi dSj
-
f=l
f=l
j=3
/-y
sj
:/1
Ni t5'f + d J,I/
+ i=1
:% Hold bonds to snancethe residual such that In order to delta-hedge 7, set Ni 1 = 0. The same analysis used to derive equatioh (21.11)leads to the following PDE =
(2l
.46)
F The tenu in braces is the same as equation (21.45),except that is replped with 77 = W' is the same as P' except that $ is replaced by ?2. With this - bpao'c. replacement, fromequation (21.45),thetermin parentheses equals rF. Equation (21.46)
becomes -
1'
+ br
-
8Q) + 1(y2 bb 2 Q
-
1) +
1-J!/' =
-
.
I,y
)o'Lbb
?'
1). Thus, with the 32 and $a) 1' bl' This equation is satislied if The parameter z4 J* in Proposition 21.1, the candidate solution solves equation (21.z14). er-*s? = For a European option, set is set so, at the point the option is exercised, X c(r-J*)r = condition. boundary ternainal solve the to yt 'h
=
APPENDIX 21.C: SOLUTIONS PROBABILITIES
Ni Si + F
+ = l
tt
(21.44)
.
-
-
-
-
11
/
=
?- IZ
J.n this section we will verify the solution in Proposition 21.1. We begin by assuming that we have a derivative plice IZ4,5', t7', ?', F t, J), that satisses ?./' l5'J/y + 1c25'2 J/'SS J/ + (r (21.45)
t*
(21.43)
+ o'idzi
F(5'I Sz, The pairwise con-elation between Si and Sj is pq. Let the claim, of consisting the value of this claim. Consider a portfolio J'P, such that
I
=
'rhus,
APPENDIX 21.A: MULTIVARIATE ANALYSIS BLACK-SCHOLES
St. =
-
=
K?
-
(Note that this payoff can be negative.) Should you accept the offer?
dSi
J'k +
21.B: PROOF
APPENDIX
i't
for free the payoff
F0,w(P)Jx mlxlo, 5'(F)
-
8it
-
i= l
-0.2.
receive
(r
7/ +
.tj
.v
For the following four problems assume that S follows equation (21.5)and Q follows = 2, ?' = 0.06, & = 0.02, equation (21.35). Suppose & = $50, Otl = $90, r Use Proposition 21.1 to find solutions 8c = 0.01, c = 0.3, tr c = 0.5, and p = the problems. Optional: For each problem, verify the solution using Monte Carlo.
21.1.3. You are offered the opportunity to
for J/':
FOR PRICES
AND
Appendix available online at wwmaw-bc.com/mcdonald.
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where dz is dened in Table 22.2. Equation (22.3),multiplied by the strike price, #, is the second term in the Black-scholes formula for a call option. If you were to be paid if S > K, you could value this as cash-or-nothing options'. -'r
To see how the naming scheme works, considerthe cash-or-nothing option, aclaim thing option pays the holder that we introduced in Chapter 21. One kind of cas h conditin under wlzich it pays off, The K. stock price than is greater F if time the $1 at Sp > K, is like that for arl ordinal'y call option, but it is not an ordinaz.y call because it call'' cashcall), pays $1 instead of S. ff.We will identify an option like this as r. t cond all-when i Sp > k. the cagh under c on as a t that same i.e., a contract pays Some options make payments only if multiple events occur. Forexample, consider Sz has not been $1 only if Sv > K and the barrier H > tcash'' a cash-or-nothing call thatkpayst:cash because call'' and this refer out will We to casluocallll up hit. as a rises to the stock price if the does and out'' because not occur payment it pays $1, ttcall'' because payment requires Sp > K. Similady we will use the terms barrier, and options to refer to options that pay off that pay off in shares and refer to ttasset'' to only when Sp < K. To simplify the formulas in this chapter, we will ude the notation -or-no
lash
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You could also have a seculity that pays $1 if S is le-s ///tw? K. This is equivalent to a security that pays $1, less a security that pays $1 if Sv is greater than K. Such an option is called a cash-or-nothing put. The value is Cashputtk,
A-, c',
?-,
F
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t,
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-
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;
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''up
i'put''
in Table 22.2.
/ xamp Ie 22 1 Suppose S $40, K $40,o' 0.3, ?' 0.08, F t 0.25, and =.a0. The value of a claim that pays $1 if S > K in 3 months is $0.5129,computed using equation (22.3). The value of a claim that pays $1 if S < K is $0.4673,using equation (22.4).The combined value of the two claims is ,-0.08x0.25 $4).9802. k. .
=
=
=
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Cash-or-Nothing Options Recall from Chapter 18 that the risk-neutrftl probability that Sp > K is given by Ndz) lisk-neutral from the Black-scholes fonnula. We ltnow f'rom Chapter 21 that discounted probabilities are prices of derivatives. Thus, the pricefora cash-onnothing call-which
We lnow that equations
(22.3)and (22.4)are correct since, as discussed in Chapter
21, both formulas satisfy the Black-scholes equation (equation(21.11)) artd the appropriate boundary conditions.
706
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.).
An asset-or-nothing call is an option that gives the owner a unit of the underlying asset if the asset plice exceeds a certain level and zero otherwise. As discussed in Chapter 21, Propostion 21.1, the price of an asset-or-nothing call is obtained from the price of a cash-or-nothing by replacing the dividend yield, &,in the cash-or-nothing fonnula with tr2, and multiplying the result by the forward plice for the stock. The result is
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0.25, 0.08, F t $40, o' 0.3, r Example 22.2 Suppose S $40, K > K in 3 months is $23.30, if S share value claim that of $ The 0. one pays and a computed using equation (22.5). The value of a claim that pays one share if S < K is %. $16.70. The combined value of the two claims is $40. =
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46 44 38 40 42 36 Stock Price at Expiration ($)
34
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Finally, we can conslzuct a gap option using asset-or-nothing options. Consider a if S > Kz. The value of this is call option that pays S .:-1
-
Assetcallts', Kz, We buy an asset
c,
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t,
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Figure 22.1 graphs the maturity payoffs of cash and asset calls. 0.25, t 0.08, r 0.3, ?' Exam'ple 22.3 Suppose S $40, o' $40, K Using results price of an ordinary call is an asset call less 40 cash calls. 0. and 22.1 and 22.2, the price of the ordinary call is $23.30 40 x $0.5129 in Examples $2.7848. (ffI ) if the stock is greater The price of a gap call in which the owner pays $20 % than $40 (#2) at expiration is $23.20 20 x $0.5129 $13.0427. =
=
Ordinary Options and Gap Options
=
-
=
=
-
We can construct an ordinary call by buying a European asset-or-nothing call with strike price K and selling K European cash-or-nothing calls with strike price K. That is,
BSCZIIS', K,
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Delta-Hedging All-or-Nothing Options
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All-or-nothing options appear frequently in wlitings about options, but they are relatively rare in practice. The reason is that they are easy to price but hard to hedge. To understand why, think about the position of a market-maker when such an option is close nightmare scenario for a market-maker is that the option is close to to expiration. 'l'he
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%.Exo-rlc
OpTlohls:
11
ALL-OR-NOTH
BARRIER
22.2 ALL-OR-NOTHING
OPTIONS
Banier options, introduced in Chapter 14, are options in which the option comes into ban4er.z There are or goes out of existence if the price of the underlying asset hits a stock pl-ice hits a barrier price if the down-and-out options, which become worthless up-and-in options. down-and-in, and below the initial stock plice, as well as up-and-out,
/M?'?'cr options. We can construct options such as these using all-ol'mothg > paying Sp K, cash-or-nothing call but modify it by $1 if Suppose we take a expiration requirement will if the stock has it only that pay $1 at adding the additional < .S'(0), this is a If the option. H the life sometime during of also hit the barrier H this be Table 22.1, would notation the in - down-and-in cash call. Using a CasI2Dlcall. digital options, options ordinary from Just as we were able to construct we will also be barrier options. options from'digital able to constnlct barrier We will examine three different kinds of barlier options:
o A contract that pays reached
on either a barrier having or not having been barrier optiollsj.
$1 contingent
cash-or-llothilg
o A contract that pays a share of stock worth S contingent on either a barrier having l75l'?-Ik?' optlls). or not having been reached asset-ormothq e A contract that pays $1 at the time a barrier is reached rebate optiolls) or that pays $1 at expiration as long as the barrier has been reached during the life of the option options). ' deferl'ed ?z'Jnd#c By valuing these pieces and adding them together we can price any standard barrier option. The assumption that the stock follows geometric Brownian motion malej it possible to delive relatively simple formulas for tese options. There are 16 basic linds of all-or-nothing barrier options. First, consider cash-ornothingbanieroptions thatpay $1 atexpiration. Such options canknock-in orltnock-out; they can be calls (pay cash if Sv > K4 or puts (pay cash if Sv < A-); and the barrier event can occur if the barrier is above the price (up-and-insor up-and-outs) or below the plice (down-and-insor dosyn-and-outs). This gives us 23 8 basic cash-or-nothing barrier options to value.. By the same reasoning there are also 8 basic asset-or-nothing barrier options, for a total of 16 all-or-nothing barrier options. =
Cash-or-Nothing Barrier Options options. To anticipate the We srst consider the valuation of barrier cash-or-nothing value how will section, in this first to we results one particulltr barrier cash-orsee formula this call. From down-and-in cash option, we will be able to value one nothing a cash-or-nothing deferred rebate options. options and remaining the seven
(1991a),Rubinstein
%.
711
Assume that the option is issued at time 0 and expires at time F. Let t denote the greatest stock price between times 0 and t (wheret S F) and let j denote the lowest stock price between times 0 and 1. Suppose the barrier is below the initial stock price, i.e., H t < &. A cash down-and-in call casllDlcalll is an option that pays $1 if two conditlons are satished. First, at some point prior to maturity, the stock price drops to reach S, i.e., a.w (! S. Second, at expiration, the stock price is greater than the strike plice, K. probability that Wecan analyze this option by frst examining the lisk-neutral this joint event (a.wS H and S.r 2: K) occurs. This probability should satisfy three conditions:
1. Once the barrier has been hit Sv
K
(the barrier at this pint
2. If at time F, 3. If at time F,
j.v .w
jh
:s
S) the probability eqals
:j
H and Sv A: K, the probability equals 1.
>
H or Sv
<
the probability that
is irrelevant). K, the probability equals 0.
Assume that H S K, and consider this expression: Probtw
S H and Sp
>
K)
=
S
IT
-
-
l
Nd4l
(22.6)
The terms #1 through ds are defned in Table 22.2. In Appendix 21.C, which is found we saw that an expression of this on the book's Web site (www.aw-bc.com/mcdopald), fonu solves the backward equation. We also want to verify that it satisses the three boundary conditions described above. First, at the point where St = S, equation (22.6)collapses to Ndz), which is the risk-neutral probability that Sv > K. (rfhisoccurs because when S = St, #1 = ds. You should exnmine equation (22.6)to verify that this happens.) Thus, once we hit the bnrrier, the barrier value H drops out of the expressiotl because it is irrelevant. Second, if at expiration j.w :G S and Sv > K, ten equation (22.6)equals 1. Yhereason is that the probability equals 1(46) once the barrier is hit, and if Sp > K, N(#a) = 1. Finally, if lw> H, i.e., St never reaches S, then at expiration Hl < SpK (recallthat S S K) and equation (22.6)collapses to 0. Thus, equation (22.6)both satisfes the backward equation and obeys the appropriate boundary conditions. Equation (22.6) assumes tat H S K. Why is this important? The answer is that if H > K, the boundary conditions may be violated. Consider the case where at expiration $55, K = $45, and H = $54 (thus violating the condition H S K), and the boundal'y has not been hit. In this case a correct expression for the probability 111(542/45 x 55) = 0.164, will evaluate to zero at expiration. However, 3nHljSvK) equation mattlrity will equal when the 1 has event not occurred. (22.6).at so equation (22.6), you might ask why it is necessary to As a snalcomment on (S/5')2(r-O/J2-l The answer is simply that #(#4) by itself mult.ip 1y Ndz) by the tenu does not solve the backward equation, whereas equation (22.6)does solve the backward kw
=
=
.
z-fhree comprehensive discussions of barrier options are Rubinstein and Reiner and Reiner ( I991b), and Derman and Kani ( 1993).
I NG BARRI ER Op-rloNs
equation.
,
k. Exo-rlc
712
To handle the case where H (22.6). When H > K, we have Probtiw S S and Sv
K4
>
version of equation
K we need a more complicated
>
X(#2)
=
-
Nj
s
+
zp
1(#8)
y
c, r, F
e-r(r-?) d
,
t, $, H)
-
jx
-r(F-J)
=
1 (#,)j tj )2V x
s s
K
y
K
(22.8)
-
N Ldoj+
Ldz) -
>
Equation (22.8)gives us the value foracash down-and-in call when uo > S. There three elosely related options we can now price: Cash dnwn-and-out calls casllDoare Call), cash down-and-in puts casllDlplttl, and cash down-and-out puts caslzDopkt. We can value each of these using only the formulafor the cash down-and-in call, equation (22.8). ln addition, we can value a defen'ed down rebate option. We first value a deferred down rebate, which is aclaim the barrier has been hit over the life of the option. The time F as long as that pays $1 at depend claim this does not payoff to on a strike price: It pays $1 as long as the banier rebate'' claim hit. We will call this has been a deferred do&vnrebate. lt is a because and if reach the barrier, it is the payment is at becaus it pays $1 we vtlue obtain of this claim barrier. the the We time reach than rather the expiration at we > always Sv result is a equation Since have the setting K in 0, by $0 (22.8). we have3 Thus, Sv S. that F long :j claim we as pays $1 at as Deferred
down rebate
ty,
r, F
K, o', r, T
cash put lf you buy a down-and-ip cash and Sv < K. If you buy reached bnrrier is the receive $1if reached and Sv : K. Thus, if bnrrier is receive $1 if the long receive as the barrier is hit. $1 as call and put, you have thus, defen'ed rebate; we Down-and-in
CashDlputts',
option
K, c.
r,
T
-
t,
S)
(,
=
DlkDeferredtu,
t,
-
r,
-
(5,
t5)
(22.10)
H)
put with strike price K, you a down-and-in cash call, you you buy both a down-and-in This is the same payoff as a c
,
-Casl1DICa1l(5', K, a,
?-,
F
?-,
T
r, 8, H)
-
(22.11)
t, (T,S)
-
Buying down-and-in and down-and-out eash puts creates an ordinary cash put. Thus, the value of the down-and-out put is Down-and-out
(.t)2S-l NLdj s
Cashcallts', K,
=
(22.7)
Equations (22.6)and (22.7)give us expressions for the probability that the barrier is hit and Sv > K. What is the value of a claim that pays $ 1 when this event occurs? To answer this question we can use the result from Chapter 21 that discounted risk-neutral probilities are prices of derivative claims. Discounting equations (22.6)and (22.7),we have
#,
H)
,
-CashD1Call(5',
cash call
CashDlcalltu,
t,
-
j
-
Problem 22.3 asks you to verify that this equation satisfies the boundary conditions. = Ndbj', the formula again reduces to Ndz). Note that when S = H, N) Down-and-in
#, o', r, F
713
a down-and-out cash call is
pay $1 if S,;m> K. Thus, the value of CashDocallt5',
opa-loxs k.
BARRIER
Auu-oa-rqo-rlqlxc
11
OpTloxs:
Casloputtq,
cash put
K,
o', r, T
t,
-
,
H)
=
Cashputtu, K, o',
K, c', r, T
Casllllllptltti,
-
r, T
-
/,
-
t,
(!)
(22.12)
J, Hj
As a linal point we can compute the risk-neutral probability that we reach the banier. The deferred down rebate option pays $1 at expiration as long as the barrier is hit. Thus the price of this option is the present value of the risk-neutral probability that the barrier is reached. Therefore, e
rtf-llDlkDeferredtu,
0, o., r, F
is the risk-neutral probability that the bnrrier is
-
t,
(22.13)
()
during the life of the option.
reached
ftdown
Edeferred''
=
DltDeferredts',
c, r, F
-
t,
(!,
S)
=
CashD1Call(S, 0, o', r, T
-
t, 8, Hj
(22.9)
0, the value of the deferred down rebate does not depend on Note that since we set K strike price. the Now we can compute the value of the remaining three options.
Suppose S $40, c 0.3, 1, 0.08, Example 22.4 value of a cltim that pays $1 if the stock hits the barrier H computed by setting K $0 in equation (22.8):
We can create a synthetic cash call by buying down-and-in cash calls with the same banier; this combination is guaranteed to
cash call
and down-ad-out
3In peforming this calculation, to avoid a zero-divide error it is necessafy to set K to be small, such as ' K = $0.00000 I rather than exactly $0. ,
=
t = 1. The 0, and F the next year is $35 over -
=
CashD1Ca1l($40,$0.0000001,0.3, 0.08, 1, 0, $35) -r(T'-/)
=
Down-and-out
=
=
=
=
)
d
-
NLdoj+
S
=
2(r-J)/c2-
I
N Lds)
-'
=
$:.5,74
probability that the stock will hit the barrier is the undiscounted value e0'08 = 0.622. of this claim, or 0.74 x 'rhe value of a daim that pays $1 if the stock hits the barrier, $35, and then is also greater than K = $35 at the end of the year is
The risk-neual
e
sr(F-/)
S -
2(r-t)/o'2-
1
NLd4) $0.309 =
714
%.ExoTlc
ALL-O R-NOTH
11
OpTloIqs:
tk This is the value of CashDICall($40, $35, 0.3, 0.08, 1, 0, $35). The risk-neutral proba() (s rli) 0.335. x. r) bility of hitting the barrier and being above $35 is 0.309 x e jk
'
=
This example illustrates an interesting point. The value of the claim that pays $1 at expiration when the stock at expiration is greater than $35 and has hit the $35 barrier ($0.309), is approximately one-half the value of the claim that pays $ l at expiration as long as the stock has hit the $35 barrier ($0.574).The reason is that once the stock has hit $35,it subsequently has about a 50% chance of being above orbelow that value. This observation suggests that the probability of being above $35 conditional upon having hit $35 is 0.5 x 0.622 = 0.31 1. The actual probability is greater than that, however. The 0.5&2 = 0.035, which is positive. Thus, after reason is that the lognormal drift is 1' dfifts higher. stock the having hit $35, on average verify this To intuition. suppose we set the lognormal drift equal to zero. We can setting the risk-free rate to 0.045, which gives us 1' 0.5&2 = 0.045 0.5x do this by 0.32 = 0. We might expect that the value of a claim paying $1 at F if the balnier is hit jtti price at is one-half the value of a claim paying $1 at F if the barrier is hit and th = 0.5c2, probability 1* the differently, When barrier. Put than the expiration is greater of hitting and ending up above $35 is half the unconditional probability of hitting $35. The next example shows that this intuition works. -
0.3, ?' = 0.0X5, = 0, and r t = 1. The Suppose S Example 22.5 $40, = $35 over the next year is value of a claim paying $1 if the stock hits the banier S (y
=
t?-r(F-J)
j
-
2(r-J)/o'2-
H
NLdo)+
I
NLd)
T
=
.
Given equation (22.14),the procedure for obtaining the prices up rebate the cash-or-nothing three options when H > Sv is analogous to that before. of other First, by setting K = cxl in equation (22.14),we obtain the price of a claim paying $1 at expiration as long as the banier is reached..'f Deferred
UlkDeferredtu, o',
r,
r
t, J, S)
-
Cashulputtu, x,
=
c, r, F
t, &,H)
-
(22.15)
With this equation, we can solve for the price of the other cash-or-notlzing options. Up-and-out
Buying up-and-in and up-and-out cash puts gives an ordinary
cash put
cash put; hence, Cashuoputts', K,
o', r, F
-
t,
S)
t,
=
Cashputts',
- Cashulputts', Up-and-in
K, c, r, F
K,
(r,
r, F
r, )
-
/,
-
S)
,
(22.16)
Buying an up-and-in cash call and an up-and-in cash put yields
cash call
the same payoff as a deferred up rebate. Thus, we have
$0.6274
Cashulcallts',
K, c, r, F
t, J, S)
-
,0.045
The corresponding risk-neutral probability is x 0.6274 = 0.6562. The value of a claim paying $1 if the stock hits the barrier and is then greater than K $35 at the end of the year is =
This is one-half of 0.3 137 = 0.328 1
715
-
-
=
Q
If you compare this formula to equation (22.8),you will see that Ndz) is replaced with #(-#2), Ndz) with N-d4), and so forth. We know from Appendix 2l.C that these terms also solve the Black-scholes equation. The effect of these changes is to reverse the effect of the di terms. As a consequence, equation (22.8),which prices a down-and-in cash call, is transformed into an equation pricing an up-and-in cash put. Problem 22.4 asks you to verify that equation (22.14)solves the appropriate boundary conditions for an llp-an d i n CaS h P ut
-
-
I N G BARRI ER OPTI ONS
$0.6274. The corresponding risk-neutral probability is
,0.045
=
UlkDeferredts', o', r, T
- Cashulputts', Umand-out
cash call
cash call; hence, Cashuocallt,.
K, c, r,
-
r
-
t, J, H) t, 8, H)
(22.17)
Buying up-and-in and up-and-out cash calls gives arl ordinary
K, o-,
F
?-,
-
t,
,
H)
=
Cashcallts',
- CashUICall(5', K,
x
K, o-, r, F tr, r,
T
-
-
t,
t, J) ,
S)
(22.18)
.
Asset-or-Nothing Barrier Options
cash put Now we consider cash-or-nothing options when the barrier is stck price. First, consider the following formula for an up-and-in the above current which when Sv > H and Sv < K.' cash put, pays $1 Up-and-ln
Cashulputts', K,
c, r, F
-
r, $, H)
=
(#.)2S-l x(-#4)
e-r(r-?) s d
-r(w-J)
jx(-#a) -
s-ds)
H yy K +
x(-#s)j t.?slzr;-i-l
u
<
K
We now wish to find the eight pricing formulas for asset-or-nothing options col-respondoptions. Fortunately, there is a simple way ing to those for the eight cash-or-nothing to do this. lf we view asset-or-nothing options as cash-or-nothing options denominated in shares rather than cash, we can use Proposition 21.1, dealing with a change of numeraire, to transform the pricing formulas for cash-or-nothing options into formulas for
(22.14) 4To evaluate equation (22.14)at K
-
x, we simply set .N(-#a)
1.
716
%.Exo'rl c
op-rloxsk
BARRIER
11
OpTloNs:
c'2,
and we multiply the asset-or-nothing options. In each case, we replace $ by Szer-ntT-tt for stock. For example, forward price the the by cash-or-nothing formula have we Assetlxcallts, K, c', ?-, F tt J, H) (pa.lq) . Sc(r-d)(F-J)CaslaD1Call(:', K, c, r, T t, J (r2, H) = The other seven asset-or-nothing pricing formulas-zd-'clfltitltz//, AssetDllhtt, Assetbe created in exDOPut, Assetulcall, Assetuocall, Assetulpllt, and Assetlop--c actly the same way. -
717
The up-rebate formula is symmetric: URIS,
r, T
(r,
J, S)
1,
-
111
S
=
N(-ZI)
-
S
lt:!
./
+
N-Zz)
-
S
(22.21)
-
where all variables are defned as above for the down rebate. Ifwe let T x, the fonzlulasfor up and down rebates become the barrier present value formulas, equations (12.17)and (12.18),disussed in Chapter 12. (Problem 22.5 asks yc to verify this.) The rebate formulas provide the value of $1 when the stock price hits a barrier, and this is exactly the calculation performed by barrier present value calculations, only for the case of ininitely lived claims.
-
-
Rebate Options Rebate options pay $1 if the barrier is hit. We have already seep how to price deferred rebate options, which pay the $1 at expiration of the option. Ifthe option pys at the time the banier is hit, we will call the claim a rebate option (orinlnlediate rebate optiol. ' We have already seen in equations (22.9)and (22.15)how to price deferred rebates. The formulas for rebats paid when the bartier is hit are more complicated because the discount factor for the $1 payment depends on the time at w hich the barir i it In effect there is a random discount factor. The fonuula for the plice of a down rebate when S > S is
-->.
22.3 BARRIER
OPTIONS
At this point it is easy to construct the barrier option formulas from Chapter 14 using the preceding formulas. A down-and-out call, for example, can be valued as CallDownoutts',
K,
(r,
r, T
t, J, S)
-
ty,
r, F
t5,
t,
-
111
H
S)
=
S
a
gyg
X(Zl)+
-
-
S
X(Z2)
=
LnH/S)
Zz
=
qn(S/5')
1-
9
1
?-
-
hz
=
G'
-
')
-
')-
+
tjj/w
T
tjj/w
T
?-
-
2
2
&
-
t.o'
(22.20)
1 2 -
-
1 2
-
-
o.
-.
o A rebate call, which pays the $20 when the stock hits $120prior to * Aknock-out call with a strike of $100,which knocks out at $120.
t
-
-
2
')
+
+
--,r
o,-
2?-
(,.2
-,
-.>.
=
0, d'ssil
.%T c'
t)
r
-
t
'''$
w
k!ll'
-,%T
?)
-
:::::::
'''''''''''
o'
F
-
Thus, the formula evaluates to 1 when the banier is hit.
t
S)
t,
t,
S)
11.'
expiration.
If the stock reaches $120 prior to expiration, the rebate is triggered and the call knocks out. lf the stock has not hit $120 prior to expiration but is above $100, the
2 1-
This formula satisfies (as it must) both the Black-scholes equation and the boundary conditions for a rebate option. Suppose that 4he barrier is not hit over the life of the F. At the point when the barrier option. Then H < S and both terms go to 0 as t = distribution is symmetric around normal the Because ln(S/& S and 0. H is hit, -
(,
-
e)
-
')
2
t
-
o'l
1
ts
o,
1-
+
-
2
-
-
gT
-
-
=
-
+ gT
Zl
j
-
-
c, r, F
t,
down-and-ins, and so forth are all constructed analogously. See Table 22.3 for a listing of formulas for barrier calls and puts. As another example of the use of all-or-nothing options as building blocks, capped options are single options that have the payoff of bull spreads, excepy that the option is exercised the first time the stok price reaches the upper strike price. An example of an American capped option is an option with a strike plice of $100 and a cap of $120. When the stock hits $120 the option pays $20. If the option expires withut the stock 100, 0). This option can be priced as the having hit $120,then the payoff is maxts'w sum of the following two options:
2
1 l.
=
-
-
Up-ad-outs,
where, letting g
K, o', r, F
- K x CashDOCall(5', K,
.
DR(5',
AssetDocalltu,
=
;''''''.'''. -'j.;r' -,' r' ''''''. j' (' t'''k'''rd -.--'.. 15)4,* ('' ,;' y' j'(' t';y---' ).1* q' ' jd yrd y' j'q!jjtqjd (' )'
.'''
.'..' ...
t''f' q'-'tf ,'---LL;' )'. y' tt)f t'-. r' ..';:.' ii2iL:-,k-.' )-' r;d i'. (jjjy'r ''t''l'; t't-' '-' qlkihitslik--.'
t'tf ''.' (i( j:k(qik((.y ' ('.' ;)(.''.. .('. k. g.( ;( 1)* ytj(y7yty-y.--;yyj-yyy..y-y-.-.y--y..y.y.y.-..--j.g.-..-:-...;yt.:q.--,)...-.t-'yy.yj.yy.yyj..y qq---ttyty yy-.j...yr'..-jjjjjg(.:r ',)-y-.jjjjjjjy)). li).'t.i ) yyyy.. r..-)-:-; t-:i2kpiiir'7.. .', r iiIItIiI...-. Ii.IIiEii:. ; !. i.l'i tlr:t!!;i tl:::!!;: t '. .(..: Ir..I:::,. ,r,. jr..,r..,j jy::y I..ll lr..,r..lk Ir..ll Ir..llea I..1.as Ik..,r pr.,,r 14 11 1/..,,a Ir,.!g((2I .It:. . )..;r k.... ... a Ir..Ir..i; Ir..Ir::,p a Ir..ll iiil.11 a ,!!;.jg:-,t:::,r ; r; s IL . i . ', '
.
;EE:. E
-!
.
-
i .
.
.
'
'
EEj'j ((i
i. -
(
.
E''' ' (' ( (j (g i.'.'( . .,:.-k: k.- -
-.'
. -
..... ... .. .
E
' ' ': ' g. .. .
jE :
;
-'.
--
...
E EE
j j .(jgi('g !(E : g. . .:
. ..
.
:
: : . .. . '
'
'
-
. .
.
'
:
.
.
....
. . .. . .
. . : .
'
.
7)4 ' :
'
. . . .. .. . .
.
-
.
..
.
:.. --
--.--
(- -.
- -.. .. .
-- .
-...
.
. . . ..
.
.
-
..
Up-and-ln
Up-and-out
Down-and-ln
Down-and-out
'.l
'
:
.
'
'
.
.
.
.
y...((. ' y.'(.E......E.... ' . '.E.''.' '.... ... . . . . ..
. .
'. -
'
.
:
. . . .. .
.
..
.
.
: ' . '. . . . . . . . ...
:'
.
.
.. ..
Assetulcall
Assetuocall AssetDlcall
AssetDocall
'
:
.
. :
j'. . ..
)36'.i.i.'3. 411::1k,. '''.'''''''''.
' ' - . . . .. . ..... ....
-
-
-
-
.pIj! ' .. .(E.. .. E'. ... . . .1111 . .' . .. EE. . '' . E. E
l ... ..... ..
''Ik''''''.''''k'. .
-
. . . .. . .
.
... ... .
-
.
.
'::::.
..
for the functions are (.S,#, o', r, T
(.. E (.. .' .... (.' E... .. ..E...: .(. j)...:. (..(q(. . . E .. ' . . .. .. . '. . ' ' .' ' . : E . . E. E . . . . ..
: . . . ': . : . . . . . . . g . . ' ' . ' E E ' ' . E. E . .. . . . : . ' .. . . .. . : . . . .... . . . . .. . . . .. . . .
d:::jll ,::::
d!!,r
j:gy,j
-.
' . . ( . ..; (. y(. E. .. . (. .. . . ... .. . .'.. . . ( ; ' EE E. .. ' ....'. ... . '' ........ . .. . . . . . ...
' ' 'E ' ' : . . . . :
(E E' ' j'q' q ''r(7(''EqF'qqt(Fq' . E((E. j(.E(j( E;k .
: ' ' ':'' E' E( E.. .g'(-.,-3.3,,,,.,..-3.)3333.. ..''. .' (.
'rljjyyyt-tjjjjjj.
-
-
t,
dt:::
. .. ' (. . .. .. . . ' . . ( E.. .q. . .. :. . .E . : .. .. .. .. .;. . .. E. ... : . . .E ... E... . q ' .. ..:. .. . ((. ' . . . . .. . . .. . . . . . . . '' .. . ': . '..' '... E . ' E' ' ...... . .. ..' . .- . . ' E. '' .E . .' .. . ; ... : . ... . ... . : . . . E E ..... : .... ..'.'E. ...EE .......'.. E.E- .i. .. .. . E E. E... .. : : i' .: . .. . ..
.. .. . : . . .' ' ' E ' 'E - . . . : . .. : ... .. . . . . . . . . . . . . . .. . ..
'.. ''. ....
'
.
'.'' ''l ......' .
'
. .
:
.:,11rt::111:
' .. .
.
-;. . .
.
K x Cashlxcall
K x Caslrtput K x CashDoput
. .
. .
.
.
.
.
.
1.1.''.1...- ' ' ' . :: . . - . .
.
K x Cashtrut K x Cashuoput
'
'
:
:
K x Cashulcall K x Cashuocall K x CashDocall
d:i,r
--.
d!!!l
' rjjr... (. (.. (1j,.11,2. . E EE. E( ':'. (. l l '. . . . ; .. . . .... . .
-
dli!,h
J, 1h.
. . . ..
.
E .
: . .. .
. . : . . . . j(. . . j ..... . . .. ... . .. . . .. .E . .. . E ' y. . . -. . ' q.. . (.;.. ... (:.. EE'. '..'.-. . !'...'.'. .. ... . .. :
. ;..' .
.
. . - .
'
'
'
..
i...(. ......
.
.
.
.
. '
-
.
.
.
-
Assetufput
-
Assetuoput
-
-
'. ' .. .
AssetDlput AssetDoput
'
.
k. Exo-rlc
718
QuAxvos
11
OpTloNs:
Nikkei by directly investing in the cash Nikkei index or by investing in yen-denominated futures, such as a Nikkei futures contract trading in Japan. Both strategies have a payoff denominated in yen. lf the Nikkei appreciates but the yen depreciates, the f//l/srt??' call lose ???tw?el' despite being correct abtt #?' ntovelllel't ofthe Nikkei. You could try to reduce the problem of exchange rate risk by hedging the Nilkei investment using currency futures. However, the quantity of yen to be exchanged is high when the index has a high return and 1ow when te index has a 1ow return. Thus, there This price uncertainty creates is ?2t? wJy to 177/:$7i'?7advance /251:/many )?e??to quantity uncertainty with respect to the yen exposure. We could imagine a synthetic Nikkei investment in which the quantity of currency fomards depended upon the Nilkei's yen rettlrn. Such a contract would permit an investor in one currency to hold an asset denominated in another currency, without exchange rate lisk. This contract is called an equity-linked forward, or quanto. For reasons that will become clear belom a quanto is also sometimes defined as a derivative having a payoff that depends on the product or ratio of two prices. The Nikkei 225 index futures contract, traded at the Chicago Mercantile Exchange discussed in Chapter 5, is an example of a quanto contract. This futures contract is and marked-to-market daily in dollars, even though it settles based on a yen-denominated price.s There is also a yen-denominated Nikkei futures contract that trades in Osaka. Both f'umres are based on the Nikkei 225 contract but they differ in currency of denomination. We will see in this section how their plicing differs. The box on page 728 discusses Nikkei put warrants, which were another example of a quanto contract. Table 22.4 lists the symbols and specific numbers used throughout the examples in this section.
knock-out call pays S $100. The table below illustrates the payoffs, assuming that the option stlike is K, the cap is S, and the option expires at F: -
H Hit
H Not llit
Purchased Knock-out
0
m?lxto, Sp
Rebate
H
Total
H
-
-
-
K at Hit
0
K at Hit
maxto, Sv
-
K)
.5-/70?-1.
Kj
The table shows that we can price the American capped option above as a straight applieation of the rebate formula together with a lcnock-out. The hplder owns a 100strike call with a knock-out of $120,and a rebate call with a $20rebate fayablet $120. Note that a European capped option is much simpler to price. Since the payoff does not this option is just an ordinary vertical spread (buy a loo-strike occur until expirtion, call and sell a lzo-strike European call with the same times to expiration).
t' ) Example 22.6 t)
.'
. . yjythat S
=
$100,ty
)') , an up-an d -Ou
=
Consider the capped call discussed in the text above and suppose 0. We can compute the price of 1, and 0.3, 1' 0.08, F t =
t call as
lt) jty )i'i'( Callupoutts', K, y.'). , t)
.. .. . .
c',
?-,
F
-
t, J, f)
=
! :
j)E
Assetuocalltu, K x
li.
lE. j1iI i'l'E
=
=
-
K,
o-, r, F
cashuocallt.
-
t,
K c r F
,
S)
-
H)
t .
''C'' :';' q' 77* :'' 't' );' i t)E );'q!fq(E' $' )1* (' q'' )' r' .'y' (r'!q))i)' '(' ! (i Eii.;'ti(Eql(E' t'qf '')'
'
)
tt: E
..,....ttbi?bbi.. .-
-
$100,0.3, 0.08, 1, 0, $120) callupouttsloo,
20 x $0.5649+ $0.4298 .) l.q.l) The price of a European bull spread for the same parameters would be j
=
=
$11.73
.
.
)i.l )t) BSCall($100, $100,0.j,
jj
ljtj ttij
) The capped .t. jlt tjt and end
.
'
0.08, 1, 0)
-
BSCall($100, $120, . 0.3, 0.08, 1. 0) =
$15.7113 $7.8966 $7.8147 -
=
. call is more expensive because all of the stock price paths that cross
$120
up lower result in the maximum payout on the capped call but a lower payout 17lil, spread. bull the i % on
' : ''
E: . .
..j.;.
LL(?.1.f()Lj36-
-
.jEE.
'''
'
!
.:
(EE.
.
@
('.'(. ( (':7: . .' (' ' ;.;; ;(' i(i((;. (.) (. k;i (.') ';)((( ;.jL.....'..; .y;y.yy.y..(y..y...y-yyj.'-'.y. .. . ltiqr). yytyy-y yyyt.'y.yyj(.yj..,y....y.j.: j-y .tyj IjI--l. .jj....).j...y yjtjllrriiprilllytgfitfitjlyilp lllilili y jr..jr..jj.,4g, jj..jr kyyjj jr.j kjjjs jr.,tjy;r tjyy lq.ti:i1i!:.. iliihl k.,yyyry......-y?j...q.......;q.... yj j y .yy ( jy:;j. iltj.li .(.##!r..:.,-, j-:.,.;;.jjL)..j;,.j.. gjgjjgr;. .
The plice of the capped call is 20 x UR($100, 0.3, 0.08, 1, 0, $120) +
k
..
EE: : E :)E !''E''.:''y.-..E.-jjyj..j'y.y.yy'..yyyqj...E.-.-yj . iE i:.'':' ' E': E'':. .-..)..(.()y...kE.Ey.g..)t.ty:.yy-.....'y,.y.yy''y(jjyyy.yjy'.y''y'y). E' : r.': ' E' ' :' E'' :''' '' E.i.E ( :(i. .: ( .. E .E'.... : y .. y yjy.syvys,yysssss-y( y yj y .,...y.... .y .. 'q!q'q'7('7;':
.
-y.(
'
:
'
.
.
,jkjjjj;.
-
...'
Dollar-denominated
A U.S. investor wishing to invest in a foreign stock index can purchase the foreign index ciirect1y or hold futures based on that index. However, the investor then bears two risks: The Iisk of the foreign index and currency (exchangerate) risk. For example, suppose that a U.S. investor wishes to invest in the Nikkei 225 index, expecting that it will increase over the next month. The investor can tale aposition in the
jtjj,r jkyj,rjyjjjjtj jjj;y jjj;y jj jy jjjr..;j jj..;j
j:ygjj jzyj!r
yjjj.tyjjj,,
.jjy
interest rate
.jyg,
,,
0.08 rg
0.04
Current Nikkei index
Ql
720,000
Nikkei dividend yield
$c
Nikkei volatility (Y)
(a'
0.02 0.15
Current exchange rate
($/Y)
jj jyyg;j
jyygyy jr..jjjgygyj jgjgjs jr..yjjrgy;jjjkygyg jr..jy..jj jyjryy yj,.jy k;,jgy kyj;j
Yen-denominated interest rate
-p
c
0.t100
Exchange rate volatility Nikkei-exchange Timp to expiration
22.4 QUANTOS
jkyg,l jkyyjr
rate ($JY) correlation
p F
1 year
5To illustrate dollar settlement, suppose the Nikkei 225 is at 22,000. Under the terms of the CME contract, a one point move corresponds to $5, so the notional value of one contract is 22.000 x $5 = $1 10,000. If instead the Nikke had been 22. 100, the notional value of the contract would be $1 10,500, a difference of $500.
%.Exo-rlc
720
itf t'. jji;L3j3L'. .'t'y' -Lj:;iL3t'.
Op'rloNs:
QuAxTos
11
.'.') yg'i.l. !''';'r' -y'.
: E:. . : : . . :
-
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-. -
:
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:. '
-
.
.
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:
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.
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:
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.
.
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.
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-. . . . .
, ..
-
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'E .
-
. .
--
. .................
:' ' '
;. .
k illi!!t!tyyty '
E E.
.Binomial .
......;...;...,z
.
E
i
;:.
:
-
.
:
-
-
-
.
:' :
.
y-
-
.
jy...j -
-- -.. . -. .
. . .
.
721
and that for te exchange rate is
4* )' ,'j' :'':' '17''17( ' trjf '!'f' 7* E(E ;'p' E .r' p'. @' )' 4* l'''lqr'' 71* i'iif (' (E(E(()!EE!E(Ei(E(qTE : !pi)i ( ; ( i'''(E.EE (.'.E(iEE F(E.Eji(EE... i'(..'E.E'.'E'...'('E.'.'... (((E 17r'('' jqf j)'r.r:' y' )'kk;., ;'(' jlf (r7l' ((' ($: jtf ''jyd ?' 4* rqf .q . .;.''.;.((''.'('(' . (.i'j y' t-ft-ftl'-yy). )f j'(y' t')'q' tyryyyyf )'. k',j. ..' jjd lllif 1IIiEiqq'. t'. )' r'l'f . ;fEE; .E..'..''. .(.. . y.. E.i. . .j.;.. . yyy.. '.g(r.Ey p. y ( qri E. .. EiE ( (;i(ii 'i'lr:iriik.y. E. E'. . ( (! . ( y.( r )- yg(.. . y. y y .k ..jj.(yyy.jy '(.j!(!ti . -j''Iii-L:LIII!' j qkj. j';.....(. yy y y . yj j y.. ty. jj EEi . .. j j.yj . j.i .. ).tt;jilp;,.. jy. . . ! ---E 1i .Ei.E .). : .. jjjj; ,rt;. y.. .,..qy;.q.... (q q .y '.. .. '. . r Ii'!!:pili: ..... .. .,.,. y.:( ;j--. . r . . ..... . .., . .. . . . . .. . ...' ,'
%
#0,l (1/1)
trees for the dollar and the Nikkei index from the perspective of a yen-based investor. Both are forward trees constructed using the parameters in Table 22.4. The risk-neutral probabilities of up moves are 0.4750 in the dollar tree and 0.4626 in the Nikkei tree.
=
Y100d (0m-0.o8)x l
We can also compute the folavard prices on both tree we have Binomial
Binomial
Tree for the Dollar
Tree for thr Niltkei ,
:23,706.10
$106.184 y2(),o(o
$1(0
Y17,561.91
$86.936
F(),I
'
(1/1)
=
0.4750 x $106.184+
(1
-
v96 (j,79
=
.
ees as expected values. For the dollar 0.4750) x
$86.936 $96.079 =
For the'Nikkei tree we have6 Fa,I
(:)
=
70.4626
x
Y23,706.10 + (1
-
0.4626)
x 117,561.91
=
720,404.03 $
'I'he Dollar Perspective Now we consider yen and Nikkei investments f'rom the perspective of a dollar-based investor. yen fonvard price is given by rfhe
Exchange rate forward ($JV): Fo r(a')
The Yen Petspectiv The yen-based investor is interested in the yn price of $1 and, hence, faces an eyhange rate of 1/-v()= 100V$.Because the Nilckeiindex and the yen price of a dollar are both denominated in yen, we use the usual formulas to find fonvard prices fpr the yen and Nikkei. For the Nikkei, we have Nilkei forward (Y): Fo w(:)
Qzerf-&otl-
=
,
For the exchpge rate, the dollar-denominated forward prie is
(22 22) *
interest rate is the yield on dollars, so the
Exchange rate forward (%$): Fo
--erl'-r
w(1/-Y)
#'
*
x
*
=
1
)w
< . ..
-,'tr (Tl
These will be the forward prices observed in Japan. A yen-based investoi would construct binomial trees for the yen and Nikkei in the usual fashion. Figure 22.3 depicts trees for' the dollar and Nikkei. The nodes on the dollar tree are constructed as (1/-Yz')
=
and on the Nikkei tree as
Ur Example
22.7
=
Quer.-sct'zko'cn
Given the parameters in Table 22.4, the l-year yen-denominated
(:1224) .
However, from the dollar perspective, the forward price of the Nikkei is not so straightAs discussed above, any Nikkei investment entails a combination of currency and dollar-based investor buys e .scv un y(s ojtj)e xj.kkei and holds it for F years. The actual steps in this transaction are as foltows:
index risk. To se why, suppose a
1
Exchange :oa:()e-d(!W the index). e-QT
-vf
doars
into yen
(this is enough dollars to buy
units of
units of the Nikkei index and hold for F periods.
3. Dividends are paid continuously over time and reinvested in the index; after T years e&QT shares. we have an additional 4. After T years sell the index and convert back into dollars. The time-F viue of the investment, denominated in dollars, is F(F)
(1/ab)d(r/.-rlrclaxff
xoer-rfb-
fonvard.
2. Buy
(22.23)
=
=
xvov
The payoff is a combination of yen and Niklcei lisk; we will call this investment the currencptranslated index. Here is a point tat is crtlcial for understanding what follows: From the perspective of a dollar-based nve-ftpl; the dollar-trallslated price ofa asset, Yv, isjust like L-ln)? other #t?//t7r-#v?t7??/f?)5l?# asset. However, yw-#d??t7??2?7t7lc# of is plice dollar-based investor, because there is no simple not for the asset an a Qv lisk. risk of without also bearing obtain the to currency Q way
forward price for the Nilkei index is F0,l (U)
=
1 V20,000/ (0.04-0.()2)xc:Favpozm4 ()3 ,
.
6The calculation shown here uses rounded numbers and therefore does not exactly equal Y20,404.03.
.
722
k
ExoTlc
QuANTos
Op-rlolqs: 11
If you are not convinced that Yv really is like any other dollar-denominated asset, consider the following thought experiment. Suppose you leftrn of a new stock, traded The price on a U.S. exchange, called ttA.s Amel-ican as Apple Pie, lnc.'' IAAAPII. investigate the stock decide stock. You to like other domestic is in dollars, just any American Depositary actually is the surjrise, lenrn company that an and, to your you Receipt (ADR) for the Nilkei index.? The sole asset of this company is shares of the Nilkei, translated into dollars at the current exchange rate, and held in trtlst. The value at any time is i'; = xt Qt and it has a dividend yield of 8 j This ADR trades just like a dollar-denominated stock, because it is a dollar-denommated stock. Had you not investigated, you would never have lnown that you were holding a cun-ency-translated yen-denominated stock. This thought expeliment is important because it tells us that, for a dollar-based investor, the forward price for i' is given by ) .
1') F0.w(
yctr-dt'lr
=
(22 25) .
Since we can trade shares of AAAPI, we can undertake arbitrage if the folavard price is anything other than equation (22.25). Similarly, the prepaid fonvard price on the cun'ency-translated index is Fo# (l') w
=
-
vtlptlc-t?r
(22 26) .
In order to obtain Nikkei risk without currency Fe need to combine the dollarlnmitively, we want to invest position with forward contracts. in Nikkei translated yen a risk. exchange Let V) w(F: a'; ; F) represent hedge and rate 1/ eontracts to in use currency of the Nilkei. The boundary dollar value the maturity, of claim that, price at the pays a security is this for condition
.:
This giveslo
# 1.20(1/1)
Vv,vLYv xv ; ,
r)
Fz -'ltr :!-
1
z)(w-,)
Ab
1 dt
--Ldx
'(#x))(#F
-
-
E (#F))
=
pso.q + s
Lemma, tat
:t
(22.28)
hen'ce, when we compute te forward ln apply ing Proposition 21.1, we have b with $c replace bc price for Q we (-1)(p'c c + -1.,
=
.2).
-
Putting thls all together, using Proposition 21.1 and with J r ,
(
I'J ,
xt - F) ,
1
xt
=
1 we have
e(r-J(?-p.(r:-x2)(F-r)
2)(r-?)
.-erf-r''s
=
.k
Qt x #
(22.29)
r.f-&o-pscrcjlr-)
= :,c The dollar-denominated fprward price for the Nikk'ei index is the snme as the yendenominated prepaid quanto index fomard plice, With a covarimlce correction. rfhe
(22.27)
where is an arbitrary exchange rate determined in advance. The boundary cpndition says that we will receive Yv, the dollar-translated Nikkei, convertit to yen by multiplying (Since the only purpose of by l/-tr, and then convert. back t dollars at the rate nd is arbitrary the value is to convert yen to dollars, we can set it to 1. ln practice Prior rate.) exchange to time F, the parket-maler the contract can call for any fixed -i
-?.
.
7In simplest terms, an ADR is a claim to a trust containing a foreign stock- ADRS are a common means for investors in one country to buy a stock trading in another countly SWe saw n Chapter 20 that 'if S and Q are asset prices, then the fonvard price for SQ containd a covariance term. You may be wondering why there is no such covariance term in equation (22.26). ln general, if an asset with price sb is traded and can be held by investors, then its fonvard price is s'actr-dlr This is a no-arbitrage result. ln the discussion in Chapter 20, we assume that S and Q are bolh prices of assets that can be held directly by inveslors. ln that case SQ does not represent the price of an asset. ln the discussion here, a-Qis the price of an asset that can be held directly by investors, but Q cannot be held directly. In general, S, Q, and SQ cannot all simultaneously be the prices of traded assets.
'-erl-r'
=
We also need to know the covariance between x and Y. Using It's covariance is
,
.
723
hedges the value of the security using both the dollar-eanslated Nikkei and exchange rate coneacts. Equation (22.27)makes it clear why quantosare said to be derivatives that depend on the ratio or product of two assets.g We can price this cotract by using Propbsition 21.1. That proposition implies that we obtain the fonvard price for Ypjxp by multiplying the fonvard price for i' by the forward jrice for 1/x, and adjusting the dividend yield to take account of the covariance between and F. To apply Proposition 21.1 we need to know the forward price of 1/x for dollarinvestors. We can obtain tlle forward price for 1/.x using Proposition 20.3. #c?ztp?/lfntzrc#
lisk,
=
%
f OI'War d PIice
is tlms p F(j,z' (Qj =
U0dfry-c-pcs-rjp
(gg go .
The role of the covariance term in equadon (22.29) is inmidve. Consider an onverts yen back to dolinvestor who buys the cash Nlkkei index and ultilately lars. Suppose the index and the exchange rate (measuredin dollars/yen) are positively ''
'
' '
'.
.
91 the market-maker hedges a claim with value / using and F, the claim value must satisfy the mulitvariate Black-scholes equation. ln this case, the equation is .)7
l0In this calculauon we start with the fonvard price for f'romthe dollar perspective and convert it to the fonvard price for l/-r. As a result, the fonvard price for 1/-r given here is different from that given by equaton (22.23),which is the appropriate fonvard price given a yen perspective. -:'
i
k
724
ExoTlc
Op-rlolqs:
QuAxTos
11
correlated. When the index does well, there are many yen to exchange. If p is positive, on average the exchange rate is favorable when there are many yen to exchange. When the index does poorly, there are fewer yen to exchange so the decline in the exchange rate does not matter as much. Thus, other things equal, the positive correlation systematically benests the unhedged investment in the Nilkei relative to a contract with a sxed exchange rate. Consequently, if the exchange rate is fixed, as in a quanto contract, the price for the index settling in dollars will be lower in order to compensate the buyer for the loss of benescial correlation between the index and exchange rate. )) E-xample 22.8 forward price is (1)j!) )i) j 1)
.
Using equation
.j((jj
-'r
wand Q are lognormal,
jjjj
''!jj (.( The .
=
-
her-rf?
jjj.js-jyjjyj
0.0 1Sj''e
=
.p,
('(
E
: ..
.
i
F t) I
(( Finally, )( t'.( E.
(jyjE..j
),,.)
'tEEE
yjyj
@.t lt'j t)
't This
-
.
lsij
=
-v()
=
QIt
e
(r-ryl-fxa//k-zl
(Lvjga)a) .
U0t?r-scllt-vcrtxlliz
=
(ar,jajjn) .
and Z2 binomially, so that However, we want to induce corfelation between Zj and Z2. We can create correlation by using the Cholesky decomposition discussed in Chaptef 19. Begin by rewriting equation (22.31)using equation (19.17):
ln the siandard binomial model, we simply approximate =
ZI
+l.
-
v0 :()e(r-'(?)?=
using equation
=
jyjj j jyyjyyyys .
Nikkei is
forward pr ic for the currency-translated
1',' '(
they evolve like this:
'
F 0, I
()
(22.29)and the information in Table 22.4, the yen
725
same answer either way. Here we will model the yen and Nikkei. Problem 22.17 asks you to jointly model the yen and dollar-translated Nikkei. The basic idea underlying the joint binoluial model for and Q is as follows. lf
Zi
..' ('
%
,
-
:=:
-
(22.29),the quanto forward price is
F(),l ($200,0.01$/ y
)
=
=
is lower tlaan the yen-denominated -'
()(oe(0.04-0.02-0.2x0.l x0.15)
vpp ,
=
Qh
=
.Y0d
(:u aaa) .
x/i/l
U0dry-cjh-bo'c
ZI
=
E:1
I -/J2)
p-ez
(g)g,jggj;; -
-
-
Y20, 342.91 in Example
-
k.
u 22.7.
-Yc
=
w't.
=
Jr D
For eacla yen move, tlaereare two
A Binomial Model for the Dollar-Denominated lnvestor
Vzt
=
As another way to understand quanto pricing, we can construct a binomial tree that simultaneously models the currency-anslated index and the exchange rate. In addition to this prticular applicatipn of tWo-variable binomial trees, some options have prices that depend on two state vriables. In Figure 22.3 we costructed separate binomial trees for the yen-based investor. For the dollar-based investor we need to construct a tree that takes account of the con'elation between the Nikkei and the yen. We can do so by hrst modeling the behavior of the yen in the usual way, and then, collditioltal 1f/p(?n the J'd?l, model the movement in the Nildei.l l Since for each yen move there are two Nikkei moves, the tree will have four binomial nodes. We will denote these as (I/lf 1/#, #1/, ##), with thesrstletter denoting the yen move and the second the Nikkei move. We have a choice of constructing ajoint tree for the yen and Nilkei or the yen and dollar-translated Nikkei, but we obtain the ,
l lrhis is similar to the two-variable binomial model in Rubinsten (1994).See also Boyle et al. ( 1989) for a procedure to generate a-asset binomial trees.
-
-
-
Nikkei forward price of 120,404
(r-ry)/,+,x4kl
1 p2c2. By construction, this ZI and Za have and Z2 = p/ l + correlationp. Now we construct the binomial tree by setting cl = +1 (theexchange There are four rat moves utj or down) and ez = +1 (ihe index moves up or down). :i= :z = = = 1', l), C 1), B 1', will label which A 2 (/1 (/l we possibl'utomes, z:2 :i= (E = = and E::t 1). D l 62 1 1), I ; (6 1 ; Fr a dollar-basd investoc the jossible yen rovs are
Thus,
$212 367 v2o o0()e(0.0s-0.02)
() 01$/v x
Xh
QB
=
QC
=
QD
=
B
-Y()e
=
tr-ryl/l--xxff
=
(r-?.y)-,x/J' -Y0d
=
=
lt
(22.33a)
vY0d
(22.33b)
.10
xilckeimoves..
Qze(ry.-J(?)/7+(reW(#+ l-p2) Q0c(ry-5:)+(r(?W(pry-slll-ovqxll-p,. 1-p2) U0d l-/a2) ry-&cjlt-o.cxll-pU0d l
-p2)
=
=
=
=
g 0z Q0 p U0c Q0 p
(jg g4a) .
tgg.jxgyj); f
(p,p,g4cl .
tgu.g4tp
probabilitie: associated with the nodes. Finally, we have to determine th As in Chapter 10, the risk-neutral probability fr an u!j move of the currency is lisk-neutral
p
=
e
r
J
tt
-
-
#
t'J
(2:,.35)
where If and# are impliedby equation (22.33a)and (22.331$.The risk-neutralprobability for an up move in t.he currency is 0.4750.
Recall that the risk-neutrtl probability arises from the requirement that an investment in the asset earn the risk-free rate. Speciscally, for the cun-ency, we consider an investment in the yen-denominated risk-free asset, hedged to remove currency risk when the investment is t'urnedback into dollars. This investment earns the dotlar-denominated risk-free return if the probability of an up move is given by equation (22.35).
%.Exo-rlc
726
CURRENCY-LI
11
OpTloNs:
'Fhe forward price for the urrency-lanslated
i ' i?f j''k' Ef (lt:jsjyjjyjsjyjyjjjjjjtylyl'. t'jlf (' j,,j!.jj,r'!l))ty)jt!'.. jiitfhppjrkjgr.ty.jjtjtj'jt!!yk. .-,111111::k*.* y' E E'E i 7': ;'(Eq! E :Eq! q' (' i'q?'.qL?1Ib'6t'11L'LLLLiqqkj6k. (r'tji k' r' 92* 774:* ,!!!F.1I1Ii,t', jtr...'t):kyj(jjj'.' grd Ep;: ( E q' E ;'pl'rqjqqiji'r:qi Eqf qt' jjjjyrjjtj('. lr' (q;7r71r* ('qq' ,.,..tb;;Ljjtjj:'.j)jjj. 11117-11112)* !1I1i....IIp'' .;' )' j'y)'. '(.i.'.. .i(;:.'... t:..E (Ei ! jj'i!). EEp' @pi (;(li jii!qjjf r' rf jr:' j')':' y' t(;'@(i 'j ' t.')t.' jjf )t'jyjy'! j'('! 'y' )j)'. . q! . .. . ;'.7t. ttf @(. (tjjjyj)jtj'. tq'jy'lk t')'.('('t'. (E E E EiE @. ! !y . j jj( r. .i rj j.i.'(..q.( .E(q ;r.ty. yyjj. j: ..jy.j(:),;.kjyj.E.;;.gyj).qj;.jt;yj y yjyy. y )E. i ! .jj!. y).. ...t;, ! t'...-.......1111::::kE q 11111* jy r yt.(jr 'Tpr.. ljjy .33..31. y( '. .,', . y ..' .. k . ':77i??q'J/i' .yjjjjjjj'))r.f ..' .'-,-3:36633333*1 -;'' --.'(' -r' ,,,jkjj,,;' r' ,,,,,,,k,,-'. s'jj ;':',.....'. '')''' t'j' didr'. ;)t'. ;'$8,(44/;*.1t12*:@t64* r.'' F' y:)' yyk'..y)jjj'. )' :'-'2.* 6* ;'!
' i E' : . :
'E !: ; : .
'
.
'
..,;!
-
:
5.
:
::
:
'
'
. .
:.
.
::
.
'
:
-k,j,,!,,,;..
E
.2,!4447;14444(..
.
:
-
. .
.
' :.
:. . .
: :.
:
. :::
:,,,,,j;,,15
'
'''. '-'.
?'' ''
'
-
..-...-.......,-,-...
.
.-
.... q..
.
., .
:. E- i
'
.
:.
., . .
..
.
Fa,1 @:)
Risk-Nual
Time 1 Nikkei, Q1
Jttl
%23,706.10
$272.68
0.2194
Y18,647.88 722,325.56
$214.50 $210.25
0.2556 0.2,425
717,561.91
$165.39
0.2825
Fo
We need a similar argument for the Nikkei. Since we cnnnot oFn the Nikkei index without bearlng currency risk, we model an investment in the dollar-translated Nikkei. Let #* denote tlle probability of arl up move in te Nlkt-ei, condidonal on te move in the wesnent earn the dollar-denominated yen. Werequire that the dollar-anslatedNikkei lisk-free rate. This gives us zl
pp* + xu
Qs#(1
kdPc(1
#*) +
-
1
xdon
+
ptp'
-
# )(1
-
-
#*)
.X0
=
Qfter-8osh
(22.
g6)
Solving for p* gives x,, psp xoptyetr-dt?l'' -
p
*=
#Af(Qz er-sqlh
QBt
-
-
+
jjsp
-
=
pjxdoc
(1 dpLj -
xdvstl
-
-
-
p)
Po)
(22.37)
p)
-
D) '#lf (A B) + (1 pjdc Tls expression is a generalization of the ope-variable formula for a risk-neual probability, taldng account of the two up and two down states for Q. Figure 22.4 depicts te blomial tree constructed using equations (22.33)and (22.34), and probabilties of each node constructed using equations (22.35)and (22.37). 'l'he quanto forward price can be constructed as the expectation E (FI/.x1). -
-
('
-
Using Figure 22.4 we carl compute forward prices for the yen, the j Example 22.9 1' dotlar-translated Nikkei, and the quanto Nikkei. risk-neutral probabil f an up yj fomard price is 0.4750. the yen is yen ytqmove in 'l'he
'l'he
)!'.') .
tlt) F(),1@)
=
$212.367
Finally, the quanto forward price is
The binomial process for the dollar/yen exchange rate (x)and the Nikkei (Q).The Iast two columns contain the value of the c'urrency-translated Nikkei and the risk-neutral probability of each node, computed using equations (22.35)and (22.37).
Q
0.2194 x $272.68+ 0.2556 x $214,50 + 0.2425 x $210.25+ 0.2825 x $165.39
=
I
(:)
=
0.009418$/Y
AI
Nikkei is
0.2194 x
$272.68 $214.50 + 0.2556 x 0.011503$/Y 0.011503$/Y
+0.2425 x
$210.25 $165.39 + 0.2825 x 0.009418$/1 0.009418$/7
'
Y20,000 0.01$/Y
=
=
Probability
AIQI
0.011503$/Y
Qo
727
'
Yen, xl
=
$
-
trq!)--!1!F5q...t.-..,... ,. '''-?':''
N KED OPTIONS
0.4750 x 0.011503$/Y + (1
-
.. 0.4750) x 0.009418$/Y
=
0.010408$/Y
720,342.91
=
All of te prices computed f'rom the tree match those in Example 22.8.
'l'he eee in Figure 22.4 can be extended to muldple periods. Rubinstein (1994) shows that in general, wit.h n steps, there are (n + 1)2 nodes; for examplet WithtWo steps there are rtine nodes. To see why. if we add another binomial period to the tree, DD). The order there are 42 16 combinations of the up-down moves (AA, AB, BA. This equivalence e nziates of the moves is irrelevant, so, for example, AB CD. Because n x (n 1) 6 nodes. leaving 10. Further, f'rom equauon (22.34),AB 1)2 9 unique nodes. this leaves + 2, (n n =
.
.
.
,
=
-
=
=
=
=
22.5 CURRENCY-LINKED
OPTIONS
There are several common ways to constnlct options on foreign assets, for which the retllrn has an exchange rate component.'z 'I'he different variants permit investors to assume different amounts of cuaency and equity risk. ln this section we exnmine fotlr variants and their pricing fonnulas. We will continue to use te notation and numbers from Table 22.4. recall theresultfromchapter Before wediscuss pndl'culrcuaency-liAedoptions, 12, equation (12.5),and Chapter 14, equation (14.16),that an option can be pdced using only the prepaid foeard prices for the underlying asset and strike asset, and the reladve volatility of the >0.13 The intuion forthis result is that a market-maker could hedge an option position using the two prepaid forwards, neither of which, by definidon, makes any payouts. In the discussions to follow we will use this result to sifnplify the valuadon of seemingly complex options.
section draws heavily from Reiner (1992),in palticular adopting lzel'11is kinds of options. different
Reiner's terminology for the
l3To convince yourself of this. note that
sscallt,s
K.o.,r,
r.&) sscallt-e-dr Ke-rv
This equality will hold for any inputs you try.
-
.
()
w ())
k. Exo-rlc
728
'kii!::d .1::t111.* dl::::''d )';' '..(' 1* t'E j'j' (y'. ;-'E $' ;1iiip:4ki:'. t'kIiik)lIkI''',' k' y'j-E j(' q' j')' j';'41::EE:11* 1* ltii!;;d 'Iii:!)' ;j' ,(' j't' )' ytf jyjd jjd lllE''''llizjzr:d il!:Ed .1::E::.,* I!E!l;' ll:q')lkd ',1..:.1:* y' ;')'. yj'. ,'yy'. (' :'. t'i;f )y'. j'.-)'.' yj' -i;isiiI,' ''J'q' jyd )4j4* qqd j((((' kpjjii!l;'d ry.tjjrf qs'kf ?', 9it'. g' ;-'. t''1111r--111.* ' ijf j!d (' r' .11!:2* t't' j)',' jyf )' y' q'q t',j'.y'f' j''.yyj)'. iii1k'. y':'1iIi!!)'. ''('j;jk)d tyd ( '!.( ; ) 'jtt.yy ('j k ' ; ( 'j . ' (111* y' (llir::l..d (1(44::..* (' j)'j;-.);,(L))jf'. y'@j' ..' @i1iik.idiiiik-iIii''. .t' )j'. IIii;-kl22.'. yt)'. p'. 444*4j4* yttj':,d (yj' .''g' ((('jjjy' (44*. jj('. jtd kj'jjg ( ' yji'j j ' jg(' (( ' . ( y(' (;'jj j'ij; y'. tjy'. )' @' t'. ((' (jj('. y(' tjd (y' 44* k' i ('' 1iki,kk.I'q jg'j j j jj (;j(j'.'' jjj i '('' I. ( j 't.' ''. jjj4(j tjjy'. ty'. iiii?k:;'. ' (' yq('jlg'...j't-q ('(''! ( ' E (. (jjE. jE (j. (jjk((. jjtj 'jyg.j'.(j.!-... t'.y t't' '.y( iiIi;kkk.' . ''j. (' j. y y( .ji-...'...y. . yjj j(j ( j.'' (jE.'.'..;. Ey.E. ... . . .. . j.y .(.r.. .j.(. E.. (. y. r.(y'jjy gy''kyj . :. )..jj .y t'y'g'4* .'((jy' 4yj (' ;jy' ((. .t.q ''j'' (g.. .. (.j. .( .... :. ... .. .k.)).. i1ilkii.;iiiI. 'r. @. . .! E ) i.(;(.jE.lj.jtjqj,.j.y'j.'Ej;j... E.....jjE i ; (.j(t.y...j:jj'...ky'qg.'.yj ' .. ...i1Iii(k:. ;j..... y. ;y...(....y.y.... y.. y.jk q. q. .. ......... !... q;.y ...:. .q...j q.... qgq. ..y(:.yj, rk yjjjl j));y.. ,.j y jj '...ggj.. g.. ,y yq;y.y(. gy( ,(.j..y);.y.''.iIiI2:i!i!ii. ....(yj-yy ;.y y(!.. y... tjyyti qy (y..yj... yy.y .. ;..tt.. '. ( ' y . .. r ' ; q. ) '- . . (.. l!iikkil g(gt. . .r. g . . . .. r..g yj,ly j)j-.. y.,.; g '. j . ) . .. ,.r.y y..y ..y .. ; j y , . r y 'l.y k j . . . . y j . ., . . . .( ( .. .( . . j,ijy . . . . ..!!!!! j y . . . r j . . q . . . . . .y.j.).. ( ( . . '(-' yy . . . . . . q . ..... .. . . .. . . ' .( . . .. r' . i .kj.t)).. . . . .. ! . ... .' . .. . . r. ' ' .. . .. . . . . . . . .. ( . .... ..( '.. . . . ...( . . E.. . .... . . . E( .. . E ' . ( . .. ( ( . ( . '. ' ' ' . ' y . jy ; . (. j ' ( ;' . y j '. y y. q j ' ( . j'. g ( . .
.' :''F' ':'-' -'( 'kiii5iik:'' ' ;''C' :'jyd :jj' ;'-'-'fl:i,r:;d -' --t' -'. -y'r --' -,' (' y' !' -' ''. ''' ':'
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' i kjE '' '' E
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Using the parameters in Table 22.4 and assuming strike price of ' ' . . . . wepdceiecebyusingeBlak-scholesfo=ulaudseing S = Y20,000 t)v19,500, l! volatility in Nikkei index price), K = 719,500, o'c = 0.15 ttheNikkei )i1 .q . tthecurrent = = = yield theNikkei). dividend and ?.y 1, We obtain 0.04, r n itt'yen), sc 0.02 tthe $tk acall price of BsCa11(Y20,000;719,500., = dollar 0.15.,0.04., 1', 0.02) Y1632.16.
q Example 22.10 jij
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PZce iS $16 32 .
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-. . - . .. .- . . . . y
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-
Foreign Equity Call Struck
Suppose we have a call opdon to buy the Nikkei but we denominate the strike, K, in dollars. If we exercise the option, we pay K dollars to acquire the Nikkei, which is worth xp Qp. Thus, at expiration, the option is wrth
FtxwQr, F)
(
j. .. . j . . j. ..
. .. . y..g.y..... y... ...
Domestic Cllrrency
l
:
maxto, xvov
=
-
K4
In orderto price tllis option, recognizethat F(r) xp Qv,the currency-eanslatedindex, prepzd forward price for the currency-anslated is priced lile any domesic asset. Qze-&QT. The prepaid forward price for the strike is index is, from equation (22.26),n:tl Ke-rT. The value of the option will depend upon the distribution of xv Qw',thus, the volatility that enters te opon pricing fonnula is that of the currency-anslated index. =
'rhe
Foreign Equity Call Siruck in Foreign Currency If we want to speculate on a foreign index, one possibility is to buy an option completely denominated in a foreign currency. The value of this option at expiration is
Z(Vw,F)
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Thus, we price this option by using the Black-scholes formula with inputs appropriate for the agset being denominated in a different cuaency.
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rqKED Op-rloxs
CURRENCY-LI
11
OI>'rfoNs:
=
maX(0,
Qr
-
where Kf denotes the strike denominated in the foreign
#yl
The voltility
of xt Qt is
v
+ sl + kpo'cs
Using this volatility and the prepaid forward prices we have
C
=
artlpod
-sT
Ndj)
-rt
-
ln @0 Qfje-bTje-rK)
currency.
we might have a l-year call option to buy the Nilkei index by paying 119,500. An investor based in the foreign currency would use this kind of option; thus, it can be priced using the Black-scholes formulafrom the perspective of the foreign and the Nikkei volatility currency. Only yen inputs-the yen-denomipated intezest rate dollar price The pricing the formula. can be obtained by and dividend yield--enter rate. exchange the current converting the option price at
&2
=
As an example,
#rt
=
d1
-
tl
Xt
e
sxt# :t )
(22.39)
+ 1p2f :?
vJ't
You can interpret tllis formula in terms of prepaid forward prices or as the Black-scholes formula wit.h ab Po as the stock price, &Q as the dividend yield, w as the volatility, the domestic interest rate ?- as the risk-free rate, and K as te strike price.
i
% Exo-rlc
730
Ol7TloNs:
11
)' ) Black-scholes fonnula. We obtain
%' :'
j'q'. 14*
1q Example
ri
22.11
Using the parameters in Table 22.4, the volatility is
.j.j'.; (
(
E t'El
:
,tt ).. : '. ijrqand q
p
assuming
2
0.15a + 0.1 + (2 x 0.2
=
-sv
t,
lt1 jjj
=
:
x 0.15 x 0.1)
$195,we price the call
a strike price of
)(j kjt'y B SCall (.'t% Qje
0.1962
=
.E;
using prepaid forwards as
() g, BSCall(0.01$/Y x Y20,000c-0'02 $195 x e-0.08 () 1962 0, 1, ()) Ke
p
,
,
,
5
@
'
3
(p)
(),..r
=
$24..:719
k
j.
t) l!j Problem
Suppose we have a foreign equity call denominatd in the foreign currency, but with the option proceeds to be repatriated at a predetermined exchange rate. This is a quanto option, analogous to the quanto fonvard, with the value of the option translated into dollars at a sxedexcliange rate. Let J represent this rte. The payoff to this ption with stlike price K.f (denominatedin the foreign currency) is =
x maxto,
.
= maxto, kov
QT -
-
K.f)
kKy)
Once again we an construct the pricing formula by thinking in terms of fonvard prices
for the underlying and strike assels. From a dollar perspective, the underlying asset, .#Qv,is a quanto index investment. The sile asset is simply a fixed number of dollars, 1. Since is just a scale factor, we can set translated at the rate -i
.i
=
v.
yhe volatility that affects the value of the option Because the exchange rate is sxed, that of the foreigp-cuaencyvenoenatedforeign indx, o'c. We can obtain the pricing is by using the prepaid forwards for the underlying and strike asset: formula C =
FP 0,r 1n(FP
.dt
=
dz d I FvPpQ)is given =
-
(:)1(:/1 ) 0,? o.
-
e-rTK
f #(#2)
,
o' Q
O
r
,
,
,
0)
=
() (uozxlxj
x().15+().os-a.()m
'
,
'
(22.40)
Q)!e-rTK 1 j + 1(y2 F 2 q
(; ijyyv x vjq, 5(se-().08 .
,
0. 15, 0,. 1 0) ,
y
Foreign Exchange Call
. $15.3 187
=
22.6 asks you to verify that you obtaih the same answer with . an appropriate choice of the dividend yield.
ltljtjjy. ! . .. underlying asset and
Equity-toked Fixed Exchange Rate Foreign Equity Call
V(P?',F)
Kye-''T ,
(
o
,
Bscall(FP
tj) (ti BSCMl1(0.0l$/Y xY20, 000 x eljj!j. (ytt
-rp
,
op-rloxs k.
cuaaExcy-t-lxxEo
-p
.
Qo as the
%
.
lf we invest in a foreign asset, we might like to insure against 1ow exchange rates when we convert back to the domestic currency, while still having the ability to profit from favorable exchange rates. Buying an exchange rate put is insucient because the quantity of curreny to be exchanged is vncertain.What we want is an option that guarantees a minimum exchange rate bvhen ;$?' convert l asset VJJIIC bqck to the fftppid-rccll?-re/?cll.Such an option must therefore protect a variable quantity of currency. '..t-c/?p??,gE' This is an eqltitplinkedforeigl option, which is another example of a quanto option. Let K be the minimum exchange rate. Thetl the payoff to such an insured position would be
Qvxv+
QT
maxto, K
.Yr)
-
=
QvK +
Qpmaxto, xv maxto, Qpxp
K)
-
(22.41) QvK) = QvK + Theexpression to the leftof the equal sign in equation (22.41)is te unprotectedcun-encytranslated Nikkei investment plus Qv exchange rate puts with strike K. The equivalent expression on the right is a quanto investment with the Exed exchange rate equal to K, plus Qv exchange rate calls. Either way, the protection entails receiving the payoff to a random number of options. All cash flows are denominated in the home currency. There are at least two ways to value the payoff in equation (22.41):(1) By using Proposition 21 1 and (2) by using the prepaid forward approach. Using Proposition 21.1, we value Qv x maxto, xv K) as a currency call with a change of numeraire. We will pursue that approach', Problem 22.7 asks you to derive the same formula using prepaid forward plices The forward plice for Qvis the quanto folavard price, equatioh (22.29).Th put is standard currency call denominated in the home currency. Since it is a curlreny option, a the dividend yield is the foreign interest rate. To change the numeraire we subtract po's, the covariance between Q and -Y, from the dividend yield, ry. Thus, we price the payoff K) by multiplying Qjerf-bo-pn'T the quanto forward price, times Qv x maxto, te Black-scholes currency call formula with rg replaed by ?'y po's. This gives -
'
.
&c Wf
-
c ,/F
.
in equation (22.30). Note that all values are dollarThe formula for denominated since implicitly multiplies all prices. By substittlting for FvP:, equation (22.40) is the Black-scholes formula with Qtlas the stock price, q + pso.c + r r.. as the dividend yield, the domestic interest rate r as the risk-free rate, K.f as the stlike, and o' q as the volatitity. .
-
-vw
-
-
c
))1Example
22.12
Using the parameters in Table 22.4 and assuming a strike price of 0.0 l$/Y, we price the call by using the t' Y19,500 with a fixed exchange rate of -:i
=
u:z
=
goelr; .0
-p(r.) -(j?
-J Q F
O()c
F
g
X(#1 )
ye-lj.-poj'T
-
N tt./j ; .e-rT
-(r+Jc+pcJ-ry)Fx(#
K O()d
S' y 2
)
tyzjj
(22-42)
k: Exo-rlc
732
Op-rloxs:
OTHER
11
ln@a/#) + (r
1Y IL:::=
ry + po's +
-
First, consider the ptice of an option with
0.5,2)r ''''''''''''''''''''
snc-r
s VF Tltis is the price of a call option with ab :0 as the stock price, K Qoas the sttike price, liskfree rate, 8c as the dividend yield, and s as the volatility. ?-+ 8q + po's ry as the lt is perhaps surprising that only the volatility of the exchange rate matters. This occtlrs because the underlying option is a cuaency option and the change of numeraire does not affect the volatility.
d2
#l
=
-
J*
andthe
=r
+ J
J
()2
= q +
-
c
-
2
(22.44)
(7'
2
g p o.g :
(:2:.45)
-.u
Its plice is
-&sT-t)NLj1)
=
S/QE
Jl
=
ln(&:) + (r
e-rtNLj2 )
-
J *+ 0 5.2)(r
-
F
Jz
-
-
JI
=
T
-
(pa.46) tj
-
.
(22.47)
t
-
Etitt
(22.48)
t
-
''
-'
.
l). To do tltis we apply Now we want to value the opdon with payoff Q x WCSIQ, (r2:. covariance betveen Q and S1Qis pco'p applying Proposidon 21.1. Proposition 21.1, we replace 8 with rfhus,
'l'he
-
BSCa1l($200, $195,0.10, 0.063, 1, 0.02)
=
$15.7287
%.
Dh
J* Finally, multiplying J7 by
22.6
underlying asset. We can
c
c WCS/Q,r).
by
WCSIQ,?)
.xopt)
l,j
733
c2 + pao'c
'using
t,)
$
variance is
Denote the value of this opon a stlike price The price of = = 0.01$/Y underlying is strike the l) Y20,000 the price is $200, asset x t.(4 = risk-free replaced tlae by 0.00975$/1 x v20,000 $195, rate is r + c + po's r.f = = = li () 08 + () 02 + o.2x 0.15 x 0.1 0.04 0.063, and the dividend yield is ta 0.02. ii):t. )l) The vlue of the option in equation (22.42)is
SjQ as the
OPTIONS
price tltis opion using the Black-scholes formula with the lease rat: for SjQ in place of the dividend yield and the vaziance of jnsjoj as the variance. Proposiuons 20.3 and 20.4 imply that the lease rate for SjQ is
-
j Example 22.13 Using the parnmeters in Table 22.4 and assuming 11of 0.0097$/Y, we price the call by the Black-scholes formula.
MULTIVARIATE
OTHER
MULTIVARIATE
Quantosare aparticular
kind
OPTIONS
V(S,
claim with apayoffdependent on tlle price of two assets. 'l'here are many ot her op tions for which the payof depends on two or more assets. In this section we exnmine several kinds of multivariate options that can be priced either by modifying the Black-scholes formula or by using the bivariate normal distribuion. We will also see how to price some of these options binomially. Throughout this section, we assume that the assets S and Q fotlow the processes given by equadons (22.1) and
-
-
:e(r-&)(7--J)
Q, tj
=
)f
(22.2).
pco.c
Jj
=
(7.2
)
+ &
r
=
ln(5'/:) +
&Q
-
-
Qe-sqT-t,yLdgj 8 + 0.5 F
=
sc
gives
Se -&T-t)yLd j)
z Jl
-
F
-
-
2 p )(
-
(22.49)
tj
-
t
This is the formula for an exchange option f'rom Section 14.6. 'I'he risk-fre rate is replaced y Je nd the volatility by American exchange opuons can be valuedusing a two-state variable binonzial tree, 22.4. However, it is also possible to value an American exchange option Section in as using a one-variable binorizial tree. Rubinstein (1991b)shows that a standard binomial tree can be constructed setting the volatility equal to the dividendyield equal to 8, and the risk-free rate equal to 8c. This result can also be demonsated by using argments based on Propositions 20.4 and 21.1. .
Exchange Options We saw in Section 14.6 that exchange options, in which the strike price is the price of a risky asset. can be priced with a snple modification of the Black-scholes formula. Here we will use a change of numeraire to see why this is so. At mattlrity, an exchange option with price Vst Qt, t) pays
,
,
VCST,
Pr, F)
=
maxtur
-
PF,
= Qp x mzxspjop
0) -
1, 0)
(22.43)
This payoff is like receiving a random number, Qp, of options in which the underlying asset is Svjov and the strike price is $1. We can pricq tis option by applying Propositions 20.4 and 21.1.
Options on the Best of Two Assets Suppose an investor allocates a portfolio to both the S&P index and the currencytranslated Nikkei. Mocating the portfolio to the index that the investor believes will obtain the highest remrn is called market-flming. A petfect market-timer would invest in the S&P when it outperformed the Nikkei and the Nilkei when it outpedbnned the
734
% Exo-rlc
Ol7-rloNs:
11
OTH ER
S&P. What is the value of being able to infallibly select the portfolio with the stlperior Pedbrmance? We can answer this question by valuing an option giving us the greater of the two returns. This option would have the payoff maxts'w,Qv),where S is the S&P index and the Q the Nikkei index. Note that
maxts,
Qv)
Qv +
=
maxts'w
N
then
a,
z2 < b', p)
b; p)
(22.50)
where zl and 7:2 are standard normal random variables with correlation coeficient p. You may recall that we used the bivqriate normal distlibutionin Chapter 14 to value compound options. The formula for a rainbow option is Rainbowcallts',
qNdsc)
pc-Jc(F-?)
LNdcsj
= +
Q, A-, t7', s, p, &, tt?. F .c-J(F-J)
t)
-
(u), dsc, (po': c'l/J.jj (y())/j) NNL-t'IL (p), dcs, tpo' pj + Ke-rT-bNNL-dzsj, NNL-dL
-
-
-
where
#1(U)
LnS/K) =
+
@ 8 + 0.5c2)(r -
o' T LnQjK)
+
=
(r
(u$')
d,
=
dk (Q) = c' Q
#2(5-) #2(V)
-
o-
T
t)
This is equal to maxtk, Rainbowcallts,
((
+
r 2
z
o' + o'c
-
-
t)
t
-
lpo'o'c
.
Q, K) K, Whichhas q, K, o-, o,c- p, s, tc, -
-
ft'q
the value r,
w
-
,)
-
Ke-rv-',
S ome op tions that seem as if they migt be valued using the fainbow option forlul, howevr, cannot be. For example, in Chajter lt we discussd the valuation of peak-load electricity plants and encountered spread options, which have the jayoff
maxto, S
-
Q
K)
-
While there are approximations for valuing such an option (seeHauj, 1998, pp. 59-61), more exact solutions reqire Monte Carlo or two-state binomial trees.
-
-
Basket Options Basket options have payoffs that depend upon the average of two or more asset prices. Basket options are frequently used in currency hedging. A multinational (11711 dealing in hedglng currencies, for example, might only about the multiple care average exchange rather than each exchange individpally. As another example, rate rate, an option on S&P off if might only the S&P outperfonns of the index the pay an averace currency' translated Nlkki and Dax (German stock) indice. With equal weights on the Nlkkei and Dax, the payoff to such an option would be -
-
t)
t
t
-
(22.51)
k.
o
8c + 0.5cn) (F m
-
T
-
t
-
cc T =
ln(:/5')
t)
-
-#:?(:),
#l(S)
=
t $Q + 0.2)(F
-
-
-
mflxtu, Q) maxEo,
NNa,
=
&+ 0.5 :.2)( r
-
this daunting formula by recognizing that, at maturity, the option You can.understand worth either be S, Q, or K. By setting t = F, you can verify that the formula must satisfes this boundary condition. The fonnula for an option that pays minlu, Q, ffl-a rainbow put-is obtained by putting a minus sign in front of each EW''lrgument in the normal and bivariate normal functions. Certain related options can be valued using the rainbow option formula.ls For example, consider an option on the maximum of two assets with the payoff
:w)
This option, called a rainbpw option, has no simple one-variable solutiop. Instead, this opi)n requires the use of the bivariate normal ltstriution.l'l The bivariate ltlipg yq normat distlibution is desned as <
ln(5'/Q)+ ((?
F
= -
Probtzl
=
Q.:
:w, 0)
-
Thus, an option on the best of two assets is the same as owning one asset plus an option to exchange that asset for the other asset. As discussed in Chapter 9, maxts'w Qv, 0) can be viewed either as a call on S with strike asset Q, or as a put on Q, with stlike asset S. An investor allocating f'unds between the S&P index apd the Nikkei index might also want to inclde cash in the comparison, so that there is a guaranteed minimum return. lf K represents this minimum return, the payoff for a perfect market-timer is
maytff, Sp,
dsc
$
M U LTI VARIATE O PTI O N5
t
''lstulz (.1982)hrst valued a rainbow option. See also Rubinstein ( 1991c) for a discussion.
'*'
-
.
*''
'
'
.
o
maxlo, Ss&p
-
0.5 x
+ 5bf,xlj sxikkei
You may be able tt? guess the problem with deliving a simple formula to value such a payoff. The arithmetic average of two indices does not follow geometric Brownian
lslkubinsten (l991c) provdes a thorough discussion of these related options, as well as discussing which options callnot be valued as rainbow options.
%.Exo-rlc
736
OpTloNs:
motion. (1nfact, if an index is art arithmetic average of stocks, the index itself does not follow geometric Brownian motion. We have been maling the common, yet inconsistent, assumption that both stocks and indices containing those stocks follow geometric
Brownian motion.) Because the payoff can depend on many random variables and there is no easy formula, Monte Cado is a natural technique for valuing basket options. Moreover, basket options provide a natural application for the control variate method to speed up Monte Carlo. A basket option based on the geometric average can be valued using Black-scholes with appropriate adjustments to the volatility and dividend yield. This price can then serve as a control variate for the more conventional basket option based on an arithmedc average.
It is fossible to build new delivative clnims by using simpler claims aj building blocls. lmportant building blocks include amor-nothing options, wllih pay eithf tash or ap asset under certain conditions. Assuming thatplices are lognormal with constant volatility, it is straightomard to value cash-or-nothing and asset-or-nothing options both with and without baniers. Cash-or-nothing claims carl be priced s discouted risk-neutral probopions. abilities, and a change of numeraire can then be used to price jjet-or-nodttg
'
These claims can be used to create, nmong other things, ordiary optins, ga# options, and barrier options. While these optiofls are straightfonvard to price, they may be quite difcult tq hedge because of discontinuities in the paypff created by the all-or-nothing
characteristic.
claims for which te payoffdepends on tlle product or quotient of two prices. They can be priced using arguments developed in Chapters 20 and 21. Quantos can be used to remove tlle currency risk f'rom an investment in a foreign stock index and thus are used in international investing. It is possible to construc i bivariate binomial trees to. price quantos. Internatlonal investors can also use currency-linked opdons to tailor their exposure to currency. The standard cuaency options can be priced using prepaid fonvards and change of numeraire. Other options, such as rainbow and basket options, have payoffs depending on two or more asset prices. Some of these options have simple priing formulas; others must be valued binomially, using Monte Carlo, or in some other way.
Quantosare
FURTHER
P Ro
11
READING
Mark Rubinstein and Elic Reiner published a series of papers on exotic options z Risk magazine in the eady 1990s. These provide a comprehensive discussion of pricing formulas on a wide variety of options. Some of the material in this chapter is based directly on those papers, which can be hard to obtain but are available on Mark Rubinstein's lngersoll (2000)also provides exnmples website (hap://--in-the-money.com).
BLEM s
k
of the use of all-or-nothing options as building blocks. An alternative approach to twostate binomial plicing is detailed in Boyle et al. (1989). If you are interested in more pricing formulas, Haug (1998)presents numerous formulas and discusses approximations when those simple foffnulas are not available.
Wilmott
(1998)also
has a comprehensive
discussion emphsizing
the use of partial
differentii equations (which,as we have seen, underlie a1l derivatives pricing). Zhang (1998) andBliys andBel1ala(1998) discuss exotic options, including many notdiscussed in this chpter. In practice, the hitting of a bnrrier is often determined on a daily or other periodic basis. Broadie et al. (1997)provide a simple correction term that makes the bnrrierpricing formulas more accurate when monitoring of the barrier is not continuous. One class of options we have not discussed is lookback options, which pay out based on the highest (orlowest) price over the life of the option. nese are discussed in Goldman et al. (1979a)and Goldman et al. (1979b)and Jtre covered in Problems 22.13 and 22.14.
PROBLEMS N.
22.1. A collect-on-delivery call (COD) costs zero initially, with the payoff at expiP if S A: K. The problem in valuing ration being 0 if S < K, and S K the option is to determine #, the amount the option-holder pays if the option is in-the-money at expiration. The premium P is determined once and for :11 when 20%, F 1 5%, tr the opon is created. Let S $100, K $100,r t year, and & 0. -
=
-
=
=
=
=
=
=
a. Value a European COD call option with the above inputs. Hint: Recognize tat ypu can constnlct te COD payoffby combining an ordinary call option and a cash-or-othing call.) b. Compute delta and gamma for a COD option. (You may do tllis by computing the value of the option at slightly different prices and calculating delta and gamma directly, rather than by using a fonnula.) Consider different stock prices and times to expiration, in particular setting t close to F
.
c. How hard is it to hedge a COD option? 22.2. A barrier COD option is like a COD except that payment for the option occurs whenever a barrier is struck. Price a barrier COD put for the same values as in the previous problem, with a banier of $95 and a stn'ke of $90. Compute the delta and gnmma for the paylaterput. Compare the behavior of delta and gamma with that for a COD. Explain the differences, if any. 22.3. Verify that equation Probxw
H and Sp
(22.7)satisfes the >
appropdate
boundacy conditions for
K).
22.4. Venfy' that equation (22.14)(for both cases K boundary conditions for an up-and-in cash put.
>
H and K
<
H) solves the
738
%.Exo-rlc
OpTloNs:
22.5. Velif.y that as F cxnin equations (22.20)and (22.21)you obtain equations discussed in Chapter 12. (12.17) and (12.18), -,+
22.6. Verify in Example 22.12 that you obtain the same answer if yo use abQoas the stock price, c + pso' (? + 1. ry as the dividend yield, ?' as the interest rate, and o'c as the volatility. 22.7. Consider the equitplinked foreign exchange call in equation (22.42).ln this problem we want to derive the formula for an option with the payoff maxto, Qvxp QvK4.
'tupside,''
a. Explain in general how this buy-back strategy could be implemented using banier options. 0.08, t 0.3, r 1, and & 0. 'Fhe prernium b. Suppose S $50, tr of a wlitten call with a $50 strike is $7.856.We intend to buy the option back if the stock hits $45. What is the net premium of this strategy? =
-
a. What is the prepaid forward price for Qpxv b. What is the prepaid forward plice for QVK, where K is a dollar amount? '*'
-'
'*'
.
.
c. What is the formula for the price of the option with the payoff maxto, Qpxp QvK) Verify that your answer is the same as equation (22.42).
=
=
=
=
A European lookback call at maturity pays Sp ..Xw.A European Iookbak put at jb are the maximum and minimum prices mamrity pays Sv Sv. (Recall that p and over the life of the option.) Here is a fonnula that can be used to value both options: -
-
G
-
1;1/*(u% t c r, T ,
,
-
,
22.8. The quanto forward price can be computed using the risk-neutral distribution as E (F-v-1). Use Proposition 20.4 to derive the quanto forward plice given by
-)
a. Construct a one-step tree for the Nikkei index. b. Construct a one-step tree for the exchange rate (yen/dollars). c. Use the trees to price Nikkei and dollar forwards. Compare your answers with those in Example 22.7. 22.10. Suppose an option knocks in at Sl > S, and knocks out at Hz > Sl Suppose knok-out'' that K < S2 and the option expires at F. Call this a option. Here is a table summarizing the payof'f to this option (notethat because Sa > Sl it is not po'ssible to hit Hz without hitting Sl ): .
J(z.-?)
t: &, rz?) = olstej e-rLT-t)
#()#0) j
x(u)#?) 6
2/-
t
2/'
-
N-oldsj
J)
-
,
st 1-2V NLcodsj a.. St
2
G
2
-
G
-
equation (22.29).
22.9. ln this problem we use the lognormal approximation (see equation 11.19) to draw one-step binomial trees from the perspective pf a yen-based inyestor. Use the information in Table 22.4.
739
22.12. Covered call writers often plan to buy back the wlitten call if the stockprice drops and, suciently. The logic is that the written call at that point has little written the position could sustain loss stok the call. from if the recovers, a
-
t
k
PROBLEMS
11
8j
(22.52)
t
Where
d
=
Lknstjt) + (r
h
=
d
dq
=
+ (?' (ln(J?/.%)
Jj
=
dq
(y
-
c'
-
F
F
-
-
-
J + 0.5o'2)(F
-
&+ 0.5c'2)(F
-
tjjo-
T
tljc
T
-
t
t -
-
t
t
The value of a lookback call is obtained by setling lookback put is obtained by setting Jt = S: an d ) =
jh
=
t -
and
jl
=
1. The value of a
1
.
GEknock-in,
H Hit
H3 Not Hit 0 What is the vale
maxto, Sv
K)
-
a. What is the value of a lookback put if St gives you the same answer.
22.11. Suppose the stock price is $50, but that we plan to buy 100 shgres if and when 1, and 0.3, 1* 0.08, F / the stock reaches $45. Suppose further that c =
-
..v.
22.14. For the lookback put:
0
of this option?
=
a. What is the value of a lookback call as St approaches 0? Verif'y that the formula gives you the same answer. b. Verify that at maturity the value of the call is Sp
Hz Hit
S2 Not Hit
22.13. For the lookback call:
-
=
= 0. This is a noncancellable limit order.
a. What transaction could you undertake to offset the'risk of this obligation? b. You can view this limit order as a liability. What is its value?
=
0? Verify that the formula
b. Verify that at matulity the value of the put is v
-
Sv.
22.15. A European shout option is an option for which the payoff at expiration is maxto, S K, G K), where G is the price t which you shouted. (Suppose you have an XYZ shout call with a strike price of $100. Today XYZ is $130. If you shout at $130, you are guaranteed a payoff of max($30, Sv $ 130) at expiration.) You can only shout qnce, irrevocably. -
-
-
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.
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''''''
80
90
100
110
120
130
140
150
160
.
,:
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.
The top panel rdplcts the oId VIXfrm 1986 to 2004. The bottom . panel compares the new and old VIXindices during 2004. .
1
o.270
170
volallity (%) lc() 80
60 40
20 0
1987
1990
1993
1995 Date
1
s..
Implied Volatility
1 year g yazry
VXO
18
0.6
16
0.4
14
80
90
100
110
120
130
140
150
160
! '
l
170
02/23
see volatilities plotted in a three-dimensional graph, with time tp maturity on one axis and strike prices on a different axis. Such a plot is called a vola ti ty su rf ace. The pattrn of ilplied volatilities g:nerally ij referred to as the olatility skew. However, specisc patterns are frequently observed. If you pse your imginadon, the pattern implied volatility plot ip Figure 23.1 rejembles a lppsided grin or a smirk. When the ptot of implied volatility in the kure is someiims called a voltility agaist strike jrics toks like a smile, it is clld a volatilit smile. Volatilityhnwns may also be observed. rf'he
.
kolatiliWmay
'
.p?if/'/c.
.
seem like a namral way
'
'
.
io measure the volatility tat is
expectedtoprevail overafuttlreperiodof fime. However, thefactthatimpliedvolatilies are not constant across strike prices and over time raises at least two issues. First, it is using the Blakuxchols model, whlch ssuGes common to meastlre implied volatil that volatility is constant. The volatility skew may reflect pricing or specificadon enor in the Black-scholes model, which raises the question of whatimplied volatility acmally measures. Second, since there is no single measure of implied volatility (for the same asset, implied volatilities differ across strikes and across option mamrites), how should we interpret the implied volatility numbers? Should we look at volatility at a pnrticular t'moneyness''? ls there some way to average the different volatihties? We will see later in this chapter that some theoretical pricing models are able to account for implied volatility patterns such as those in Figure 23.1. .
.
.'
.
.
.
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-
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.
12
'
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06/02
-
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07/22 Date
Soltrce: Optionmetrics
Implied
2004
(%) %
70
.
2001
20
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02
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lmplied
vojatuity
.
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743
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VOLATILITV
09/10
10/30
;
12/19
source:cOE ln addition to exnmining thepattern of impliedvolatinties atapointintime, itis also possible to ack implied volatilities over time. Since 199 1 th e Clzicago Board Options Exchange (CBOE) has reported an index of implied yplatility for near-term S&P 100 indx options. This index is called the ICVlX,''afterits tickersymbol. Usipg this peasure, rijinally, te csci tqmputed over tiuae in implied volatitity. can tiackchags extracting volatility f'rom near-the-money options, much by implied implied'volatility This is called the V1X,'' with ticker symbol discussed in Secon 12.5. index as we rld volatility beginning 2003, the CBOE began computing implid based in VXO. However, on a new fonzpla that we will desclibe later ih this chapter. Figure 23.2 plots the old '71X index f'rom 1986 to 2004, and comples te new and 'WX for one yeqr. The spike in the VIX in 1987 occurs on fotlr days, October 19, old 22, and 20, 26, when the WX exceeded 100%. Tlzis period corresponds to the October decline over 20% on one day.l 1987, market crash, in which the Dow Jones 19 'I'he bottom panel in Figure 23.2 compares the new and old VIX during one year, showing that while the two measures are not idetical, they are generally quite close. At
we
tndex
lplotting implied volatility over time raises the question of whether implied volatility is an accurate forecast of acmal future volatility. Generally, implied volatility is a biased forecast of future volatility (e.g., see Bates, 2003).
'
:'j' (' ;y'. t'1* !' $' (li.l.qiqllql'llhissqsst:l (!(5r' (!' '1* J' q' pd tiid ii)d ''.-' )yrk'k' r' ljd ;).jjjL'. @1* 't' tqd .'1I1Ij:illIi.. 1111:*. -'''1IIi::;2;i-'-' ).g).))'. ;r,$)'.)' :Ei5!1l:-'. j,,r'. yyjt'ttjjd jyd .jj' p'y y'. #(j?'. tEEijdIll'; j;trd i'. i' lt'y
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744
k. VOLATILITY the scale in the top panel, it would be difficult to detect any difference at a11in the two series.
th We will see in Section 23;3 below that the new VW measure natly snesses o? considering prices aII pr0blem of which particula.r option implied volatility to use by out-of-the-money options with a given time to mamrity.
23.2 MEASUREMENT
AND
BEHAVIOF
OF VLATILITY
(. !
dst/'t
(a
=
8jdt + o'st,
-
.YI
,
tjdz
(23.1)
where a is the continuously compounded expected return on the stock, is the conXI t) is the instantaneous volatility. tinuously ompounded dividend yield, and o'st loking equation is subtly different than the process we assumed in eadier This tainililr chapters. In equation (23.1),volatility at a point in time can depend on the level of the stock price S, other variables, X'. and time. By comparisqn, the standard Black-scholes Equation (23.1) is an example of (ro, a constant. assumption is that o''t Xt 1) which in volatilit'y, instantaneous volatility can stochastic with stock price process a randomly.z change Given a series of stock prices that we observe evel.y 11periods, we can compute continously cmpounded returns, 6: ,
,
,
,
=
fr+/,
=
1nSt+I,IS;)
We will assume throughout this section tat 11is small, and therefore that we can ignore the man rettlrn
I'Iistorical Volatility The namral starting point for examining volatility is historical volatility, wlzich we compute using past stock returns. Suppose that we observe 11 continuotlsly compounded stock returs over a period of length r, so that 11 = T/?l. Under the assumption that volatility is constant, we can estimate the historical annul variance of returns, ir2, as
4.2 H
=
-
1
1
l
(?? 1)
t
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ENT AN D BEHAVI OR O F VO LATI LITY
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E.E'' IEE tE (E'EE' E' E' ( (. E . i'i)! . li(: 'i'!Ey'..!.'i(.'(E.('.i.(.:EE.'ii i. ''( (.(.((E(i .. (.'...(. (.'y(i!..'EqEE.;!.(::76:1. . . ti. '! . E E ( .E . !. . i. iE ! :E:. E . E E .EE .yE .
i:.i.;. !k!.t : . . ;. !.) yI E. Ii ; ! t:y:.y..8.j; . :y.jyy. . jyyy.jjy. . 1iI!EiEi:.. j..((y...rq.yk.ii:!..y!.j;.r.jjy. .. .; j1lIi-:-.iIl'. ))j . jjj.jyjj. :' .IIIE:qdq.. . ..E; .q. ;; riji.yi)! ..i . ;tj. . '-)): -----LLj;;, .. !... ..;F.. !.)-. '. i . . . r . .. .. ........ . . r. (. ..r . .. rl . . . . . ... . . . Sixtpda# volatility estimtes for IBM nd the S&P 500 indexk The top pane I j hows volatilityestimates using equation (23.2), whereas the bottom panel uses equation (23.5), with b = 0.94.
rttj
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e xl.l yj r I s .J ! sv
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,,
1998
1 1
uj l
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Year Volatility
O.8
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S&P 5O0 Index IBM Ii
,
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%
1996 Year
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1998
2000
Ssuppose
?-f is computed daily. lt is important to annualize aher computing the standard devation of the daily returns, as opposed to annualizing the dailyreturns and then computing the standard deviation. To see why, let (?n l be the set of nonannualized daily returns. The annualized standard deviation is
252 x
Varlt/-f))
If instead we Iirst annualize the returns by multiplying them by 252 and then compute the standard deviation, we obtain va(252 x
zNote that there are no jumps in equation (23.1).and we will assume that the volatility, CLS, X, t), also does notjump. It is possible to permitjumps for both; for example, see Dufse et al. (2000).
......
.
The multiplication by 1//) annualizes the varimce estimate in square brackets.3 T calculation differs f'rom the usual formula for variance since it assumes the per-perio'/plyt (.( (. . ' E' ' '; (ii. i'. This assumption makes little differene'' iil . mean is Zlr. 1!811I:E1i5:1L ElEi'' $rE11. 11L t!i;i lL. yy(.,y,. ( ( j The top panel of Figure 23.3 displays the historical 60-day volatility for IBM and t the S&P 500 index from 1991 to 2002. Each day, the preceding 60 trading days are used (( ; to compute the standard deviation of the continuopsly compounded daily return. Since ( ( (' '
E 11.,; 'E.l :E'i ';i)f;'.fE 1( ( !
ln this section we exanzine different ways to characterize and measure the behavior of volatitity using only historical information about the asset price. In our examples we will concentrate on stock plice volatility. We take as a starting point the lognormal model of stock prices. Suppose the stock price follows this process:
t(l)Eyii
l' g((' EE.(i(.''.E.(.i . '. . ' ..'' .'. .'.(.('( g'.'.iEE@ E .i (.'' '.. .' ' ..'. . . ...j.E.j . . . .). . .r ..E, E..k...(.(.:'j.:.(..).yi..g'.g.(.q.(r'.(y..'..... . .. E.EL..(.., .. . ...tljjt4j .
(rf))
=
252 x
Varttrf ))
Annualizing returns before computing the standard deviation creates a return series that has too much . volatility-
746
k. VOLATI
MEASU REM ENT AN D BEHAVIOR
Ll'rv
there are approximately 252 trading days in a year, the resulting standard deviation is multiplied by 252 to produce an annualized standard deviation. The use of overlapping 60-day intervals induces smootlmess in the series, since each day's returfl affects the next 60 days of voltility calculations. Even so, there is a great deal of valiability in the standard deviation. For both selies, volatility appears to have risen toward the end of the 1990s. lt is natural to estimate volatility using daily returns for a11trading days. It is important to recognize, however, tat not all calendar days, and not even a11 trading days, exhibit the same volatility. For examplej if all days were te same, ret'urn volatility over a weekend (fromFriday's close of trading to Monday's close of trading) should be ,/-3 times the weekday volatility. However, French and Roll (1986)showed that returps
fromFridaytoMonday were signiscanyly less volatilethanreturns ovetllreeconsecgtive weekdays. More recently, Dubinslcy and Johnnnes (2004)showed that individual stock t.11:.11 price volatilities are greater on the days when firms make enrnings nnnouncements nnnualized provides estimate of volatility, wltile equation other days. Thus, (23.2) an on the volatilities on individual days can vary.
OF VOLATI LITY
$
beconaes 11
2 ENVMA,I
j.. j (1 b4b (1 = b)b ''-l./,-1
..
-
=
?1
f==l 11
=
(1
b4b j
-
...
j
.j.-
j
c2 t
-f
C2
(23.5)
-i
1
i'= l
bn '
-
= Becaus E''izzzl (1 @p)!?f-l 1 bn the weights again sum to one. There is also a simple updating formula, analogous to equation (23.3),in the case of a moving window estimate. Each peliod we add the latest observation and drop the oldest observatio. Equation (23.5)is equivalnt to -
-
;
2Ewxlx-,
:, EwMA,,-I
=
Example 23.1 Suppose b term in equation (23.6)is then
=
1
+
0.94 and
-
b
z b g6,-t s,, -
j
11
-
=
,,
60. We have 1
6,-1-,,1 z
-
(rqa6) .
b'l
=
0.9756. The first
(1 0.94) 0.0615 0:9756 = This compares wit.h a Weight of 1/60 0.0167 for each observatio in the equalweighted estimator in equation 23.2. Subsequent (earlier)observations hake weights of . C 0 0578 0 0543 0 051 1 etc -
Exponentially Weighted Moving Average Because volatility in Figure 23.3 appears to be changing over time, it is natural to try to take tlzis variation into account Fhen estimating volatility. We rnight reason that if volatility is changing, we want to emphasize more recent obselwations at the expense of more distant observations. One way to do tlzis is to compute an exponentially wer/dtf moving average (EWMA) of the squared stock returns. 'Fhe E formula computes volatility at time t as a weighted average of the 2 estimate, sqvMx,i-j and the time t 1 squared stock plice change, time t 1 E Thus, we have 62 t 1 -
-
,
.
-
2EvMx,/
=
(1
-
2 y)e?-l + y).zEyvMA,?-l
(p,aa) -
We can lgg qpgtipn Fhere b is te weight applied to the previous EWMA estinpt. 2 right-hand side of the substitute resulting expression and for into (23.3) te mvvxr-j esimator as a weighted equadon (23.3).Continuing in this way, we obtnin the E average of past squared remnzs:
=
The bottom panel in Figure
13.3displays
the E
stimptefor b
2
ENVMA,I
=
J-'lg(1 !7l#-lj -
-i
(23.4)
i=l
'Fhe term in square brackets in equation (23.5)is the weight applied to historical returns. The weights decline at the rate b, with the most recei fttlt'n receiving the peatest weight.Because E=fz.j (1 !7)!7f-' = 1, the weights on past squared rettlrns sum to one. volatility. For lt is also possible to use a moving window when estimating E ?1 of equation might previous days data. In this case, use only the (23.5) example,we -
0.94 and
standard histodcal volatility estimate. 'his addttional vaiiailit'y oecurs because the most recent observation has four times the weight in the E estimator as in the standard estimator. Thus a pgrticulayly lyrge rettlrn will crate a large effect on the estimate. This eflct will then decay at the rate b. %. There are two problems with the E estimator, one practical and one concepFirst, if we use the EWMAestimator in equation (23.3)to forecast future volatility, t'ual. a constant expected volatility at any horizon. The reason is that the forecast of we obtain 2 E t2 i S t j S o that all forecasts of future volatility would equal (72 Thus the E t j estimator does not forecast patterns in future volatility. Seotzd, the EWMA estimator is not derived from a formal statisticl model in whih volatility can vary over time. ARCH and GARC, which we discuss next, address both problems. -
,
.
-
62 t
=
,
Time-varying Volatility: ARCH A casual examination of data, such as looking at historicl volatilities (Figure 23.3), or looking at tlie behavior over time of implied volatilities (Figure 23.2), suggests that
748
%.VOLATILITY
M EASU IIEM t)'.'yj )' ('El' !' tlf 77* (' r-k;'j','jj'' ;:y5,j.' j.jyf y' r' !j' 7r:7r'* 'q' yyqyj;f ;')' y' 1* rsf r(' j','j;f r,,::.f y' trf r'' r' .(:::jI(.' jj'' ytt'j,',)('. .'('i' :E ('. 'ii E!.' ij;.lj'i(!1i .I..t..j'. (.( i (.'(.'.' q((!'.(((q'.( (t.jr.E;. '... ('.;'.' q'' i(''r7'(7.qr:' .jjjjjjkkjjjj't)' jj'. tt,f j'kkEI!iii;;;;'. ?' y,.kf ( (.(.((('!'j'( );'. tf ijji!j!f . .'y .. ' .7(4'rq .r' tl-). . . p.''y (l'' ,(tt'-!i:.. qqy.ry i.; ' (''.(j.qy, 'q. j,. . ;;.q. y j;jj'.j jy.. q . . jy E:. ) . r yjyyyj.q..jj..yyyy...qyy.y.(j(.. . j. ..jliti:tlkii. . j jy j. .jy . j j ;q. ). iijiiiq. .jyyg... jj;;kyjjjj;. y.( . .. y jj; . .j.j.yy.jyg . .. . . qy y y yy y j r y . y y y . . y . ... . . ....... .... ....
.'.'
ENT AN D BEHAVIOR
oy
VOLATI LITY
$
749
-'.' --' -' --y' :':'' 5* .(' F' ;-'
volatilities are not constant/ What do we do once we formally accept that volatilities change over time? Ideally we would have a statistical model that permits volatility changes to occur. Such a model could serve both to provide better estimates of volatility and also to provide a building block for better pricing models. Research bn the behavior of volatility shows that for many assets, there are periods of turbulence and periods of calm: high volatility tends to be followed by high volatility and low volatility by low volatility. Put differently, duling a period when measured volatility is high, the typical day tends to exhibit high volatility. (High volatility could in principle also arise from an increased chance of large but infrequent price moves.) Figure 23.4 displays squared daily rettlrns for the S&P 500 index and mM. At a casual level, this figure exhibits tis effect, with peliods in which many of the daily squared retul'ns are large, and periods when many are small. This is called volatilit.y clustering.
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Sqtlared daily returns the S&P 500 index (top panel) and IBM (bottom panel) from january 2, 1991, tp October 24, 2001.. Or1
sap s()()Index
0.01 0.008 ().(06
0.004 0.002 0
1992
1994
XL K.-.A,'A 1996 1998
-.-..-.=
..
2000
E
If volatility is persistent, a volatility measure should weight recent remrns more
difference in weighting is exactly how an E heavily than more distant remrns. volatility estimie differs from the ordinary equally weighted volatility measure. ARCH and GARCH models also give more weight to recent returns. rfhis
Squared Ret'urn
0.03 0.025 0.02 0.01j
The autoregressive conditional heteroskedasticity (ARCH) model and the subsequept GARCH (Generalized ARCH) model of Bollerslev of Engle (1982) models thatattemptto capture statistically andwidelyusedvolatility important (1986) are volatilitys fact Engle ebb and in flow of the won the 20013Nobel Prize ip econornics for is that 750). The basic idea motivating.Rcl-l the box this in is Work area (see on page Engle be high high today it is likelier than volatility to tomorrow. (1982) if is average provided a statistical framework for modeling this effect. A statistical modl for asset prices could take the form
The ARCH model
Lnst/vt-h) In this specifcation,
(ce
=
0.5c 2)11+
-
-
E:t
(23.7)
the error term would have variance
var(6;)
=
c lh
(23 8)
4We can perform a back-of-the-envelope calculation by assuming that continuously compounded returnsare normally distributed. ln that case, the ratio of variances drawn from independent time periods hasthe the F distributin. If %ve estimate two annual volatilites usng 252 observations, the ratio of thetwo estimated variances is distributed F@, 251 251). The 99.5% and 0.005% confidence lekels F-l (p, 251 251), where p 0.995 or p 0.005. At a l % signilicance levelr ihe are obtained from two bounds for the ratio of estimated variances are 1.386 and 0.722. corresponding to volatility ratios of 1.177 and 0.849. Thus, if volatility in one year is 15%, a subsequent measured volatility outside the range 12.74%-17.66% rejects the hypothesis of constant varance at a 1% signi:cance level. ,
,
sBollerslev
=
=
the literature on ARCH and its vafiants. Two recent accessble introductions were written as a result of the 2003 Nobel prize: Diebold (2004)and Royal Swedish Academy of Sciences (2003).Nelson (1991)proposed exponential GARCH, which models the behavor of 1n(o-2). et al.
(1994)surveys
.
IBM
0.01 0.005
1992
1994
1996
1998
2000
If (7.2 is constant over time, we say the error term, 6:, is homoskedastic. Based on Figure 23.4, however, a more reasonable specitication would be to assume that the variance of varies over time, in which case it is heteroskedastic. t lf the time interval in equation (23.7) is short, then (as we saw in Chapter 20), 0.5c2)/1 is small, and /f2 is essentially the squared return. We will the mean, (a continue to assume that 11is short enough so that we can ignore the drift in equation -
-
(23.7). Let kfq denote the information that is available up to and including time t, and therefore information that we have available at time t. The idea behind Engle's ARCH model is that squared remrns have a variance that changes over time according to a statistical model. Speciscally, let qt be the conditional (upon information available at time t 1) value of the rettlrn variance, i.e., -
qt
EEE
The ARCH model supposes that we cap
2Itp ;-I E (6?
)
write
(23.9)
;
k
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2003 Nobel Prize in econrfticg was awardedto Robert F. Engle and Clive Grager their work in statistical naethods in economics.Engle was cited for his work in studying the behavior of volatility. This from the Royal Swedish Academy o? quote, Science press release announcing the awtrd ot the2003 economics prize, describes Engle's '
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conditional hetAt this point we cap understand the meaning of depends point time value in at a on past eroskedasticty.'' Alltoregressij'e means that the valtles. Hetelnsketlasticity means that variances are not equal. The unconditional varicollditional variance is ance is the variance estimated over a long period of time. The upon'') taking into variance recent estimated at a point in time, account (ttconditional the volatility. Thus, alttol'egt'essive collditional hetelnskedasticity essentially means that the level of variance depends on recent past levels of variance. This is the behavior captured Gtautoregressive
by eqtlation
(23.9).
forecasts An important practical question is how many lags we need ARCH volatility equation estimate order (23.9). To better understand the behavior of an ARCH in to 0, i > 1. The volatility where we set /71 > 0 and ai single consider lag, let's model,
a
::s
(t'I()
+ a I :2,-
(u
=
+
ao +
??-period-ahead
tRI
)
j
E(c2,+l j+i-jl
t7.1
+ a + a (,,0 gt-o l
j
:,2-
j
)j
forecast, we have /1
1+J l ) .-
-
av
ai
1+
l
+ anel l t
-
I
(23.11)
j .. .
l . .. .
.
Equation (23.11) implies that unconditional volatility long-run average) is a
(thevalue we would estimate (23.12)
,
.
+ aI
(/?a.1'l-I)
squared return today implies larger squared returns at al1 futtlre dates, but the effect decays per period by the factor t7l Shocks to volatility are expected to die off at a constant rate. RS
???. Equation (23.9)is an ARCH(m) model, signifying 1 where at > 0, ai a: 0, i that there are /?? lagged terms. ln order for volatility to be well-behaved, we must have S)'=jai < 1 This model states that volatility at a point in time depends upon recent observed volatility.
(23.10)
This predicted pattern of volatility persistence is vel'y specifc and inflexible. A large
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7 51
where aj > 0 and t?I < 1. Equation (23.10)is an ARCHII) model. Suppose we forecast volatility at time t + 1, t + 2, etc., using only the information we have at time t. Equation (23.10)implies that for a one-period ahead forecast of qt we have
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ENT AN D BEHAVI OR OF VOLATI LITY
Thus, with
stipates of ao and tz1 we can compute the unconditional volatility. ln prctice, if markets become more turbulent, they may remain more turbulent
for a p'eriod gf time. Equation (23.9) with a single lag cannot account for a period of sustained high volatility. As you might guess, more than one lag-generally many lags-are necessary for ARCH to fit the data.
The GARCH Model The GARCH model, due to Bollerslev (1986), is a variant of ARCH that allows for infinite lags yet can be estimated with a small number of parameters. The GARCH model has the fonp
(23.13)
752
MEASUREMENT
%.VOLATILITY This model states that volatility at a point in time depends upon recent volatility as well as recent squared rettlrns. Equation (23.13)is a GARCH(m,n) model. GARCH(1,l) is frequently used in practice. The GARCHII,I) model is
qt
+
t70
=
a +
J11 6;- l
bL
ql-
(23.14)
l
lt is instructive to eompare the GARCH(l,1) model to the ARCHII) model, equation (23.10). To do this, we can rewrite equation (23.14)to eliminate tzf-l on the right-hand side. Lagging equation (23.14)gnd substituting the result for o-1 on the right-hand side of equation (23.14),we obtain 62,-, + t-lj c2/-:t+ Jpjo-2,l + bk az + 51
(c()
qt =
AND BEHAVIOR
OF VOLATI LITY
$
753
For a GARCH(1,1), qi is a function of e, 471 and bj The maximum likelihood stimate the of is set pazameters 5a, a I and 871 that muimizes the probability of obsrking the remrns we actually obselwed. Typically it is easiest to maximize the log of the liielihood function, in which case maximizing the likelihood is the same as maximizing ,
.
,
l
g-0.5 ln(o)
0..5c/Iqij
-
(23.18)
i= l
We omit the term ln(2;r) since it does not affect the solution. The muimization of equation (23.18)can be performed in statistical packages or even using Solver in Excel (see the chapter appendix). -0.5
Continuing in this way, we obtain We can forecast volatility in the GARCH(1,1) model as we did volatility foretasts ARCHII) the model. To understand the calculation, recognize that since qt in f(c2ItpJ-l), fr j k: 1. Thus, using equawe then have E qt 1%-.j) E (c2I%-.j) / I tion (23.14),we have =
=
(23.15)
E (:/+11lJ-1 A GARCHII,I) model is therefore equivalent to an ARCHIXI model in which the lag coefcients decline at the rate /71 Notice that the last tenu in equation (23.15)can be rewritten in terms of an EWMA volatility estimator (23.5):
)
=
aft + 51 1 bt 1 bI -
-
2
swM.,k
(23.16)
in the EWMA expression in equation (23.16) (23.3),but is estimated as part of the GARCH
bL
It is important to note that the parameter is not arbitrarily chosen, as in equation estimation procedure.
= aj +
Iillihood
estimation
varian'ce q: and mean zerp, we an estimate a GARCH model using maximum likelihood.6 The probability density for t conditional on q;, is ,
fEt
; qt4
1 e lz'qt
=
-fj.szifqt
lt
11
i= l
) + bkEqt l%-jl (f71+ bj )f qt I%-l ) + /71)(e + ctl6?-2
(JI
+ b 1:1-1
)
The goal in this calculation is to express the forecasted value of t.n+I in terms of what we can obselwe at time t namely, 6,-1 and (h-1 Following the same procedure, we obtain .
E (t.n+2I +,-1
)
az +
=
e(t?I
+ :1) +
It.n) -
i= l
l eln'qi
k
E (t?,+k 1+/-1) = az + az '
(t7l+
bL
)2(:70+
(7I
cf2-j + fplth-j
)
t
-
As we let k go to insnity, we obtain an estimate of unconditional GARCHI1,l ) model:
volatility in the
X
.
() seg: '
(t71+ lall + (t?l+ l1l)'(t7l62 j + pplo-j )
f= l
(p,y jg)
Since the e? are conditionally independent, the probability of observing the particular set of 11 returns is the product of the probabilities, which gives us the likelihood function: fEi
E (6,2Itp ?-l
t'll
For a k step-ahead forecqst, we have
of a GARCH model Given the assumption that conditionally normal, with distribution that is continuously compounded returns have a Maximum
+
afl +
=
.
qt
t?0
=
'2
(t7I
aj
=
i
+ bj )
:70 =
1
f=0
Using equations
in terms of the E
(23.16)and (23.19),we can estimate
-
aI
.z
express
bj
(23.19)
the GARCH(1,1) equation
of volatility:
'
q:
Jt'i
2 + (j
-
J)
4.2 EwxjA
(za po) .
where a (1 t71 bj )/(1 bq). Thus, the GARCHII, 1) expected volatility at a in time int is a weighted average of the unconditional valiance, #2,and the current po 2Ewua. estimated EWMA volatility =
6Alexander (2001) discusses the estimation of GARCH models and is replete with examples.
=
-
-
-
!111)* (jj' (.j4* j'((jy'. k' 't' t'(' ttd tl.ij:llllid
k
754
''.' .' ''.
VOLATI LITY
j' ))' Example 23.2 t'(r'r'
r' E' ''t:.
''' .'
Estimating a
GARCHI1,1) model for IBM using daily return data
1 .) from Janum'y 1999 to December 2003 yields the GARCH volatility estimate '#jj :.,r ') 0.000001305 + 0.0,4466/-1 + 0.95520-j qt tEj ljjtit The implied estimate of the unconditional annualized volatility is .)) .t .qgrjy 0.000001305 tyt 1.5318 x 252 )t) 1 0.0446 0.9552 ( yj The historical volatility during this period was 39.85*. An estimated unconditional 1') 'i..'olatility of 153% suggests that the GARcH(1,l) model has trouble fitting the data. In lt t')(tllis case, it ttlrns out that the problem is caused by large returns on days during which y
=
'
=
-
-
-
'
E ' .
:
.
l DM announced enrnings. .j.jj
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Comparison of GARCHII,I) volatility and EWMA volatility for IBM, January1999 to December 2003. The EWMAvolatility . estimate set.s b = 0.94, while the GARCH are parameters '
estirnqted.
..
.
E (
:It:q. i:!:,.. ii.ii.
0.8
-
-
GARCH EWMA
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t
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=
Mar99 Oct99
Aproo
.,.
Novoo
Mayol
Dte
Decol Jt1n02 Jan03 Aug03
0.4229
-
k
E E
y
QuadraticVariation
We saw in Chapter 20 that the quadratic variation (the sum of squared increments) of a Brownian motion from t to F is F t. That is, suppose we frequently sample diffusion a process', aZt). Letting lt (F tl/ll and Zi) Zt + ih), we have -
=
(23.20).
7In order to obtain an unbiased estimate for nonearnings announcement days, the correct procedure wouldbe to eliminate all earnings announcement days, notjust those which, after the fact, exhibited highsquared remrns.
k.
-
=
/1
(c'Z(f + 1)
=
17 tion equa
i
Quadraticvariation
over time.
1
li
.
y
0
=
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jt ..
.
:
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.
;
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.
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.
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'l) tion to eliminating earnings announcement days, one could permit a fatter-tailed rettlrn t.)t distribution (e.a., see Bollerslev, 1987). ),!) l estimate 1tj Figure 23.5 compares the GARCH volatility estimate with an E lt ' although estimate EWMA different, the )) where b 0.94. The two are not dramatically 1ilexhibits 0.9462 would be even more extreme behavior. An EWMA estimate with b )) tj ll! closer to the GARCH estimate, because of the relation between GARCH and EWMA in
..
.
-
0.1
Realized
,
.
.
.j The other parameters do not change much, and this unconditional volatility estimate of tlj t )tt42.29% is more reasonable. This example illustrates that a GARCH model estimated li) ') using normally distributed returns can be sens itive to extreme data points. In addi-
-
g
:
.
''
=
.
.
0.5
.
-
-q,
-
i i
.
0.6
.
j . These parameters imply an unconditional volatility of j( ' . tjlit ().()()41()2:/3 i x 252 '@7 1 0.0507 0.9462
. .
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,1
ov
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.
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. .
j:
(1, 1)
:;45 1:2/
'fhe
:
pj:rrr,,-
(() (i4
1. 1 During the 1999-2003 period there were four days on whih the abtolute one-day (j t return exceded 12%. On each of these days (April 2l, 1999; October 20, 1999; July 19, k....y . 153% volatility illustrates lij 2000: and October 17s 2000), DM announced earnincs. (.. . the GARCH model's dificulty in explaining these large magnitude returns under the '' ion that returns are nonmally distributed. If we omit these four days from the .j assumpt ' (1,sample, the estimated GARCH model is7 )t a + 0.9462:,-1 0.000002203 + 0.05076,-j qt ) .
.
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%
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.:
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ENT AND BEHAVIOR
:
.
j(k . .j@Ijiii:q. 1Iji!Ei!k.. j tj((jjjjg j ..j. )-. ... . r .-.. (tt,.ljj)-. . .,. r.r.. (.,...r... .. . '
.
'
.
=
-
c'z(j)j2
;4s
0'2(w
-
t)
1
therefore provides an estimate of the total variance
of the process
Suppose stock retulms are generated by equation (23.1),in which volatility varies time. Consider what happens if we compute the quadratic variation of the log stock over price. In ordef t do this, se would need to observe the stock price at high frequency, a for example, using multiple prices over the course of the trading day. Suppose we observe continuously compounded rettlrns from time r to T every 11periods. To simplify notation, let St + ih) = Sij, and c' i4 = o' L&+ihXk-vii t + /l). The realized quadratic variation of the stock price from time t to time F is then the sum of squared, ,
,
.
756
k. VOLATILITY
$
HEDGI NG AN D PRICI NG VOLATI LITY
757
''''j!.
#'
cpntinuously compounded returns:d
.
:
::.
():
l
1)) 2
jnLzil/ui
:
.
.....
E :
E
!
' E'E''
Ijl::z 121
...
:
'
.( :
.
14;::: .....r.. .
..
.
...
.
*
a -lI ri,a for IB anua .
(23.2 1)
'
.....:;(;.....
.
!!! i!;
.s... : ... . . i : ' ; ' r: ' : : j
:.. ::
.: : ... . . : :
a tili 1999 to ece . er 2002, Io ed a ainst 1 1 volatll. lt R .. esti ate . . I-I;r(?
i= l
'
.
...,.....; ,(
...
.
:.....:.
:..(
!
V )a t.j)j
r
!/()1
-
-
1
-
-
-
:
.
j '
..'j.
(
j
: r :. p'l f..Ei
.'!
Kt
h
.!1 '!.
::Q
Q
When 11is small, the drift term in the sumination is small relative to the diffusion term (see Section 20.3.4), j that the squared. change in Z(/) dominates the summation.g The light-hand side is an estimate of the total stock price variance from time t to F.
.
! ':
,
Etz
1E)tii .
7 F, l
S.A! 3 ;1.. ux
.1
1.
t p .tJ .y g . %)! .@ I Z)SI'C pg p i eq j; IF . 9*4. 1E... 1t i.
!
! ! r.l
l
rr
.
.4
One well-lnown difculty with using high-frequncy data is that some observed Price movements' occur simply because yransactions gltemate between customer purchases (made at the dealer's offer price) and customer sales (made at the dealer's bid bounce.'' pric). The resulting up and down movement in the price is called intraday when they with data derrinstfate deal use curAndersen t al. (j003) a way to data to compa.re reiktized quadratic variation with other vazipnce stirpptes, such rency 3o-minute as GARCH(1,1). They calculate realized quadratic variation as follows. At following the and immediately ask preceding prices intervals, they observe the bid and and'ask pliees impute price bid the averaged at to a 3o-minute mark. They interpolate contin3o-minute the price this tcl measure imputed 3o-rninute intervals. They then use uously compounded return, from which they construct realized quadratic variation. In compm-ingforecasts based on realized quadratic variation with other methods of forecasting volatility, both in- and out-of-sample over one- and ten-day horizons, they find that realized quadratic variation is generally at least as good as other estimates.
-. . jt e aljzed vlolatlll R (1 1)
1.5
,
c ;';j-
J) :k).
.r
.:
ilE!; .fvy
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.
.
. f.il '!j ?Jg jC''i . l f. .-/F.4yr. -.if?'i%L. v
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.,
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r.-c
6-
i:
t't' tj. ' R!
: kkJrr ! Jly : . . )? $ .;,r, 4 q . . .JJ:k.....#.>g lkti2 $LAX. ' qlx ! .s3?tSjuvrlhsk'' I $ 5 r tr C?'mz 1.''.. ' T@ 4(q. #% ;; :p!: To j ! 1ETirs . .sl,.l./jij q/tk. ? k ;ty i tf' tj tvflkr:k ? .t' t. y ..j.... :J9)'; .c 1.t.jy. s rr 17 ; y .1. ik';D 1 p(j, t. :';j' x&l4t . t; :k:;;....J! !;r$'' y h. 1 pgr ok. :f ;k'S .k .t) 7r qi F:5' q ! : t x. r Ii .1. . :f13.p : A' : # 4./1 ' k'F .z y .q ?T Q m''. 1!Z .I'. ssCsr%s.rosru 4 jj! qt(? j $:: # t..J w 'F' . 1 :! w/y: ; 9.?1 u : 9.j : I: AJ jrikrk!r sbrs v qtth...tr. ;d w a : gk ,. t.j)1)) & r;h rlf j: fjzrl i7'.y:jthr rj''gtTs :.f :.- j' r V:'? '.'l ! J4u > y%' a k(.L.' .;#4't.: ZJ@ t.!. f.1 :.y.Ir Ln' I tb3T.Cg j,, i.h@. ' .. 23. trs:. J't1.jj' TaJ1 tf/xt x!prly,u E 'a Jk 4: w py :; s.:.. ; t$ lp:dl.J. v ;rj . Ju :').14 XSCCJ. hk Ey). (krrqtrly.. t rsj.t;. k ': ft j y lhqak'):sz. ?x 12 $.)y':> 7rsYJ7-'y?;. . . sr e. k jqijtjg. x xtw k 2 g! j!! tjt jr jytitjqv . a x yc . :ky. 4T. glkrjrl:.>;a.. ;r' y j j bujvnq.. 1 j; g. . au sI . . . tj tr yyyjk . 0. r,'jn s z. a rx1+ -
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ry .
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ar99
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Figure 23.6 plots daily realized quadratic variation for IBM (usingreturns computed at five-minute intervals from open to close) against the GARCH(1,1) volatility estimate using the parameters computed in Example 23.2. The GARCH estimate is a forecast of volatility, while reatized quadratic variation measures actual volatility on a day. Thus, realized quadratic variation is considerably less smooth than the GARCH forecast. (Realized quadratic variation is sifiother, on the other hnd, than the squared daily returns depicted in Figure 23.4.) In interepreting the graph of realized quadratic variation, it is important to recognize that if the priee bounces around a great deal during the day, then realized volatility will be high even if the rettll'n for that day is not high. The empirical question is whether the magnitude of this intraday bouncing around of the price predicts volaiilit'y on subsequent days. The evidence in Andersen et a1. (2003)is tllat it does.
'volatilitl'-''
e'realized
'trealized
Aproo
Soltrce: ndersen et al. (2005)-
.
gUnfortunately, there does not appear to be a standard terminology for the realized quadratic valiation lt would seem consistent to refer to the sum of of an asset price'. lt is common to call (y' the volatility'' would then be the square root of variance,'' qnd tlze squared returns as the of squared the returns is sometimes called (he however, In realized variance. practice, sum the volatlity'' See, for example, Andersen et al. (2003).Moreover, the realized quadratic vmiation only c'2, under certain regularity conditions. Thus, for measures the variance of the stock price diffusion, quadratic variation,'' which s unambiguous, albeit clumsy. clarity, we will use the term 9You may wonder about the difference between estimating historical volatility and realized quadratic variation. Recall the histo/ical variance estimate, equation (23.2). Nlultiplying by 1//1 in equation (23.2) is the same as multiplying by nl T. Thus, the historical variance estmate can be rewlitten as
c199
4trealized
ltrealized
aH
1
= r
1
J1
:?
-
(?2 lj'
'
-
farl
Apart from the ?7/(/) 1) tenn, this appears to be the same as annualized realized quadratic variation. thistorical volatility'' usually implies the use of daily or. less frequent data over In practice, the term tquadratic variation'' implies the use of intraday data over short medium to long horizons, while -
horizons.
'
23 3 HEDGING *
AND PRICING
VOLATILITY
In this section we discuss delivative claims that have volatility as an underlying asset. We begin by discussing volatility and variance swaps (includingone contract based on the V1X). We then look at an example of pricing a variance swap. Finally, we discuss the constnlction of the histol'y of the VLX volatility index reported by the Chicago Board Options Exchange (CBOE). ln this section we will let JZ denote measured volatility and 72 measured variance.
- t).))) ! ,..). j.) ..qj .j
758
k
VOLATILITY
H EDGI
Variance and Volatility Swaps
l,eprevents
,
,
(
-
# 0,F
(1S))
X
1
by
i
l 1 l tt
volatility
j
N
'i
where F(),z(F) is the forward priee for volatility
* Whether returns are continuously compounded or arithmetic ( the valiance is measured by subtracting the mean or by simply squaring whether . the retulps .. .
noct of' time over which variance is measured e ' How to handle days on which, unexpectedly, trading does not occur The
Most of these design issues are straightforward, but the last deserves some comment. Most futures contracts settle based upon a final observable price. A variance contract, by contrast, settles based upon a series of prices. Therefore, failing to observe a price (for example, because the market is unexpectedly closed) creates a problem for closed on day r, then measuring the realized variance. If the market is unexpetedly will will have which be tvo-day measured return, the next ret'urn a greater expected a how following example shows The than one-day variance return. one contract deals a this issue. with
)j Example 23.4 The Chicago Board Options Exchange volatility index, the VJX, is li(lr the basis volatility for f'utures contract that trades on the Chicago Fylmres Exchange. jtt a ('' (( Unlike the variance futtzres contract, the volatility fumres contract settles . based upon the ('. VLX index. The payoff is .)() jE
j
'
E
.
E
j't ji
'
)
'
1000 x FlXz.
-
Fo,p (F)j
1 ).
ln comparing the volatility f'uttlres contract (Example 23.4) with the variance futures contract (Example 23.3), note that the two contrcts are based on volatility measured over different periods of time. The variance contract settles based realized on quadratic variation overthe periodfromo to and thus thefumres pricereCcts volatility r, expectations from time 0 to time F. The VLXcontract, since it is based tti VI.Xindex on e measures volatility expetations f'rom time F to time F + 30 dys. Thus, the kplatility contract measures volatility going forward from the settlement date, Whil the variance contract looks backward from the settlement late. There are at least two rasons that the valince contract is in some sense more ''natural'' than a volatility contract. First, we will see below that it is possibleto plice andhedge avariaceforwardcontracttgiven someassumptions) using optionprkrs. The pricing of a volatility forward contract is more complicated dtle to Jensen's inequality. Because the volatility is the square root of the variance, Jensen's inequality implies that the volatility fonvard price will be less than the square root of the variance forward price of the vazianc. Second, variance swaps arise namrally from dealers hedging theiroption positions. Recall from Chapter 13, in particular equation (13.11),that the prost of delta-hedging a dealer depends on the squared stockprice change. Dealers can hedge this lisk in realized variance by using variance swaps. For example a dealer with negative a gnmma position could enter a swap that pays the dealer when the stock has large price change. a ,
Three-pwnth S&P 500 variance futures traded on the Chicago FuExample 23,3 tures exchange are an example of a variance swap. The payoffis based on the anpualized 2. sum o f squared, continuously comgounded daily returns over a three-month period, . as IZ a x 10,000, and by desnition a one-unit change in The masured price is quoted
this number (calleda varialtce pt?/?2r)is worth $50. For simplicity, we treat the payoff as if it were a fonvard contract, settling on one Qi be the continuously compounded return on day i. The payoffat expiration is
day. Let
?1:, l
Ei
i= l
??c 1
:?
---
$50 x
l0, 000 x 252 x '
-
-
Fo,w(JZ
2
)
?lr, is the acttlal number of S&P plices used in constructing 72 (hence 1 returns), and lle is the number of expected trading days at the outset of there are the contract. Thus, in the event of an unexpected trading halt, the sum of squared returns will be divided by a number ltrger than the number of squared returns. The reason for this is that the trading halt will not necessarily change the total variance over the period (if the trading halt is on a Tuesday, for example, the Monday to Wednesday return will
In this formula, la
-
l
t j
k
'
* How frequently the return is measured .
a
this.
A volatility swap is like a vmiance swap except that it pays based on rather than variance. The payoff is
-
where N is the notional amount of the contract. There are numerous measurement details that we have to specify in order to write the contract for a variance swap:
.
. 759
%.
typically reflect two days of volatility). Dividing the sum of squared returns by na would mechanically increase the measured variance when there is ading halt. Dividing
A variance swap is a forward contract that pays the difference between a forward plice, 2, F 0,T (72) and some measure of the realized stock price variance, over a period of payoff variance is notional The to a time multiplied by a amount. swap w(J/2)j 72 Ftl x N
(
N G AN D PRl CI N G VO LATI LITY
Pricing Volatilie We will see in this section one way to determine the fair price for a forward contract on
variance.
:
760
k. VOLATILI'I'Y
HEDGI Nc
Consider a variance contract that pays the sum of squared price changes from time 0 to time F. If price changes are measured over an interval of length /1 F/??, the contract would have the payof'f =
lt
Payoff
$lt Sih 5'(f+1
:!
-
=
(23.22)
Slt
f= l
Note that, since arithmetic and continuously compounded returns are close over small intervals, this is the same as the realized quadratic vmiation measure discussed in the previous section. As /? gets small, this equation (23.22)becomes F
Payoff
o'st
=
.
Xt
,
1) ldt
,
(23 23) .
0
where o's, .Y, r) is the diffusion coeficient in equation (23.1). We want to answer two closely related questions. First, how is it possible to replicate the payoff to such a contract? Second, how should the contract be priced? As you might suspect, replicating the' contract yields a wayto price it. In principle, the price of a forward contract on variance will be the expectation of equation (23.23)under the liskzneutral measure. ln practice, how do we compute such an expectation? The following section will provide one answer. The Iog tontrau
# euberger (1994) pointed out
that a'forward contract that pays
(23.24)
- knupjb)
could be used to hedge and speculate on vmiance. A claim with the payoff in equation (23.24) is a log contract. As of early 2005, there is no exchange-traded log contract in existence, but for the moment, suppose such a contract does exist. Assuming the stock price follows equation (23.1), we can use It's Lemma to charactelize the process followed by the log of the stock price. (Note that we assume that the stock price does notjump.) Equation (23.1)permits a wide range of processes for the volatility, but the prospective volatility over the next instant is known. Applying lt's Lepma, Fe hay tgln
&)j
1 =
y
ds
05
-
.
)
y.
df
Thus,
0.5c(r)2#r l ds =
Intepating
T
jv
c2(r)#?
=
j
T
j
ydS
%.
-
j
T tglnts'f
ljdt
The integral on the left-hand side is quadratic vmiation over 0 to F. Let 42 denote
761
annualized realized volatility from time 0 to time T We can then rewrite this as r j 0.5F 2 IS jnsvjh) (23.25) 0 The right-hand side of equation (23.25) is the cumulative return to an investment in .
-6
=
-
1jS shares, less the return on a contract paying the realized continuously compounded return on the stock price from time 0 to time F. The left-hand side is anntlalized realized quadrqtic variation, which fromequation (23.23)is also the payoffori variancecontract. Equation (23.25)demonstrates the connection between the payoff to the log contract and
volatility.
Take expectations of both sides of equation (23.25)with respect t the lisk-neutral expectation of 6IS/S under the risk-neutral distribution is
stock price distribution. The rdt. Hence we obtain
T
j2j
*
0.5Ff
1
E*
=
y#5'
LnSp(%)
-
0
= l-T E* llntu$'w/uVll (23.26) This expression seems to be of little help in pricing volatility. There is trick, however, a for pricing the log contract using other instruments. -
Demeters et a1. (1999)and Cal-rand Madan (1998)indevaluing the Iog contract pendently showed that it is possible to use a portfolio of options to replicate the payoff on the log contract. Note frst that
j,
-%U
.-ll/
K
g,
sv4dlc
In(r)
=
-
+
=
In(,)
In(u) +
-
a
.-J.S -
&
t'
Use this to obtain the following identity, for any Sv (see Demeterfi et al., 1999):10 % l X ST ST S+ 1 ln maxtft0):/A- + + maxts'w K, 0)#:a S. S. K K1 o -
=
.%v,
-
-
-
.
s.
Notice that if we take expectations of both sides with respect to the risk-neutral distribution for 5'w,the integrals on the right-hand side become undiscounted option plices, and the expected stock price is the forward price. We can add and subtract ln(&) to the
#glnt&ll
-
this equation, we have 0.5
AN D PRlcI NG VOLATI LlTv
l0To interpret the expression on the right-hand side, notice that for any given Sv, the value of the first integral is zero for K below Sv, and the value of the second integral is zero for K above Sv. Thus, the effective integration bounds are not 0 and co, but they instead depend upon Sv. For example, if 5N Sv m K : S the equation becomes =
*
1
t-
- ln Ss
t=
2
-
&
25--
+
.9
l Ka
mxtc
-
S, Q)dK + 0
;
762
k
VOLATILITY
EXTEN DING TH E BLACK-SCHOLES C' 'yy'. $ii1:tt:::kr'. S' F' (' f' )' 1'* 1:* yjjd !..))' (pyyjj.d (:' (' t'jytj)'. E' j'ytryd J)' (i' ipjjjy;yd @' yjf jlljjitqqd y' jj'. !' )' r' t';' @1ijtj--jIj('. j'k'yjf (' y' E y'y j;;qj,:?;;'. t'. t'tyd rij!!!j:i,d i (i;(g'!' E('' '((E..((((;' i EE'E..EEE;.E. . EE(jg ' E.EE'.. .' (('':'.:!' q. E(('EEE.:'(q '':( '!i;tj.'.i' i(i'.i !ij:'I.Et!(!)' .(E ri. . '... .... .. . ( . . . !.(... r'.' 'i.l..1'7. (: ..jjtjltjr:y ').i.r' . . ...j! ((i.(:!i)EJ:E)j;i(j:(.(.;(jj..() . ..(lly.'j.y... .r. y.(. (.-Ey.yyEyEgy....t..(-j.y...y..yygy.yjyj(jyy...y.jy.yyy..i,t..;.yjjj gk ..j.j.jy yy...y.yyyjyy .yy...y y g . yy.y . y j ; j y jy Ci EEE. j . jj y j . .. . y :. .jL3LL3j3:. jiji!!!!jt (j, .g. .j. yj .gj(. lE' )y' j'1111: ..jjyy ' ... . jljl:i;lil:, .y.;;:yjj;E... .jyyj,. r . ik. .jr.yyr..y..qy). y.y.yyyjyy..q...g..y.jy.y.. . . .jy . .., .. . . .
.'
Q
MODEL
763
'r:(' -k;-'. ..14.,,;,-.* -;' ,':)' j')':''r' 7* :'C' ',yj'.
lft-hand side and then take expectations, to obtain '.r
E* ln -
s
.
'f;
,
=
.
F 0, F
-
E
- .
S*
+ ln
'
' :
s) s. -
S.
.v
+ e
%
?
()
x
l K :j
PKjdK
+ s.
in equation
Us this expression to substitute for F*glnt5'w/&ll
1 K2
CK)dK
(23.26).The result is
(23.27)
:'
' ;
.
-
:
. . .
. -.. -.
.
-.
-
,. . .
gy
y
''
' ' : . ..
' : ': .:
..
,-
:
. . ... . . . . . . . . .
.. ..
:
'tq
':
!
rj
-
y .
. .. ... .
-
..
- ..
. .. ....
..
--- -. , .
''' :-.
( ((( .y-.
.
,
y
solid Iines depict vegas of options with (from Ieft to right) strikes of 25, 30, 31, 40, 45, 50, and 55. The dashed Iine is the weighted sum of the vegas, with each divided by the squared strike price, times 600. The calculations assume ()- = 0 30 r = 0.08, T 0.25 ? and 0. .
.
.
. E . .. . y . --- . . . ..
.. . ..
..
-. .. . .- . ..
'
.
.
.
-
-
.
'i
.
, ,
-
.
=
F0,w(5').the first three tenns on the right-hand side of
.
:
-
. .
' r. .
:
'
'
-
- -.. . . ..
jjzEq!r
() y.:a '
0.1 ..---.-
2
Fr
oz.enrT
=
w
()
K
:t
s
l
0.06
N
Ao
tr
sk
,
f:
0.04
,
'
Nx
R$
,''
xN
:
=
N. 4
0.02
X
N.
X,
cxa
1*
xs
?
Finally, note that if we set S. equation (23.27)vanish, and we have =
z
0.08
PLKIIIK +
,.w
) K :j
0 10
(rs g.jg)
C( K )#w.
.
Remarkably, this formula gives us an estimate of expected realized variance that we compute using the observed piices of out-of-the-money puts and ealls! (lrut of the money'' here is with respect to the forwtrd plice rather than the current stock price.) It is important to note that we have not assumed that options are pliced using the BlackScholes formula or any other specifik model. One important characteristic of equation (23.28)is that the variance estimate can be replicated by trading options. It is possible to buy the qtrip of out-of-the-money puts and calls, weighted by the inverse squared strike price, to create a portfolio that has the
value of 2 .
To get a sense of why equation (23.28)works, we can examine the vega of a portfolio of options held in proportion to 1/#2. Figure 23.7 graphs vegas for a set of options and also displays the vega of a portfolio where the option holdings a'e weighted by the inverse squared strike price. The resulting portfolio has a vega that is not zero andis constant over a wide range of stock prices. lf you hold such a portfolio, you make or lose monef depending on volatility changes. CBOE'S
Computing the VIX We can now explain the formula used to compute the volatility index. The calculation is based on equation (23.28). ln practice, option
new stlike prices are discrete and there may be no option for which the stlike plice equals the index forward price. The actual formula used by the CBOE is a discrete approximation to equation (23.28): 2 Fo,r LKi rwcaljtsj; 2 2 Ll-i a e + e Puttfl = T T 14% F Kz A'z K >Kz KsKh (23.29)
20
30
40
stockPrice .
g
.
.
. .
.
y.
50
60
70
.
where Kfj is the flrst strike b:low the fonvard price for the index and LKi (A-/+1 f-j)/2.1l The last term is a con-ection for the fact that there may be no option with a strike equal to the forward price. =
23
*
4 EXTENDING
.
.
'
'
.
TIIE BLACK-SCHOLES
In thi! section we examine three priing
MODEL
models that are capable of generating volatility
skew patterns restnbling those obserkd in ontion markets. The coal is both to understand how the Black-scholes lodel can be extended and also to gain a sense f how these xtensions help us to beqer understand the data. We consider three models: (l) the Merton jump diffusion model, which relues the assumption that stock price moves are continuous', (2) the constant-elasticity of vaziance model, primarily due to Cox, which relaxes the assumption that volatility is constant', and (3) the Heston stochastic volatility ntodeli which allows voltility t follow an 1t pmcess that is orrelated with the stock price process. These models have al1 been siynifiantly generalized, but we can use them as touchstones for better understanding the economics of departures from the
Black-lcholes
lognormality assumption. Atthe outst nbtethattheBlk-scholes model easily accommodates tilelkarying ! volatility if the volatility pattern is deterministic. Speciscally, Merton (1973b)showed
.-1
,.w
.j
-
-
.
-
o
1lFor the highest strike, lh'i is the difference between the high strike and the next highest stlike. LKI for the lowest strike is defined as the diflkrence between the low strike and the next highest strike.
gj.)' j'yjd y' (.jd i,jlj!ilrkid td t';'ytj)d (' j' rlll)liqqd 1**
% VOLATILITY
764
.'.' ''
that ifvolatility is a deterministic function of time, then it is possible to price a European T 2 t) in the option with r t periods to maturity by substituting j t:r sjt' l s for c2(F Black-scholes formula. We can think about this result in the context of a delta-hedging market-maker. As long as the market-maker knows the volatility at each point in time, the delta-hedge will work the same as if volatility were constant. What creates a problem is a l'altdom change in volatility. -
-
.....l..;,,..:r;i;,..!):qyrji.k;.j(.;y!.q..q
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MODEL
lmplied volatility in the CEV model When p < 2, the CEV model generates a BlackScholes implied volatility skew curve resembling that in Figure 23.1: Implied volatility decreasej with the option strike price. To understand why the CEV model generates this volatility skew, note from equation (23.31)that when p < 2 and the stock price falls, volatility increases. Thus compared with the case of a constant volatility, an outof-the-lpopey put pptipn has a greater chance of eyrcije apd is likly to be deeper ifi the money when it is exercised. The only way for the Black-scholes model to account for this higher price is with a higher volatility. As the strike price increases, less of the option value is due to the stock price behavior at low plices, and volatility therefore need not increase as much. Figure 23.9 plots implied volatility curves generated by using the Black-scholes formula t compute implied volatility for prices generated by the CEV model. The top panel shows that, for the given parameters, the implied volatility curve is unaffected by changi'ng time to maturity.
l3Cox (1996) originally derived a pricing formula in termg of insnite series for the case p < 2. Emanuel and MacBeth (1982)generalized Cox's analyss to the case where p > 2. Schroder (1989) showed that both cases could be expressed more compactly in terms of the noncentral chi-squared cumulatve ctistribution function. Davydov and Linetsky (2001)derive ricing formulas banier and lookback options undet a CEV process. lilrfhe pricing fonnula is sometimes written in terms of the conlplenlentaly noncentral chi-squared distribution, which s 1 Qa, b, c). The noncentral chi-squared distribution, unlike the chi-squared distribution, is not typically a standard function built into spreadsheets. However, it is available in softwm'e programs such msMatlab and Mathematica. -
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=
0
*=
34
'
tzzz0..s C 1
=
.,,2,
' () a
.
=
''
-.
-
-
-
'
60
lmplied
yclfi
=
vttdt
-
+ o'v
(23.15)
vtlclzz
We assume that E (#Z(#Z:t) p#?. The interpretation of equations (23.34)and (23.35)is famililm Equation (23.34) for the stock is the same aj equation (21.5)except that the volatility t)(1), is random. The equation for volatility, equation (23.35),has two noteworthy characteristics. First, the instantaneous variance, is mean-reverting, tendipg toward the value f, with a .. . . . speed ot adjustment given by Ic. Second, the volatility of variance, cs t)(r), depends t)(r)a the variance of and is therefore said to follow a square lnotprocess. square root on Suppose that the risk premiun for the lisk o'v vtjdzz can be Wlitten as vtjpv, where we assume pv is constant. This assumption that the risk premium is proportional to the level of the variance is analytically convenient. Given this assumption about the risk premium, the risk-neutral volatility process is 'p(r),
''
0 28 0.26
769
=
=
....j.
$
follows the mean-reverting process
1?(r),
dvtj
t= 0.25
0.32
=
=
kyy,j,
0.36
CEV model. Both panels assume that 0.30, $1O0, o': . r 0.08, and T 1 ln the top panel, p 1, while in the bottom Panel, F = 0.5. =
tjjs rjt,
jlzjgry jjr.:jjg,rjj, jjjs
.
OLES MODEL
80
70
90
11O 100 Strike Price
13o
120
140
Volatility
8 0 7 0.6 0.5 0
0 X
Jl
*F v.p
,. ....p . .
* .
0.2 0 1 ' 60
@
*
@ *
@
70
* . *
* .
@ @ @ .
Z **3 .Q.W I'
80
90
E
'**
*.*
u
xexrawe..
twtr
120
11O 100 Strike Price .
(
.
.
u
lztzyrr
--4
= =-2
=
dvt)
()
=
.p,
=
:J
.
130
* *. * *. . .
140
'
(7('Ei l*
where &.*
l?(/')j
-
-
gfi''r(?)1 -
r(r)#t,)
dt + o-v
r(?)#z.l
(23.36)
-
dt + o'v z?(/)JZ
= Ic + pv and f* = fv/(z + pv). This model of stochastic volatility is the Heston model. Let JZ(5'(/), r(/), 1) represent the price of a lrivative on the stock when the stock price and volatility are given by equations (23.34)and (23.35). Suppose we proceed with the Black-scholes delivation, in which we hold the option and try to hedge th resulting risk. We immediately encounter the problem that there are rlt/tpsources of risk, ('IZL and dzz. A position in the stock will hedge #ZI but what can we us to hedge risk resulting from stochastic volatility? Apart from other options, there will typically be j6 no asset that is a perfect hedge for volatility. In that case, we rely on the equilibrium approach to pricing the option. The PDE for the deriyative k'(k(r), u(r) r) is then: ,
The Heston Model In the CEV model, the instantaneous volatility of the stock evolves tochastically with the jtock prie, but volatility is a nonsyochastic function of yhe stock price. A more general approach is to permit volatility to follow a stochastic process. The Heston model (Hston, i3) allows vlatility to va.i'/ stochastially but still to be correlated 15 with t1)e stock. This generates a di f'feren t op tion prici'g model than the CEV process and also implies that market-makers must hedge both stock price and volatility risk. ln the CEV model, market-malers need only hedge with the stock since volatility depends
on the stock price. be the instantaneous stock ret'urn variance; hence, Let Suppose that the stock follows the process 'p(1)
ds
=
@
-
ldt +
'vtjdzj
t)(1)
is the volatility.
(23.34)
'White lsEarlier papers that modeled volatility as following a stochastic process included Hull and (1987), Scott (1987).and Wiggins (1987). The Heston model has been generalized signiscantly by Duffie et al. (2000),who allow bothjumps in the asset price andjumps in volatility
,
kvtlfvss 2
+ 1(7'2.p(r)J/rp + pvtlcvsvsv 2 t!
t&-lii This equation is the multivariate +
(r
-
&)SVs +
-
r(/)j
-
vtlpv
;
) J/r +
P)
=
1, JZ
(23.37)
Black-scholes equation, desclibed in Appendix ZI.A. The third term is due to the covariance between the stock return and volatility. Since there is no asset to hedge volatility, the coecient on the P'r term has a correction for the risk premium associated with volatility. Heston (1993) shows that equation (23.37)has an integral solution that can be evaluated numelically. Given this solution, we can see how implied volatility behaves when volatility is stochastic. Similar to the analysis of jumps in Section 21.5, we price optons for different strikes and expirations under the stochastic volatility model and
l6lt might be possible to use other options on the same stock to hedge volatility, but the option would then be priced reltltibe to the price of the o'ption used as a hedge.
%.VOLATI
770 ' C' j';'tj' jd 9* j)d E' 1* lf Ef 4* ?' 'C' )' j'tjd 1' ('' (J' 131@* pill!f (' )' t'?':'j'' j')' ''$'77'** r' ''.)'i.';';''()' r' jqd y' jy'jyd jjd j'tld
EXTEN DI NG TH E BLACK-SCHOLES
LITY
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)' ''.' y' r' LjjLL;jjjjj'. tt'jj( ( 1:(!: ' . '
'
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.
!
E' E :EE'' E :''' E !( Fi 'E.':' ((;''' E E : ' 'E ' ' '' E' :E'EEE '.' C!'!1'5* ': ' ' E :i 'LE.' iE'.' ... . ' .'.! E'.'EE'E'..'..''.:'E'!..:7* '... . i:'. .. E 'i ''E.. i('''': '( ' . . .' (I;E :' : . ' ; . : E! ( (: (: E. :.. EE' E: E k!: : ' i (E iE;E 4(j '(j'((.EE'..:..:. ..(.(.EE. E iE j . .ti . .y:.(.y . ( . ... . j y (. .q.. i.. (jg . ):',o,t,.#,,,t;'EjpjE,.,'.,,,!l..,E',,.,,ri.!'.''Ek','(.r g . 4: .. . . y . .j. . '''
'
'
lk. '. . '' j'' . . y j( .(j . . . XltA. ..,l')t.GljItt't (E!'' ;'i.''i
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.
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j..'. .. j ;. '.. 'j(
.
. .
. .
:
lmplied Volatility (r,, 0.10) p
Blackcholes
=
0.36
Blackcholes 0.36
vear
t.().:zs -'
0.32
.(' .
.
t t
--
''x
=
=
0.32
= =
0.28
e=o>u>->->->o=m
E
l
45 40 35 Strike Price (5)
30
Black-scholes lmplied Volatility 0.509 p
(r,, =
0.36
t t '-' t
--.
0.34
->
0.32
= = =
I
.
.
w-ooo
=.
'=
50
.m'
=
Blackcholes
lmplied Volatility ok, 0.50; p =
0.36
().2s year 0.5 Year 1 Year
-->w*
O.a4
=
=
=
-
'
50
-0.30
=
tczz()!'zs year l 0.5 Year t 1 Year =
.
.
0.3
O.3
1
..
=
'x
0.32
....1
45 40 35 Strike Price (5)
30
O
=
l ....... .........
1
I
...
A
''Nw
.%
w
*'>
** **
0 ' 28
,..
*0
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a.
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.=
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'a'
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.s 'h.
..
0.26
0.26 11
30
1).
.............
...11 ............
1'
45 40 35 Sike Price ($)
1
l
10
30
1*
I
*>
..
'=
11
40 45 35 Strike Price (S)
*=
==
'w
1.
50
when prices
are lmplied volatilities computed usng the Black-scholes formula strike maturity, different different to times model for three computed using the Heston that volatility, The panel assumes of o'v. top volatilities qnd prices, two different panel assumes that cv 0.5. In both panels, v* 2.0, o.v 0.1 0, while the bottom 0. v(t) 0.32, k* 0.25, r 8%, and =
=
=
=
=
=
The main challenge for an option pricing model is to match the observed volatility skeml; The literature investigating ways to do this is too large to adequately summazize ' tead we will sketch the nature of fndings in the literature and highlight issues here. ns that arise when trying to match models to data. The pricing models in this section illustrate ways il which modifying the BlackScholes assumptions can enable a pricing model to better fit observed option prices. For example, al1 the pricing models we have discussed are cappble of gnerating higher implied volatities for tn-the-money (low-strike)calls. The Merton jupp model and the CEV model in the examples above both generate implied volatility curves that are :atter incfeases.l8 of the models, juch as alleston model that combinations astime to maturity also allows jumps, seem able to rproduce qlalitatibe fatpres of Figure 23.1. However, matching models to data is a more involved exrcise than just a visual comparison of ,
womo.
0.26 1
l
l
light-hand panels of Figure 23.10 arises from assuming a negative con-elation between volatility and the stock price.
Evidence
0.3
. 0.26
=
implied volatility curves. To illustrate the issues, suppose you want to match the Heston stochastic volatility modl to data. There are a number of ways you might proceed. First, on a given day, you could find a set 9f model inputs that best matches the volatility curves for that day. This entails tindina a retulm. variance (tl(r)), a volatility of volatilitv (cu). a mean reversiol rate (v*),a long-run risk-neutral variance (J*), and a coaelation between kolatility and the stock return p), t flat ma tch the data for a particular day. Matchipg implied yolatilities across a set of options on a given day is a crosssectional test of the model. Once you admit mpltiple days of data, the model has time-series implications as well. Equations (23.35)zand (23.36)imply that volatility evolves over time in a specific way. lf you look at the evolution of volatility over time, does it match equation (23.35)2Are the pararpeters that enable the model to 'lit the crosssection consistent with those implied by ihe vo l a tility tim series? W'hn there is a lisk premium in the equilibrium pricing model (asin the Heston model), it is potentially easier
l'Cerhe
that the stock price then use Blaek-scholes to compute implied volatilities. We assume ranging from $30 pl-ices strike with is $40 and compute implied volatilities for options months to l year. to $50 and with maturities from 3 values of o'v Figure 23.10 shows the result of this experiment for two different volatility, current the than less and p. In the ligure the lonprun volatility, 5*, is 25+, with time to volatility decreases 32%. Because volatility reverts to the mean, implied = is almost = there no skew, maturity in evel'y case. In the panel where o'v 0. 10 and p 0. and 50% p = 0, the although the mean reversion in volatility is apparent. When o'v asymmetric skew in both The exhibits both symmetric skew and mean reversion. E
.=
figure
771
-0.30
rasyear 0.5 Year 1 Year
.c
%
=
.
.
0.3 .Nggg
g
0.34
tzzz0.5 Year '=' t 1 Year
0.34
lmplied Volatility (r,, 0.10; p =
0
=
MODEL
true model should give equal implied volatilities for options at different strikes and maturities. For example, if the Heston stochastic volatility model were true and option prices were conkistent with equation (23.37),then Black-scholes implied volatilities would exhibit jkew, but if the Heston model were used to compute implied volatilit'y, then the options in Finure 23.10 would all have implied volatilities of 32%. lSlt
is not obvious how to interpret the change with respect to maturity in implied volatility curves such s those in Figures 23.1, 23.8, 23.9, and 23.10. The issue is that the economic distinction between a$50 strike and a $55 strike is greater with one week to epiration than with one year tp expiration; therefore, other things equal, we might expect to see iatter volatility curves as time to expiration increases. Bates (2000) corrects for this effect by scaling strike plices by o' F. effectively measuring distance between strikes in deviation unit-s-'' tstandard
'
772
% VOLATI
FU RTH ER READ l NG
LITY
to mconcilethe behavior of the stock with option prices because there is an additional parameter. However, as Bates (2003)emphasizes, a risk premium must be plausible. Bakshi et al. (1997)and Bates (2000)both asked whether option pricing models incoporating jumps and stochastic volatility can generate realistic volatility skew for options based on the S&P 500.19 Both studies find greater volatility skew at short maturities than at long maturities. lf you compare Figures 23.8 and 23.10, you can see that this pattern is generatedby thejump model but not as obviously by the Hest6n model. This explains why, although Bakshi et al. (1997)foundthatthe stochastic volatility model provided the best overall explanation of prices, they added jumps to account for skew at short maturities.zo They also found that permitting stochastic interest rates (which can be added in the same fashion as stochastic volatility) helped explain prices at longer maturities. Bates (2000)found thatjump models (as in Figure 23.8) tit near-terr ption prices betterbutfound thejump parameters implausible: the stockprice does not appeartojump also as often as implied by the estimates necessary to explain implied volatility. Bates voltility of explain skew, the volatility model orderforthe to stochastic that in concluded developed Duffie al. However, implausibly large. et had be plicing (2000) volatility to procedure that permitted jumps in both the asset price and the volatility, and noted that allowing jumps in volatility potentially addressed the problem of an implusibly large volatility of volatility. with jumps in returns clrastically Broadie et al. (2004) conclude that in offer a significant plicing overall pedbrmance jumps improve (andq in withjumps model cross-section unless the remrs ij llowed in improvement a pricing words, In other in at least some volatility-of-jump risk premium.'' jumps have a to dimensions are important, and risk premia can be important as well.21 To add one more layer of complication, casual observation suggesis that in some enrnings, cases volatility changes deterministically over time. When a flrm announces for example, volatility will be higher than on ordinary days. You can show that this is true by comparing the volatility of returns on earnings announcement days against that on other days. Dubinsky and Johannes (2004)show that this effect is also apparent in ption prices, which imply a higher volatility before an enlmings announcement than after. 22 This Ending suggests that in addition to the use of increasingly sophisti'tmodels
'volatility
.
.
.
'
lgBakshi et al. (1997)examined European options on the S&P 500 index while Bates (2000)examined optons on the S&P 500 futures contract. zocarrandWu (2003)formalize the intuition thatjumps mattermore if there is ashort time toexpiration. 21It is possible to measure volatility Iisk premia directly by looking at the remrns on portfolios that lind negative retums are hedged against stock price risk. For example, Coval and Shumway (2001) written straddle loses the S&P indexes. S&P and Since straddles 500 written the l0O zero-delta on on m' attributable to a volatility risk money when volatility increses, this finding ay be at least in pal't premium- Bakshi and Kapadia (2003)lind smallerrisk premia associaled with delta-hedged individual stocks than with index options. 22111fact, %%'e saw the effect of earnings announcements in Example 23.2 when we estimated a GARCHII l ) model for IBM and found the estimates sensitive to the inclusion of four earnings announcement days. ,
Q
773
cated mathematical pricing models, careful option pricing requires data sets that identify anticipated days Of tlnklstlal VOlati1ity.23
For options on a given underlying asset on a given day, implied volatility generally varies across.option strikes and mamrities. Implied volatility also varies over time. As a result there is great interest in measuring volatilities and in plicing options when volatility can Valy
Methods of measuring volatility using past data include historical volatility, exponentially weighted moving average volatility, ARCH, GARCH, and realized quadratic variation. ARCH and GARCH estimates are based upon a formal statistical model in wilich volatility is random. Realized quadratic variation exploits high frequency data to obtain a reliable volatility estimate using data from a short time holizon. Both varance and volatility swaps permit hedging and speeulation on volatility. The valiance forward price can be obtained as a weighted sum of the plices of traded European options, acalctllation that is the basis for the VIX measure of implied volatility. The Black-scholes model does not perfectly explain observed option prices; there is volatility skem which means that implied volatility varies with the strike price and time to expiration. Two modiscations to the model are to permitjumps in thestock price and to allow volatility to be stochastic. Both changes generate option prices that exhibit volatility skew and that better tit the data than the unmodised Black-scholes model. ' Attempts to explain prices of traded options suggest that it is important to account for jumps in both the asset price and volatility, and that risk premiums on one or both jumps may be importantk
FURTHER
READING
Early studies of stock remrns (e.g., Fama, 1965) found that continuously compounded rettlrns exhibit too many large rettlnas to be consistent with normality. In recent years, research has focused on speciing stock price processes that give theoretical option prices consistent with observed prices. lntroductions to GARCH models include Royal Swedish Academy of Sciences (2003),and Bollerslev et a1. (1994).Alexander (2001)is a readable text for less technical readers. Realized quadratic variation as a measure of volatility is presented and applied in Andersen et al. (2003). Demete
et al.
(1999)present a clear and well-written discussion of volatility
hedging, and the paper also develops the volatility measure VIX. See also Chicago Board Options Exchange (2003).
used now to construct the
23This is not just an issue for individual firms. Governments make economic announcements on prespecihed days at set times. aftd these announcements sometmes generate large moves in pfkes. For example, Hanweck (1994)shows that implied volatility in Eurodollar futures options is greater on days when the government announces aggregate employment.
(t' (
. : r. .. . .
. -. . y... .
774
% VOLATI
' .
Ll'ry
PROBLEMS %.
tirst papers to suggest alternative assumptions about the stock price for option pricing were Cox and Ross (1976),Cox (1996),and Merton (1976). Merton noted in his paper that the jump model had the potential to explain volatility skew patterns noted The
by practitioners at the time. The first stochastic volatility models were proposed by Hull and White (1987),Scott (1987),and Wiggins (1987). The Heston (1993) model has been generalized by Duflie et al. (2000),who develop a prieing framework that can accomodate jumps in volatility as well as in the stock price. The empiricl literamre examining the ability of option pricing models to fit observed prices is rapidly evolving. Well-lnown papers include Bakshi et al. (1997),Bates (2000), and Pan (2002). Current research (whichinclude citations to numerous other papers) include Andersen et al. (2005) and Broadie et a1. (2004). Dubinsky and Johannes (2004)examine deterministic volatility changes, such as those due to earnings
775
a. Verify that the Black-scholes price is $50.0299. b. Verify that the vega for this option is almost zero. My is this so? c. Verify that if you compute the option price with volatilities ranging f'rm 0.05 to 1.00, you get essentially the same option price and vega remains about zero. Why is this so? What happens if you set tr 5.00 (i.e., d. What can you conclude about difculties in computing implied for very short-term, deep in-the-money options?
23.7. Use the same inputs as in the previous problem. Suppose that you observe a bid option price of $50 and an ask price of $50.10.
PROBLEMS For many of the srstffteen problems you will need to use data on the CD accoinpanying t is book. The b contains stock prices, option prices, and interest ratej for' kariety of maturities. If an interest rate you need is missing, use the rate fpr the nearest available
c. (Optional) Examine the code for the Bscalllnlpvol function. Explain why changing the starting volatilty can affect whether or not you obtain an
maturity.
23.2. Compute daily volatilities for 1991 through 2004 for IBM, Xerox, and the S&P 500 index. Annualize by multiplying by 252. How do your answers compare to those in Problem 12 23.3. For the peliod 1999-2004, using daily data,
compute
the following:
IBM'S volatility using all data. a. An EWMA stimate, with b 0.95, of 0.95, of IBM'S volatility, at each date b. An EWMA estimate, witl b of data. days previous 60 only the using =
=
Plot both estimates. How different are they?
23.4. Estimate a GARCHII, 1) for the S&P 500 index, using data from January 1999 to December 2003. 23.5. Replicate the GARCHII,I) estimation in Example 23.2, using daily rettlrns from estimates with on IBM from January 1999 to December 2003. Compare your four without the largest returns. and 23.6. Use the following inputs to compute the price of a European call option: S $100, K = $50, r = 0.06, o' 0.30, F = 0.01, J = 0. =
L
=
.
ty .j-
' . ..
.. ti g
)
1.
1
j
l
l 1 i
d.
=
=
=
j
l
j
interpreting =
$50,
=
a. Verify that the Black-scholes plice is zero. b. Verify that the vega for this option is zero. 'Why is this so? c. Suppose you observe a bid price of zero and an ask price of $0.05.What answers do you obtain when you compute implied volatility for these P
d. Why would market-makers set such prices? e. What can you conclude about difticulties in computing and interpreting implied volatility for very short-tenn, deep out-of-te-mney options? 23.9. Compute January 12 bid and ask volatilities (using the Black-scholes implied volatility function) for IBM options expiring January 17. For which options are you unable to compute a plausible implied volatility? Why? 23.10. Compute Janual'y 12 bid and ask volatilties (using the Black-scholes volatility function) for IBM options expiring February 21. a. Do you obselwe a volatility smile? b. For which options are you unable to compute a plausible implied ity? Why?
h '
t
23.8. Use the followinginputs to computethepriceof aEuropean call option: S K $100,?- 0.06, o' 0.30, F 0.01, J 0. =
'
1
ansWef.
What can you conclude about difculties in computing and implied volatilities for deep in-the-money options?
(q
volatility
a. Explain why you cannot compute an implied volatility for the bid price. b. Compute an implied volatility for the ask price, but be sure to set the initial volatility at 200% or greater. Explain why the implied volatility for the ask price is extremely large.
23.1. Using weekly price data (constructedWednesday io Wednesday), compute historical allnltal volatilities for-lBM, Xerox, and the S&P 500 index for 1991. through 2004. Annualize your answer by multiplying by 52. Also compute volatility for each for the entire peliod.
.
=
announcements.
-
..
y-
implied
volatil-
J
776
k
APPEN Dlx
VOLATILITY
23.11. Compute January 12 implied volatilides using the average of the bid and ask prices for IBM options expiring Febl'uary 21 (usethe Black-scholes implied volatility function). Compare your answers to those in the previous problem. Why might someone prefer to use implied volatilities based on the average of the bid and ask prices, rather than the bid and ask volatilities individually? 23.12.
ln this problem you will compute January 12 bid and ask volatilties (usingthe Black-scholes implied volatility functiop) for pne-year IBM options expiring the following January. Note that IBM pays a dividend in Marclt, June, September, and December. volatilities ignoling the dividend. a. Compute implied b. Take dividends into account using the discrete dividend correction to the Black Scholes formula, presented in Chapter 12. For simplicity, discount future dividends at a 2% continuously compounded rate. all How much difference does this correction make in implied volatility? .observed
adividend yieldformM based c. Take dividends into accountby computing this dividend yield on its annualized dividend rate as of January 12. Use in the Black-scholes model. How different are the implied volatilties from those you obtain in the previous part?
d. Do you observe a volatility smile? 23.13. For this problem, use the implied volatilities for the options expiring in January 2005, computed in the precedlng problem. Compare the implied vlatilties for calls and puts. Where is the difference largest? Why does this occur? 0. Use the Black30+, F 1, and 8%, o' 23.14. Suppose S $100, 1' ranging from with strikes the and call prices put generate formula to Scholes $40 to $250,with increments of $5. Compute the implied volatility from these 13.29).What happens to your prices by using the formula for tle VIX (equation estimate if you use strikes that differ by $1 or $10,or strikes that range only from $60 to $200? =
=
=
=
=
23.15. Explftin why the VIX formula in equation (23.29)overestimates implied volatility if options are American. The following three problems use the Merton jump formula. As a base = 8%, o' = 30%, F = 1, and = 0. Also assume case, assume S = $100, r and cg = 0.30. that = 0.02, a.l =
c. How is the implied 0.50?
volatility
plot affected by changing
23.A O'J
Q
777
to 0.10 or
23.17. Using the base case parameters, plot the implied volatility curve you obtain for the base case against that for the case wher there is a jump to zero, with the same 0'.20, and in part (b) consider expected 23.18. Repeat problem 23.16, except let a.t alternate jump magnimdes of 0.10 and 0.50. The following two problems both use the CEV option pricing formula. Assume in both that S $100,?' 8%, o'o 30%, F i= 1, and 0. .
=
=
=
=
=
23.19. Using the CEV option pricing model, set p 1 and generate option prices for strikes from 60 to 140, in increments of 5, for times to mtgrity of 0.25, 0.5, 1.0, and 2.0. Plot the resulting implied volatilties. (Thi should rproduce Figure 23.9.) =
23.20. Using the CEV option pricing model, set p 3 and generate option prices for strikes from 60 to 140, in increments of 5. for times to maturitf of 0.25, 0.5, 1.0, and 2.0. Plot the resulting implied volatilties. =
APPENDIX
23.A
Here is one way to set up aspreadsheetin
likelihood using Excel.
orderto estimate aGARCH modelby maximum
1. Enter daily prices in column B, beginning in B l0. 2. Compute continuously compounded returns for a period in column C, begins-yea.r ning in row 11. Leave cells A1:C7 empty. 3. In column D compute the squared continuously compounded be 6 2
return.
4. ln cell E1 1, enter the variance of the continuously be your starting value for q.
returns. This will
This will
.
compounded
5. ln cell E12, enter the formula =$B$1+$B$2*D12+$B$3*E1 1. Be sure to pay attention to which cells are absolute and which are relative references. Copy this formula down the length of your data.
-0.20
23.16. Using the Merton jump formula, generate an implied volatility plot for K 150. 50, 55, .
.
=
plot affected by changing a.i to
-.-0.40
or
-0.10:7
b. How is the implied volatility plot affected by chapging
=
7. Suppose that
.
volatility a. How is the implied
6. In cell F13, enter the formula lntf 13) D13/f' 13. Copy this formula down. This is your log-likelihood function for each observation.
to 0.01 or 0.052
-
-
'your
SUM(Fl3:F1200).
last return is in row 1200. ln cell B4, enter the formula This is the log-likelihood function for your data.
=
8. ln Solver, set up the following constraints: B 1 R: 0.0000001, Bl k) 0.00000001, #3 > 0, Bl S 0.99999999 S3. -
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% VOLATI
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11. Set up Solver to maximize cell 84 (the likelihood) by varying cells B 1:B3 (the parameters).
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ur goal in this chapter is to understand how to price derivatives that have bonds and . . rates rather than stocks a: the underlying asset We begin by seeing how the uterest ;;(,,yqk(.(t!E'((li(i;jj.E; ' '';EEEE i i yi ! .:..j (yj ( j sjxsscholes approach to option pricing, discussed in Chapter 21 applies to bonds. ' ('. Ei !.i;j.yjjy j.: yjy; jy .;jj.y j .y E ;!iEi; ri..,ry)q (4 y.yj j y ),tt,,,j.y, ;'''i'i!!t).... y As with stocks, there is a partial differential equgtion that characterizes the behavior ..iEEE'.',''.I'E(l!i;;.r'y.jjijEi..,:j,k!jyj,ygjy(yj .( jyyy . . . .j; ttyjtjy, . . j ojr jnoyz prjces and other functions of interest rates. The Vasicek and Cox-lngersoll.. ''...''.)'!;i';i.(':q'q.(qjijqgij.(y'kky;)yIyy;qj;.y'.yyyr'r.iy;yiy.j.,k(ll ' E i iil'j R. ,, oss mo de Is illustrate tlie p rocedure for deriving bond p rices from an assumed model jtyjyyltylq, (EE))Ij(il!:yj!,j: yjjjjj j. .'E.' j(ytjjjijj o tjye sjjoyoterm interest rate. Next we examine the pricing of bond and interest rate jyjjyyyyyy opons using the Black model (thename for the version of the Black-scholes model .. E. .E ! gl,,y jbr wlch the underlying asset is a futures contract). The Black model ! ! i ( . k ; g . ! k i ; yjjjr ;y. q.. .'.r.. !jy jy.E.:.qpi)Ir,,,;.!iI.,),r rjyjyjy. gj. jj assumes that .. ..E'''! jttttyr ':(...' j.(..!i.(jE)(;; . j ilyty, iE q interest rates are lognormally distributed and can be used to price interest rate j(j j.. smard E jytjil ( E ; ;i. oap s as well as bonds. Flally we examine binomial interest rate models, in particular .(.;..jj!q;);y (jjgjjjyjjj..,j yy .(r...y,.,!'(,E.Eij.!ljky!jjj;j;j;,E;.)ji; E ttje sjack-Dennan-rfoy model. .... ... li.)!. jj;jj;yjjj;jy!j.jI( ijj;tjgjj:i..j ! ; :.j! .k i;j.jj!!Ij.j.ijj.kk . .' ;E.i;..((i.iEi.y!.;!q i j.i ljr;jy .' .'r !.I.j(jj( yyj.., ljy : . .
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we examine market-maldng in bonds in order to better. understmad how
. the Black-scholes option pricing framework applies to bonds. We begin by examining . k .. qj(.!(jyyjjt;j;yyyy,).j;jy j.yjyy;jq.(yj.... ...,((;')j,'ir;;'.)j..i.y).j..k'.'t;;;.r..: the hedtring of bond EE...'.i!i'!'y.(!#iI:t;klyki'g2i:;;)ti)!k(!k..'y77,;l'.'.i( with another. one o ' ' E!i.j Er'pjrr jyjy.(gy of the option pricing model characterizes the fair op-j derivation . ..y.'...'q...(j..5!((.yr..ygi.j(jjg.jj.;rrjj.gr(.; Elijrt' . yj . . iEE --
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market-maker. Vasicek (1977)used the same approach for on price for a delta-hedoing o skcug. iil jnoyjds. 'i q rr t,@iiitt,iiq delta-hedcring bond portfolio manager, like the market-maker p :l i E y a E E; i@ o consider ;jj(;y y.( jjj.. ' ,@'ii;,ii()E;E)i2)tk)l)'r!k:j1y;;Ikt'j'tj;,jyj.g r.rpp; r:j.j., ryjjtyt,ljyl who delta-hedged options in Chaptrs 13 and 21 Speciscally, suppose the .. . manager ;.'i;(.''';i.!.!i(tj.j)it;)j!yr, y.jq(j.I(;(y; ! ryttyyjtl owns one bond witll maturity ra and hedges this bond by buying N bonds with maturity qg rjyj;yj(rjgyyy. jjjy. j.jyy. ... ..E,.;yyl'()i!;)))),)y:: . jj gy.!yy . . . using short-term bonds paying r. Hedg'.' ;'.Ei,'El)i',..,;r:(;j;.tirI(;jEt.;)jy;(Ejj'jk E ii. (! . t.jytyy.,t.q r 1 (x can be negative). The position is snanced ;y q. . y .... q).qjj(y;yp.!ty yjyjj(gj j.lytl,y,.t ing one bond with another is often called duration-hedging rather ihan delta-hedging. '.'.'iy'i,(q';.,!yyi')jy:,iky;!,.,;j,j,jyry:,jj,; EE ! : ilj i!!i!).(i. E j;ig .. q (''i'ti'.)'..@j;. vjwutentof duration- and delta-hedocringis the same, but as we will see, the two are . .. .jyy yy;ytrrrtj. i ( j p g enerally not the same if we use the standard de:nition of duration from Section 7.8. .;!r...j.y. . yljtjjyt (. ..E.',E,EEi.!,,!l,i;.jjrjtIt.)jk;t'j')li. . . ...(;iij!'rLI!I;.l.I)l@i? ljijl. of the Vasicek approach to pricing bonds is identical to the Black-scholes Tjw joctic j li IILLLLLLLLL, o i. (iijjjtjij ''E'.q'liltpjlpyijyjr?jijlljll.k 4r i.: . jlrytt !(!l approach analyzing options'. We think about the problem . faced by a market-maker to ( ! k jl yyyqj ! qE ' '..;,'''.!;iirill.l,2).pp)pIi,it,, yy . . . yjy..y . .'.i.. )j y! itktt .; . and what it tells ! about bond price ! ' will focus on pricing zero-coupon behavior. '.'.E'!,iiE,h.iq'!qyp,:#i(1,qj(j$t,;:I'! see us ij.i'kit'),E jj;( .y '.i (()j)j.jjrq we . . 7, they bonds since, as discussed in Chapter '.' '.''( (jj.2'i(jE(E jj. .;j.jyy y(jyq... building block for all sxed-income are a ! j qjij I ( ) (yy jIjyjj.j ;y ;!yj jjryg ..'.q..E;@E(!!.y'jiy;gy(()T.. ;iyyj rljjj;, q.; (114(2:1,j yj .yyy;;yj.jy :.ty . j ... :;(!kj)y. ;
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k. IXVEREST
RAVE MooEus
MARKET-MAKI
The Behavior of Bonds and Interest Rates Before discussing how a bond market-maker would delta-hedge, we Erst need to specify how bonds behave. Suppose we try to model a zero-coupon bond the same way we model a stock, by assuming that the bond price, Pt, F) follows an lt process:
dp
ar,
P =
(24.1)
t jdt + q (r, tjdz
In this equation, the coecients a and q cannot be constants and in fact must be modeled bond satisfies its boundary conditions. For example, that the carefully to ensure rather maturity Also, the volatility of the bond pric should be worth must bond the $1 at maturity-a given change in interest rates affects the the bond approaches decrease as thr of a short-lived bond. Neither of thse than price lopg-lived bond of price a more re:ected in equation is automatically (24.1). ln order to acommodate such restrictionj of the interst riit ad time. specised functions carefully and be ptttst behavir a An alterhatite to beginning with equation (24.1) is to model the behavior of the interest rate and solve for the bond price. If we follow this approach, the ond price will alttonlatically behave in an approp' liate way, as long as the iterest fat jiocess is
reasonable. Suppose we assume that the short-term interest rate follows the It process dl- t7 rldt + c' l-tdz (24.2) =
This equation for the behavior of the interest rate is general, in that the dlift and standard deviation are functions of r. Given eqpation (24.2),what is the bond plice? We will see that different bond price models arise from different versions of this interest rate process.
An Impossible
Model
Bond Priclg
We will srstlook at a bond pricing model that is intuitive, appealing in its simplicity, and widely used informally as a way to think about bonds. We will aisume that the yield curve is tlat; that is, at any point in time, zero-coupon bonds at al1 maturities have the same yield to maturity. lf the interest rate changes, yields for al1 bonds change uniformly so that the yield curve renaa'ins flat. Unfortunately. this model of the yield cul've gives lise to arbitrage opportunities. It can be instructive, however, to see what doesn't work in order to better appreciate what does. To analyze the flat-yield curve assumption, we assume that the interest rate follows equation (24.2).The price of zero-coupon bonds is given by .e-,.(w-J)
p
(p,y a)
t, w)
,
=
N P t Fl ) + P t F2) + F ,
By It's Lemma, and using the fonnula for the bond price equation (24.3),the in the value of the portfolio is Nd P t Fl ) + d P t F2) + d J'P' dl =
,
= N + .
(F2
,
-
-
tlpt,
FI)#r
j (Fl 2
+
t) P t, F?,)#?- +
-
t) atr z Pt,
-
=
change
Fl ldt +
?'#(r,
Fl ldt
Tzldt +
l-pt,
Fc)#/
(24.6)
'
j
(F2
-
2
,)2c'2#(/
-
,
+ rvdt
We pick N to eliminate the effect of interestrate changes, 611., on the value of the portfolio. Thus, we set (F2 1) P t Fa) N = (24.7) (F, t) P t, Fl ) The delta-hedged portfolio has no tisk and no investment', it should therefore earn zero: -
,
-
-
dl Combining equations
=
0
(24.8)
(24.4),(24.6),(24.7),and (24.8),and then simplifying, gives us 1 (F2 Tl )c'2 O (24 9) -
2
=
.
This equation cannot hold unless FI = T2. Thus, we conclude that the l7tp?/Jvalltati' ilnplied %'eqltations (24.2)alld (24.3)is ilnpossible, in the sense that arbitrage ???t7#c! possible if the yield curve is stochastic and always iat. is This exnmple demonstrates Jhe diculties of bond pricing: A casually specifed model may give rise to arbitrage oppo>nities. A crucial feature of bond prices is the' nonlinearity of prices as a function of interest rates. a characteristic implicitly ignored in equation (24.3).The same issue arises in pricing stock options: The nonlinearity of the option price with respect to the stock price is critical in pricing options. This is another example of Jensen's inequatity. The example also illustrates that, in general, hedgillg a bond polfolio based on #lt/'drt?ll does ?lt?l result l a pelfect hedge. Recall that the duration of a zero-coupon bond is te bond , s time to mattlrity. hedge ratio, equation (24.7),is exactly the same as eqution (7.13)in Chapter7. The use of duration to compute hedge ratios assumes that the yield to matul'ity of a1lbonds shifts by the same amount, which is what we assumed in equation (24.3).However, this assumption gives rise to arbitrage opportunities. The use of duration to compute hedge ratios can be a useful approximation', however, bonds in equilibrium mllst be priced in such a way that duration-based hedeing does not work 'l'he
W
exactly.
=
0
An Equilibrium Equation for Bonds Let's consider again the bond-hedging problem, only this time we will not assume a particular bond pricing model. Instead we view the bond as a general function of the short-term interest rate, r, which follows equation (24y2).1
Since J1/ is invested in short-term bonds, we have
d I/P
781
,
-(Fl -
%.
-
In this specifcation, every bond has yield to maturity ?'. We now anlyze the delta-hedging problem. lf we buy one bond maturing at time by buying N bonds maturing at time FI and fnace th difference at the hedge F2, short-term interest rate, the bond portfolio has value I
N G AN D Bo N D PR1 CI N G
?' Wdt
l'rhediscussion in this
sect-lon follows Vasicek-(1977)-
782
k
INTEREST
RATE MODELS
MARKET-MAKI
First, let's see how the bond behaves. From It's Lemma, the bond, wlzich is a function of the interest rate and time, follows the process 1)# 1 lP l# dt d? + #?. + dpr, t, T) = ( a )/' 2 alol.)2
-
=
t7(?')
p# pr
1 :2#
+
cll'j z +
-
)?-Z
2
-
ar,
t,
r)
q (r, /', F)
=
a (r)
P (r, t, F)
#
l
=
G
P (h.t, T) 1),-
We can now rewriie equation
dt +
ot
p# Pr
(24.10)
ctrllz
'P 1 02# .2 + cr (?) + l)/' nl- 2 l- n
p'
(24.11)
-
(24.12)
(r)
' #(r, t,
'
(24.13)
By using equations (24.11)and (24.12)to define a and q, equations (24.1)and (24.13) are the same. Note that a and q depend on both the interest rate and on the time to maturity of the bond. Now we consider again the delta-hedged bond portfolio, the value of which is given by equation (24.4). From It's tzemma,we have
ln order to eliminate interest rate risk, N=Note that by using the delition
we set
P (r, t F2) q (r, t F2)
rl ) q (r, t, ) of q, equation (24.12),this can be rewritten J'r (?',t, ra) N=Pr r, t, Fl ) FI
P (?-,t,
lf you compare this expression to equation (7.13),you will see that #r(r, t, F) replaces duration when computing the hedge ratio, N. Substituting equation (24.15)into equation (24.14),and setting dl 0 (equation (24.8)), we obtain =
-
,
,
Substituting equations
(24.11) and (24.12)for
1)2# 1 2 l ) + Et'l(? ) -o. )?-:t 2
.)4(/.
t)- l
,
(24.17)
a and q then gives us
pP
/.)j
pP
..).
-
)?-
ot
rp
.()
(24.18)
When the short-terminterest rate is the only souree of uncertainty, tlispartial ltf/rnzlf eqttation ???I/./ be satisjed /J.)/ al' zero-coupon bond. Different bonds will have different maturity dates and therefore different boundary conditions. All bopds solve the same
PDE, however. The Black-scholes equation, equation (21.11),characterizes claims that are aftmction of the stockprice. Equation (24.18)is the analogous equation forderivative claims that are a function pf the interest rate. A diference between equation (24.18)and equation (21 1) is the explicit appearance of the risk premium, c (r, /)/(?', t), in te bond equation. Let's talk about why that
?-
=
a
(1-,t,
T2)
-
r
(24.16)
q (r, t F2) ratio the /5$// bollds is equal. Since both bond This equation says that the 5'/7t7//, random prices are driven by the same term, #Z, they must have the same Sharpe ratio if they are fairly priced. (We demonstrated this proposition in Chapter 20.) ,
w/b?-
,
happens. ln the context of stock options, the Black-scloles problem entails hedging an option with a stock, which is an investment asset. The stock is expcted to earn its risk premium, which we will call $2c. Thus, for the stock, the dlift term, which is analogous to c(l-), eqtmls '' + 4/(r The Black-scholes delta-hedging procedur eliminates the risk premium on the stock. By subtrqcting the risk premium, we are left with the lisk-free rate, r, as a coeficient on the ) V/OS term in equation (21.11). Fc ltet-est rate, /J.: contrast, is ,1:)1 the price of (kpl westment asset. The interest rate is a characteristic of an asset, not an asset by itself. The risk-neutral process for the interest rate is obtained by subtracting the risk premium from the drif't. The lisk-neutral process for the interest rate is therefore .
dl-
,
,
FI ) a (r, t Fl t (r, ) q
783
.1
et?-, t, Flt'/r + t.?(?-, t, F)#Z
=
F)
$
-
'
-
(24.10)as
dpr t F)
Denote the Sharpe ratio for #Z as /(r, l). For any bond we then have ar ' t F) ?' = 4(?-,t) qlh F)
.
(24.1),but we can desne terms so that it does.
This equation does not look like equation Let
1
-
p#
NG AN D BON D PRICI NG
=
(t-l(r) -
(7-(?-)4(/-,
tljdt
.#
o'rldz
(24.19) .
The drift in this equation is what appears in equation (24.18).You can also confrm that equation (24.18)is the same as equation (24.17). Given a zero-coupon bond (whichhas a terminal boundary condition that the bond is worth $1 at mattlrity), Cox et al. (l985b) show that the solution to equation (24.18)is
PLt T ,
'
r(/.)q
=
E*J
-&tJ'rlj
(24.20)
where E* represents the expectation taken with respect to risk-neutral probabilities and Rt, F) is the random variable representing the cumulative interest rate over time: T
.R(/, F)
(24.21)
=
I
Thus, to value a zero-coupon bond, we take the expectation over all the discount factors implied by these paths. We will see the discrete time analogue of this equation when we examine binomial models.
784
%.INTEREST
EQu I Ll BRIU M SFI ORT-RATE
RATE MODELS
the bond payoff Keep in mind that it is ltot correct to value the bond by discounting
by the average interest rate,
#
=
E* g.R(/, F)):
Pt, T,
?-)
+ e-k
on single-variable models in this chapter.
Delta-Gamma Approximations for Bonds lt's Lemma, One interpretation of equation (24.18)is familiar from Chapter 21. Using of the interest distribtion risk-neutral the under bond plice in the change the expected rate, equation (24.19),is DP p/7 82# 1 l ?)j o-ll-jlr, (J(r) a + E*(dPj = + )?-Z )?-f 2 nt .
-crf?-l
-
-
-
'
-
Equation (24.18)therefore says that 1 -f*(##) dt
=
MODELS
Q
785
We discussed bond duration and convexity in Chapter 7. For a zero-coupon bond, dtlration is time to maturity and convexity is squared time to maturity. Conceptually it seems as if duration should be the delta of a bond and convexity should be gamma.
However, this is true only in the bond plicing model of equation (24.3). For any correct bond plicing model, duration and convexity will be different than Prjp and Prr/P. We will see examples of this in the next section. 'impossible''
different Beeause of Jensen's inequality, this seemingly reasonable procedure gives a equation (24.20). bond price than details Different bond price models solve equation (24.20),differing only in the lisk premium. the of modeling and the of how ?' behaves with apodel of To summarize, a consistent approach to modeling bonds is to begin differential equation that the interest rate and then use equation (24.18)to obtain a partial eqation), Blackcholes the the really is equation as same describes the bond price (this conditions, boundary with together the PDE Using time-varying interest rate. but with a lf this seems familiar, it should: It is exactly the we can determine the price of the bond. procedure we used to price options on stock. of The derivation of equation (24.18)assumes that bond prices are a ftlnction a allow bond prices single state variable, the short-term interest rate r. lt is possible to empirical for having bond support is and there variables, additional state to depend on estiprices depend on more tlan one state variable. Litterman and Scheinkman (1991) that a three-factor model typically mate afactor model forrfreasury bond remrns and find bond's variability return. They identify the three in of the than a 95% explains more single most imporfactors as level, steepness, and curvature of the yield cuve. The of the movement in l'ates, 90% almost for interest accounts of level tant factor, te of interet rates explains why level of the ovemhelming importance The bond returns. duration-based hedging, despite its conceptual problems, is widely used. We will focus
-
BON D PRICE
'
'
-
(24.22)
/-#
lisk-neutral distribution, This is the same as equation (21.31)for options'. Using the lisk-free rate. bonds are pliced to eal'n the The fact that bonds satisfy equation (24.22)means thaq as in Chapter 13, the deltajp/creur approxnatiollfor #7echaltge n a bondprice holds awc//y gtw?7//lcl-lpM exactly the bond for are not a rate ???tp1?,u one standall dejtiation. However, the Greeks convexity. same as duration and 'fthe
24.2 EQUILIBRIUM PRICE
SHORT-RATE
BOND
MODELS
ln this section we discuss several bond pricing models based on equition (24.18),in which all bond prices are driven by the short-term interest rate, r. The three pricing models we discuss Rendleman-Bartter, Vasicek, and Cox-lngersoll-lkoss--differ in teir specifcation of J(?'), 0'(?'), and $r). These differences can result in very different P
Iicing
implications.
The Rendelma' n-Bartter Model The simplest models of the short-term interest rate are those in which the interest rate follows arithmetic or geometric Brownian motio. For example, we could write
#?'
=
adt + O'CIZ
ln this specihcation, the short-rate is normally distributed with mean o'lt. There are several objections to this model: ance
(24.23) ?'a
+ at and vari-
shorto The short-rate can be negative. lt is not reasonable to think the llonlillal would cash be since if it investors negative, prefer holding under can rate were, a mattress to holding bonds. . @The drift in the short-rate is constant. J.f a > 0, for example, the short-rate will, drift up over time forever. ln practice if the short-rate lises, we expect it to fall; i.e., it is lneal-revertillg.
@The volatility of the short-rate is the same whether the rate is high or low. ln practice, we expect the short-rate to be more volatile if rates are high. 'l''heRendleman and Bartter (1980)model, by contrast, assums that the short-rate follows geometric Brownian motion: dr
=
at.dt + o'rdz
(24.24)
While interest rates can never be negative in this model, one objection to equation (24.24) is that interest rates can be arbitrmily high. ln practice we would expect rates to exhibit mean reversion', if rates are high, we expect them on average to decrease. The Rendleman-Bartter model, on the other hand, says that the probability of rates going up or down is the same whether rates are 100% or 1%.
%:INTEREST
786
'.
EQUI LI BRI UM SHORT-RATE
RATE MODELS
--jjjiilijkky' --' -' :'')' '''''--?'''';;iii..f -l::::::f (S'E -y)' dl!!!!,,f ,j:::::' j;:!,,f ;'-y' '(y' qjf )' E ill:jilll.d llij!i:d dl:!i!,f dl:!!!!,f (' ' EE. : (E jjE E' j').' yl):f ,'y'-' y' y..y' ;'y' y'. r' qltlE: i E .:. ' E ''':'E (::ziiij-..:-'''!jlll);lp.d .iI222!:' t'':';'''q' Ir..'f y' dl:::E::f t,f (' yy' E (E.' 'E . .: .'. ' tE ''.;;!;' (q' ..kyi (E.(E;(;'j!( jq.;.(.''.(.'.'(..E''''-'.'(.''..q.((': .. : ii.. :rr':.E(..k:((. ... ..! 'y.(E..!.E. .. . r(('.' ( .. . - i - ' -. -E.. . . ..E-. .- .jE E . .E . .:;.; ,. . q.
E( ' '. ' E ( '7* jyf r' r'rf .11:::* jjjjjj!jf kj'i ygf ''r( q(Eyj jtf jyf jjq:yg:,,f j'(' 117-$::.* jjjkg!j;,j;f llr...:l.f tjf rs'yyj ;'tr'(yyy' )' l'tt-r y.;')' ( . (EE.. j (..g(.' ' @' (;;;'. 'yk-kj:jjjjrj'. ;'rt-r;(j(r:jr'. 'j qq ; .q -..y. E.;-.y.'.-. E.'. 1. ' :'r'jy.' )ii.(r. .?-).. '. . 'y.j2jj;jjjjj;. 'ii..('EErq;li1 ; g.r . . y- ...(y j. (kjj-yr y()r. .jyy ..yyyjy.jy....y.yjjyy..jrjyy-y.. yj.. ((. : j y..y . jjjjj-k;k,. ;yyyy jjyyts l.'.' yy;y,jy. .. .jt.j.. ... .y.rt;,ijjjj. . .. , ..... . . ....j.. -t;'' '(' '''
t')'r'i
.' . -
.
.
. - -
-
........
'' '
.
. .
' '
-
.
. .
.
.
- .
. . -.
.-
...
. .
. .
''y; '( ''
'
'.
.
..
.
.
. .
.
:11.
..
''r
:
. .
.
:
'. '
'
.
.
. .
.
.
.
...
.
.
. .
..
,
....
...
.- -
,
-
.
-,y .
-
-, .
-
-
. --.
-
.. -
-. . ..
.
., -
''j ... .j
.
.
-
- . ..
.
-
,1:1,,,. jjyyyg, Ir'... Il'...pl.s iiie .1Ir::. .1It::. ,!(((2II .tIk::.ii'Ir...!.l .1It::. k:,,,,:y, .,1t2:, ii'Il'...rli ,I::::rI1. li....:ll e e I!iii;. a e iiiIr....,'. a I;....,l-
model. Msumes a
=
0.2, b
=
0.1, and
o'
=
0.01
1 a lP tr )?-a + Lab
.
i'
(IIi2r:1IL. iliEi!j,r ljjijj!;;q i:jq!ilt. 'rlllj'illi. !. (.... y'. (i.. qq (. (...y.. .. (qllliil::;. . lljjtk- :tii. 1i iEgik qigl. tt.'l.. (k kkk .. ;.) ..' !li) 1lj qlj IjEFjZI .j iz ilsj:.'. 11) )' #..klr)l tj. kjiii w.'' 0.01 5% E E' ' .. -. i .. ;. q....- . .( . . ..E. -
E .
.
.
'.k
10% l5% 20%
;' .'. .. .... . .y .
. ... . . .
-. .
.
E. .. -
.
.
't
' E ' . . . .. E ... .- . : . .- g. E . . . . 't..
.E
!11!E!2E'
,111:(((((2:2.
..
.
.).
r)
-
p.p
c'/)
-
p?-
')... ... E .
P g/,,F,
0
-(/.))
?
At
=
$
787
bP
?'#
-
at
=
0
subject to the boundm'y condition
r),-f('.F)r(?)
(24 26) .
where
-0.01
-0.02
At, F)
=
F)
=
Bt
e
FBt,Tt+t-T)-Blo'lj4tt
(1
e-*l--blja
-
-
0.5c2/472
with being the yield to maturity on an infnitely lived bond. FJ
The Vasicek Model kasicekmodel
MODELS
,
P = b + c4/c
The
+
The bond price fonnula that solves this equation #(F, r, r) = 1, and assuming a /: 0, is2
.E - .E E .
. .
). E.( E.
PRICE
(24.25), equation (24.18)becomes
E
-. -y-
-
..
-. - -.y .
BOND
incorporates mean reversion:
dr
=
t7
b
-
rltlt
(24.25)
+ o-dz
rldt terpz induces This is an Ornstein-uhlenbeck process (see Chapter 20). The ab = = These parameters 1%. 10%, and. 20%, b o' mean reversion. Suppose we set a basis points. The is 100 short-rate for the one-standard-deviation imply that a move > short-rate th If b, interest which shortlterm level revert. rates r to parameter b is the illpstrates rise. Table is expected 24.1 < 1the short-rate If to /p, is expected to decrease. mean reversion. The parameter a re:ects the speed with which the interest rate adjusts to b. lf = 0, then the short-rate is a random walk. lf a = 1, the gap between the short-rate a 20+, we exlect the rte to decrease in and b is expeeted to be closed in a year. If a 20% of by the the first year gap. N,ote also that the term multiplying #z is simply o', independent of the level of the interest rate. This fonaaulationimplies that it is possible for interest rates to become negative and that the variability of interest rates is independent of the level of rates. In the Rendleman-Brtter model, interst rates cold not be pegative because both (jj the mean and variance in that model are proportional to tjye level of e j n (erest rate. Thus, as the short-rate approaches zero, both the mean and variace also approach zero, and it is never possible for the rate to fall below zero. In the Vasicek model, by contrast, rates can become negative because the variance does not vanish as 1' approaches zero. Why would anyone construct a model that permitted negative interest rates? Vasicekusedequation (24.25)to illustrate the more general pricing methodology outlined in Section 24.1, not because it was a compelling empirical description of interest rates. The Vasicek model does in fact have some unreasonable pricing implications, in pallicular negative yields fot long-term bonds. We can solve for the priee of a pure discount bond in the V.asicek model. Let the Sharpe ratio for interest rate risk be /. With the Vasicek interest rate dynamics, equation -
The Cox-tngersoll-Ross Model The Cox-lgersoll-lkoss rate model of the form
(C1R) model (Cox et a1., 198518 assumes a short-term interest
=
=
dr
ab
=
rldt
-
+ cvflz
(24.27)
The variance of the interest rate is proportional to the square root of the interest rate, instead of being constant as in the Vasicek model. Beause of this subtle difference, the CIR model iatisties all the objections to the earlier models: 0, the drift in the rate is @It is impossible for interest rates to be negative: If ?positive and the variance is zero, so the rate will become positive. =
* As the *
jhort-rate
rises, the volatility of the short-rate also rises.
short-rate exhibits mean reversion.
'l'he
The asjttmption iha t the variance is proportional to V-/also t'tlrfls out to be convenient analytially Cox, Ingersoll, and Ross (C1R) derive bond and option plicing using this model. Th lisk prrium formulas in the Cm model taked the form
4(,-,r) /,/--r/(y =
z'When a
0, the solution s equation (24.26),with 0.5rz4(r-J)2+tr2(r-J)3/6
:4 B
When a
=
e -
F
-
t
0 the interest rate follows a random walk; therefore, F is undelined-
(24.28)
788
%.INTEREST
RATE MODELS
EQU I LI B Rl U M SH O RT-RATE
With this specification for the risk premium and equation (24.27),the C1Rinterest rate dynamics, the partial differential equation for the bond price is 1)2P 1 -0-2,+ g/t/p :/-2
.
pP
-
?-)
-
2
-
r/)
+
Dl-
pP -
)r
?-'
=
0
The Cllk bond price looks similar to that for the Vasicek dynamics, equation with A(/, F) and Bt, Tj dehned differently:
#(/, F,
?-(r)j
=
At, T)e
(24.26),but
'j' j'(' .t'. t'))!:(T(' 'yjf t;f Eptjrtf (' .jjyy' gjjyy;jj,.;jjd y'g r' jgf jqf jjj'jjy (td y:lf );' .j' j.' j'y'y' jjjd jjyyjj---r'yjjy. yjj.d (' )' ytjf r' tqf ;)' p' k' 'y('. jt(jt' yjf j))'(j;'. t';',,t#'g;'. q'. y' (tf l'q q' E'''' ( ' E (.(j' y ( ' E' ' . E'' '. ; (.' E(. Ejj (g;'ij' yq!(qj: (E'.(ij 'E.''' . . . .(( r . . j.. j g. . . y)( ('qy('j7y(('!((q( . . tj 'E.(.'' j(g.'.j)(j .' j. !' ql'l!l!ryrq :'E E..Ei jEEi! . jE. E.ii.. E-.. EE.E E!. . . !.1.E. yE .. E . .) .f.; . )! . . . .I.. . . .... ..qq.gj.!...ri .. .q.E;. . .j: ! j ; . !E(gq. .jy. qy.iq! .(.$. E. E . y j., q y..y q 2 ; . r . . 1I1Ik,--1III. . y.( . E E gy... yyg.yyj.yg ..jyy. . jy. .)j.. E g.yjg ;..y . .. .LLkk ).. ;..y ....... .: . .. :...
.'.'y. ' .. . . . ..
y''l!I::::11kr-. )(' (' );' t('. tt;'r''ilII:.-' 11IEiEiE,'. 1111F14!7*t''' tjg.' r' ttt'. .. ri . ttj!. . y yr . k-y@.. :-1:::1E!1:... .j y.rqj !IIi::;lit:. . 1111). ;;;kq))-)))-. . . ' ' . . . . . . .. . . . .. . Yieid curves implied by the Vasicek and CIR models, assuming that 0 * 05 a 0 2 r b 0.1 In the top panel, c 0.02 in the Vasicek model and c 0.02/ 0.05 = 0.0894 in the CIR model. In the bottom panel, o' 0.10 in the Vasicek model and o' 0.10/ 0.05 0.447 in the ClR model. In aII 0. cases, $
'. *(). ''::
:
:
:
.
-
: !'
i
'
:. ..
.
(,4. r,lj)) .
where
: ' ' ' ::
.
- E.
.
.
.. : .. . . .
:
.
'
.
:
.
-
..
-
.. . . . .
-
.:
:
..
.
-.
. :
' :...
'' :
:
.
-
.
.
. .
'''
.
-
..
..
:'
:. --.. -. . .
.
---.
. . . . .. .. .. . .
--
.
:.
: .- : .- E . . y
-. ..
zablo.l
A t, F)
=
2Y -
a+ $+
)z)(dvtT-')
p,(,1z(T'-/)
Bt, F) ?=
=
-
a+
$
-
1) + 2/
+ 1z)(c7?(7'-''
.
789
.
- . .
.,,jj(j,,.
jjjs
.. .
.
.jj)rggyj;,,,, kyyyjjj rjry,.
jjrjyrr
clR
() (s '
.x.
0.08
vasicek
xv..=x>xx>>.x<>,,>>'*'*'.* -.--w.-.<-
'-
e--e
...''
'
O 07 '
=
=
1) + 2/
=
a + $-)2 + 2&2
With the CIR process, the yield on a long-term bond approaches the value F $ + )z) as time t mattlrity goes to insnity.
0.06 o.t)s0
l
S
10
15
20
Maturity
25
(years)
=
zabja
Yield to Maturity o ork -
---.......
0.05 . 0.04
*.... ''...
*.
0 . 03
+
*>.
e** *.
**.
0.02
cIR =u. Vasice k
0-01 ()
0
*a.
Comparing Vasicek and Cl'l How different are the prices generated by the Cllk and Vasicek models? What is the role of the different vriance specifications in the two models? Figure 24.1 illustrates the yield curves generated by the Vasicek and by the Cllk models, assuming tat the current short-term rate, r, is 5%, a = 0.2 and b = 10%. Volatility in the Vasicek model is 2% in the top panel and 10% in the bottom panel. The volatility, tr, has a different interpretation in each model. In the Vasicek model, volatility is absolute, whereas in the Cl.tkmodel, volatility is scaled by the square root of the current interest rate: To make the Cm volatility comparable at the initial interest o'clR.x/-?' = cvasicek, Or 0.0894 in the top panel and 0.447 in the rate, it is set so that lisk premigm is assumed to be kero. interest rate bottor panel. eyhibit The two mpdels can vel'y different behavior. The jjotp!u pane j jjas a voltllity. For short-term bcinds-with a mattllity extendine to about 2.5 relatively high years-the yield curves look similar. Tlzis is a result of setting the Cllk volatility to match the Vasicek volatility. Beyond that point the two diverge, with Vasicek yields whereas below Cllk yields. The long-run interest rate in the Vasicek model is Vasicek 24.1 the in Figure evident is difference 0.0463. as This that in the Cllt model is approaching yields approach zero (in the long run W'hat accounts for the difference in medium to long-tenn bonds? As discussed earlier, the pricing formulas are based on averages of interest rate paths, as in equation (24.20). Some of the interest paths in the Vasicek model will be negative. Although 'he
-0.025,
*
*
....
**>.
**4..... vv...os....p..j
I
1
5
10
I
1
15 Maturity
-0.025).
%
=
j.) -
ELS
.
.
..
*
=
-
.
. :. :: :: . . . : -...
-
=
e(t'+i+p)(F-J)/2
M OD
' : . .
.
-:
.
-. .
=
#
'
''
' ' :
--i .
.
:
.
.
.. .
=
=
-s(J,w)r(l)
BON D PRl CE
-'','-;' .2:166333*. jj:yyjjjd --' -' ;'' IEIIX :')' i-f y' ''jy:yy;j,d r' 5* yrf j' F'
0
25
(years)
the typical path will be positive because of mean reverjion-rates will be pulled toward will be paths oh which rates are negative. Because of Jensen's inequality, lol-there these paths will be disproportionately important. Over sufciently long horizoris, large negative interest rates become more likely and this leads to negative yields. I the Ctlt modelvthis effect reslts in the long-rtln yield decreas tng w ith volatility. Negative yields are impossible in the Ctlk model, however, since the short-tenn interest rate can never become negativ. ln the top panel, withrelatively low volatility, both yieldcurves are upward sloping. q
tn
The effect of mean reversion outweighs that of volatility. the long run, the Vasicek yield exceeds te Cllt yield because volatility increases with the level of the ipterest rate in the Cllk model. Consequentl th Jensen's inequality effect more pronounced in the Cm model than in the Vasicek model. We mentioned earlier that hedging in the context of this ldnd of interest rate model is different frpm duration hedging. ln the C1R and Vasicek models, delta and gamma for a zero-coupon bond are based on the change in the short-tenu rate. The following example illustrates that the resulting hedge ratios can differ from duration and convexity as traditionally measured.
ts
'
% INTEREST
790
Bomo Opn-loxs,
RATE MODELS
'('
Consider a s-yearzero-coupon bond priced using the C1R model, 24.1 (',1Exmple )l' 0.2. The ond prie 0.08, $ 0, and tr 0.1, 1, 0.2, b )). and suppose that a .j.,! . lli is $0.667. Because it is a s-yearzero-coupon bond, duration is 5 and convexity is 25. 5.5 18. 1 l 8 and Prr '1i)However, in the C1R model with these parameters, Pr and 5) of -prj.P 2.876 (instead short-term sensitivities the rate The implie d are to ' jtl Prrlp % 8.273 (insteadof 25). .-..
=
=
=
=
=
CF,
...
.9
=
li' pi')
.
,
t
jE
=
-
AND THE BLACK MODEL
If we assume that the bond forward price is lognormally distributed with volatility c, we obtain the Black formula for a bond option: a
''r
(: '
CAPS,
,17(0,
F), o', T)
.P(0, F)
=
FNtlLj
KNdzt
-
%
791
constant
(24.32)
..
.
=
where
=
,-/1 =
dz
CAPS, MODEL BLACK THE AND
distributed.
We will begin by seeing how the Black model can be used to price an option on a time-l price of a zerozero-coupon bond. As in Chapter 7, let Pt (F, F + s) denote the = F, time lf then Pv (F, F +J) is r +s.J) t coupon bond purchased at F and paying $1 at Without a subscript. lf t < F, the spot price of the bond and we will write #(F, F + will also which represent as Ft.vLPLT, T + we then Pt (F, F + is a fonvard price, expiring strike at time r, on a zero-coupo price K, Consider a call option with option this payoff of at time F is bond paying $1 at time F + s. The ')).
.)
=
mtxgo, #(F, T +
(24.30)
f-j
u) -
We can price this option as an exchange option (see Sections 14.6 and 22.6). Recall that the time-r forward price of the bond deliverable at F is F:,vLPT,
T' +
.)1
=
Pt,
T +
(24.31)
T)
sj/pt,
x P (r, F) 'l'he prepaid forward plice of this bond at time t is Fr,w(#(T', F + The time-l mat-ulity bond. Pt, T + s), which is just the time-r price of the F + s prepaid forward for the strike price is Kpt, T). The appropriate volatility for pricing this exchange ption is the volatility of the ratio of the prepaid forward prices for the undrlying asset nd strike asset: .))
=
Var(ln(#(r,
F + s)/KPt,
o,.i
-
and where F is an abbreviation for the bond fonvard price &,w(#(F, F + J)1. Since P(0, F + s), this formula simply uses the price of the F + bond #(0, F)F s as the underlying asset. The formula for a put can be obtained by put-call pality. This use of the Black formula to price bond options is intuitively reasonable. The price of any particular bond varies over time. However, the value of a bond option depends upon the volatilit'y of the ratio in the prices of bonds with different maturities. If today is time t and the option expires at time T the interest rate from time t to time F affcts the dlscounting of bth the underlying asset (thebond miring at time F + ad the jttike Irice (fromtime t t time T). Since the option price depends on F)),. only the volatility of the bond forwardpl-ice affts ilie price ln(#(r, F + sljpt, of the option. T'he Blck formula can be extended to price options on interest rates. lmgine that a :oating rate bonrwer wishes to hedge the interest rate at time F fr loan with time to mamrity s (thereforematqring at time F + s). We saw in Chapter 7 that the foovard interest rate from time F to time F + s, 1%(F, r + s), is =
We encountered the Black formula for pricing options on f'uttlres in Chapter 12. In this section we se how to use the Black model roprice interest rate and bond optios. The idea behind using the Black model in ihis context is that the forward price for this forward price is lognormallf a bond is the underlying asset, and we assume that
Call option payoff
cJF dk
=
24.3 BOND OPTIONS,
ln(F/A-) + 0.5.27-
F)j)
=
=
Vartlnlpl/,
F)J
F + s)(Pt,
Var(1n(Ff,r(#
(r,
F +
*''
''
*'
7
'
.
7
.)
Rfj',
T+
')
=
pp, w) #(0, F +
-
,)
1
(24.33)
Notice that in quation (24.33),R is not annualized. lf you invest $1 at time F at the fol'war rate, af4er s peliods yu will have 1 + Rtln, T + s). One way for th borrower to hedge interest rate risk is by entering into forward a rate agreement (F1kA), receivig at time F + s the difference between the spot and the forward rate, R(j(F, rate, Rr(F, F + r+ x-period
.),
.):
Payoff to FRA
=
Rpln, T +
J)
-
&(F,
.)
F +
As an alternative to he dging w ith an FRA the borrower could enter into a call option on an FRA, with sttike price KR. 'I'his option, which is also called a caplet, at time F + s ,
'))))
That is, the volatility that enters the pricing formula is the volatility of the fonvard price for the bond, where the forward contract calls for time-r delivet'y of the bond matuling at F + s.
aNote that we can write the option in terms of the bond prices as #(0, r -sjNdt ) KPCQ,T)Ndz), where dj (1n(#(0,F +sj(KPQ, T')) +0.5o-2F)/c Writing the formula aj in equation (24.32) r. emphasizes that the relevant volatility is that of the forward bond price. -
=
%.INTEREST
792
RA'I'E MODELS
A -'LLf.jL )' r $j))'. Lk-'.Likt''t-;k (,11(:;1,:.* k::;;2!!!::*-* k' jj'j q' )' ir!f 1I1Ii'-' t'.' y(' yyt'jy'jj 1*j'j'. rjrf r'qf ' r' t'('y'y' r)ll-i::::f r:l:!-'-';ir.f tyf '.' .(. j j j.tjjjjjj i. jj--k::iIid1i.' ).! j..y j.jj ..jj. j.jyj tj'. I1I!:itiIii.. 1I1iiiE!!.. ;.yjg( !9iI::::;2i!.. 1i1IFiiFi. ))'. .@(i ' . . .'. . . . . ... .r. . . . . ..... :''i11k-2-.i1p'. i')' (' y:'tE jjd ;'!' Ipd j'j' r' 1* ;l7T' q' ' E :'F' .ll:.:)lr.f i;!!l..f jjyyjy'tyyy'jj jyd yykd )' jjjd yyjf y' 5* q' 'iJ' r' '' : i!i i '' : .' E(':' ::'.''. i '(E jltl'fttr; ?' jiizf (' y' t'lyf )' 7rr8:7:r* ;'(@' lit':)f q' 'j'E (@: q.tf ;' (E!Ei;'I.j '(: (if:'E(i ll!t jjr E: )@' ((.!E 'E'.E.'E( E.i'' ! j.' 'r''. ('.j'E.j)i ((.!.. . . !i (.'..(. .' ': : : EE (i!;E. 11qq441:1 .'.. 'j ( . . g .tjj E! E ( !2E( : (. j:; : .E Eiii(: E i !.;(;; i . ;y .(y''ii( ijjg y ( . j .. ( g.. g y j, . : . ;g yyg...jy j:g... y yj. gy g g . . .. .; ! j j j jg j j. g.j.j gjj . 'E' y --. j -' :..- i-.-- .- -. -.--- - -. @- -.-.:g-.g i;.. ! E. :L...;. :: . ..-. . ...-.q . . . . (t. ---)-,'. .'-'')*:!!:74::7:'7777)/:* .....' .-' ----' -' t','!E'(;' )' -' ';'k'. -lki:,!f -'-' y' '''pt'iE : )! yttE E '' .'.
pAys
tFX ''. '
.'' .
.
'
:
.
.
=
maxlo, Rw(F, F +
KRj
.) -
(24.34)
The caplet permits the borrower to pay the time-r market interest rate if it is below KR, but receive a pafment for the difference in rates if the rate is above KR. lf settled at time F, the option would pay
1 1 + Rwtr, F + x)
maxlo, Ar(F, F +
J)
(24.35)as
Let Sw be shorthand for Aw(F, F + s). We can rewrite equation
(1+
KR$
g0,
+XwF
max
(j
-
ILR
w)(1+
'
KR)
j
= (1 +
KR)
max 0,
(24.35)
f'al
-
1
1 -
1 + Kp
1 + Rp
.
: '''
...
.
..
.:
.
.
. .. . .
- .. .
.
:
'
.
.
.
.. .
..
:
,
.
..
:
,. :
: .
:
:
.
.
'
. . .. . . . .
:
.
:
.
. .
.
..
. . .
.
-
.
.1Ik::.:,I(:-. ()1(. s Probability
.
.
Three-period interet rate tree showing the evolution of the
/72
continuously compounded l-year rate. The state abeach node (representedby time index i and Ievel index jt are in braces at each node. The rsk-neutral probability of an up move is p.
rl0z 01
rddfl,01
.
,
.
.
.
(1 p)l -
,
Cap payment at time p+1 = maxlo, Rti ti
,
p+I)
KRj
-
(24.37)
The value of the cap is the summed value of the individual caplets.
24.4 A BINOMIAL
INTEREST
RATE MODEL
We now examine binomial interest rate models, which permit the interest rate to move randomly over time/ O approach is to model the short-term rate, where the delinition of short-term is h, the length of the binomial peliod. ln this example we will mbdel the l-year rate; hence, a period is 1 year and /1 1. To conseuct a binomial tree of the l-year rate, note that we can observe today's l-yemrate. We assume the l-year rate moves up or down the second year, and again the third year. This behavior gives us the tree in Figure 24.2, which is drawn so that it need not recombine. The notation required for interest rate trees is a bit more complicated than the interest rate notation we have been using. We have nv t, T) as the forward interest rate at time rtlfor time t to time F. This notation accounts for the fact that at a poirt in time, there is a set of fonvard interest rates at different future times (r) and covering different times to maturity (F r). When tz t, rt (r, r) is the set of current spot interest rates for different times to maturity. =
''''' ''''''
'.''''
24.2 One-year and z-yearzero-coupon bopds with a $1 maturity value r Exmple tll have prices of $0.9091and $0.8116. The l-year i lied forwcd l-year bond price is mp yy .t. )) therefore $0.8116/$0.9091 $0.8928,with an implied forward. rate of. 12.01%. Supprice of a l-year put option to pose the volatility of the forward bond price is 10%. The =
l(E '. .
)ksell the t)! jj)jy
. t)
l-year bond or a price of
$0.88is
.
-
BSPut($0.8116,
793
(24.36)
Note that 1/(1 + A?.) is the time-r price of a bond paying $1 at time F + s. The side of equation (24.16)is therefore the expiration payoff expresslon on the right-and KR). The bond option model, bond options with strike price 1/(1 1 kp + put to therefore price caplets. be used to equation(24.32),can An interest rate cap is a collection of caplets. Suppose a borrower has a :oating ll. A cap would make the series rate loan with interest payments at times ti i = l of paymenis ,
$
''
.
:
RATE MO DEL
,
..''. ..
'' '' '' ' ' . . . . ..
.
:
..
'-
Payoff to caplet
BINOM IAL INTEREST
$0.9091x $0.88,0.1, 0, 1, 0)
=
$0,0267
-
=
In practice, the implied volatility from the Black formula is convenient for quoting
prices of caps and caplets. For example, the statement that caps are priced at about a 10% volatility (using the Black formula) gives a general sense of prices, even though there is likely to be a volatility skew across strikes.
4Early binomial bond pricing models include Rendleman and Bartter ( 1980) and Ho and Lee (1986). Heath et al. (1990)derive a general m'bitrage-free binomial model that includes the Ho-l-ee model as a special case.
794
k
INTEREST
A
RATE MODELS
BINOM IAL INTEREST
P
()(0
l 0) '
,
e-rl'
=
,
(24 39) .
The two-year bond is priced by working backward along the tree. In the second period, the price of the bond is $l. One year from today, the bond will have the price c-r,' with probability p or d-r' with probability 1 p. The price of the bond is therefore -
-r/)
/$(0, 2, tj; m o
/J
gpc '-'gg
g
= e -rh I)p# l (1
,
Thus, we can price the prices.
z-yearbond using
(j
+
p jg
.
2. 1) +
(1
,
either
.''-&
h
j
p).p!
-
(gy yg j (24.41) .
(1,2. (p) ,
.
the interest rate tree or the implied bond
Finally, the 3-yea.r bond is again priced by traversing the entire tree. The price is $1 after 3 years. After 2 years, the price will be $1 discounted at ruu, rttd, Or rad. Continuing in this way, the price is ?-fJIt
Bond Prices
170(0, 3-,0)
At time 0 we can determine a bond price on the binomial tree in much the same way we determined option prices in a binomial stock-price tree. The one-peliod bond price at which is giyen at any time is determined by discounting at th: current one-period rate, each node: Pi i i + 1. j ) = e ,
,
-ri
(f,f+1ijjh
e-r
=
Lpe-r'
(JJd-&'''
+
(1
+
(j
tpe-rsu
gj e-rd
-
..y.
(j
-
)j
gj e-rrp/
(24.42)
terms in equation
.
P 0 (0 3- 0) ,
We can value a two-period bond by discounting the expected one-period bond price, one period hence. At any node we can value an n-period zero-coupon bond by proceeding in this way recursively. Beginning in period i + n, we value one-priod bonds, then in period i + ?2 1 we have two-period bond values, and so forth. Because the tree can be used at any node to value zero-coupon bonds of any maturity (up to the remaining size of the tree), yhe tzee also generates implied forward interest rates of all maturities and volatilities of implied folavard rates. Thus, we can equivalently specify a binomial interest rate tree in terms of interest rates, zero-coupon bond prices, or volatilities of implied folavard interest rates. 6
,
iuc
plte-r-bmu-ntus + ptl
+ (1
p)e-(r+r,,+r,ta)
-
p ) p e-(r+ra+ra,,) +
-
(j
-
plle-r-bra-rdd)
(24.4g)
This version of equation (24.42)makes clea.r that we can value the bond by considering separately each path the interest rate can take. Each path implies reallked dijcount factor. We then compute the expected discount facto using lisk-neutral probabilities.
-
Denoting this expectation as E*, the value of the zero-coupon bond is (g-(r()+rI
g,v
More generally, letting
n
AIl 17-7# valltation
??7t?#c/.
A-rzjh
)
represent the time-f rate, we have E
suggested n Rendleman (2002). 6The central role of volatility in a term structure model is emphasized by Heath et al. (1990)and Heath ' et al. (1992).Their model is discussed in Appendix 24.A.
,
ple-rudj
-
The 3-yea.rbond calculation can be wlitten differently. By collecting (24.42), we can rewrite it as
(p,y ag;
sAny interest rate is reached by one (or more, if the tree is recombining) combination of up and down in movements. One convenient way to characterize a node is by numbering the nodes at a given point time, beginning with Oat te bottom node. lf we are at node j in period i, then we can move to ei er node 2 x J- or 2 x j + l in priod 1-+ 1. Forexample, if we are at node l in period 1, then we ca move this scheme' to assign to each to node 2 (2 x 1) or 3 (2 x 1 + 1) in period 2. It is also possible to use 2f numbered l is (20 0), 01 node the the + (1, 0) node is numbered 2 + j. Thus, node the number (0, (2l + O), and so on. In effect this uses a binary representation to number the nodes. This scheme is
795
Using the tree in Figum 24.2, we obtain the following valuation equations. For the one-period bond we have
llode When we have a binomi/ tree as in Figure 24.2, at eacll we have a large set well Thus, we need different maturities, rates. forward for as as of spot interest rates will let rto t, F; j4 We under node discussion. identify notation the the expand to to quoted where the F, from is prevailing rate ar tirfie tj < t interest t rate the to represent quoted, the which period is binomial tells the at rate is the you can Since tv and us state j. point Similarly, in time time the at height along the telling tft. of at any tree think j as us bond Ptvt, F; j). prices, implied forward and both of is zero-coupon spot r(),there a set Using this notation, the one-period rate at the th time and level j is l'i i, i + 1', j).5 /*0(0, 1; 0) and the rate rdu, for example, is In Figure 24.2 the initial one-peliod rate is the nodes such timing represent one-period rates observed The is 1). that tinal ?'a(2,3', 24.2 Thus, today. the Figure from in periods can price bonds up to 3 years in tree two probability of an up move. We will assume that maturity. Let p denote the lisk-neutral rates are continuously compounded in this example.
Zeto-coupon
%
RATE MODEL
*
(c-
rj
Zs2n
ilttplicitly calcttlate
(24.44) 4
..
eqttation
(24.zI.zM.
t' /4 Example 24.3 Figure 24.2 constnlcts an interest rate tree assuming that the current l)tl . lzyear is 10% and that each year the l-#ea1- rate moves up or .down 4%, with probrate q; ability p 0.5. wecan use this tree to price l-, 2-, and 3-year zero-coupon default-free it? ' .j bonds. . ' ''
.
: ;.
E E
=
796
k, INTEREST
A
RATE MODELS
(24.39),the price
From equation
One-year bond:
I ! r;): ( J)q;q:( '.q' t'. ;'!11I:::::;;j.. 11j1y*. 'rql::rq!:qqf q1I1k-k,jijt': (' yjjj.f J' j'jIIpC:qFq-'. jj.yjjd yjjj,j;;y)f y' ;'y'. )' t'.-*--..::111142*. .!((( . qql. (jjjy'. tyy'tjjytjyd r' '!j' (. ((.((j jj t' t(4j j.j.)!. j j j. jjj.ili (.k 4jj.. (g' j j.y!'..ji. yk y. (j. .(.,. yy ' ;y;i l lj;j:;jil:. . ' illi!!s!k. kr yt..;yj jj(.E(. q(y 11.. pj j.y '. q t.,L. . .qq. ty.. .q ypjjy . ........ ... .r .. .
;;gjEjj(.' )' yjjjyyyjjry:,jj.f .r' ;'..(' @q' .yjj,k;jj,' (EI.!)Y jd jld /r17r:7,* 7!:7f1:(r7* q' (' frf r' y' 4* j'j-'( @' ;)' '(i'i.'q)' q' /'(7* (' 7q?*r1r)t1t)* j(' ))' t(' p(' $'. t.(;'(i!!!jiIIk'. jy'tt' .jjI:yyrjjr:,.' j'.y' jjj(j4('. jjd sj)d 4* 1* t.' q (''.''!.l i' i.EE'i'.'E.E.(. '((iE iE'ij'iE.EElt('i !(E( '.ii q:q' t)ili(. @: (!:(j!(! .'h! . . @ E E! !! i E q . (!.(rj.tjjtr q:iEE;.. E..E.'' '.E 'EEE.j j'jijjjk'.E('.; k(EE!!!! E;k.' jti jly g(j k.E-kkfi. jj; .' !jjIjjyjy,E .k.
-jjj-syf ?' '7* jjjyyr,f (' E' :)' CqIIE' j'j''E:z)d .jjj,y;jj.' !ky))jj'.jy;trg;,yj-. t?rd (' F' ;'y)'. i'()' f'I i TE 'J'()F' r' E' . . E ;'
.''.
,-jj',)j;()jy't;' .' ..,.,' :
of the l-year bond is
Po,
/?)
P (0 2) .
=
e
=
(24.45)
$0.9048
.
is
.
= $0.8194 Three-year
z'(0 3)
=
Finally, from equation (24.42),the price of the 3-year bond is
bond: c-0.l0y
5e-0.l4(n.5e-0.18+:.5e-0.l0)+g.5,-0.06(:.5e-0.10+:
E E .
..
''
iE
.
.
'
.
'
::
'
:'
' .
E-E
. Three-perid ........
- ... .
. . .
-
..:,
...
.....
..
.
..,..
..
..
. .
E
.
.
:..;
' :'i ' . E: E E :
,,........:.
.
:
-
.
......
'
RATE MODEL
. E.:
.
E
...
797
rrrF;g
E E .
gy
E g
.
:.,E
.. : ' . . .. . E ! : EE : E : r -y . g j. - . . EE . . . . . y y- . yy E y -:.r .
:
. . ..
,--.
.
.
jjjzy,rr k!!jj;z jjjjgy;j::.ytjjj,
jjjj, k;y!jj,, jjjj,.jjj::y ryjr:?,,jkjj,. Factor oiscount
.
.
:i
ihterest
0.18
up or down 0.04 each year. The risk-neutral probability of an up move is 0.5.
,jj:yyyyyy:,
probability ().2s
.,,jk,;,,,
.!jj.,:jjr.
0.4:
() 14
5e-0.02j
= $0.7438
Equation (24.44)also gives
%
''
':
'
: -:t:;1!!!:(.
-.
.
.
: 'E .. . .
:
'
' E'. g.
.
'
-
:
rate tree assuming that the ihterest rate moves
(05c-0.14 + () 5e-0.06)
c-0.l0
' : :
'
. .
.
(24.40),the two-period bond plice
From equation
Two-year bond:
'
:
jjEE
-y.
=
BINOM IAL INTEREST
0.26
$0.7438as the price of the three-period zero-coupon bond.
k
We should note that the volatility of the bond price implied by Figtlre 24.3 is differentfrom the behavior of a stock. With a stock, uncertninty about the futtzrestockprice increases with horizon due to the fact that the volatility of the continuously compounded retttrn grows with the square root of time. With a bond,'the voladlity of the bond price initially grows with time. However,-as the bond approaches maturity, volatility declines because of the boundary condition that the bond price approaches $1. Just before maturity, volatility of the price must be essentially zero for a default-free bond. The binomial model in Figure 24.3 produces this behavior of volatility as a matter of course since it models the intefst rate, not the bond ljrice.
0.02
0.25
0.18
interest rate tree and repeating the bond valuation.) This is another illuseation of the effect of Jensen's inequality tht was evident in the Vasicek-cm comparison.
Option Pricing Using the binomial tree to price a bond option workj the snme way as bond pricing. Suppose we hake a call ption with stn'ke price K on a (F fl-year zero-cupon bond, '' . with .the .open expirig itl t . tz periods. The . xjiration value of the option is -
Yields and Expected Interest Rates
-z
In Figure 24.3, we assume that p = 0.5 and the up and down moves are symmetric-the interest rate follows a radom walk. Consequently, the expected interest rate at each node is 10%. The yields on the two- and three-period bonds, however, are not 10%. 'Fhe yield on the two-period bond is
- 1n(#(0, 2)1/2
=
-
ln(0.8194)/2
=
0.0996
The yield on the three-period bond is
- 1ngP(0, 3)2/3
=
-
111(0.7438)/3
=
0.0987
inYields are less than 10% on the two- and three-period bonds becaus pf Jpsn's equality: The average of the exponentiated interest rates is less than the exponentiated psing the expected average. Tltus, as we discussed earlier, we cannoy price a bond by tluul expected t'Ilpcm'e ftntzcr the interest rate. Uncertaitlty causes bond yields to be ?w?c. and yields The discrepancy between average interest rates increases with interest by constructing a different reltttionship this volatility. (Problem 24.7 asks you to verify
0(/. j4
=
maxgo, Pt,
F;
j)
-
ff)
(24.46)
To pdce tlie optiop we can workrecursively backward through the ireeusing risk-neual P ricing as with an option n a stock. The value one period earlier at te node j' is ,
O t
-
h, j'j .
=
Pt-it t
-
x
/,
f';
g.p
j'4
x O t, l x
.j?
+ 1) +
(1
-
2 x p) x O (?/3,
.j/lj
=
(24.47)
The calclation here assumes there is a nonrecombirling tree. Since ech nodgenerates two new nodes, if there are J nodes in one period, there will be 2 x J nodes the next period. Thus, if we are at nod j, we can potentially move to node 2 x j or (2 x j) + 1 in one period.? We continue in this way to obtain the option value in period 0. 1.rtthe
?For example, from node 0 we move to node 0 or 1- from node 1 to node 2 or 3 and so forth.
.
% INTEREST
798
TH E BLACK-D ERMAN-TOY
RATE MODELS --'-k ' 111::1111.* 411:2:1,1*' q(' ''j' t!'.r:)F'1q.j:) s' t':'k' qf :'llii.f jjy'.i)(;-..' ' ' EEEE'i E iE E ?' :1itEE' (iii.d ilIIEr:iIk.'' iI(:EtI,,'. ,r' ,''r;rlf jf 'F' Eiji)d t''(i.: ;''i(E'';iE';)'1;(!(it:(iE!T!ii'li E;E!((E'' E('E(((E'i'i'Eq( (' .iIIkif.1i(.Ei1l(.' q' ?' )' k' r' ( 'ii'(.E.('EE' EE.''q'i . q('i'' ' E'q( (7))7F:r7:747:7: t'(' )' (111)* ((jIlEi!i!ll;'. p' jp'yjyjjqqf (klqll'dki'-''.!.lltl' j'('jjij(jkjk'yyf y' )'.E.'''('E''''.iE(..(.;'-.' E'(!(' . .. . . . E.;E.E' (.!. (E@. . ;!,: - -. - -- ; - ; ; : r?. ' .--! !-jjjjjE!jj;y. E . . --.i. - . - 'r '. .) ) : - q ; r . - . ... - - . . ; . , . .. . j . j.g yjy.. EtE . . . :kE.-.y.. Ej ( .:t;. j y-Er..; ..- - Er..yjy . gk. . .) y j-lypotiaeticrl . ( E . .;.:i..(. rE.... q!'i.!!. . . ...;.. . i... .i.. . .
$
MO DEL
799
'.-11111,-;;-''*. -'
y'. yy'yyytf llii)ltiii)d )'j,jg')q ))y' )l yl: ' (-;,jj,jjjjg't'-',jjjiLqj. t'. trf tpf ..jjjjj!!j:y' r'tttlililtyk )'. t,' y. '-. 11 . '' ... 'ty . )jjjjjjg,. . .-.'-y. t. . . LLLLLL. -' ---' ,'.-'
same way, we can value an option on a yield, or an option on any instnlment that is a function of the interest rate. Delta-hedging works for the bond optionjust as for a stock option. In this cae the derlying asset is a zero-coupon bond maturing at F, since that will be a (F zl-priod un bond in period t. Each peliod, the delta-hedged portfolio of the optlon and underlyipg asset (the bond with F tfj to expiration) is financed by the short-term bond, paying whatever one-period interest rate prevails at that node. E
.'.' :: :
E.
:
l
-
t)E . E' liE : :
'
:
-
.
-.
.
. -
.
-
.
-
.
...
-. . . . ..
--
.
.
.
-.- . . ..
.. . .
-
k
'' ::
'
.
'
E .
( ..
'
i :.
E
' E' .
:
:
-.
'
:' ' ' ' ..
'
'
:''E''
''
:
'
-
- .... .-
-
. .
-
-
.
-
.
-
..
. .
-
. -
. .. . . .
.
. . .... . . . ......
-
. .
. .. ...... .. .. .
-. .
...
..
.
.
-...-..
. .
-
. . .
t.,jyyl . -
. .
'.
'.:.... . '''''' ''''.'
(.t' ('
Suppose we have a two-year put on a l-year zero-coupon bond and lt Example 24.4 :( .@) price is the sike .. $0.88. The payoff in year 2 is jlj tt . maxgo, $0.88 P (2, 3, 2. .j))
Ejty
-
option price is computed based on the l-year bond price in year 2. From Figure 24.3, there is only one node at which the jut will be exercised, namely ) ( jg E, .that where the interest = $0.8353.Using rate is 0.18 and, hence, the bond price is e i j. j probability of 0.25 risk-neutral accounting the and along the for interest rates tree, t the tljtllreaching that j one node, we obtain an option price o . . . j:jit .$().()()gg 1l) x () :2,,.,5 :s. ($0.88 $0.8353)c k. j(
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24.5 THE BLACK-DERMAN-TOY
MODEL
At any point in time we can observe the yield curve and the volatilitiej of bond options. Thus farwe have ignored the important practical question of whether aparticular interest rate model 5ts these data. For example, for any interst rate model, we can ask whether it correctly prices zero-coupon bonds (in which case ttwill correctly price fonvards and swaps) and selected options. Matching a model to fit the data is called calibration. Yield curves can have various shapes. The models we have examined, however, are not particularly iexible. For example, the binomial random walk model has two PafanRe ters: The starting interest rate and the kolatilit'y genetatig up and dow moves o-) and fnodels and generate yield four Vasicek have The C1R and parameters (a, b, r, match the shapes that particular stylized dgta. These models are may not curves with assumptions. the real world, however, ln with their world consistent arbitrage-free in a oppormnities, in the arbitrage will generate apparent sense that observed prices they cqncluding choice of either that zerothep theoretical W hayt prices. will not match a models enough that the priced bonds incorrectly or to capture ar nol accurate are coupon .
reality.
,
q
Some models attempt to provide a lich characterization of the yield curve and yield th curve volatility. Notable papers desclibing these models include Ho and Lee Black et a1. (1990),and Heath et al. (1992).We will focus on the Black-Derman(1986), model Toy (BDT) to illustrate how calibration works.
The basic idea of the Black-Derman-rfby model is to compute a binomial tree of short-term interest rates, Fith a :exible enough structure to match the dara. We will matches these data. We begin with sample data and demonstrate that a particular then explain how will to construct the tree. We assume in this discussion that the length of a binomial period is 1 year, although that is arbitrary. . Table 24.2 lists market information about bonds that we would like to match. We follow the Black-Derman-rlby paper in using effective annual yields rather than copppvpded yields. Sipce the table contaips prices of gerpriwppn bppds, continuovsly stnlire inferthe of impliedforward interestrates. There is also informatlon term we can the volatility of interest rates. The column headed about in Year 1i' is the of deviation the natural log yield that the standard bond 1 year hence (We could, of for r . if we wished, convert. this into a standafd deviatin . of th bond piice in a .ylml The volatility for the n-year bond tells us the uncertainty abot the yeapl yield n an ll 1)year bond. The volatility in year 1 of the z-yeaibod is 10%., thij tellsus that the l-year yield in year l will have a 10% volatility. Similarly, the volatility i yar 1 of the zyear bond (whichwill be a 3-year bond in year 1) is 14%. While the matches obselwed yields and volatilities, it makes no attempt to capture the evolution of the yield curve over time. The yield curve evolution is of course implicit in the tree, but the tree is not calibrted with this in mind. The BDT approach provides enough Eexibility to math this data. Blck, Drman, and Toy desclibe their tree as driven by te short-term rate, which they assume is lognormally distributed, The general structtlre of the resulting tree is illustrated in Figure 24.4. We assume that the risk-neutral probability of an up move in the interest rate is 'ee
'Evolatility
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-
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50%.
For each period in the tree there are two parameters. R;l can be thought of as a level parameter at a given time and o'i as a volatility parameter. These parameters rate used be to match the tree with the data. ln an ordinary lognormal stock-price tree, can
?
k. INTEREST
800
TH E
RATE MODELS
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p' (.11:F:.--* ))' Ell:.-,f tl')r yy' yg.iyyy.tyyyl'. ('! 1lIIF:1lIl:)'. r' r;prtl)qqljljpf )' (jy'' tiil.f ..'tt't' ,111::-111.*. y' t'j;jy(jjj('.' 11111*. r'1'jg t..' r' rjllrq:.-'f r' l (j.y)jyjr. yj lIji5iEE.. (,r'k .... . (.y' .. j jq: itit--j. 1III:!!-.. ' q:. .. . .. .. '.. .. .. . .. . . . ..r. . . .. . General form of a Black-Derman--lby interest rat tree. Th probability of going up or down from each node is 5094.
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c2cW = ratio between down node is Agzx'jvtke-zl Figuf 24.4. The volatilitis i Table 24.2 are measured in the tree as follows. Let the time-/l price of a zero-coupon bond matring at F when the time-r short-term rate is r() be yleld f th bond is PLh,T, ,-(/7)1. rjxe
theratio of the up node to the adjacentnodes is the sm i
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At time h the short-term rate can take on the t'wo values ?-,,or lognormal yield volatility is then Yield volatility
=
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13.66%
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we have seen thus far. Unlike a stock-plice tree, the odes are npt necessarily centered on the previous peliod's nodes. For example, in year 1, the lowest intrest rate node is above the year-o interest rate. If we track the minimum interest tate along the bttom of the tree, it increases, thelt decreases, then increases again. The maximum interest rate in year 3 is below the maximutn rate in year 2. These oddities alis: because we constrticted the tre to match the data in Table
24.2. Although bond yields steadily increase with mayurity, kolatilities do not. In order to match the pattern of vlatilities given the structure of the BbT treet rates must behave in what seems lik an tlnusual fashion. Notice that in periods 2 and 3 the ratio of adjacent ntjdes in the sa tiriodij the srri. #or exmpl, niuujrauu 20.03/15.68 15.68/12.28 rduujrddu Now let's verify tqtt th tree in Figure 24.5 mptches the data in Table 24.2. To verify that the tiee matches fe yield curve, we ped to compute the prices of zerocoupon bonds with mattlrities of 1, 2, 3, and 4 years. To velify the volatilities, we need to compute the pricesof 1-, 2-, and 3-yearzero-coupon bonds at year 1, and then compute the yield volatilities of those bonds. =
=
(24.48)
We multiply by 0.5 since the distancebetween nodes is twice the exponentiated volatility. The tree in Figure 24.5, which depicts l-year effective annual rates, was constructed using the data in Table 24.2. The tree behaves differently from binomial trees
.
=
,
% INTEREST
802
TH E BLACK-DERMAN-TOY
RATE MODELS y'... jjjjj'. -j'. F' f' :'(' 11E 5111:*(:(Eqq(14E i il ''' ji.EIEE ) E;'E' g' :'. 'ftqf k' )' tf r' :E' E '' ' 'E gjjt:-,f yjjryy,f q' t)' y' )r'..t)jy;t))j)'-' @' $' lrtf (t.!l E j'tyyf Ijjij-jjjjj('. ytjz'y .y)'. rqf jf' : '(;:). prf (' lf 91* q' i':)F' ')' ).!.qi:' ; ii' i.''((;;''.(iE(.Ei' ! ):(i.Ei.(E i.i ()( ' ! 'it';i y' jjyy'y,;,yjjyj.. ;'j---j;,:;jjj;,'. t'(''','!'ql' r' . .E (qi. ..i..( .(.' (. i:(.(('.' E .'.'.'... j. ''.i!'i.;(( ; tEi. ;.:.; ;- ;..E..!E. E.. ! - . !E : . ;;rr : ,., ,i;-...i ....- !;i.;. E yyi . : . (jj .. . g. . y ( j . ;...:. . jjj.ry.g(y..yy ;ql-E yy r.gy . .y y. yy yyy.yjy ..yyyy. .tL.....(............:.. .ty. . y y y..... g..y ..
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j'(' p' 1* )' Illi)f pIi1k-iiIr'. ().k)-'-'. jy'jf j'! r' jjjjjjf )'. ip!piip4!py'. Ilii!!i!:f rti''i y..tr,'yjj)y'' (q' t'. r,-'.' t)jf 'y' ti-f t'. illi:)f . ' jj jy.(;ti. '.'(. L-jj,jLj,j, ';j((k,. jg .j..'.yjj..y r'.).j ytyyy .tj t,. ;jy..ty . )LLfLL-. . 1. t r . . t-. ' ... .... . .... . ... . r. . Tree illustrating the evolution of the 3-year zero-copon bond, based upon the prices in Figure 24.6.
j'!' (' :';f .* .y' E);tf )' q--,' t'1;* r.'.)' 'jy',,,j,jjjjjj,.-. ,'y' q' ''qqq'll!f j'p' ;')' (' jtE E i i ' ''E (t'' r' (' y' (y' ;j($,-'.. E ' E 77!4: j'(Il.')tt)t' r,lr:r,f r,lr:r-f ::;!!:.' r,lr:r.f tl?f pf t',111:::122;.F*. 7(((q (r.r.i';qgl i')((. (' q;)i 11)444))4! y: !t)jpl ( 'j.'r(.'E. j.EE.(.'(.(..(((E.ti.q.E'!'q(:i.(.;t..((.'IE'.' q( ' 'rtqj. .''( ( i('(!'( ( '(( j (. ( '@ . y;)l . 'E!- E . i. E. EE: E E i; E: !.i .p; rtEE;jj. !.E jjjjjijjjjy,. !. :y klE' . ... iyjy(qgyj . .; .rl gjjy t q j. - y rEyq.y.-(.; y.j.;!y.qqy-.(yyy.jqgy; k: fl r . . i . i (.E :. i .. E; !.. ;E !- .-..-.-. - ---; . (r.. k, .Eil. E . . ' -
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Vertfy'ing Yield s The rate at the first node is 10%, which corresponds to the current l-year yield. We can compute the price (and thus yield) of the z-yearzero-coupon bond by starting in year 2 and worling backward. It is slightly more convenient to use the tree of will be worth either $0.8832 l-yearbond prices ilzFigure 24.6. ln year 1, tlw z-yearbopd 10.8219. of Thus, discounted expected plice yield the yield of 13.22%) (a or $0.9023(a time is 0 at
$0.9091x (0.5x $0.
83 2 + 0 5 x .
$0.9023) $0.8116 =
Figure 24.7 illuseates the eee corresponding to this calculation. The price of the 3-year zro is cbmputed in a similar way. Worling backwards from the year-3 nodes we have
$0.9091 y
$0.8832)<; (0.5 x $0.8321+ 0.5 x $0.8798) + 0.5 x $0.9023x (0.5 x $0.8798+ 0k1 x.$0.9153))
(0.5y
=
$0.7118
$0.9153
Figure 24.8 illustrates the tree showing the evolution of the 3-year bond. Problem 24.8 asks you to verify that the tree in Figure 24.6 generates the correct zyear zero-coupon bond price.
Verifying Volatilities Now we want to see what volatilities are implied by the tree. The volatilities in Table 24.2 aeyield volatilities. Thus, for each bond, we need to compute implied bond yields in year 1 and then compute the volatility.
804
TH E BLACK-DERMAN-TOY
RATE VODELS
%.
INTEREST
For the (24.48) is
z-yearbond (l-year
equation bond in year 1), the yield volatility using
0.8832-1 0.9023-1
0.5 x ln
l
-
$0.7 1l 8 0.l
=
1
-
7
bond) will b worth will be a z-year l From Figure 24.8, the 3-year bond in year (which 0.8099-1/2 1 :zF: 0.1112. = 0.7560-1/2 1 0.1501 or $0.7560,with a yield of
1 1 + 0. 10
(0.5x
1 = l + 0.10
0.5 x
=
P (1 3, ru) + 0.5 x P ,
1
0.5 x
l 1322 .
M O D EL
$
805
(1d3, ?u)) 1 .
l + Rzeqo'z
+ 0.5 x
1 Rzelo'l 1+
-
either The yield
-
+ 0.5 x
is then
volatility
0 5 x ln .
0.1501 ()-jvjjg
=
l 0.5 1.1082
1 1 + Rzell
x
+ 0.5 x
l 1 + Rz
0.15
(24.52) (24.53)
Both yields match the inputs in Tab1e'24.2. the correet Problem 24.9 asks you to verify that the tree generates volatility.
zyear
By iterating, it is possible to solve Rz and tya. In the same way, it is possible to solve for the parametefs for each subsequent period.
yield
Pricing Examples Constructing a Black-Dermnn--fby Tree
Tabte 24. 24.5 is consistent with the data in We laave verilied that the tree in Figure the generate t'ee in the data, how did we Now we turn the question around Given the nodes, later the worked and to early nodes first place? The answer is that we started at building the tree outward. Therefore the l-year bond The first node is given by the prevailing l-year rate. price is
In this section we use the interest rate tree in Figure 24.5 to compute several examples.
.
$0.9091
1
=
(24.49)
1 + Rfj
caplets and caps As discussed in Section 24.3, an interest rate cap pays the difference between the realized interest rate in a period and the interest cap rate, if the difference is positive. To illustrate the workings of a cap, Figure 24.9 computes the cap payments on $100 3-year loan with annual interest payments, assuming a 12% cap settled annually. The payments in the fgure are theprdzn? value of the cap payments for the interest rate at that node. For example, consider the topmost node in year 2. The realized interest rate is 20.173%. The cap payment made at the node, 2 years from today, is therefore 0.12) $100 x (0.2017 1 + 0.2017 -
0.10. bond is P (1,2, For the second node, the year-l price of a l-year satistied'. We require that two conditionj be
Thus, Rz
=
/',,)
or P (1,2. rd).
(24.50)
0. 10
=
=
0.5 x ln LP ( 1 2, ,
0.5 x
ra)-
l
-
11/I.P (l 2, rd),
l
-
1))
Cap payment
(24.51)
ln(A1:2c/AI)
equation = 0. l and this value enables us to solve the first The second equation gives us (y to obtain Rj = 0.1082. conditions, but concepttlally we are It is a bit messier to solve for the next set of the match two inputs (the 3-yea.r yield and still fitting two parameters Rz and o'zj to the two possible prices of a z-yearbond at z-year yield volatility 1 year hence). The
-
=
$6.799
'
=
Since 20.17% is the observed l-year rate 2 years frm today, 3 years frm today the borrower will owe an interest payment of $20.17.The $6.799payment can be invested at the rate f 20.17%, so the net interest payment will be
$20.17
-
($6.799x
1.2017)
=
$12.00
In the same way; we can compute the cap payment at the middl node in year 2, $1.463. The payment at the bottom node is zero since 9.24% is below the 12% cap. We can value the year-z caplet binomially by wo-king back through the tree in the cacutation ij way. The
usual
$0.9091x (0.5 x $0.8832x (0.5 x $6.799 $1.463)+ 0.5 x $0.9023x (0.5 x 1.463 + 0.5 x 0)) $1.958
Value of year-z cap payment
+ 0.5
x
=
=
..
'
.
The value of the cap is the value of the sum of the caplets. Problem 24.10 asks you to verify that the value of the cap is $3.909.
k. INTEREST
806
Contract B is a forward agreement
y'. y'j ttfyjy'y'yf Ij'.i y' ;'T' !' t'js j.jtyjy'. jqqjjjjjj'. i'yjjjjjjjjjjj';f p'. ;':'j'I-'-' j', yy'. y' i'lit'ti ;j;)).'. f1* iEq tiE E EEEE 7'7E5r'7t17'7E '' ' EEE F!!T'T!!!)!:IEETET?S !':'7' (' ' ' IE gjjrgy,f gjjygj,f yjjygj,f gjjygy.f iykjyjj,f .j' yj'yyyf y' yjg.kjyyyg;yyj'. ((' y'y q(;'?i i)f )'E)/EiEq.F!!(('iE!! (. ilrirqri;?ll ll';: ;':''17)* i''f' (i' E !::(E j'gyf t'ryf ))t.4444,:2j,g;ky*1 g' Tttqdtf q' (' y'y'y.r'.rt'gy,:y,t( 11* if frf E ' '('E!E EE ; i' (. ' E''E'E(E('!;'!iEEE';i!E. EEIE li'';. (.FE(; E'''.yg. '.i ( gr;jjjrg,f t'tj('. r' y' .. t. . . y gy ..jg. '(..ty. y y E )jj i EIE'q iT.( i:q. @ ( ;...E y.y g r,ygy,jg rtr .Ejjjjjgjjjyi .. E yyg jjj ..g.... ..j..gg.j.j.jy.g.. j.j.y..y.g.y . i g @ ; ... t ! r jjy. . . .. g t;jyjg .yjj . . y;. y.jg..yjg .tjj ?F.-.,... . . yg.. g . j g...jgj.jjj . j.j g j..jy y . ., .. . . .
,,jyjj$jj,,' s,jyjjjgyj,,jjkyjyryf syjyjjrjyys'yjkyjyyyf :y' j''''
y' rj'.f ,k:jy'jy' :'r'y'tyy'.jf y;kk::jjjjj,'. .yg'j ('''' jyf E : : . . ! : .
yyf j(' )' (j' r'( (;'. (' tjj;f y' rpf .jyj'y' ttf ytf .yjyjjjjjygy;''' (jj;;;jjjr;'. .;yj k (( jj.. (y,,., jjjjjjjjjrj,i .. j .-i..!rti..t.-.!1!q5..-.lrI7..,., .lq'.-i); . . jy.j j.
.' .'.'
'
''
!
:
::
.
.
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.
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. .
E
'
.
'
'
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:
g :
: E;' : '
E
'
' ''
:
:
:
:
:' .
'
::
. .. . Tree showing the payoff '
i
.
E
.
r.
.
E...
: .... . . : .. : .
.
.
, : , ..
.
ryjs
,
........
...
.;.
g
......-....)-:.'...........-..-.-.........-.(.--.r.
....y..
y
,.
y
..
.y y
;
........
TH E BLACK-DERMANWOY
RATE MODELS
. .. g g g y :. :.
,.
,
y
g
g
y
g
g
g .
MO D EL
$
807
that settles on the borrowing date in year 3:
yyyy
jjtgjg;jjg
kyjjgjy;y
g
a,jyjjjgyj..jjkyjyrg
.jgjy,.. kykjgjj;y
jjyjgggg
tgjjjgj;y
k;jggjj;s
Contract B payoff in year 3
56.689
to a 12% interest rate cap on a $100 3-year Ioan, assuming that interest rates evolve according to Figure j4.5. Each amount is the present value of the cap pagment made at the interest payment date.
?.(3,
4)
-
i
(24.55)
This scond contract resembles a Eurodollar fumres contract. There is no marlingto-market prior to settlement, which would occur with a real f'utures contract, but the ttming of settlement is mismatched with the timing of interest payments. The correlation betweep the contract payment and the interest rate liscussed above and in Section 7.2 is therefore present in contract B. We can compute s by tnking the discounted expectation along a binomial tree of r(3, 4) paid in year 3, and dividing by #(0, 3).
$6.799
$3.184
$1.078
=
We can value both contracts A and B using the interest rate tre in ))I Example 24.5 li')Figure 24.5. The ' 17(0, 3)/P(0, 4) 1 0.7118/0.62431. rate on contractAis L )j) :r1: ' 0.140134. CC') coneactB can be valued as follows. supposethe fbrward rate on B is Fb. In year tk t) FB We can value on the tree th: pa#rtit 3, B makes the payment /'(3, 4) .. .. ?'(3, 4); lr! .' the time 0 value of FB is simply Ps x 17(0, 3). Figure 24.10 depicts the fayinentfor =
$0.00
=
=
t)
'
'
-
-
.
i . .
. . .
.
.
.
.
:
.
'
'
.
.
'
.
.
=
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p' qjf )' jyf Iiliddi:f lkk':1iiiI.'. 1* y'. j'('. y'q (' r' )' jjjrtjjjyjj'. yyjf y' t';'jjj,, j;,gj(jj'. jj(j;gjjjj('. ((y*7* ) yq '.'.. r'. ( 1' ')l'( jl' IE11-. . yj .' . )..r '.. '.. IliE::lii: ... -' ---' ;'g' '
t'iiiiy::t('. :::' : :: ' . . .
:'. .
.
.
--
-
.
.
E
-
'
'
.
:
' E. :... .
''
We discussed in Chapter 7 two different styles of settlementfor aforward contractbased on interest rates. The standard FlkAcalls for settlement at maturity of the loan, when the interst payment is made. (Equivlently, the FRA can be settled to pay the present value of this amount when the loan is made, with the present value computed using the prevailing interest rate.) Eurodollar-style settlement, by contrast, calls for payment at the time the loan is made. As we discussed in Chapter 7, the two settlement pfocedures generate different fair forward interest rates. We can illuseate this difference with a simple example. Consider two contracts. Contract A is a stadard fonkard rate agreement as desclibed in Section 7.2. If /'(3, 4) is the l-year rate in year 3, the payoff to contract A 4 yearsfrom tday is Forward
rate
agreemvpts
Contract A payoff in year 4
=
,'(3,
4)
-
L
(24.54)
by taking the This is a forward rate agreement settled at matulity. We can compute discounted expectation along a binomial tree of 7.(3, 4) paid i.n year 4 and dividing by #(0, 4). Since it is an implied forward rate, we can also value L as #(0, 3)/P(0, 4) 1. -
-.-. .
.-
-. . ..
.
E' g
:
.
.
. ..
................
.
. .. ... ..
:'
.
. -.. ..- . .. .. .
'4:7'76. '
:
.
' :
.
E
.
. ........
.-. -.-..
:
E
.
. . ..
.
. ..
.. ,. .
-
.
--
.
..
.
-
'
.
:
.
-
E
:
-..
.
:
'
E :' . . 'y
( j '. '. '. jy.jjy yyy ...j..j..j... .y .jj. ..yy. ..... . ..
. Tree depicting .......
'' :
'
:
y
.
-
' ...
.
'
.
--
'
--
:
,
.
-.
.
value of that contract pays the a revailing l-year qlnterest;rate 3 years from today. Interest rates are from Figure 24.5. If the contract value at any node is $/(r),the amount at each node is (0.5'x y(ru)+ O.5 x y(rd))/(1 + r).
.
. - -. .
.q'li'lrr-..-liii,,r
ilEsl...
(I!El...:::11)..
:;!i..-.::;i!!:.
a :!i!!lk-
$20.03
$14.86
511.99
$15.68 J
$10.03
'2*)1()),,*! @;E'
%.INTEREST
808
@)4 $100notional lit $100. We then
amount. In the final period. we receive the prevailing l-year rate times discount this payment back through the tree. The time-o value is $10.03. ti) ) The implied rate is $10.03/P(0, 3) = $10.03/$71.18= 0.1409. yt.'st ' Thus, Eurodollar-style settlement in year 3 raises te fonvard rate from 14.01% to jj 11114 09%. Problein 24.11 asks you to verify using the binomial tzreethat FA = 14.0134%.
yjjjt
ltii
%
PROBLEMS
RATE MODELS
809
PROBLEMS Fpr the rst three problems, use the following inforlpgtion: Bond maturity
(jars)
-'
0.9259
Bond price
,illpplpr-''-l:)):::,::..
bYear forward price volatility
24.1.
Derivatives that are f'unctions of interest rates can be pliced and hedged in the same way as optinns. . As with derivatives on stocks, prices of interest rate derivatives are charactelized by a partial differenttal equation that is essentially th same as the BlackScoles quation. The Vasik and Cox-lngersoll-Ross interest rate models are derived using tlzis equation by assuming that tlle short-tenu 1n terestratefollows pnrtilarmeansrevertingprocesses. Thesemodels generateteoretical yieldcurves but are toorestrictive to match observed yield curves. Under the assumption that the forward price for a bond is lognormally distributed, the Black model can be used to price bond and interest rate options, and therefore interest
rate caps.
The Black-Derman--fby ee i-s a binomial interest rate tree calibrated to match yields and particular set of volatilities. This catibration ensures that it a zero-coupon matches a set of obselwed market plices (forexample the swap curve) but not necssarily the evolution of the yield curve. Valution of interest rate claims on a binomial interest rate tree is much like that on a stock-price tree.
FURTHER
READING
Classic treatments of bond pricing with interest rate uncertainty are Vasicek (1977)and term structure models, Cox et al. (1985b). These are examples of so-ciled discussed more generally in Due and Kan (1996)and Dai and Singleton (2000). Binomial treatments include Rendleman and Bartter (1980),Ho and Lee (1986), and Black et al. (1990).Heath et al. (1992)have been exemely intluential insofar as they provide an eqilibrium characterization of the evolution of forward rates. See also tlnfne''
Brace et al.
(1997)and Miltersen et al. (1997).More in-depth treatments of interest rate
derivatives can be fond in Hull (2000,chs. 20-22), Rebonato (1996),Jarrow (1996), and James and Webber (2001). Litterman and Scheinkman (1991) is a classic study of factors affecting bond returns. Bliss (1997)surveys this literamre.
0.8495
0.7722
0.7020
0.1000
0.1050
0.1 100
a. What is the l-year bond forward price in year 1:/ b. What is the plice of a call option that expires in 1 year,' giving you the light to pay $0.9009to buy a bond expiring in 1 year?
c. What is the plice of an othenvise ideptical put? d. What is the price of an interest rate caplet that provides an 11% (effective annual rate) cap on l-year borrowing 1 year from now? 24.2.
a. What is the z-yearfomard price for a l-year bond? b. What is the price of a call option that expires in 2 years, giving you the right to pay $0.90to buy a bond expiring in l year? c. What is the price of an othemise identical put? d. What is the price of a interest rate caplet that provides an l1% annual rate) cap on l-year borrowing 2 years from now?
24.3. What is the price of a 3-year interest CaP rate?
rate cap with an 11.5%
(effective
(effectiveannual)
24.4. Suppose the yield curve is flat at 8%! Conjjder 3- and 6-year zero-coupon bonds. You buy one 3-yeaT bond and sell an appropriate quantity of the 6-year bond to duration-hedge the position. Any additional investment is in short-tenn 'tzero-duration) bonds. Suppose the yield curve can move up to 8.25% ordown to 7.75% over the course of 1 day. Do you make or lose money on the hedge? What dat-yieldcurve model discussed does the result tell yu about the (impossible)
in Section
24.1?
24.5. Suppose the yield curve is flat at 6%. Consider a zyear bonci. A1l coupons are annual. an 8-year 7%-coupon
5%-'coupon bond and
a. What are the prices and durations of both bonds? bond and duration-hedging by selling an b. Cortsider buying one zyear quantity of the appropriate 8-year bond. Any residual is fnanced with bonds. Suppose the yield curve can move up short-term (zero-duration) 6.25% down 5.75% the to to or over course of l day. What are the results from the hedge?
'
810
k. INTEREST
APPEN DIX 24.A:
RATE MODELS
=
=
=
=
and a. Compute the prices, deltas, and gammas of the bonds using the Cllk Vasicek models. How do delta and gamma compare to duration and convexity?
b. Suppose the Vasicek model is true. You wish to hedge the z-yearbond using the lo-yetr bond. Consider a l-day holding peliod and suppose the interest rate moves one standard deyiation p or down. What is the retul'n on the duration-liedged position? What is the return on the Vasicek delta-hedged position? c. Iepeat the previous part, only use the Cllk model in place of the Vasicek model. 3-step (8 terminal node) binomial interest rate tree where 24.7. Copstruct a zperiod, the initial interest rate is 10% and rates can move up or down by 2%., model and your tree after that in Figure 24.3. Compute prices and yields for 1-, 2-, 3-, maturity? Why? with yields bonds. Do decline zyear
24.8. Verify that the zyear 24.6 is $0.6243.
zero-coupon bond price generated by the tree in Figure -
MODEL
Q
811
Tree #2
24.6. Consider two zero-coupon bonds with 2 years and 10 years to maturity. Let 44.721%. The 10%, and o'clp 0.05, o' vasicek 0.2, b 0. 1, r a consider will risk is each prrnium a position case. We interest rate zero in which with a will hedge value we consisting of one $100 par z-yearbond, position in the lo-year bond. =
TH E HEATH-JARROW-MORTON
0.08000
0 08111 0.09908 .
24.12. What are the l-, 2-, 3-, 4-, and trees?
0.08749 0.10689 0.13060
0.08261
0.t0096 0.12338 0.15078
s-yearzero-coupon
0.07284 0.08907
0.10891 0.13317 0.16283
bond prices implied by the two
24.13. What volatilities were used to constnlct each tree? (You cotnputed zero-coupon bond prices in the previous problem', now you have to compute the year-l yield volatility for 1-, 2-, 3-, and zyear bonds.) Can you unambiguously say that rates in one tree are more volatile than the other? .. . .
24.14. For years 2-5,
..
compute
.
'
the fllowing:
a. The fonvard interest rate, 1..f, for a folavard rate agreement that settles at the time bol-rowing is repaid. That is, if you borrow at t 1 at the l-year rate f' >nd repay the loan at t the contract payoff in year t is -
i
-
rg)
b. The fonvardinterestrate, rc, foraEurodolla-styleforwrdrateapeement that settles at the time borrowing is initiated. That is, if you borrow at t 1 at the l-year rate and repay the loan at 1, tlze contract payoff in year t 1 is '',
year zero-coupon bond price gen-
24.9. Verify that the l-year yield volatility of the erated by the tree in Figure 24.6 is 0.14. 24.10. Verify that the price of the l2% interest rate
cap in
24.11. Using a binomial tree like that in Figure 24.10, rate 3 years hence in Figure 24.5 is 14.0134+.
Figure 24.9 is $3.909.
velif'y that
the l-year forward
For the next four problems, here are two BDT interest rate trees with efective interest rates at each node.
Tree #1 0.08000
0.07676
0.10362
0.08170 0.10635 0.13843
0.07943
0.07552
0.09953 0.12473 0.15630
0.09084 0.10927 0.13143
0.15809
annual
-
-
c. How is the difference between r.f and re affected by volatility (you can compare the two trees) and time to maturity?
/
24.15. %ouare going to borrow $250mat a floating rate for 5 years. You wish to protect yourself against borrowing rates greater than 10.5%. Using each tree, what is the price pf a s-yearintlrest rate cap? (Assume that the cap settlos each year at the time you repgy the borrowing.) APPENDIX 24.A: THE HEATH-JARROW-MORTON The Black-Dermgn-rlby
MODEL
model illustrates one particular way to construct a binomial
tree from data. There are other ways to construct trees, such as the Ho and Lee (1986) model, which we do not discuss here. The Heath-larrow-Morton model (Heath et a1., 1992) is notable for proposing a general structure for interest rate models, one which contains othermodels as aspecial case. Theirbasic insightis thatno-arbitrage restrictions
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812
% INTEREST
RATE MODELS
require that the evolution of fonvard rates (or equivalently, forward bond prices) hinges in a specife way on bond price volatilities. When you adopt a specisc volatility model, forward interest rates. you implicitly adopt a specisc model for tlae evolution of IiI'IIC between volatilities and forward rates, suppose the singleTo understand the and we have two zero-coupon bonds with state valiable is the short-term interest rate, r, FI ?') implied forward zero-coupon bond > The r) nd Pt, F2, with F2 plices Pt, Fl price between F1 and Fa is .
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proposed assessing capital using the 99% lo-day VaR (seethe box on page 815 fpr more details). lkiskmetrics'' (seeJ. P. Morganmeuters (1996:,developed by J. P. Morgan in the mid-1990s, is one comprehensive proposal for a value at risk methodology. Much ofa the discussionin this section, especially forbonds, follows thevskmetrics methodology. Before we discuss how to compute value atrisk, recognize thatthe ideas underlying liskiness of bank portfolios. risk assessment matter in contexts other than measuring the and capital can pursue one of two investment For example, suppose a firfn has $10min million. One year, investment A rettlrns $12 million for opporttmities, each costing $10 :
$0 sure, whereas investment B rettlrns $24 million risk-free and the B is idiosyncratic risk of the investment probability one-half. Suppose the same rate is 10%. Standard investment theory will assess both projects as having 11r111 will its entire lse time half of B, the the with investmpnt positive NPV. However, additional make inkstments, ordr capital. In of to its investment and therefore all must raise additional capital, a costly process. Once we accout for the costs the (11-111 associated with 'losing a11capital, A and B may no longer seem equally attrctive. More much of a Erm's capital is at risk with a generally, managers will want to lnow how affect project selection.z terefore given project. Risk assessment can level as well. Suppose you are the personal of Distributions outcomes matter at both how much to save nd how to decide will need You to planning for retirement. For any stratejy, a key other and assets. stocks, bonds, allocate your savings among this strategy probability thatby following you will fail to achieve the question is: Whatis retire'p This is not the the time savings by retirement level of you a desired minimum pnnilegs--t of probability leaving lzigh with you ask, but a only question to a sategyconsideration. careful We call grounds-should for other desirable on no matter how lisk underlying but chapter, the ideas planning in tltis personal will not discuss snancial decisions. assessment can be used in making personal decisions as well as corporate regulators mentioned, First, risk. of value can at as There are at least three uses Second, instimtions. Enancial managers requirements for capital use VaR to compute risk-management decisions. Third, can use V:.R as an input in maldng risk-taking and bank's models. For example, if of quality the the also use V:R to assess managers can trading operation will lose $lm that particular 5% is chance there that a the inodels say a 20 days (5% of the t'ime) the ading then l-day horizon, on average once evel.y over a frequently, the models are this size of lf losses should more lose occur $1m. operation such lf losses activities. bank's risk the little occur less frequently, the to assigning too much risk. assigning too models are Most of the examples in this section use lognormally distributed stocks and linear normal approximations to illusate VaR calculations. Currencies and commodities can be modeled in this Fay as well. Although for long horizons it might not be reasonable with
with probability one-half and
2See Stulz(1996) foradetaileddiscussion of the linkbetween
nvestmentdecisions
andriskassessment.
3Bodie and Crane (1999)for example use Monte Carlo simulation to examine return distributions the suitability of financal products for retirement savings. assess
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to treat commodities as lognormally distributed, for short horizons this is generally a reasonable assumption. We ignore the possibility ofjumps. We discuss bonds separately.
Value at Risk for One Stock
is the dollar return on a portfolio over the holizon h, and f @, /?) is the Suppose of distribution retunas. Define the value at lisk of the portfolio as the rettlrn, x (c), that Probty such S xh (c)) c. In other words, xIl (c) is the c quantile of the ret'urn horizon h. the distribution over .%
=
Value at risk measures the loss that will occur with a given probability over a specified period of time. Notice that the defnition of value at lisk requires that we specify both a holizon, /7, and a probability, c. Suppose a portfolio consists of a single stock and we wish to compute value at risk the horizon h. If the distribution of the stock price after h periods, S, is lognormal, over
.
816
k
VALU E AT
VALU E AT R1s K
Rlsu
i;
we have
With 95% probability, the value of the portfolio over a l-weekhorizon
;(' E( . tgt ()..k
will exceed
:
knsl,jsol
-v'
hl-ka
-
0.5c2)/,,
-
As we saw in Chapter 18, if we pick a stock price Plice will be below u is Probts
J),)
<
ln(u$,) -
N
=
-
(r2/Tj
(25.1)
ljt
%, then the probability that the stock
jtl r) j
(a
-
-
-
-
0.5c2)/7
(25.2)
k
(
where N-
1(
(0.05)
-
j
-
-
sjy
We can solve for v (c) by using the inverse cumulative probability distribution, .N'-1 Applying this function to both sides of equation (25.3),we have
j
N .-
solvingfor
u$,(c)
l
(c)
ln(J;F,(c))
=
lntsbl
-
-
(a
J
-
0.50-2)/7
-
=
(c)
%:
%
(25.5),N-3 (c) takes
25.1 Suppose we own $3mworth of stockA, whichhas an expected rettlrn 30% volatility, and pays no dividend. Moreover, assume A is lognormally 15% and of a value of the position in 1 week, 7, is distributed. The Example
V
=
$3m X
e
+().a.&
(().
&
=
(1
zcx-fl) + ah +
(25.7)
We could also further simplify by ignoring the mean: =
(1 z(rvF?l
vbb
(25.8)
+
(25.7)and (25:8)become less reasonable as 11grows.
Example gives
25.2
Using the same assumptions as in Example 25.1, equation
$3m x
j.j-().5x().a2)
'
(25.5)
This expression should look familiar from Chapter l8. In equation the place of a standard normal rando-m variable.
'
.
In this case, we would say the 95% value at risk is
%
-s-z.so.ljll+o..-'
se
$p,gge/p,m
If the assumption of lognonuality is valid and if the inputs are correct, a l-week loss of this magnitude occurs on average once evel'y 20 weeks. In practice it is common to simplify the V:.R calculation by assuming a normal rettll'n rather than a lognormal return. Recall from Chapter 20 that the stndard lognorma1 model is genergted by ssuming normal returns over very short horizns. We can therefore approximate the exact lognormal result with a normal approxiriiation:
Both equations
J,(c)
=
-$0.1928m.
o'VF7
gives
.645)
'
-1.645.
=
.
'
x (- l +0.3./F
x().g2)
.j.().rj
=
-
ln(&)
-
ygm x s((). ..
.. ) $2.8072m $3m
The complementary calculation is to compute the h (c)corresponding to the probgbility c. By the dehnition of (c), we have 0.50-2)/7 ln(J),(c)) lnt5'al a (25.3) c= N a
z
V:R is therefore
1+
0.15 + 52
$2.8033m $3m =
0.3 52
x
(-1.645)
-$0.1966m.
=
$2.8033m
lgnoring the mean, equatipn
=
(25.7)
(25.8)
gjves $3m x
1+
0.3
x
(-1.645)
=
$2.7947m
.6;
(p,5
Xo, 1). Given this assumed stock plice distribution, a 5% loss will occur if Z satisses
where Z
%. 817
VaR is $2.7947m -
$3m
-$0.2053m.
=
ew
$3m x e
Z=
(0.l5-0.5x0.32)i+0.3
ln(0.95)
-
(0.15 O.3 x
z
=
() 95 x $3m .
.1-
-
0.5 x 0.32) :,2 =-. .UL
j (.48J.,..5
normal with meqn, and norFigure 25.1 compares the three models-lognonual, holizons of day without mal to one year. As you would expect, the mean--over one 25.8) is less accurate over longer horizons. the approximation ignoring mean (equation In practice the mean is often ignored for two reasons. First, as we saw in Chapter 18, means are hqrd to estimate precisely. Second, as we saw in Chapter 20, for short horiztms the mean i lss irfottant than th diffusion tenn in an It proes.
32
VaR for Two or More Stocks
We have NormSDist(-1.2815)
=
0.1000
Thus, we expect that 10% of the time there will be a weekly loss in excess of 5%.
When we consider a portfolio having two or more stocks, the distribution of the future portfolio value is the sum of lognormally distributed random variables and is therefore
%.VALUE
818
.'.'.
AT RlsK
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vAu
.y' yyjyjttjd )j)' j'))'))' )y' t'' )1 Ellqq:,,f ylsli.ltqy,f (' )' jj'. jjy': .;,!!Eii5i)'. '11!11*. liltiE!j-f ?11Ik---1iii'. q-'. 1IiIE:qFq'.' )#' rtiqfr7-:lllt'---iq'. r:'' j g (('j'.jl (';.jE..jE'qij.1: 'LikttL' ytyg jyjltg 444.2 qjjj () 1iIi:itiI::(jjg((yj((g. gyj.. j..jjj....jjj.j.jgj...j.jjjj.yjjj..g.j.jj.gj: .jyj;jja yk (. g. j. . E g.. jg . .lr@- '. '.'. 1L-.'. ;k: ik-i-.. (... .. . .. . t.t:y! .. '. ;: tty. . r(i ... ; (.
E AT RlsK
k
819
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Example
.
: :. .
,
y
.
-.
-
.
ii. ()1..
. .
. .
.ll,,;ll:. (:1:.
dl::::;.-
.
..
. Comparison of VaRfor a single stock over diferent horizons using the Iognormal solution .
'
E ' E . . . ! . :g r .y j .. g . g . g. g. g y j. g g . g - .-. . -. - . . -.. . -;. - -.-.:
:
,
.
. ..
c2
of 5)
(Oillions
Exact lognormal Linear, with mean Linear, without mean
100 95
(equation (25.4:, normality with a Positive mean (equation (25.7:, and normality qssuming a zero mean (equation (25.8)). Msumes the Same Parameters as in
=
>-r.
0.15, o'j 0.3, Fl Suppose we have (rl 0.18, $3m, (:2 and The annual of the portfolio 0.4. $5m, return is p l,c mean
k *N .
N.
Bcel + J72a2 IVl + 1P2
=
=
1247.2 l l +
F20.2 + 2 2 WI
*
N.
80
*+ **..
75
=
+. *+..
*.
70
**. **.
o
0.16875
=
*
.
*A
*o.
*
.q
o
*
qr
h
* mt 4.
*#*.
**o
0.1
0.2
0.3
0.4
*vo
$8m x
**v
I
0.5 0.6 0.7 Time (years)
0.
0.9
1
(25.7),there
(1 +
t notlognonnal. Since the lognormal distribution is no longerexact, we can use the normal approximation or we can use Monte Carlo simulation to obtain the exact distlibution.
Let the annual mean and standard deviation of the realized ret'urn on stock i, (-f/, be ai and o'i, with te correlation between stocks i and j being pq. The dollar investment in stock. i is lW.The value of a pprtfolio containing ?7 stocks is tt
IV =
Jl
The ret'urn on the portfolio over the holizon /7, % is ,
Portfolio return
=
Rh
=
(-f,
F
Jlj
i= l
Assuming normality, th.e annualized distribution of the portfolio return is 11
ai /7Wk ,
i= l
1 z
11
11
1
1 + 0.34216 x
x
The l-week Val ignoring the mean is therefore
jjl
x
(-1.645)
=
$7.40154m
-$0.5985m.
(-1.645)
=
$7.3756m
$7.3756m $8m -
This example illustrates the effects of diversifcation.
-$0.6244m.
=
Although stock 2, which.
constitutes more than half of the portfolio, has a standard deviation of 45%, the portfolio standard deviation is only about 34%. Problem 25.5 asks you to consider the effects of different correlations. assets, the V:I.R calculation requires that we specify at least the (and possibly the mean) for each stock, along with ttll pairwise cor-
11
VaR for Nonlinear Portfolios lf a portfolio contains options as well as stocks, it is more complicated to compute the distlibution of returns. Speciically, suppose the portfolio consists of 11 different stocks Jlj There are also Ni options worth Csi ) with oi shares of stock i worth o)i Si vtlue The portfolio for each stock i. is therefore J'P 5)'=1 l(tnSi + Nici (&)). We cannot easily com'pute the exact distlibution of this portfolio; not only is the sum of the =
cf o'j pu/7 W J'Py i= l jzzzl
0.34216 x -
relations.
11
l
+
The l-week 95% VaR is therefore $7.40154m $8m = Using equation 91% value which ignores the chance that the of the portfolio (25.8), mean, we have a will exceed
If there fe standard deviation
i= I
is a 95% probability that in 1 week, the value of the porfolio
(j (0.16875 x
$8m x
'
+ +2
($3m x 0.3)2 + ($5m x 0.45)2 + (2 x $3m x $5m x 0.3 k 0.45 x 0.4) $3m + $5m
Using equation witlexceed
*wx *o
0
ziyjyyao.jgapj,z
''''*-
50
1
$3m x 0.15 + $5m x 0.18 $3m + $5m
*<<<.
**o
55
.V-
=
= 0.34216
.....-
*+
60 :
e'w
=
The annual standard deviation of the portfolio return, c/?, is
Gp
65
Rl,
=
'kk
8s
Example 25 *2
=
=
=
=
av
k
90
25.3
0.45, +2
.
=
%.VALUE
820
vAuu
AT RIsK
lognormally distributed stock prices not lognormal, but the option price distribution is complicated.
in 1 week
We will explore two different approaches to handling nonlinearity. First we can create a linear approximation to the option price by using the option delta. Second, we can value the option using an appropliate option pricing formula and then perform Monte Cado simulation to obtain the return distribution/
E AT RIsK
821
will exceed
$2,666,507 x
l
1 + 0.084343 x
1
+ 0.16869 x
x
(-1.645) = $2,568,220
Value at
k
lisk
using the delta approximation is therefore
(25.12)
$2,568,220 $2,666,507 -
=
-$98,287. We can compute the exact value at risk by srstdetermining the stock price that with will chance, exceed and then computing the exact portfolio value at that 95% a we will Wit.h probability, exceed the stock price price. 95% we
lf the return on stock i is :1., we can approximate the ret'urn Li-i Af is the option delta. The expected annual retul'n on the where on the option as then portfolio is stock and option Delta
approximatlon
,
$100 x e
(25.10)
2 p
tl
l
=
?1
Sf Sj
JF a
()f + Ni Af)()y + Njhjjciajpij
($93.574 x 30,000)
(25.11)
With this mean and vmiance, we can mimic the n-stock analysis. First, however, we will compute an example with a single stock for which we know the exact solution.
$:g
c::z
=
xsgy
$9.5913kand the
kale
of
$2,576,438
$2,576,438 $2,666,507
-$99,069.
=
-
Figure 25.2 compares the exact value of the portfolio as a function of the stock
price 7 days later, compared to the value implied by the delta approximation. The delta approximation is close, but the Valk derived using delta is slightly low. The delta approximation also fails to account for theta-the time decay in the option position. Because the option is written, time decay ovqr the l-week horizon incrses the return of the portfolio. increasel return is barely perceptible in Figure 25.2 as the exact portfolio value exceeds the delta approximation when the stock price is close t $100.
,
'this
1E1
.
-64.5)
($9.5913x 25,000)
-
The exact 95% value at risk is therefore
1 i= l jz:zz
t'yd ('t Example 25.4 Suppose we own 30,000 shares of a nondividend-paying stock and 'lf have sold los-strike call options, with 1 jtj year to expirat ion on 25,000 slaares. The stock price is $100, the stock volatility is 30%, the expected ret'urn on the stck is 15%, and il .jj the risk-free )'j.tq rate is 8%. The Black-scholes option price is $13.3397and the value of ))) the portfolio is lk .; ti) J'I/ 30 000 x $100 25,000 x $13.3397 $2,666,507 't'l (. 11 q y.?.) their value in .computing the value (', (Since the written options are a liability, we subeact ) of the portfolio.) The delta of the option is 0.6003. Using equations (25.10)and (25.11), 1, we obtain Rv 0.084343 and o'v 0.16869. The written options reduee the mean and volatility of the portfolio. Therefore, there is a 95% chance that the value of the portfolio @
x (- l +0.3.Ul#
If this is the stock price 1 week later, the option price will be the portfolio will be
The term oz + Ni Af measures the exposure to stock i The variance of the ret'urn is .
()'
(0. 15-0.5x0.32)
-
=
g E
..
'''
'''
E...(.
l
=
-
,
.
.
!
' ' . '.
=
=
j'gy'. y'p1Example 25.5 )' '')' we have two stocks along with written call options on those C1s tocks. Information suppose for the stocks and options is in Table 25.1. Using this information, '''
I'E
tEj
t) we .j.yj ).1
obtain a portfoli 1;7=
j!t
t.t )i)
(30,000 x $100) (25,000 x $13.3397) + (50,000 x $100) (60,000 x 10.351 1) -
-
.) $ Using 'ti.-() El
value of
equations (25.10)and (25.11),the annual mean and standard deviation are 8.392% . and 16 617% There is a 5%. chance that the portfolio value will exceed
j'; @ ' ltjtj 1I/ x E
4A thrd alternative is to use a delta-gamma approximation, which as we saw in Chapter l 3 is more accurate than a delta approximation. However, because the gamma term depends on the squared change in the stock price, the approximation is harder to implement than the delta approximation(Morganmeuters l 996, pp. l 29-1 33) discusses an approach for The Riskltlerics Techllical Dt?ct/??2d?7? implementng the delta-gamma approximation.
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To use Monte Carlo simulation in the case of a single stock, we randomly draw a set
of stockprices as discussedin Chapter 19. Formultiple stocks, we can use the appropriate parameters for each stock and use the Cholesky (see Section 19.8) to correlation the appropriate prices. have the portfolio stock Once among we ensure values corresponding to each draw of random prices, we sort the resulting portfolio values in ascending order. The 5% lower tail of portfolio values, for example, is used to compute the 95% value at risk. We will look at two examples in which we compute Valk for a position using Monte Carlo simulation. First we will examine a straddle on a single stock, and then a straddle-like position that contains a written call on one stock and a written put on the other.
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Msumes the position is Iong 30,000 shares of stock at 5100 and short 25,000 call options with of $1O5. a strikeatprice risk is the value between the diference
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entlies in Table 25.2. If the stock price declines, there is a 0.9% probability that the If the stock price lises, there is a value of the position will be less than will be less than Thus, in total, there 4% chance tat the position value is a 4.9% probability of a loss in excess of about $942,452,which is the average of the boked numbers. The l-week 95% VaR is therefore approximately Even in this one-stock example, calculating the VC).Rfor (-$685,776) this two-tailed positin is not as simple as computing the stock prices that are exceeded with 2.5% probability. Monte Carlo simulation simplises the analysis. To use Monte arlo we randomly draw a set of z .VIO. 1), and construct the stock price as -$942,266.
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Table 25.2 shows a jubset of the values plotted in Figure 25.3. Examine the boxed
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Msumes the risk-free rate is % and that neither stock pays a dividend. The correlation between the stocls is 0.4.
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Example 25.6 Consider the l-week 95% value at risk of an at-the-money written 100,000 shares of a single stock. Assuming that S straddle on $100, K $100, 0, the initial value of the straddle is 30%, ?8%; F 30 days, and J o' -$685,776. Because the underlying asset is a single stock, we can compute the VaR of the position directly without Monte Carlo sipulation. Figure 25.3 graphs the exact value of the straddle after 1 week, compared with its initial va1ue.5 The expected ret'urn on the stock is 15% in this calculation.
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creases or decreases, wltich is not a situation suited to a linear approximation. Because of losses from stock moves in either direction, we need a two-tailed approach to VaR. Monte Carlo simulation works well in this situation since the simulation produces the distribution of portfolio values.
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E AT Rlsu
.
$6,789,740 $7,045,440 -
The delta approxnation
=
-$255,700
k
We compute the Blackrscholes call and put prices using each stock price, which gives us a,distribution of saddle values. We then sort the resulting straddle values in ascending order. The 5% value is used to compute the 95% value at risk.
can workpporly for nonlinearport-
folios. Forexample, consider an at-the-money written saddle (a written call and written put, both with the same strike price). The saddle suffers a loss if the stock price in-
sne
increase in value of the straddle if the stock pn-cedoes not change is due to theta.
%:VALU E AT
824
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j'j' '; (( ( j'.t' 1II1i.).)iI1' LLLI'.LCL y' yjjf t'rcl!iqf tilf rqf 'gjj,f Illi!ij!ld Illliillll:d ll;)d yyf k)j': (' r' t'gjjjgjjyjg'. . . . y ;.' . . 7( .j j...y j.g jjjjy.yy. j.. yyjjj gj .jg ( . rj.. r.jjyyj r.jjy. yj j. jj.jg.yj y y.y gj.g y . . . j j i.. . . j ..!. gjyg jyjy gj jg ( 11111. ;.. q#!r(:.. y.i.1IIIq!!1i. ....... . gg.g . k. ;..g .. The value of a poitfo jjo as a function of the stock prie, contain ing 100,000 writlen call options with a $100 strike and 100,000 hritti, rl tl t ) glticlrlS hh/itl r) a 100 strik, Msumes 3t%, r 8%, o' t 23 days and J 0.
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Figure 25.4 plots the histogram of values resulting frbm 100,000 random simulations of the vtlue of the straddle. There is a 95% chance the straddle value will exceed This -$943,028; hence, value at lisk is (-$685,776) result is very close to the value we inferred from Table 25.2. %
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-1045788
0.980
-873152
0.018
2.10
109.35
-2.1063938
-859958
0.020 0.023
2.15
109.58
-1082345
2.20
109.81
-834263
0.026
2.25
110.03
0.982 0.984 0.986 0.988
-821779
0.029 0.032
2.30
110.26
2.35
110.49
-
0.036
2.40
110.72
-
2.45
110.95
2.50
111.19
-2.15 -2.10 -2.05 -2.00
91.82 92.01 92.20
-1.95 -1.90
92.39 92.59
- 1.85 - 1.80 -1.75 -1.70
92.78
-886566
-846992
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92. 93.17 93.36
-809547
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- . . . .... .
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in the z column are standard normal values with the corresponding cumulative probabilities in the N(z) column. Over 1 week there is approximately a 5% probabilitythat the stock price will be outside the range $107.77. The option values are computed using the Black-scholes formula with tr 3091, r 8%, t 23 days, and 0. The stock price movement assumes a 1594.
-$257,252.
-$943,028
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the same underyling stock, they have different, correlated underlying stocks.
)') Example 25.7 Suppose that there are two stocks. Stock 1 is the same as the stock . .yj.( 25.6. Stock 2 has the same parameters and a correlation of 0.40 with stock in Example l,y
. 1 l's y
Because tlae stocks have the same volatility and dividend yield, the initial option Based on '' values are the same and the written straddle has an initial value of ;l'.' 100,000 simulatedplices forboth stocks, theportfoliohas a95% chance of having avalue ! 135,421- (-$685,776) )rlgreater than $1,135,421.Hence, the 95% value atrisk is ).)1 Figure 25.5. this calculation The histogram is in for . )) %. (( .'
-$685,776.
Egj)
-$1,
=
-$,449,645.
A comparison of the results in Examples 25.6 and 25.7 shows that writing the straddle on two different stocks increases value at risk. If we exarnine the distlibutions in Figures 25.4 and 25.5, we can see why this happens. Notice
srstthat
in Figure 25.4, the value of the portfolio never exceeds about
-$597,000. The reason is that, since the call and put are written on the same stock, stock price ?ntpvd. call ndvcr indltce #7d rwt?options to appreciate together. They can appreciate due to theta, but a change in the stock plice will induce a gain in one option
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Histogram from Mnte Carlo simulation of the value of a written straddle after 7 days. Both the call and the . . put' are writtn on tbe same stock. Assumes at-the-money calls and u:.sare Writlen on 100,000 shares, with .5' $10O, (7' = 309% r 8% T 23 days,
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gjjyygjjryyyjtj .jjjygyyjrjjjy, yjyyy-jjkrjj, rjrg,, Observations jjyjjgr, jtyyyy;jj
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r 11. ii. ('' ))t;. )t.t! jy .yjjjjyj. . gy . j rjyy. jy. jL:;3Ljj,:. .. .G jjjjj:;jj;t, . EFr.lTr)E.1Iiqq.. . ... ty jj l . .. . . . . The value of portolio, as a function of the stock price, containino .. call 100,000 written options on one stock witha $100 strik and 100,000written put optionson a diserent stockwith a $100 strke. For both stocks, assume c 30%, r= 8%, F= 23 days, and $ 0. The correlation between the two stocls is 4094.
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Value of Option Portfolio
and a loss in the other. The sape effect limits a loss, since the two options can never lose money together. 'Whe the options are written on different stocks, as in Figure 25.5, it is possible for both to gain orlose simultneously. As aresult, the distribution of prices has a greater vmiance and increased value at risk. As a final commenk all the value at risk calculations in this section assumed remlm for the stocks that was positive and different from the risk-free rate. expected an Because the horizon was on'ly 7 days, the results are not too differentfl'om those obtained assuming the drift is zero or equal to the risk-free rate. For longer horizons the particular assumption about expected rettlrn would make more of a difference.
(5)
0.5 x 106
swap reduces to one of decomposing the claim into its constimtent zero-coupon bonds and assessing the Iisk of these. The risk of the bond or other claim can then be measured lisk of a portfolio of zero-coupon bonds. a; tlle With zero-coupop bpnds and other pite maturity clims, the historical volatility of the claim is not riecessarily a good measure of the future volatility of the claim. Other things being equal, bonds become lesj volatile as they approach mattlrity. A naturai solution tp this problem is to characyerize risk in trms of the bond yield rather than the bond pric. Yield uncertlkinty implis price gncertainty. of how to measure VaR for a gro-coupon Here is an eampl bond. Suppose that bond tine and haF price gnd annualized yield F #(F), that the matures at a zero-coupon volatility of the bond is o'v. For zero-coupon bond, duration equals maturity. Thus, if the yield chapgs by 6, the percentage change in the bond price will be apprpximately 6F. Using this linear approximatipn based on dulption, and ignoring the mean return on the bond, yer the horizpn 14th bond has a 95% chqnce of being worth more than
VaR for Bonds In this section we see how to compute VJtR for bonds, using information about the volatilities and cot-relations of yields for bonds at different mamrities. At any point in time there are numrous interest rate sensitive claims, including bonds, FlkAs, and swaps, all of which can have different mamrities and be denominated in different currencies. We can simplify the problem of risk modeling for interest-rate sensitive claims by recalling from Chapter 7 tat all of these claims can be decomposed into zero-coupon bonds. Thus, the problem of assessing the risk of a bondj FRA, or
-
::.
1
0 -2.5
0.5 x 106
)')t?(.')X.t.y).J. r:1$. p..y .
ti.yt.j.y-.. (. ).
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: !: E ..E
'
#(F)g1
+ cVT W/2 x (=1.645)J
Suppose a bond has F 10 years to maturity. Its yield to maturity 25.8 r@Example iii'E 5.5% and the annualized yield volatility is l%. The one-week VaR is on a $10mposition =
'.
% VALUE
828
vAl-u E AT
AT RlsK
rltin these bonds is )) )t) $10m x jll EE
il!..)):.
1
1 + 0.01 x 10 x
x
(-1.645)
-
$10m
-$228,
=
%.
120
Now suppose that instead of a single bond we have a portfolio of zero-coupon bonds. In particular, suppose we own J7l of a bond mattlring at T'1with annualized yield volatility o'r, and W2 of a bond maturing at F2 with annualized yield volatility trwa Let volatility p represent te correlation between the yields on the two bonds. tTl1is yield using volatilities.) using implied As Mstorical data estimated could be ipformtion or approximation, of only here delta instead the of stocks, portfolio use two with a we can correlated syock rettlrns we have two correlated bond yields. .
0.012 p 0.01, cwz jpExample 2s.9. Let rl l0, r2 15, o'., invested in the lo-year the portfolio is 60% B'1 .= $6m atad Jn $4m. since ))) I) 40% invested in the ls-year bond, the variance of tlae bond portfolio is =
=
=
=
l'E!1
=
ttt
E)1 E;.
jjr. :
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0.985, bond and
=
0.01729 )! . $( () 0.1315. The 95% one-week V:R for this portfolio is lijjThe volatility is 0.01729 C'i) )1 therefore li) k. $10m x (1 + 0.1315 1/52 x (-1.645)) $10m lj E
=
-
'
The go.l is to nd afl interpolation procedure to express any hypothetical zero-coupon bond in terms of the benchmark zero-coupon bonds. This procedure in wltich cash flows are allocated to benchmrk claims (in this case zero-coupon bonds) is called cash flow
mapping.
Suppose, for example, that we wish to assess the risk of a lz-year zero-coupon bond, given information on the lo-year and 15-ye1tr zero-coupon bonds. It is reasonable to use simple linear interpolation to obtain the yield and yield volatility for the lz-year bondfromthose of the lo-yearand ls-yearbonds. Forexample, if the yield and volatility of the f-yetr bond are )?; and c'f linear inteolation gives us ,
-$301,638
-
=
shortcomings of duration as ameasure of bondprice the use of duration in these examples. Duration wondeling about might be risk, so you the plice change for bond for a given change used is mechanically compute to here approximation yield. This is delta to the actual bond price change. the bond's in a own The concept'ual problem with duration becbmes problematic when w use duration to compute a hedge ratio for fwt? bonds. The hedge ratio calculation asjumes that the yield to mattlrity for the two bonds changes by the same amount. By contrast, in Example 25.9 each bod has a different yield volatility and there is an imperfect correlatin between the two yields; thus, we do not assume that all yields to mattlrity change by the same amount-iae., that there is a parallel yield curve shift. (For parallel yield curve shif't = 1.) we would need each bond to have the same annualized yield volatility and p ln general, if we are analyzing the lisk of an instrument with multiple cash flows, the frst step is to find the equivalent portfolio of zero-coupon bonds. A lo-year bond with semiannual coupons is equivalent to a portfolio of 20 zero-coupon bonds. Every t'buckets'' containing the interest rate claim is decomposed in this way into interest rate clim's constituent zero-coupon bonds. A set of bonds and swaps reduces to a portfolio of long and short positions in zero-coupon bonds. We need volgtilities and correlations for a1l tese bonds.
We discussedin Chapterz4the
ylz (0.6x =
)'lc) +
(0.4 x yt5)
(25.14)
(25.15)
=
-)
829
As an empirical matter, the movement of a zero-coupon bond at an 8-year matulity, for example, is higllly correlated with that of a zero-coupon bond at an 8 l/z-year maturity. nus, fortractability, volatility and yields are tracked only atcertain benchmark maturities: In Riskmetrics, these are 1, 3, 6, and 12 months,and 2, 3, 4, 5, 7, 9, 10, 15, 20, and 30 years. lf we have a zero-coupon bond in the portfolio that does not exactly tnatch a benhmark mamrity, we want to determine the portfolio of the benchmark zerocoupon bonds that matches the charactelistic of te nonbenchmark zero-coupon bond.
'
(0.6x 0.()1 x 10)2 + (0.4 x 0.012 x 15)2 + (2 x 0.985 x 0.6 x 0.4 x 0.01 x 10 x 0.012 x 15)
k
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These interpolations enable us to determine the price and volatility of $1 paid in year 12. ,-)'I2X 12 However, We are nOt snished because these interpolaln particular, the price is tions do not provide correlations between the lz-year zero and the adjacent benchmark bonds. We need these correlations because we could have a portfolio containing 10-, 12-, and ls-year bonds. bonds The next step is to ask what combination of the 10- and ls-yearzerououpon would have the same voltility as the hypothetical lz-year bond. lf we let ) equal the fraction ayllocated to te lo-year bond, we must solve 2 2 () 2 c'1(,) cja, =
+
((1
-
.)2g.2
1()j
+ jgpjtj jsltj
-
,
upgjgo-jsj
(;5.j6)
Since this is a quadratic equatibn, there are t'wo solutions for Typically, as in the following example, one of the two solutions will be economically appealing and the other will seem unreasonable. .
Example
25.10
Suppose we have a $1 cash flow occun-ing in year 12 and that we wish to map to the 10- and 15-year zero-coupon bonds. Suppose that ),10 5.5%, 1.2%, and p 0.985. The yield and volatility of the y15 5.75%, ty 10 1%, c' 15 hYPothetical lz-year zero-coupon bond are =
=
=
=
.:12
=
(r12
=
=
0.056 (0.6 x 0.055) + (0.4 x 0.0575) (0.6 x 0.01) + (0.4x 0.012) 0.0108 =
=
We next need to find the cash flow mapping that matches the volatility. Solving equation (25.16) gives the two solutions a) = 6.2097 and 0) = 0.5797. The srstsolution maps
830
k
VALU E AT RIsK
VALUE AT RlsK
the cash flow by going long 621% of tlle lo-year bond and short 520% of the ls-year )!t j.:. . tjl bond. The second, more economically reasonable solution, entails going long 57.97% lrt lti' of the lo-year bond and 42.03% of the 15-yea.r bond. k 0.5797 is close to the 60%-40% split you might have Notice that the solution 0) weights, value at risk for the lz-year bond can be these the Given guessed at outset. the bond portfolio in Example 25.9. VaR in the for computed same way as lf you find this procedure confusing, recognize that if we had mapped the 12-year bond by assigning 60% of it to the lo-year bond and 40% to the ls-year bond, then Is'c the yield alld vtl/t/rf/ry given by eqltations (25.14)and (25.15). bvoltldnot /?n?d lnatched Because of the nonlinear relationship between prices and yields, an intemolation based on the yield will give a different cash ow map t han an intemolation based on prices. Although we have discussed cash flow mapping in the context of bonds, mapping be can applied to' any claim with multiple cash flows. =
Estimating Volatility
4
k.
831
However, independence may be reasonable over shorter horizons, where supply and demand responses do not have time to occur. Finally, we also require correlation estimates in order to compute volatilities of portfolios. Correlations can be estimated using historical data the same way as volatilities. flowever,it is important to be aware that, just as volatilities will change in times f inancial stress, correlations will change as well. 'When there is a stock market crash, for example, by definition all stocks move togethr, exhibiting a temporarily high positive correlation. lt is important, therefore, to test correlation assumptions. One way to do this is by evaluating scenarios in which asset prices undergo large moves and asset correlations become extreme.
Bootstrapping Return Distributions It is possible to use observed past returns to create an empirical probability distribution of rettlrns that can then be used for simulation. This procedure is called bootstrapping. The idea of bootstrapping is to sample, with replacement, from observed historical returns under the assumption that future remrns will be drawn from the same distribution. For example, if the stock price over a 5-day period has remrns of 0.10, 0.03), then this distribution can be bootstrapped by ran(-0.02, 0.015, domly selecting one of these retums each time a new l-day return is needed. ln effect, bootstrapping randomly shufles past rettlrns to create hypothetical f'uture returns. We have seen how the lognormal model has trouble accounting for events like the 1987 market crash. The advantage of bootstrapping is that, since it is not based on a particular assumed distlibution, it is consistent with any distribution of returns. For example, if an event like a signiscant market crash occurs once every 10 years historically, it will occur on average once every 10 years in the bootstrapped distribution. A disadvantage of simple bootstrapping is that key features of the data might be lost when the data are reshuflled. For example, if historical remrns exhibit persistence histolical remrns will not exhibit such persistence. in volatility, randomly reshued It is possible to bootstrap a distribution while preserving correlation; this is called depenent bootstrap.6 First, it is possible to bootstrap by randomly selecting blocks of data at a time. The use of blocks preserves correlation across time within the blocks. Second, given a specification for the correlated series, we can estimate the correlated process and use this to bootstrap. For example, suppose we believe that a commodity price is generated by the prpcess -0.01,
volatilityis the key input in any V:.R calculation. ln most of the examples in this chapter, with c' onstant. This calulation assumes volatility over the horizon /7is tysl, ret-u:'n volatility, change, and that returfls are independent over time. does that he not o', both ln practice, both assumptions may b violated. It is straightforward to estimate historical volatilties given a time series of rttlrns. However, volatility is typically not constant over time. As a consequence, it is necessary to update volatility estimates when V:.R is computed. ln Chapter 23 we discussed a number of ways to esrimate volatility: simple historical volatility, exponentially weighted moving average, GARCH, realized volatility, and implied volatility. Keep in mind, however, that when volatility is stpchastic, there can be a lisk premium for volatility that will affect the option price. lnfening the volatility from option plices when volatility is stochastic requires careful modeling, since it is necessary to disentangle the tnle implied volatitity and the lisk premi.um. This is an area of opgoing research. Depending on the asset, returns may be correlated over time. To a flrst approximation, stock rettlrns are independent over time. However, over horizons as short as a day, reml'ns may be negatively correlated due to factors such as bid-ask bounce. There is some evidence of negative correlation at longer horizons, though the effect is more subtle.
When rettlrns are correlated, vplatility does nlt scale with yg 3. s retvyo s are negatively correlated over time, thn high returns are followed by low returps. Thus, negative return correlation dampens volatillty feltive to the independent remrns case. If returns are positively correlated, high returns follow lzigh returns, which results in a higher volatility than in the independent returns case. With commodities, return independence may depend on the horizon. For example, high copper prices lead to increased supply and reduced demand, which eventually induces the price to fall. So rettlr'n independence is not reasonable for long horizons.
St+l,
=
97+ pSt +
t-lt
(25.17)
lf we 'Iit this process to the data, we obtain an estimate of p and a time series of errors, These errors should be uncorrelated if equation (25.17) is correctly specied. Et+;l. We can then simulate the process in equation (25.17) by randomly drawing from the
6Two general discussions of bootstrapping that also discuss bootstrapping and Tu ( 1995, Chap. 9) and Horowtz (2001).
of dependent data are Shao
832
k
IssuEs
VALU E AT Rlsu
estimated errors. This will generate a selies with the observed en-ors drawn from the data.
correlation
and with
25.2 ISSUES WITH VAR VaR provides a single number that is easy to interpret and communicate. However, VIIR is not the only way to assess risk, and it has conceptual shortcomings. ln this section we discuss some alternatives to simple VaR and introduce the concept of sltbadditive risk measures. In the course of comparing risk measures, we will also discuss the use of the trpe price distribution in computing VaR as compared to the risk-neutral distribution, which we use in pricing derivatives.
't' j,:gg:,,f :'j'.f ),t' ''';' E'('EE' EE'EE''''EE'( (l' E E:!!lEi 'EtE;i El())( if ,'t',yy-'.'E 5:y94* )' ly'.tE ykpyf (y' j'q' 85* y' $2* 1* ?)!' 'qE:E E i(;EEE :'.1t2rr:!j,-;,.' k,' (q' lq'! !' 7q'::77'* (('E'.(.(iE''i E' E.I'E(E!i'kE'.. ((' ( (( E E ''i @' ;(E'(')E'!('(' j)l)f q' r' j'j)yttyyf (' ).:rt' yj'jyyyy;',ri(jt('. Fqr7'rlrrq!r'f )' y' 'yf j'y': ;'@' E( (: -:::;?3:L'. ;'. rqf 4/2j(74*5q(54, ltq'.s jt.f ii.f j,lljf t;f ttf r. . ' . . . . .( k!. '. E Ej' (''... ((' .'E ' ' p!' .. .()Cf'E .((t!'. qyj . . .. !' . i t.EEE E. E . i E. . i ! ! Eq ! i. . ( i.;. i;.! j . . ( ( .tt'hfE:[email protected]':qq-::; y(... j. ji . ; .j.yyg(. . . ,r ; iE...(. (.j.q.q.E.E.qij.EE.(yj.y.y .qj.;.ti . ! . E .E. EiE..;........-.....:. 'i:'i')C1 ,j E . jjjy-y. .y(y.y(.. q., y. q q .:L-..-.......:............ .r.... . .., ...E:...........j; .i ....y...q.. ... y. .. ..
.'.'
,(Ij!!E5;;;r'.)j.yrrjjy'. y' I11Ik,,-j1I(' tjjy::,,f r' ;'@' )' L;j33'. rjj'-..ay-.yyyf IljE!!!!)f rjtyr:,,f r' .'. 1r,.:.. '. ; . k:k-. jjlt::;ljl:, . )t. .,111:::::::j2. j .. -L-LL).. . .. . .. . . The probability distribution for Sn the VaRprice Ievel and the tail (.(0.05)), VaRprice level (f g-rl-r ..Assklmes < the same parameters as in Example 25.1 -' ,'-'t;yj-j.(-' -t' L' :' E '' S' '-r'
:
:
'' '' '' : . : :
.
. .. . g. .
-.
:
.. .. . .
. -.
.
: .
'
:
.
:. . :. ...E
.. . .
g
-
.......,.
.......
'
.
.
.
.
..
..-
.'
.
-
.
-
::: ..
.
. :
. . .
:
'' ' :
:'
. :
.
-
.
-
:. . -:
vAR
%. 833
'7r)'
--q.
:
:
-
-.-:t; :. ,;k.Lt--.a-j.
. -
. :.. .. . -
-
:
-
. . . .
. ..
:-
:
. -(--,..--
-
... . .
-yj
.
.-
-
....
-. -.
-
. .. .
ji. .1k(gr),-
y,r:,. ;,E,j,
...-....
.
0.03 o (s5
Tail vaR
-
($68.244)
...
.$(0.05)1).
0.02
0.015
.
':
:
.
+
:
: : : : :
0.005
qs'o
.
: : : : : : : : : : : VaR :'($74.347) : : : :. : : : : : :: ::
.
Alternative Risk Measures and we Consider again th Valk for single stock. Suppose the current stock plice is wish to aqsess the risk of this investment at time F. As before, if the confidence level is c, desne J(c) as the stock price such that there is a c probabilit'y that the stock price at time F will be worth less than Zcj. ln addition to VaR, we can compute two additional risk measures based.on the VaR.:the tail VaR, which we define below, and the cost of insurance against outcomes below the VaR level.
.
: : : : :
0.01
A shortcoming of V:.R is that it specises the inillilnttl'l loss that will occur with a given probability. ln practice we would like to lnow the expected magnitude of the loss should a loss occur. This is calle the tail VaR or the expected tail loss. For the simple case where the portfolio consists of one stock, we can assess the severity of the loss by computing the averag loss conditional upon the stock price being less than
wl'rn
.
:
: : : : :
() 0
.
20
40
60
.
80
100
120
140
Stock Price in 6 months
160
180
200
($)
Tail vaR
VaR specifies the level, #(c), that will be breached with probability c, tail VaR measures the average severity of the breach. For the single stock case, tail VaR is de:ned as
uI(c).Whereas
ELSZ
-
s &(c)) &
Sv l5'w
ln Section 18.4 we discussed lognormally distributed stock, expectation.
=
calculating
-
E E5'w Iur <
uI(c)1
Sv
<
Esv6'v
=
-
(0.l5-().5x().32)x0.5-
$100c = $74.347
I
sryis
xtj.ytjx
p.5
1O0x
c0r15x0.5x(-j) '
x(-a)
+ (0.15+ 0.5 x 0.302) x 0.5 1n(100/74.347) 0.30 0.5
= J1
0.30
0,5
Tail VaR is therfore Tail VaR
=
=
=
where
l
Exampie 25.1 1 $100,a volatility of 30%, an expected return of 15%, and a dividend yield of 0. Also assume that the risk-free rate is 8%. Let F 0.5. Figure 25.6 depicts theprobabilitydensity for S., the stockplice atthe 5% quantile, and the conditional expected stock price. To compute the VaR, we frst need to compute (0.05)
$74.347)
= $68.244
-
j
<
(25.18)
the conditional expected stock price. For a (18.28)to compute the conditional
J(0.05):
=
-
The calculation of tail VaR is based on the expected stock price conditional upon ' . . . $74.347.Using equation (18.28),we obtain
we can use equation
Assume that we own stock with a current plice of
$100 $74.347 $25.653
V:.R =
=
$100 $68.244 $31.756 -
=
Figure 25.6 depicts the probability density for the stock price in six months, the V:I.R price level, and the tail VaR plice level. Q
lt is possible to interpret tail V:.R as an average of Valks with different confidence 1eVe ls We can approximate the conditional expectation of the stock price below J'(c) .
% VALUE
834
IssuEs
AT RlsK
,':'7)* ('' j'y.j-'r! ;')' )' ;p' y'. !' rf E' ' ' E E' Ei E E'((!' .jjrryy:jj,;,.' iyjjjj,d t''y'i jjjjjkj'(t'gjjy);'. ()', r' i'1* q' J' :'' tf f' )' j'f';'(f ;'811* ltf T:L'r:CF!TCF?E' 7. ('f5:7r:67)7! jjj.d 2'j(j'E !111::::;:;;*. $)' (' q' )' ?y' qr'ljjf ' yt'j!;';!'(.i ii.q.ttj. i!)!q;((@!q;r; .jjg:r;jj' j'y'yjj. r' ; :'('IE E).!r !qEEi. ' (.'.'.';'.E.'.E '.(i.(E.. ' j' ! ..' (!(.i . ;. (E(q .. .'y!.l . 'T . ''y'.:Ey . (.' ( '.' . . .L)L.?;Lj(L3'jLL)L yjj)).jI.J ;. y. y .. yj...( j y qy.; t. yg:g. ., ..E ;...'..L.E( .. . j;'!: y y j j.j .jy yj, . j j j . yj.yyj jyyyy j . .. ; . . . j.j.g. . . ( lyE.y jpE .. t;g i. E q. : ! y. y E (.jjyy.yy. ..yy.( y..y. ji;' ! .. . jjjyyjtk,. ry..gj.yy j,.(.. ... j y y ,. j .. . . . .
.'.' .' -'j;),. -y' ' t')d r' ,'-'''''')q'
y'rd )' jljlq:p:,jd 11j1(*. 11iIj,k,jIjr': yyfjyyjj ilf 'r;' y' lljjq:;ljli.d jL:;3L?33,:'. :;6;34f,3:4*,* ty.yyf (;!!!!141;)* jygjjjjy. .. .'ty ty gt, 1. jlli!!!!kt . .' . . y.. ... yr..: );;. . ' : : E
: ')'tj$E).:
'
::1
'
.
:
:
'
-
E
.
'
. .
:' . .:
.
:
'
:
:
-
'
.
. -
.
.
' i
rE'(E'...': .r.tjjt:yllj
.
.
:
' .
.
:-
.
- .
:
.
.
.
: ''
'' :'.
: i .
.
'
:
: jjk)... ': .
' . :..
:.
'
.
'
:
:
,
.
. 2.yjyttjjy. .. : .
.
.
' :
.
:
'
.y
:
.
.
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y -y
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.
The top panel depict.s quantiles from 0.0025 to 0.05 at intervals of 0.0025. The average of these quantiles is The botlom $68.809. . panel depicts quantiles from 0.001 to 0.05 at intervals f 0.001 The average of these quantiles is $68.496. Assumes the same Parameters aS in Example 25.1 1 -.
jj:yy:)j, .jjyr:;j, jjj,jjj, kjj:r, y)jjy:,,.
x
8
10-3
Es
l
w sr
<
$74.347)
:: : : :
(668.244)
6
: .
: :
i
2
: :.
0 so
55
.
60 65 Stock Price in 6 months
vfla
maxp, &(c) svj -
Thus, tlte insurance premium is the value of a put option wit strike price to expiration F. We can obtain the value of the put by computing
($74.347) :: :
70
75
($)
e
gLsF jsT <
x 10-3
($68 :,44)
::
j
: : :
.
sy4.a4.gl :
6
.
:
:
1 :
4
IF't'I.R
::
2 65 60 Stock Price in 6 months
:
-
svksp< &(c))x e-r'r
('(c)x
Prob*luw
.-Esksv6sv
< <
&(c)1 J(c))) x
prob*ls'r
<
&(c)and time
&(c)j (25.19)
where E* and Prob* represent the expectation and probability computed with respect to the risk-neutral distribution. The put price calculation appears similar to the conditional price ciculation used in computing ta VaR, but the two calulations are pot identical. One obyious difference is that tail VaR is computed using the conditional expectation of the stock price under the true distribution, while the option plice uses the risk-neutral distribution.Note that while Probls'w < J(c)1 c, Prob*ls'w < J(c)) + c untess the risk premium on the asset is zero @ ?'). Also, the put price is discounted, while the tait V:.R is pt. Valk horizons are often short, so that discounting may not be an important issue. Also, with a short horizon the difference between the risk-neutral and true distbution may not be large.? =
:
:
55
: : :
($74.347) : ..
:
0 50
-rFF*(&(c)
= .
.
undertake the business would then be the price of this insurance. The market price of this insurance provides another way to measure risk. Remrning to our example of a single stock portfolio, if we insure against losses due to a stock price below Zc), then the time F payoff on such an insurance policy will
: : ::
Probability 8
835
be
:
4
. ::
*
VAR $
wlTH
.
=
70
75
($)
by averaging the stock prices associated with Valks at lower levels lower than c. For example, suppose we were to compute VaR at a series of different consdence levels: 0.005, 0.01, 0.015, etc. We would flrst compute a se ries of stock prices: :(0.005) :(0.01), :(0.015),etc. By defrlition, each of these stpck prices is a quantile. While the probability that Sv is belokv each of t.h se stoc k p lices is different there is an equal More precisely, probability that Sv is approximately equal to any of these stock and J(0.01) is equal to the te probability of the stock price being between :(0.005) averaging the stockprices and :(0.015),etc. By 2robability of Sv beingbetWeen :(0.201) approximate the conditional expectation 5'(0.005), :(0.01),J(0.015), we of the stock price below uI(t.05),which is $68.244. Figure 25.7 illustrates this alculation. The two panels show quantiles below 5%. The average of these quantiles is approximately equal to the conditional expectation of $68.244. As we average more quantiles, we approximate the conditional eypectation more closely. ln most applications there will be no simple fonnula for the cditional expectation. It is then Jssible to approximate tail VaR by averaging quantiles. ,
,
vrices.
Assume the same parnmeters as in Exaple 25.11, and also that Example 25.12 lisk-free the rate is 8%. The price of a put with 6 monts to expiration and a strike of $74'347 is BSPut(100, 74.347, 0.30, 0.08, 0.50, 0)
=
$0.4289
,(0.05),
.
.
.
,
One application of VaR andvalk-like calculations is to compute the capital required to support a lisly business. Capital is a resource that permits the lirm to sustain losses and still meet its business obligations. As an alternative to capital, we could imagine a ftl'm purchasing insurance against a loss. The capital required to
The cost of insurance
VaR and the Risk-Neutral Distribution It might seem odd to you that we have mostly used the risk-neutral distribution in this bopk, but in discussing V&R we have concentrated on the tnle distribution. Using the foregoing examples, we can explore this difference.
'/When prices move continuously and the hedging of risks is possible, the difference between the true and risk-neutral distributions depends upon the risk premium, @ ?-)/2. When h is small, the risk premium will be small. When jumps are possible, the risk premium associated with the jump could create a signiscant difference between the risk-neutral and true distributions even over short horizons. -
836
% VALU E AT
lssuEs
Rlslt
wln'u vAR
%
837
Let's try to interpret VaR and tail VaR in terms of insurance. If we are willing to accept losses less than the VaR level, then we can think of VaR as a deductible: lt is the di/erence between tail loss we willingly sustain before insurance pays anything. The insurance', tlzis payout occurs with true VaR and VaR is then the average payout from appropriate discount rate for the probability c. Since we are using the true distribution, manifestation of the problem of unclear. another (This is a conditionalt'true''expectation is Aback-ofdenote the discountrate. option.) discountrate for an Let y obtaining a payoff, assuming value insurance of the average y = r, the-envelope calculation for the
The risk premium is compensation for the fact that when the stock earns a 1ow return, investors generally have suffered losses on their investments (theirmlrginal utility of cosumption is high). Insurance hedges against this outcome and is therefore valuable to invstorsk The Black-scholes calculation properly accounts for te role of the lisk premium, while the back-of-the-envelope calculation using V:.R does not.
is
Artzner et a1. (1999)point out a conceptual problem with VaR. As we have discussed, lisk measur is to decide how much capital is required to support a common use of a gctivity. Art4ner et al. argue that a reasonable risk measure (ormeasure of required an cpital) should have certain properties, among them subadditivity.g If px) is the risk measure associated with activity X (the capital required t support activiti X4,then p is subadditive if for two activities X and i'
-YT
(TZI V:.R
= e
-1?W
VaRl x Probls'r
-
(5'0
E g.
-
-0.08x0.50
J(c))
(25.20)
l5'w< J(c)) g& uI(c)q)x -
'
x (gy g4g
= e = 0.2932
<
-
.
-
c
$6g.p,gj4.) x 0.05 .
The value of insurance inferred from the VaR calculations is substantially less than that computed using the Black-scholes fonnula. There are two reasons why tlzis calculation gives the wrong nswer. First, 5% is the VaR probability under the true distribution, not under the riskprobability that Sp < 74.347 is 6.945%. The neutral distzibution. The lisk-nentral the probability is than tnle probability because the 8% risk-free rate greater lisk-neutral stock (i.e.,a V ?'). the return expected is less than the 15% on distribution is Second, the conditional expected stock plice under the lisk-neutral distribution: the true less than that under
15'w $74.347) $67.919 t5'w #he risk-neutral expectation
E*
because a
>
<
=
exceeds
Agaip, tle true conditional
Subadditive Risk Measmes
conditional expectation
?'.
p(X + F) :G p(X) + p')
This simply says that the risk ineasure for the two activities combined shuld be less than that fort h e two activities separately. Becausecomining activities creates diverjication the capital required to support two activities together should be no greater than that required to support the two separately. tf capital requirements are imposed using a l'ule that is not subadditive, then srmscan reduce required capital by splitting up activities. VaR is not subadditive. To show this, Artzner et al. provide an example using European out-of-the-money cash-or-nothing options having the same time to expiration and wlitten on a single stock. Option A, a cash-or-nothing call, pays $1 if Sp > H, while option B, a cash-or-nothing put, pays $1 if Sv < L. Represent th premiums of the two options as PA and Pe and suppose that either option has a 0.8% probability of out-oilthe-money is 99.1%, and P aying off. The probability that either option expires that options expire out-of-the-mney is 98.4%. the probability both Consider a snancial institution that writes such options. For either option considv ered alone, the bak is confident at a 99% level that the option will not be exercised, in which case the bank keeps the premium. Thus, VaR at a 99% condence level is -PA (for option A) or -PB (for option B). VaR is negative because with 99% confdence, the option writer will keep the premium without the option being exercised. Now suppose te instimtion sells both options. Because the two written options the have same underlying stock, they are perfectly negatively correlated. Therefofe, the probability that one of the two optons will be exercised is 0.8% + 0.8% = 1.6%. In the lowest 1% of the ret'ul'n distribution, one of the two options will be exercised. The VaR ,
'
We can change te probability and conditicmal expectation to their risk-neutral values and repeat the calculation in equation 25.20. We then obtain
e
-0.08x0.50
x
($g4 gyg
-
.
$6g g jp; x 0.06945 .
=
$0.4289
This is the same as the put premium in Example 25.11. As a fnal comment, note that the 5% V:.R will also be different depending upon whether we use the true or risk-neutral distribution. ln this discussion we continued to use $74.347--computedunder the true distribution-as the VaR stock price. when ipterpreting The conclusion of this discussion is that we need to be catios when alculations ?', Valk-styl % Economically, calulations. tail VaR and VIIR a understate the insurance distribution cojt because plice stock using the computed true of the the drift riskpremium asset. the for acomponent on as they fail to properly account
gArtzner et a1. (1999)dehne cohelmtt lisk measures as those satisifying four properties: l sttbaddi() lfl,f/y-,(2) nlolloonicity-. if X :; F, p(X) k pY) (if the loss is greater, the risk measure is greaterl; Iisk (3) translation invariance-. pX + c) = p(X) mesure is a (if you add $1 to the cayh flow, the reduced by $1)-,and (4) positive twitptEwcf/>' p(lX) = p(,Y) for > 0 (if you multiply the risk by 1O,the risk measure increases by 10). -
SIf there is no risk premium on the stock
putprice.
(25.21)
i.e., a'
=
1-
this calculaton
produces the Black-scholes
838
% VALU E AT
PROBLEMS
RlsK
at the 99% level for the writer of the two options is therefore $1 VaRt-A
B4
-
$1
=
Fzt
-
Pa
-
-PA
>
-
PB
PA
-
-
#S.
We haye
VaR(-z4) + VaR(-S)
=
This expression has the opposite inequality as equation (25.21).so V:.R is not subadditive lisk, as measured by 99% VaR, in this example. In words, te institution can eliminate by undertaling the two activities in separate entities. As adifferentexample that illustrates this point, suppose you are comparing activity C, which generates a $1 loss with a 1.1% probability. with activity Dj which generates a $ lm loss with a 0.9% probability. Any reasonable rule shpuld assign greater risk (and mquire more capital) for activity D, but a 1% VaR would be peater for C than for D. These examples highlight an intuitively undesirable property of Vak. aj a risk large measure: a small change in the VaR probability can cause VaR to change by a ,-PA, amount. For the written cash-or-nothing call in this example, ;t 0.81% VJtR is while the 0.79% VaR is $ 1 PA In contrasi with VaR, tail VaR and the cost of insurance are both subadditive. lntuitively, tail VaR takes into account the distribution of losses beyond the V:.R level, so it does not change abruptly when we change the VJI.Rprobability. To qee that it is subadditive in our examples, the tail VaRs at the 99% level for A and B are -
.
p(A)
=
pb)
=
0.8 x 0.8 x
(1 (1
-
-
P,) + 0.2 x #sl + 0.2 x
(-#A)
0.8 0.8
=
(-#s)
=
PA
-
#a
-
level, tail VaR at the 99% level for A and B together is
With the same consdence
p(A + B )
=
l
-
PA
PB
-
=
1
-
fk
-
PB
<
p(A) + p(S)
=
1
839
the ret'urn distribution of the portfolio. It is possible to constnlct examples in which V:.R is an ill-behaved risk measure. Tail VaR, which takes into account the distribution of losses beyond the VaR level, and the cost of insurance against losses exceeding the VaR level, may provide better alternatives. FURTHER
READING
The Riskmetrics Fcc/znc/! Docltment (J. P. Morganmeuters, 1996) is available from lt was distributed by J. P. Morgan-and wwmriskmetricsom. now by Riskmetricsto explain VaR, illustrate some of the calculations behind VaR, and review some of the judgments behind the particular set of calculations. The document ws in:uential and remains worth reading. Pearson (2002)provides an excellent overview of Valk'with vel'y clear discussions of relevant mathematical techniques. Jorion (2001)providej brad overview of the regulatory, practical, and analytical issues incomputingvaltl-lendricks (1996)compares the results from computing V:.R in a variet.y of diffemnt ways. Artzner et al. (1999) is an inquniial paper offering om importnt Wnrnigs abut the use of VaR as a decisionalng thej lins inclue Acerbi (j002)and Acerbi making tool. Further explrtions and Tasche (2002),and other papers in the July 2002 special idsue of the Jourllal of Stwln,g and Ff?lt7//cd.
PROBLEMS ln the following problems, assume that the risk-free rate is 0.08 Jmd that there are three stocks with a price of $100 and the following characteristics:
We then have pA + S)
k
.6
-
PA
-
Pp
Thus, tail VaR in this example is subadditive. As for the subadditivity of insurance prmiums, Merton (1973b)demonstrated that optiops on portfolios are no more expensive than a portfolio of options on the portfolio constituents. Insurance premiums are therefore subadditive.
a
tr
Stock A Stock B
0.15 0.18
Stockf
0.16
0.30 0.45 0.50
Correlation
Avith
B
Correlation svith C
0.00 0.02
0.25 1.00
0.20 0.30
0.00
0.30
1.00
25.1: Consider the expression in equation (25.6). What is the exact probability that, over a l-day horizon, stock A will have a loss? for example, in computing Value at risk (VaR) is used to measure and manage lisk riskf': price changes so-called primarily with deals capital requirements. VaR commodities. value The and currencies, interest at risk of a portfolio rates, of stocks, of the exceeded given will be time that over a specified is the level of loss percentage a lisk distribution of the approximating requires the Computing retul'n value at horizon. covariance requires and of which variance in assets the information on portfolio, turn example, only containing stocks portfolios simple for When portfolio. in the are standard portfolio risk calculations can be used to compute VaR.When portfolios contain options and other nonlinear assets, Monte Carlo simulation is commonly used to assess 'market
25.2. Assuming a $10minvestment in one stock, compute the 95% and 99% VaR for stocks A and B over l-day, lo-day, and 20-day horizons. 25.3. Assuming a $10minvestment that is 40% stockA and 60% stock B, compute the 95% and 99% V:R for the position over l-day, lo-day, and 20-day horizons. '95%
and 99% 1-, 10- and 20-day Valks for a portfolio that has $4m 25.4. What are invested in stock A, $3.5min stock B, and $2.5min stock C?
25.5. Using the same assumptions as in Example 25.3, compute VaR with and without 0, 0.5, and 1. Is risk eliminated the mean assuming correlations of 1, with a correlation of 1:7((f not, why not? -0.5,
-
-
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25.6. Using the delta-approximation method apd assuming a Slominvestmentinstock A, computethe 95% and 99% l-, 10-, andzo-day Valksforapositionconsisting of stockAplus one los-strike put option for each share. Use the same assumptions as in Exnmple 25.4. 25.7. Repeat the previous problem, only use Monte Cado simulation. 25.8. Compute the 95% lo-day VaR for a written strangle (sellan out-of-the-money call and an out-of-the-money put) on 100,000 shares of stock A. Assume the opdons have strikes of $90 and $110 and have 1 year to expiration. Use the delta-approximation method and Monte Cado simulation. What accounts for the difference in your answers? 25.9. Using Monte Carlo, compute yh' 95% and 99% 1-, 10-, and 20-day tail VaRsfor ihe position in Problem 25.2. 25.10. Compute the 95% lo-day tail VaR for the position in Problem 25.8. 25.11. Suppose you write a l-year cash-or-nothing put with a strike of $50 nd a l-year cash-or-nothing call with a strike of $215,both on stock A. each option separately? a. What is the l-year 99% VaR for b. Wlzatis the l-year 99% V:.R for te two wlitten options togeter? each option separately and tlae two ck What is the lryear 99% tail VaR for together?
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of 25.12. Suppose the 7-year zero-coupon bond has a yield of 6% and yield volatility volatility yield and 10% and the lo-year zero-coupon bond has a yield of 6.5% and lo-year yields is 0.96. What of 9.5%. The correlation between the 95% and 99% lo-day Valts for an 8-year zero-coupon bond that pays $10m
' '
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E
jjjjjjjjj, jjjjjjj,,s jj;:rjjjj,j
842
%.CREDIT
TH E MERTON
RIsK
By taking a Taylor series expansion of the right-hand side of equation
S0, as
obtain
X
B0 =
e-rT
:4
T.% *(A
0
:*(Aw; AfjjdAv
z' AjldAv + '
,
(26.1)
The irst integral on the right-hand side is the risk-neutral partial expectation of the asset value, conditional on banlruptcy. The second integral is the risk-neu'al probability that the 5rm is not bankrupt. Thus, we can rewrite the value of the bonds as B() = e -rT =
B0 =
(
wlDefault) x Prob*tDefault) +
'*(z4
(1 Prob*tDefaultlq)
x
*
c-rF(f*
(lwlDefault)
x
ProhitDefault)
+
h
(1
x
-
Prob*mefaultlql
Since
(26.2)
If we set the probability of defqult equal to zero, equation (26.2) yields the standard Equation (26.2)also illustrates formulaforthe value o f default-free bid, Bo e-rl'h that de fault introduces two new elements: the default probal/lity (Prob*mefaultl) and the payoff conditional on default E* g#wlDefaultq). Thepayoffconditional on efaultcan be expressed in differentways. The rcovery rate is the amount the debt-holders receiye as a frctln of what they are owed. Thus, issues a s lng le zero-coupon bqnd, the risk-neutral expected in yhe case where a f11-141 =
.
,
recovel'y
ra t e is
f*tlkecovel')7 rate)
E* =
(.?wIDefault)
(26.3)
h
The loss given default is the difference between what the bondholders are owed and what they receive, as a fraction of the promised payment:
'*ttvoss given default)
cF:
1
E
-
*tltecovry
rate)
difference between the yield to Finally, we can express the credit spread-the maturlty on a defaultable bond and arl othenvise equivalent default-free bond-in terms of the risk-neutral defaultprobability apd expected los; given dfault. In equation (26.2), divide both sides by and take the natural logarithm of both sides. Recall that the annual yield to maturity on the bond, p, is =
-
1 ln
F
-
1' ;4s
-
l
,q
,y
Thus, the credit sprad approximately equals the annualized product of the risk-neutral default probability and the expected loss given default.
26.2 THE MERTON
MODEL
DEFAULT
ln Section 16.1 we analyzed corporate securities as options. We saw tht owning zerocoupon debt subjectto defaultis the same thing as owning a default-fyee bond and wliting a put option on the assets of the firm. This'is an example of a stnlctltral approach to modeling bankruptcy: We create an explicit model for the evolution of the srm'sassets, coupled with a rule governing default. lf we assume that the assets of the firm are lognormally distributed, then we can use the lognormal probability calculations of Chpter 18 to compute eitherthe risk-neutral or will go bankrupt. This approach to bankruptcy modeling actual probability that the f114'11 has cme to be called the Mel-toll ??lt?#cf since Merton (1974) uted continuous-time methods to provid a mdel f the credit spread. The Merton default model has in recent lisk analyses provided by Moody's KMV. years been the basis for credit
Default at Maturity Assume that the assets of the firm, A, follow the process dA
W
= (a
-
(26.6)
8ldt + cdz
where a is the expected rettlrn on the firrfl assets and is the cash payout made to has issued a single zero-coupon bond with claim holders on the fil'm. Suppose the f1171,r1 time and inakes no payouts. Default occurs at that F matureg at Promised payment < probability of bankruptcy if Aw The F time at time F, conditional on the value of time is assets at r, ,
.
Bu from equation
(26.2):
In this equation, Ja the Black-scholes dz term with r replaced by a. The expression J2 is called the distance to default, and measures the size (in standard deviations) of the random shock required to induce bankruptcy. To understand tls intepretation, recall that when assets are lognonnally-distributed, the xpected log asset value at time F is .is
p
r
=
-
1 F
ln
1 1
-
(26.5),we
/
After some rearrangement, we obtain the following expression -
843
(Default) x E (Loss given default)
T
(26.4)
Conventionally any such measure is expressed as a percentage.
P
-prob
p
Q
-
and Prob* are computed with respect to the risk-neutral measure. Aw in default, we can also wlite this as
where E Bv
DEFAU LT MODEL
Prob s (Default) x f''tlwoss given default)
The left-hand side of equation (26.5)is the credit spread. Both the probability of default and the expected loss given default are less than one, so, as we would expect, the credit spread is greater than or equal to zero. Ifeither the probability of default or the expected loss given default is zero, the bond yield equals the risk-free rate.
E llntAwll
=
ln(A;) +
@
-
-
0.5G
2)( y
-
t)
.
844
k
CREDIT
TH E MERTON
Rlslt
Thus, the distance to default is the difference between S(1n(Ar)) level h, normalized by the standard deviation:z .E'rlntAwll
and the bankruptcy
-
-
Distance to default
-
=
-
&
0.5c'2)(r
-
r
ty
-
t)
-
ln()
t
-
At
xg-
cWrt't'T'-''
) q
gww
u(z,/)+(a-J-j.a)(w-,)
xg-
-.,,-s
Pf0b*(XF
<
; Z/)
=
X
-D-
+
(r tr
-
J F
1tr2)(F
-
-
2
-
S*llkecovel'y rate)
(26.8)
=
71.867 100
=
0.71867
=
68.144 100
=
0.68144
under the risk-neutral measure. Note that the risk-neutral expected loss given default is
This is the sam as equation (18.28). lt is important to notice that the calculations in equations (26.7)and (26.8)are perfonned under te tnle probability meastre (alsosometimes called ttjephysicalnteasktltj. Thus, these equations provide estimates of the empirically observed default probability and recovel'y rate, but we cannot use them in pricing calculations. In order to cpmpute the theoretical credit spread, for example, we replace the actual asset drift, a, with the lisk-free rate in equations (26.7)and (26.8). This gives us ln(A,/)
845
under the true measure, and
ln(zI,/J)+(e-3+.Jo'2)(F-/)
=
$
default one-third of the time. Under the risk-neutral measure, however, defaults occur almost half the time. The greater risk-neutral default probability is due to the assets growing more slowly under the risk-neutral measure.
E (Recovel'y rate)
This is J2.The default probability is Nt-distance to default). The expected recovely rate, conditional on default, is E tAwlA,z' < .: )
MODEL
Using equations (26.8)and (26.10),the expected asset value conditional on default is $71.867underthe tnle measure,and $68.144underthe lisk-neutral measure. Expected recovery rates are threfore
=
o. r t 1n(z4,)+ a
DEFAULT
t)
(26.9)
t
A'*tlaoss given default)
=
1
-
0.68144
=
0.31866
Using the risk-neutral default probabilit'y and loss given default, we can compute debt yield. From equation (26.5),the credit spread is theoretical the
1
J'
ln
1 1
=
0.4726 x 0.31866
=
0.032635
This implies a debt yield to maturit.y of 0.060 + 0.032635 = 0.092635, which is the same answer as that using the Black-scholes fofmula to compute the teoretical debt
value.
k
Md
E *(A T lA,z' <
)
=
A/dtr-ltr-l)
N N
1n(z!?7)+(z'-J+)o.2)(r-?)
gg- (26.5).
oxvus
lntzt/
)+(r-,-jga)(r-;) clr;F
We can use these expressions to compute equation
j
1
(26.10)
25%, 10%, 1- 6%, c Suppose that W $100, z/ttl $90, (r Example 26.1 $ 0 (theftrm makes no payouts), and F 5 years. As we saw in Example 16.2, which used the same assumptions, the theoretical debt value in this case is $62.928,which implies a yield of 9.2635%. Using equations (26.7) and (26.9),the trtle and risk-neutral default probabilities observe a are 33.49% and 47.26%. Thus, over a live-year horizon, we would expect to =
=
=
=
=
=
As the preceding example shows, historical data on deftlts provides different infonnation than historical data on prices. Historical default frequencies and recovery der the true measure, correspond to equations (26.7) and ratej, which are bserved If ekamine crdit spreads, by contrast, we can infer the risk-neutral ex(26.8).. we default and frequency pected recovery rate, which correspond to equations (26.9)and (26.10). Notice, however, that we would infer the same asset volatility from both sets of calculations.
=
2The Moody's KMV model uses the different expression default. See Crosbie and Bohn (2003)for a discussion.
(z4f -
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f distance to
Related Models Suppose that the value of assets canjump to zero according to apoisson process. Specifcally, suppose thatoveran interval dt, theprobability of ajump to zero is #f, and that the occurrence of thisjumpis independentof the market and of otherdefaults. We saw in Section 23.4 that when a stock can independentlyjump to zero, the value of a European call is obtained by replacing the lisk-free rate, ?', with ?' +. As before, equity is a call option If firm makes the value, Bt, of a single issue of zero-coupon the the payouts, assets. no on
846
% CREDIT
Bo N D RATI N GS AN D DEFAU LT EXPERI EN CE
Rlslt
debt maturing
at time F is
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-
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F
(26.11)
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The possibility that assets can jump to zero will J'pisethe bond yield. a There is a special case where the effect of the jump probability on the bond yield is particularly easy to interpret. When the bond is default-free except for the possibility of ajump, then the bond yield is 1' + : The yield increases one-for-one with the default
probability.
Suppose that a firm has a single issue of zero-coupon debt promising 30%, 1- 0.06, and 0. From equation to pay $l0 in 5 years, and that Atl $90. 0, the value of th debt is (26 11) when Example
26.2
=
-
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$7.408
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Bt
=
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$6.703
=
-
= The yield is then ln(10/6.703)/5 0.08: The yield increases by the default probability. When a default is likely apart from jumps to zerd, then the increase in the bond = For example, when $100,the bond yield is 10.342% without yield is lesj than jumps, and 11.588% with a 2% jump to zero. %. .
sufciently.
Equity in this model is a call option that lnocks
...
26.3 BOND RATINGS
AND DEFAULT
EXPERIENCE
Bond ratings provide a measure of the credit risk for speciic bonds. Such ratings, wlzich are provided by third parties, attempt to measure the likelihood that a company will default on a bond/ In the United States, the Securities and Exchnge Commission (SEC) identies specifc credit-rating rms as Nationally Recognized Statistical Rating The history and significance of this designation was explained Organizations (NRSROS). chairman of the the SEC in congressional testimony:s b
y
The Commission originally used the term Nationally Recognized Statistical Rating Organization,'' or NRSRO, with respect to credit rating agencies in 1975 solely to differentiate between grades of debt securities held by broker-dealers as capital to meet Cornmission capital requirements. Since have become benchmarks in fderal gnd that time, ratings by NRSROS state legislation, domestic and foreign linancial regulations and privately negotiated nancial contracts. Moody's rates bonds using the designations Aaa, Aa, A, Baa, Ba, B, Caa, Ca, and C. Within each ratings category, bonds may be f'urther rted as a 1, 2, or 3, with 1 denoting the highest quality within a category. Standard and Poor's and Fitch have a simila.rrating system, using the designations AAA, A.h, A, BBB, BB, B, CCC. CC. C. ttinvestment grade'' (a rating c)f BaaJBBB or The market distinguishes between grade'' or grade'' (a rating below BaaJBBVI above) and bond. Some investors are permitted to hold only investment grade bonds, anl some contracts have triggers based upon whether a company's bond rating is investmet grade. Forexample, priorto Enron's bnkruptcy, some of the company's deals ontined lauses requiring that Enron make payments if Epron lost its investment grad stattls. Enron's snancial difficulties were worsened when its rating fell below investment grade. I'below-investment
The models we have discussed are relatively simple: There are no coupon payments and bakruptcy occrs only at maturity. ln practice, sris Wpicallyave a mix of debt, so ttaat dbt tnaturityis not well-deuned and long-term coupon-paying short-term bankrptcy can ccur at anytime. onesolution in tblis case is tp approximate the and bankruptcy trigger as the face kalue of short-term debt plus one-half the face value of lpng-term debt (Vassalou and Xing, 2004). With b arrier option prictng formulas, binomial valuation, Monie Cado, or other numerical methods, it is possible to create bankruptcy models that prmit bankruptcy Black and Cox (1976)model is a variant of the Merton prior to a matulity date. prior to model in which bankruptcy occurs if assets fall to a predetermined level, maturity. This assumption mimics a debt covenant that triggers default if the firm's '
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Yield increase
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,
it is pssible
that goes bankrupt will typically have had ratipgs doWngrades prior to bankruptcy. By looking at the frequency with which bonds exprience a ratings change,
A company
'lcompanies pay ratings a'gencies to have their bonds rated. Some have criticized this practice, arguing that payment for ratings creates a conllict of interest. ( svv-restimonyConcerning The State of the Securities lndustly'' by William H. Donaldson Chairman, U.S. SEC, U.S. Senate Committee on Banking, Housing, and Urban Affairs, March 9, 2005. As of Moody's Investor Services, Standard and Poor's, mid-2005, live lirms were recognized as NRSROS: Fitch Ratings, Dominion Bond Rating Servicee and A.M. Best.
'
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it is possible to estimate the ultimate banknlptcy probability. Achange in ratings is called a ratings transition. Table 26.1 is a ratilgs r/'tpp.Wl(7llmatrix, reporting the probability that a lirm in a given ratings category will switch to another ratings category over the course of a year.6 Flrms ' rate d Aaa Aa or A all have about an 89% chance of retaining their pting over horizon, and almost no chance of suffering a default over that tim. They one-year a do, however, hav some chance of experiencing a downgrade, after which bgnknzptcy becomes likelier: The default probability increases as the rattng decreases. Given certain assumptions, we can use a short-term ratings iansition matrix to compute the ultimate probability that a firm wit.h a given rating will go bankrupt. Specifically, suppose we tjelieve that a ratings transition mat ri x i s cons tant over tilpe and that theprobability of moving from one rating to another in a given year does not epend on therating in a previous yar. Then we can use the matrix to compute the probability that that is A-rated (for example) will rpove to any other rating, and yhe subsequent a 51-111 probability ihat it will move from one of the new ratings to a different rating, and so forth. The following simple example will illustrate how to interpret and use a ratings tansition matrix. Suppose that scurities can be in one of three categories: Good, Bad, and Ugly. The matrix in Table 26.2 displays the probability that a firm with a rating in the left-hand colmn will, over the course of a year, move to a rating in the top row. For example, there is a 90% probability that a firm rated Good will still be Good one year later. There is a 3% chance that the Good firm will become Ugly. ,
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Notice that each row sums to one. Tlzis memas that aftr ne yttr e :I'I inustbe in one of the three categories. By contrast, in Table 26.1, thefe is a t'WittziiWri Rating'' categoly indicating that a :rm has for some reason dropped out of th Vopdy'srating tln i Verse We can use the transition matrix to compute the probability that flri raied Good recognize tat there will still be Good two years from now. To perform this calculati, are three different paths by which a 517n that is now Good can still be Good in two years.
o There is a 90% chance that a Good 5rm remains Good over one year. There is therefore an 81% (0.9 x 0.9) probability that the fil'm will be Good for bot.h years. will be Bad next year, in whih case there is * There is a 7% probability that the 51711 will be come Good the subsequent year. There is therefore 111-111 15% chance the a a 1.05% (0.07x 0.15) probability that the tirm will go from Good to Bad and then back to Good will become Ugly, and yhen a 6% @Fipally, there is a 3% probability that the $11*1,11 probability that it will become Good pgain. There is therefore a 0.18% (0.0j x () n6) probability that the firm become Ugly and then Good. .
The total probablity that a Good tirm will still be Good in two years is therefore 82.23%:
(0.90 x 0.90) +
(0.07x
0.15) +
(0.03 x 0.06)
=
0.8223
In order to perform this calculation, it may at srstseem necessary to enumerate the possible transitions. Notice, however, that the calculation entails multiplying each element of the the srst row of the transition matrix by the correspo dinj element t ihe srst column, and then spmming the results. It t'urns out that if we wish to lnow all possible transitions over a two-year period, we can construct a new matrix f'rom the one-year transition matrix, where the element in the th row and yth column of the new matrix is created by multiplying the fth row of the original matrix by the yth column qf the original matrix and summing the results. Table 26.3 shows the result.
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In order to compute the ratings distribution after three years, we can duplicate the prpdure, orly taking the two-year rpatrix pnd multiplying it by the one-year matrix. by itself twice. 'this is t e same as mu ttip ly ing the one-year ma probability horizon, 1', let that, over an denote the In general, j, t + -hi, . .. . . niri in Table 26.1 that in column The the rating i in row to j. a firm will move from probability /1 the of ratings. 2 1', 1). Over years, Suppose there are give us pi, j, t + rating is rating i moving from to j .)
0.2 0.5
0.61 4.89
Ba
1.22
0.08 0.54 3.34
5.79
10.72
B
5.81
12.93
19.51
30.48
48.64
22.43
35.96
46.71
59.72
0.07
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0.41
4.85
14.43
1.56
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4.6
6.94
Caa-c lnvestment grade Below-investment
'-year
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grade
All rated
1.48
0.98
2.08
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,
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2.44
2.74
4.87
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29.67
37.07
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4.18
6.31
21.91
33.61
42.13
47.75
10.53
13.51
16.13
78.53
Soltrce: Hamilton et a1. (2005).
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Thus, a transition probability matrix can be used to tell us the probability that a firm will go bankrupt after a given period of time. The long-term exp'erience of bonds with a given rating is reported in Table 26.4. Note that if a bond has a Aaa rating, even aftr 20 years there is only a 1.54% chance it wl have gone bankrupt. However, if a bond is below-investment grade, there is a 47% chance of banknltycy over a 20-year holizon.
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.
Soltrce: Hamilton et al. (2005).
Recovery Rates There is also historical information about recovery rates. Table 26.5 displays historical average recovery rates for different linds of bonds. As you would expect, bonds designated as more senior have higher recovery rates. When we modeled debt with different priorities in Section 16.1, we assumed thatjunior debt was not paid at a11if senior debt was not completely repaid. This rule for assigning payments i.scalled absolute priority.
tfthe banltruptcy process respects absolute priority, we expect more senior bonds to have lligher recovery rates. There is considerable variation in recovery rates across srms.Hamilton et al. (2005) report, forexample, thatfor senior subordinated bonds in 2004, the meanrecovery rate was +1..4%,but realized recovel'y rates ranged between 8% and 90%, with a standard deviation of 25%.
852
%.CREDIT
CREDIT
RlsK
26.4 CREDIT
Rvduced Form Banltruptcy Models The existence of data on corporate bond ratings, ratings changes, and defaults, suggests that we could construct statistical pricing models designed to match the behavior of bond prices. Such models are called reducedfolnl nlodels.l In order to price bonds we require risk-neutral probabilities, so we cannot directly use histolical data. To understand how reduced form models work, consider the simplest version of such a model. Suppose a F-year bond promises to pay %at maturity and ther ij a zero recovery rate in the event of default. The risk-free rate is r and constant over time. lf default follows a Poisson process with the risk-neutral intensity then the bond price depends only on time and on the occurrence of thejump. From Section 21.5, the partial differential equation for pricing the bond is
Ixs-rRuM
Ex-rs
k
853
INSTRUMENTS
The buyer of a corporate bond acquires both interest rate risk and the credit lisk of a specisc frm. A particular investor may wish to hold a different combination of these risks. There are ways to alter the mix: lnvestors can buy Treasury bonds instead of corporate bonds, thereby minimizing credit risk, or use interest rate derivatives to reduce lisk. More recently, it has become possible to use or increase exposure to interest rate lisk t'/cfvtzlvcof a speci:c Erm. ln this section, we explain credit to trade the credit these instruments.
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With the boundary condition that #(F) B (1)
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isS
Collateralized Debt Obligations cash A collateralized debt obligation (CDO) is a snancial structure that repackages the ( flows from a set of assets. You create a CDO by pooling the returns from a st p assets and issuing linancial claims to tlzis pool. The CDO claims reapportiop the reirns on tj e asset pool. Typically, CDO claims are tranched, meaning that the different CDOclaims have differing priorities with respect to the cash flows generated by the collatera lisky bonds. for example, and create nW claims, a CDO, it is possible to take a group Of some of which are less rijky than the original bonds, and others which are rislier. Given this general escription, there are many different ways a CDO can be s-ucIf tured. First, the asset pool can be a 'Iixed set of assets, in which case the CDO is instead a manager buys and sells assets, the CDO is nlanaged. Second, the CDO claims can directly receive the cash iows generated by the pool assets; this is a cath-flobv CDO. Alternatively, the CDO claims can receive payments based on cash :ows and the gain or loss from asset sales; this is a nlarket vcld CDO. There are at least two reasons for creating CDOs. First, inancial instittions will sometimes want to securitize assets, effectively removing them from the institution's balance sheet by selling them to other investors. A CDO can be used to accomplish this, in which case it is a balallce sheet CDO. Second, a CDO can be created in response to institutional frictions. Forexample, some investors lre pennitted to hold only investment grade bpnds. As we will see below, CDOs can potentially be used to create investment grade bonds from a pool of noninvestment grade bonds. This is called an arbitrage CDO. CDOs are relatively complex fnancial structures. The box included here provides arl example of the legal arguments that sometimes result. .
-(r+)(T-r)
(p.6 j g) .
Given out strong assumptions that recovery rate is zero and interest rates are nonstochastic, it would seem a simple matter to price this bond by observing r and infening from data on defaults. The problem, however, is that equation (26.13)presumes that is lisk-neutral jump probability. Thus, we can infer from bond prices but not from a default data. histolical To understand the issue, suppos: that bond defaults are idiosyncratic. In this case an investor can diversify default risk. We then expect in equation (26.13)to equal the historically observed lf defaults are correlated, however, then even a portfolio containing numerous bonds will eneounter systematic losses from correlted defaults. Defaults occur when firms perform poorly-i.e., when equit'y returns are low. Investors require a positive Iisk premium to hold such bonds, and therefore the risk-neutral default probability in equatipn (26.13)will exceed the historical default probability. A more general approach than that in equation (26.13) uses ratings transitions. Equation (26.13)does not take into account that default becomes more likely as ratings ratings transitions, it is possible to price bonds taldng into decline. With lisk-neutral paths by which default can occur. Jarrow et al. (1997)show how to the various account historical ratings transitions to infer risk-neutral ratings pl-ices and observed bond use probabilities. transition .
Berhe reduced form approach was first used
in Jarrow and Turnbull (1995).See Dufhe and Singleton (2003) for a survey. Slqote that this is the 0. same bond price solution we obtain in equation (26-11) when B < A, and o' =
'tac.
independent A simple example can illustrate how CDOs work. defaults Suppose that there are three risky, speculative-pade bonds that each promise to pay $100 in one year. Defaults are independent and occur only at maturity of the bond. Each bond probability of default and a 40% recovery rate, and the risk-free has a 10% lisk-neutral each bond is of The price is rate 6%.
A CDo with
,-0.06
g(1 0.1) -
x
$100+ O.1 x $401 $88.526
The yield on each bond is 111(100/88.526)
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and three bonds default (0.103 0.1% probability). To' compute the price of a CDO default 'anche, we can compute the expectedupayoffof the tranche using the lisk-neutral probabilities. The CDO pricing is illustrated in Table 26.7. Note that the senior tranche is almost risk-free. The only time the senior tranche is not fully paid is in the unlikely (0.1% probability) event that all three bonds default. In that case, the senior tranche receives $120, arecovery rate of 85.7%. Since itis almostpaid in 111,investors will pay $131.828 for the senior tranche, which is a yield of 6.02%. The mezzanine tranche is fully paid if there is one default, but it is not fully paid if thef are two or three defaults. The yield is 7.61% and the average recovery rate is 0.4285. Finally, the subordinated tranche receives less than 40/90 x 0.027/0.028 f'ull payment if there are any defaults. Consequently, it is priced to yield 32.96%. Note that the sum of the prices of the three tranches is $265.58.As you would expect, this is the same as the price of the three bonds put into the asset pool. This example might remind you of the discussion of tranched debt in Chapter 16, especially Table 16.1. The idea is exaetly the same, except that instead of valuing claims on corporate asset (thebonds in Chapter 16), we are valuing claims on a pool of corporate bonds. =
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if at maturity, either a11linns default or none do. A compalison of Tables 26.7 and 26.8 shows the importance of default correlation in the pricing of CDOs. Given the strucmfe of the CDO and the assumptios about therecovel'y rate, with perfect correlation of defaults, the mezzanine and subordinated eanches have te same yield. The senior tranchq becomes riskier because the probability of three defaults-the only circumstance in wlzich the senior tranche is not f'ully paid-is greater with perfect con-elation.
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ln the preceding example we assumed that the bonds A CDo with correlated defaults uncorrelated. guessed, this is an important assumption. Table might have As you were
26.8 shows how the CDO tranches are priced if the bonds are prfectly correlated-i.e.,
The previous examples showed that it is possible to pool lisky baskets bonds and create rislier and less lisky claims. A particular variant of this strategy is the Nth to defaltlt basket. Consider a CDO that contains equal quantities of N bonds. Over the life of the CDO there can be anywhere between 0 and N defaults. lt is po'ssible to create tranches where particular bondholders bear the onsequences of a particular default. The owner of the first-to-default tranche bears the most risk: If any of the bonds in the asset pool default, the srst-to-default owner bears the loss from this default. The owner of the last-to-default generally bears the least risk, since allbonds must default in order for this claim to bear a loss. Table 26.9 shows the pricing that results from this structure, assuming that the defaults are uncorrelated and occur only at time F. By comparing Table 26.9 with Table 26.7, you can observe a similarity between te subordinated tranche and the srst-todefault on the one hand, and on the other, the senior tranche and the third to default. Nth tp default
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last few years. ln addition to general growth, there has been a great deal of innovation ln the credit derivatives market in the last few years. A single name credit default swap (CDS) makes single name credit default swaps name'') experiences a credit event. The specifc when company a (the a payment buyer of the swap is the protectiol lll/yd?'.A corporate bondholder, for example, could use a CDS to buy protection against the credit risk of a company. The countrparty providing the credit insurance is the swap ik'?'2c/' or protection sellen lf a credit event occurs, the protection buyer receives ttsingle
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Financial settlement and physical settlement are economically equivalent in thethe market for a defaulted corporate bond may not be liquid, and it However, oy. difficult be to determine a fair plice upon whih to base fnancial settlement. To may avoid this problem, dfauly swaps often call foq or gt least permit, physical settlement. Nevertheless, cash settlement is increasingly common. For many firms, it is possible to trade CDSS for a variety of different expirations. The set bf prices with different Iaturities generates a crelt Jpret'lt cul-e, where, for example, you may obselwe that credit spreads are small at short horizons and substantially larger over 'live years. With this array of contracts it is possible to make sophisticated bets. For example, you could buy protection with a four-year horizon and sell protection with a hve-year horizon; this is a bet that default will not occur in the hfth year.
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* The payoffs to a protection
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o The payoff to the buyer of a single tranche CDO containing the XYZ bond (tlzisis jttst the XYZ bond). Table 26.12 compares the payoffs on the two positions. Both the written CDS and the CDO bear the costs of a default. The difference between the two positions-in the last column of Table 26.12-1s the payofto a defaultfree bond. The difference between the CDO and written CDS, therefore, is that the CDO
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Pricing a Default Swap
* Buy protection with a CDS on $100 worth of senior bonds issued by XYZ. The default swap premium is p. * Short-sell a $100 default-free qoating rate note paying ?'. Suppose that we can short-sell the bond costlessly. * Buy XYZ senior floating-rate notes paying a
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* lf the CDS mamres and XYZ has not defaulted, sell the XYZ Qoating rate bond rate note in order to close and use te $100proceeds to buy the default-free noating that both bonds are worth using floatng short sale. rate notes, (By ensure the we $100 if there is no default) * 1.fXYZ defaults, under the terms of the CDS, surrender the defaulted Qoating rate notes in exchange for $100. Use the $100proceeds to buy the default-free floating rate note in order to close the short sale. In
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Notice that we made very strong assumptions to reach this conclusion, so we should not expect equation (26.16)to hold exactly. In practice, we would need to take into account a variety of complications. including t'ime variation in the credit spread, bonds aving tixed coupons instead of being floating rate, and transaction costs (suchas costs of short-selling). Duflie (1999)discusses the effects of many such complications. An important question implicitly raised by this discussion is the denition of an ttotherwise equivalent default-free bond.'' It seems natural to use government bonds as a benchmark, but government bonds can be unique in certain rspects. Pdces of government bonds may include a liquidity premium and sometimes re;ect spcial tlx attributes (for example, in the United States, federal government bonds are exempt from state tues). Houweling and Vorst (2001)estimate a credit swap jricing model and ;nd that, empirically. credit swap premiums are more related to the interst rate swap curve than to the govenlment bond yield curve. This linkage between crdit sWapsand default-gee bond'' is not interest rate swaps suggests that the yield on an bond and in fact be directly observable. Rather the the government curve, may not default-free yield be inferred the market for default swaps as the from equivalent may reference default premium. the less the asset rate on swap ttequivalent
An invesiorwho buys abopd and a defaultswap on thebond owns asynthetic default-free bond. This observation suggests that the defalt swap premium should approximately equal the default premium on the bond. To make this more precise, suppose we simultaneously undqak th followipg set of ygpsactions:
Premium
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A credit-linked note is a bond issued by one company with te credit status (i.e.,bnnkrupt ornot) of a diffrent company. For exampl, bartks can issue credit-linked notes to hedge the credit Iisk of loans. To see how a credit-linked note works, suppose tat bank ABC lends money to cpmpany XYZ.At the time of the loan, ABC creates a tnlst that issues notes: These a'e the creditlinked notes. The f'unds raised by the issuance of these notes are invested in bonds with a low probability of default (such as government bonds), which are held in th8 trust. XYZ remains solvent, ABC is obligated to pay the notes in 111. If XYZ goes bankrupt, te note-holders receive the XYZ loans and become creditors of XYZ. ABC takes possession of the securities in the trust. Thus, the credit-linked note is in effect a bond issued by ABC, Fhich ABC does not need to repay in f1111if XYZ goes bankrupt. This structure eliminates a third-party insurance provider. Credit-linked notes an be rated and exchange-traded. Because of the trust, the credit-linked notes canbepaid in full even ifABC defaults. Thus, even though they are issued by ABC, the interest rate on the notes is detennined by the credit lisk of XYZ. An arrangement lile this was used by Citigroup when it made loans to Enron in 2000 and 2001.1 l When Enron went bankrupt in late 2001, Citigroup
credit-linked notes payments that depend
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Altman, :'How Citigroup Hedged Bets on Enron,'' Nebb'l'ork Tfl?le-. February 8, 2002,
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CDS lndices Thus, a CDS index A CDS index is an average of the premiums on a set of provides a way to track the overall market for credit. ln this sectiop we will discuss the kinds of structures and products that are traded. To a srstapproximation? it is possible to replicate a CDS index by holding a pool of CDSs.l2 As with a single CDS, one party is a protection sller, receiving premium payments, and the other is a protection buyer, maldng the payments but receiving a payment from the seller if there is a credit event. There are numerous ways in which g CDS index product can be structured and CDSS.
0.
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Jan02
Jan03
* The claims are generally tranched in various
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@The underlying assets can represent different countries, cuaencies, maturities, or industries. ' A 0-100% tranche on a CDS index can also be desclibed as a sylzthetic CDO. The rettlrns on a CDS index re:ect the performance of the bonds in the index, just as the returns on a CDO reiect the perfonnance of the bonds in the asset pool. Credit indices have had a relatively brief histoly'3 There are a number of potential challenges in trading credit risk. Histolically, the corporate bond market has been less liquid than the stock market. Trading volume for bonds is lower than for stocks, and bid-ask spreads have been higher. A large f11.11.1 can have dozens of different bond issues outstanding. ?
12The replication may not be exact since a defult index can dehne a credit event differently than a single name CDS- For example, DJ TRAC-X, discussed belom specihes bankruptcy and failure to pay as credit events. A single name CDS can have a broader detinition' j3 See Dufhe andYurday (2004b)foran interesting account of the development of creditindex products.
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lt2 201 Morgan in ihe mid lgts. singlename credit default swaps srst stanleyintroduced TkA/ERS, a bgsket ot colporate onds that investors could eade wer both fttnded apd nonfunded as a unit, much like an exchange traded funl. calle 1'tYD1. ln 2003, TRACERS products. J.'. Moran created a competing prdue Morgan Stanley and J. P. Morgan merged these products to create TRAC-X. Later in 2003, competing bnnks introduced CDX as an alternative credit index. ln 2004, the TRAC-X and CDX products merged into Dow Jones CDX. These indices could be traded in both f'unded and nonfunded products. Firmscan drop out of an index due to bankruptcy or illiquidityi but otherwise the male-up of agiven CDX offeringis setforthelife of the offering. Anew CDxis currently offeredevry 6 months with 5- and lo-yearmamrities; with the mostreentpfferingbeing the most heavily traded. For North America, there are versions of the index re:ecting The investment grade companies, high-volatility companies, and high-yield coppais. CDX North America index is an equally-weighted basket of 125 CDSS representing investment grade companies. There are also credit indices for North America, Europe, i E Asia, Japan, Australia, and emerging markets. Figure 26.4 shows the CDS premium for the sve-year onvthe-run idex. The series by' Morgan Stanley from premiums OIITRAC-X and CDX, and foryears constructed was beforeerlkAc-x, fromdataon creditdefault swaps from srmsinthe index. Creditdefault swap premia were four times larger in 2002 than at the beginning of 2005. This is due to the fact thgt a number of large companies defaulted during 20029 including Worldcom, Global Crossing, UAL Corp, Adelphia, and KMarq among others. The CDS premiums 'aled
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for these companies' would have been high prior to their defaults. ln interpreting Figure 26.4, keep in mind that after a company defaults, it drops out of the index, so tat other things qual, firms remaining in the index have a lower probability of default. Tranched CDX and has a payoffstructure illustrated.in Figure 26.5. There are both funded apd unftmded products based on this structure. As you would expect from our earlier discudsiim f UDOsand default correlation, the pricing of the various tranches is sensitive to correlation. In fact, tranche pricing is quoied using implied correlation in much the sale way quity option prmiurs are quoted usig implied volatility.
READING
Both the actual traded credit contracts and pricing theoly continue to evolve. Books with a practitioner perspective include Goodman and Fabpzzi (2002)and Tavakoli (2001). Dufe (1999) discusses the pricing of credit default swaps. Frameworks for analyzing credit Iisk are discussed in Credit Suisse Financial Products (1997), J. P. Morgan (1997), Kealhofer (2003a,b)and white papers on the Moody's KMV Web site, A debate between advocates anl critics of the KMV apwww.moodyskmv-com. proach is in the Februal'y 2002 issue of Risk (Kealhofer and Kurbat, 2002; Sobehart and Keenan, 2002). Books with a more academic and theoretical perspectiv include Cossin and Pirotte (2001). Dufe and Singleton (2003),Meissner (2005),and Schnbucher (2003).Dufe and Yurday (2004b,a)provide a blended discussion of the histol'y of some of the products, practical issuesi and pricing models. Papers examining Merton-style models include Jones et al. (1984), ttimet al. (1993),Leland (1994), Leland and Toft (1996) and Longstaff and Schwmz (1995). Empilical smdies of Merton-style models include Anderson and Sundaresan (2000),Bharath and Shumway (2004),and Eom et al. (2004). Shumway (2001) estimates flazardmodels of bankruptcy. Johnson and Sttllz (1987), Hull and White (1995),and Klein (1996)consider how it affects prices )f derivatives. Data on bankruptcies are ayailable from www-bankruptcydata-com. Simple summary data, such as the largest bankruptcies sorted by year, are available without
charge.
PROBLEMS Aparty to a contract may fail to Iriale a required fumre payment. This possibility, which is called default, gives rise to credit risk. Credit risk is an important consideration in valuing corporate bonds, where two key inputs are the probability of default and the does default. The Merton model uses option expected payoff to the bond if the 111411 pricing to value debt subject to default. Credit agencies assign debt ratings to lirms; these can be used to assess the probability of bankruptcy in the future. There are various linancial vehicles that permit the trading of credit risk. Collateralized debt obligytions (CDOs) are claims to an asset pool. The claims are often stnlctured (tranched)so as to create new claims, some of which are more and some of which are less sensitive to credit risk than the pool as a whole. The value of these claims depends importantly on the default conelation of the assets in the pool. Credit default swaps pay to the protection buyer the loss on a corporate bond when there is a default. In exchange, the protection buyer makes a periodic premium payment to the jeller. Baskets of credit default swaps are also called sjnthetie CDOs, since the holder of such abasket bem's 4hedefault experience of the firms represented in the basket,
For the rst eight problems, assume that a 5rm has assets of lisk-free rate is 8%. a = 15%, and J 0. The
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=
26.1. The firm has a single outstanding debt issue with a promised maturity payment of $120 in 5 years. What is the probability of banknlptcy? 'What is the credit spread? 26.2. Suppose the 'Iinn issues a single zero-coupon bond with maturity value $.100. a. Compute the yield, probability of default, and expected loss given default for times to maturity of 1, 2, 3, 4, 5, l0, and 20 years. b. For each time to maturity compute the p
=
r +
1 -
r
approximation
for the yield:
x Prob Default x Expected loss given default
How accurate is the approximation?
868
k C R ED IT
PRo B LEMS
RI s K
26.4. Suppose the :rm issues a single zero-coupon bond. a. Suppose the maturity value of the bond is $80. Compute te yield and default probability for times to maturity of 1, 1, 3, 4, 5, 10, and 20 years. b. Repeat part (a), only supposing the mattlrit.y value is $120. c. Does default probabilit'y Plain
increase or decrease with debt mamrity? Ex-
.
26.5. Repeat the.previous problem, only compute the expected recovel'y vglue instead of the default probability. How does the expected recovery value change as time to maturity changes? 26.6. Suppose that there is a 3% per yea.r chance that the firm's asset value can jump to zero. Assume that the firm issues s-yearzero coupon debt with a promised payment of $110. Using the Merton jump model, compute the debt price and yield, and compare to the results you obtan whe te jump probability is zero. 26.7. Suppose the tirm has a single putstgnding debt issue with a promised mattzrity payment of $120 in 5 years. Assume that bankruptcy is triggered by assets (which are observable) falling below $40 in vglue at any time over the life of the bond-in which case the bondholder receives $40 at that time-or by assets being worth less than $120at mattlrity, in which case the the bondholderreceives the asset value. What is the probability of bank-ruptcy over the life of the bond? What is the credit spread?
26.8. Repeat the previous problem, except that the time to maturity can be 1, 2, 3, 4, 5, 10, or 20 years. How does the bond yield change with time to mamrity? Forthe next two problems, use this information on creditratings. Suppose there are three forlorn, and forsaken). credit ratings, F (srst-rate),FF (futurefailure?), and FI;'F (fading, this: looks like The transition matrix between ratings
Rating To: Rpting From: F 17F FFF
26.9. Consider a 5rm with
F
FF
FFF
.9
.07
.03
.15
.80
.05
.10
.30
.6
an F rating.
a. W'hat is the probability that after 4 years it will still have an F rating?
869
b. What is the probability that after 4 years it will have an FF or FFF rating? c. From examining the transition matrix, are firms tending over time to become rated more or less highly? Why?
26.3. Suppose the firm issues a single zero-coupon bond with time to maturity 3 years and maturity value $1l0. a. Compute the price, yield to maturity, default probability, and expected recovery LEE#wlDefaultl). b. Verify that equation (26.5)holds.
%
26.10. Consider two lirms, one with an FF rating and one with an FFF rating. What is the probability that after 4 years each will have retained its rating? What is the probability that each will have moved to one of the other two ratings? 26.11.
uppose that in Figure 26.7 the tranches have promised payments of $160(senior),
and $90 (subordinated). Reproduce the table for this case, $50 (mezzanine), default correlation. assuming zero
26.12. Repeat the previous problem, only assuming that defaults are perfectly correlated. 26.13. Using Monte Carlo simulation, reproduce Tables 26.9 and 26.10. similar table assuming a default conelation of 25%.
Produce a
26.14. Following Table 26.9, compute the prices of first, second, and Nth-to-default bonds assuming that defaults are uncorrelated and that there are 5, 10, 20, and 50 bonds in the portfolio. How are the Nth-to-default yields afkcted by the size of the portfolio? 26.15. Repeat the previous problem, assuming that default
correlations
are 0.25.
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(B
.2)
Thus, the expression dehning e is the same expression used for interest compounding calculations, equation (B.1)! By sing e, you can compute a futtlre value. lfyou know how much you have earned from a $1 investment, you can determine the continuously compounded rate of return by using the nattlral larithm, ln. Ln is the inverse of the exponential f'unction in that it takes a dollar amount and gives you a rate of rettlrn. In other words, if you apply the logarithmic function to the expnential function, you compute the oliginal argument to the exponential f'unction. Here is an example: ln(drJ)
).k )!1 .j!j E
-
,0.0953
=
rt
1 j.o
=
.)i'.!
.
.
pl)i (/,1
Finally, note that
tl j
ln(l.10)
ln(e0'0953) = 0.0953
=
e
The exponential f'unction is is a cnstant approximately equal to e, where c is, if interest accnls evel'y instant2.71828. tf compounding is continltous-tbat then we can use the exponential function to compute f'uttlre valu:l. For example, with a 10% continuously compounded tate, after 3 years we will have a fumre value of
$)
:
hotlrly compounding.
.
0.47655
%.
When we multiply exponentials, exponents add. So we have
=
=
ln($100/$62.092)
Changing Interest Rates
0.1/365)1095 $1.34980with daily compounding, and
d
*
.
.
(B.l)
$1.3482with monthly compounding, =
..
1 = () 10
-
*
')1The continuously compounded rate of return is ))
'
$1.331with annual comjoundirig,
0.1x3
($100/$62092)1/5
)
;:
'jjt).
5
-,,
. ($1 + 0.1/8760)26280 $1.349856with
is
(E
W'hg happens if we let get very large, that is, if interest is compounded many times a yeltr (evendaily or hourly)? 1f, for example, the interest rgte is 10%, after 3 years we will have
=
l) return )))
yj
@
lt
0.1/12)36
$
Suppose you have a zero-coupon bond that matures in 5 years. The $62.092for a bond that pays $100. The annually compotlnded rate of
'j(k.j)
.
+
. ($1 + . ($1+
FUNCTIONS
q Example B.1 (ll price today is
Ejj
Interest rates are typically quoted as per year, compounded 11 times per yean'' As student lemms,tlzis has the interpretation that you will eal'n an every beginning snance r//7 interest rate of per period for /7 periods. 'Fhus, if you invest $1 today, in 1 year you will have
=
AND EKPONENTIAL
'''
B.2 THE LOGARITHMIC
* ($1 + 0.1)3
LOGARITHMIC
.K
d ),
.z
--).
d
=
Suppose you can invest for 4 years, earning a continouously coppotnded
return of 51
the first 2 years and 6% the second 2 years. If you invest $1 today, after 4 years you will have j2x0.05
e
2x0.06
e
=
0. 10+0.12
=
$ 1 p,4.6j .
We could of course do the same calculation using effctive annual rates. For the first 2 05 l = 5.127%, and for the second 2 years, e 0.06 = 6.184%. T jje years we earn e 0 f'uture value of $1 is '
-
f
'
2 2 1.05127 1.06184
$1.2461
=
This calculatiop gives us the same answer. What is the average apnual rate earned over the 4 years? The average annual continously compounded rate is
1 -4 111(1.24608) which is the average of 5% and 6%.
=
0.055
However, if we express the answer in terms of effective annual rates, we get 0.25
1.24608
-
j
=
5 6,gyj% .
This is ltot the average of 5.127% and 6.184%, which is 5.6554. This makes calculations with continuous compounding easier.
878
% Colq-rllquous
PRo BLEMS
CoMpoulqolrqc
b. What is the arithmetic compounded return?
Symmetry for Increases and Decreases composite index closed at 2292.89. On March 10, On March 4, 1999, the NASDAQ 2000, the index closed at 5048.62. On January 2, 2001, the index closed at 2291.86, essentially the same level as in March 1999. The percentage increase from March 1999 to March 2000 was
5048;62 1 2292.89 -
=
2291.86 1 5048.62 -
=
120.19%
The subsequent decrease was -54.60%
When computing simple rates of return, a plice can have an increase exceeding 100%, but its decrease can never be greatr than 100%. Wecan do the same calculations using continuous compounding. The continuously compounded increase from Mgrch 1999 to March 2000 was
1n(5048.62/2292.89)
=
return
%
879
over the second year? The continuously
c. What do you notice when you compare the first- and second-year returns computed arithmetically and continuously? B.3. Here
are stock prices on 6 consecutive days: $100,$47,$88, $153,$2l2, Note that the cumulative remrn over the 6 days is 0.
$100.
a. What are the arithmetic returns from the first to the second day, the second to the third, and so forth? b. What are the continuously compoundedreturns from the irstto the second day, the second to the third, and so forth? c. Stlppose you want to compute the cumulative return over the 6 days. Suppose you don't know the prices, but only your answers to parts (a) and (b). How would you compute the cumulative return (whichis 0) using arithmetic returns and continuously compounded returns?
78.93%
while the subsequent decrease was 1n(2291.86/5048.62)
-78.97%
=
'Whelt Ilsnq colztinuolts colnpoltnding, increases t-l/2# dereases t'I?-c sylmnetric. Moreover, if the index dropped to 1000, the continuously compounded retum from eak would be the P -
-161.91i
1n(1000/5048.62)
=
Continuously compounded returns can be less than
-100%.
PROBLEMS B.1.
a. A bond costs $67,032today and pays $100,000in 5 years. What is its continuoqsly colpounded rate of return? b. A bond costs $50 today, pays $100 at mamrity, and has a contiuously compounded annual return of 10%. ln how many years does it mature? compounded rate of c. An investment of $5 today pays a contiusly 7.5%/year. How much money will you have after 7 years?
d. A stock selling for $100is worth $5 1 year later. What is the continuously comppunded ret'ul'n over the year? What if the stock plice is $4? $3? $2? What would the stock price after 1 year hve to b in order for the continuously compounded return to be .(..
.
.
.
.
-500%2
B.2. Suppose that over 1 year a stock price increases from $100 to $200. Over the subsequent year it falls back to $100. over the first year? a. 'What is the arithmetic rettlrn (i.e.,(./+1 Stjj't ln(u%+1 compounded 'What is the continuously ret-urn (i.e., jstl -
*
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EXAM PLE: TH E PRICE
lN EQUALITY
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. Illustration of Jensen's inequalty with a call option. The Iine Iabeled /(') depict.s the call payoff at expiration. The option ev. Iuated at the expected stock price Iies on this Iine. The expected value of the call, on the other hand, Iies on the line connecting the point.s Iabeled (35,0)and (45,5). That Iine is always above the call payoff at expiration.
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Stock Price at Expiration
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.5431
and f (- 1) lies on the chord connecting those Graphically, the average of is below the chord, which is what points, which is the straight line in Figure C.1. f Jensen's inequality states. ./(1)
PRICE
OF A CALL
Here is an example of Jensen's inequality. Consider a call option with a stlike price of Binomialos, 45; 0.5). Then $40. Suppose that .A7is the stock price, and that wi
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which is consistent with Jensen's inequality.
THE
-
When we evaluate the call price at the expected stock price, .(F(-:r)q, we have
= 1
C.2 EXAMPLE:
maxt-'r
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=
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-
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Since 2.5 > 0, E (.J@))k fLE (.x)1,in accord with Jensen's inequality. Figure C.2 displays this example graphically. The straight line connecting /(35) and represents E (.(;r)) ; this line always exceeds the payoff to the call option. This example illusates in apurely mechanical fashion why uncertainty makes an option more valuable. ./-(45)
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module. Here are the steps formula: 1. Open a lpD?7kworkbook
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Before delving into VBA, it is helpful to appreciate what learning VBAwi1l entail. First, it. Part of the challenge it; to you will never learn VBA by reading about you must t.y use everything, it is going is that if a maero language is so powerful that it enables you to do commands. A book or t'utorial (likethis to be too conplex for you to memorize a1l the one) will enable you to use VBA to solve specisc problems. How ver, once you want comfortable out VBA by trial and error. sguring to do more, you will have to become To facilitate learning VBA, you should use the macro recorder ip Excel. When actions will be recorded in VBA. Tl'y tlzi: of you use the macro recorder, the results your the Select ToolslMacrollkecordNew Macro in Excel. Then create a simple graph using described in is graph wizard. Look at the VBA code that Excel creates. (This example if but look result is daunting when you lirst at it, you want more detail on page 904.) The point that you can starting gives recorded graphs, you a the create VBA macro to to use modify; you need not create the basic code from scratch. The main objective of this t-utoritl is to help you create your own functions. 'While of the examples here relate to option pricing, there are many other uses VBA.
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* You needto tellthefunction whatto expect as input, hence, the the name of the f'unction.
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Creating a Simple Function which will With VBA, it is a simple matter to create your own function, say BSCall, firsi and this, create open a macro you must conpute a Black-scholes option price. To do
* The f'uncon you typed now appears in the function wizard,just as if it were a builtln in Excel ftlnction. To see this, open the function wizard using lnsertlunction. Desned''. the left-hand pane ('Tunction Categoly'), scroll to and highlight luser
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A Simple Example of a Subroutine A hmctionrettll'ns a result. A subrolttine (calleda
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We ran the subroutine by clicling on ToolslMacrolMacros and then double-clicldng on the subroutine name. lf you are going to run the subroutine often, creating a button in the spreadsheet provides a shortcut to the subroutine. Here is how to create a button'. and right-click
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Functions can call functions. Here is an example.
1L.llltLtl-
We have just created and run a subroutine. lt pops up a dialog box that displays a message. The MsgBox function can be very useful for giving information to the spreadsheet user.
Creating a Button'to
it.
t??l
k.
Functions Can Call Functions
4. Retttl-l' to Excel.
5. Run the
Doltble-click
u'// pop up. Olle ofthe
vBA
Some comments: an action to supply functions when the
1. Retltrn to Jhe Vsttal Basic Editol; 2. Click
p?lt?l/,dbutton, a dialog box
wl'rH
The answer
z1..2,
(3 4:151'
enter ,
5
,
7
)
will appear.
Illegal Fnnction Names Some function names are illegal, which means that you will receive an enor message if you try to use them. You cannot use a number as a function name. You cannot use : ; + # $. % / If you try the following characters in a function name: space ''
'
:
-
.
,
.
wnzcffn.
4. The clfrx/r chaltges to a crosshaill Mtn the ??lt/l/-' to r/lc spreadslleet, hold #()lpn I'l/cnyt:ul l#ytrhngeroffthe the /'./i'???t?I/.c blttton, tw/##?w, to create a rectalgle.
3If you have examined the code for DisplayBoxz, you may be puzzled by checking to see if Response = 4tvbYes''. VbYes is simply an internal constant which VBAuses to check for a yes'' button response in the help hle are vbOK, vbcancel, vbAbort, to a dialog box. The possible responses--documented vblketly vblgnore, vbYes, and vbNo.
890
k
AN INTRODUCTION
To VISIJAL
BASIC
STORI NG AND RETRI EVl NG VARIABLES
FOR APPLICATIONS
Basic lets you lnow immediately that to use any of these characters in a name, Visual underscore where you would like to have that is can use an you something wrong. Note example, is a legal function for So BS-2, function of readability the name. a space for Here is a more subtle issue. There are function names that are legal but that you should not use. BS2 is an example. Tlzis would be fine as the name of a subroutine, which is not called directly from a cell. But think about what happens if you give this example, <'BS2(3)'',in a cell. How does name to a user-defined function. You enter, for Excel understand this? The problem is that ttBS2'' is also #?e llanle ofa cell. So if you confused and return an 4l'y to use it as a function in the spreadsheet, Excel will become BS-1, BS-3, and This is why, later in this tutorial, you will see functions named
error. SO On.
AND RETRIEVING IN A WORKSHEET
D.4 STORING
@ Rangeo.Activgte and Activecell let you access cells by using traditional dresses (e.g.,';Al'') * Cell lets you ddress cells
'tactivate''
cell ad-
using a row and column numbering scheme.
Using a Named Range to Read and Write Nlzmbers 9om the Spreadsheet
x
-l/l/?-t/lfr?pe
slbdttlel:
in
Readvariableo
RangelltTestn)
=
(Str(x))
MsgBox End
Sub
thell .f/pz?-llNl?plclD#nd, 2. Select cell A1 n sheet click OK. F(?I/ havejttst created a ntwnct range. ''qed/z
''
-
#
3. Ellter
J'/'lc vallte
fd5''
ill. r/7ccell yt?lfjttst
?27??1'#
4. Select ToolskMacrokMacros, #7cn double-click
..Test'ts- #?en
and Jype
'.lnest''
t?n
.6lleadval-iable
''.
At this point you have just read from a cell and displayed the result. Note that tx'' is a number in this example. Sometimes it is useful to be able to convert a pumber to its characterequivalent (forexamplete character 7 ratherthan the number 7.0000000 ). You can do this using VBA'S built-in ''StI''' f'unctionx lt t'urfls out this was not neessary in this example', entering ttMsgBoxtxl'' would have wprked as well. As you might guess, you can use the Range function to write as well as read. :G
Sub
Suppose that there is a value in the spreadsheet that you want to include as input to your function or subroutine. (For instance, you mighthave a variable that determines whether the option to be valued is American or European.) Or suppose you create a stlbmutine that performs computations. You may want to display the outputin the spreadshet. (For example, you might wish to create a subroutine to draw a binomial tree.) Using VBA, how do you read and write values to the spreadsheet? If you are going to read and write numbers to specifc locations in the spreadsheet, identify alocation is to use anamed you must identify those locations. The easiest way to that you examine below-require will a altenmtives-which we range. The this within and wiite read and then workbook, within worksheet the location specific or activated region. There are at least three ways to read and write to cells: '
891
You may be thinking that it seems redundant to have so many ways to access cells,
1. Elkter thefollowing
VARIABLES
$
but each is usef'ul at different times.
Sub
capabilities. Functions and subroutines are ltot interchangeable and do nothave the salue Think of subroutines as code meant to be invoked by a button or othelwise explicitly called, while functions ret'ul'n rsults and are meant to be inserted into cells (although functions can alsp be called by subroutines). Because of their different pgrpose, some VBA capabilities will work in one but not the other. In a subroutine, for example, you can write to cells of the workbook. With a subroutine you could perform a calculation and have' the answer appear in cell A1. However, if you invoke a function from a worksheet by entering it it a cell, you worksheet or cannot write to cells from within that function. You cannot activate a other hand, if the (On f'unction. such within display from anything about a the change these do worksheet, it invoked f'tom but can subroutine, invoked a not is by the function a restrictions things.) Subroutines, on the other hand, cannot be called from cells. These exist beeause functions and subroutines are intended for different pulyoses.
EET
@Range lets you addreis cells by name.
1. Enter thefollowing
Differences between Functions and Subroutines
I N A WORKSH
( t'Testz ) ( ''Number ''
MsgBox
2. Give the a
?/t7l??e
nlfpplMr
in .#./b#If/c,/.-
()
Writevariable
Range
3. Enter
Jl/lp/-t/l//f?zc
Range
=
.'Testz
copied!
''
( t'Test )
t'
)
1'
f.tpcell Sheet2.B2.
il Sheet2.B2.
thell double-click 4. Go to Ftltp/-l3rftzcrtilli-ft-lc/-ql,-
5. The ?7lfn2l7'r.//t???l
4xbucan locate the
'tstr''
gt
,,
t??l
&jlitevariable.
''
Test is copied to Testz.
functionby using theobjectbrowser,
lookingundervBA, then
'Tconversions-''
,,
892
% AN
To VlsuAl-
Irq-rRoouc-rlolq
BAslc
USI N G EXCEL FU NCTI ONS
FoR APPLICATIONS
Reading and Writing to Cells That Are Not Named
t Write '
Copy
'
Copy
End
Activecell.value
into
cell
A3
2
into
cell
A4
number
893
2
=
the
number from Ce1ls(3, from the number 3) Cells(4, 3)
$
i
=
1)
l
VBA
=
=
cell
A3
into
cell
A4
into
cell
C3
1)
cell
C4 '
1)
Sub
(Str(x))
This subroutine reads the numbers 1 and 2 into cells A.3 and A4, and it then copies will use the Cells function to draw a binomial tree.
Sub
End
number
the
Cellsll,
Ce1ls(4,
RangetnAlls.Aetivate
MsgBox
1)
Cells(3,
Readvariablezt) Worksheetstl'sheetzlll.Activate
Sub
=
the
'
You can also accss a specific cell directly. In order to do this, you 'Iirsthake to activate the worksheet containing the cell. Here is VBA code to read a variable:
x
Write
Ce11s(3,
FRO M WITH l'N
In this subroutine we first activate the worksheet named G'Sheetz.''NextWe ctivate cell 1W1'' within Sheetz. You will see thgt when you have fipished calling this the cell'' is function, the cursor has moved to cell A1 in Sheetz. This is because the instruct the lines to be first move te two cell happens cursor to the on; whatever cursor Sheetz.Al. to The active cell has properties, such as the font, color of the cell, apd formatting. All of these properties may be accessed using the Activecell hmction. #or fun, insert the line
the values into (23 and C4. Later we
Gfactiv
Reading 9om within a Function It is possible to read from a worksheet from within a function. For example, consider these two functiops: Function End
after the MsgBox function. Then switch to Shetz, ruri the subroutine, and watch the change in cell A1 We can also assign a value to Activecell.value; this is a way to write to a cell. Here is a macro that does this:
ReadTestltx.
ReadTestl
=
x
+
y) y
+
RangecrRead
In
FunctionlAllo.value
In
FunctionlAilo.value
Function
Function
y)
ReadTestzlx,
'
Application.volatile
-
.
ReadTestz End
=
x
+
y
+
RangetnRead
Function
An interesting experiment is to create the sheet named In Function,'' put number in cell and ReadTstl and ReadTest2(3,4) A1, in cells /.2 and the enter (3,4) A.3 Both f'unctions will return the valu wf/f properly Now chnge the val Readltz in ell A1 to 20. The lf/7crftpp? bvill ,/4)/. challge. ?-er!Ir?2 the vt-l/llc 27, bltt Readstl Press the F9 ky to recalculate l.W#still not recalcttlate. The problem is that Excel has the spreadsheet. Readlstl of lnowing value ip affects either function. However, that change Al the no way ReadTestz recalculates because of the Application.volatile statement at the beginning. This tells Readrrestz to recalculate anytime allything changes. Obviously this will slow the worksheet, but it is ncssary in this case. Reading from the worksheet frolp within a function is possible, but, other things equal, it is preferable to pass valus to th function explicitly as arguments. Elkead
Writevariablezl)
sub
'5''
Worksheetstklsheetzlll.Activate
:112
RangetliAlfo.Activate
x
=
Activecell.value
RangelnBlnl.Activate
Adtivecell.value
x
=
This subroutine reads the number from Sheetz.Al and copies it to Sheetz.B 1.
Using the Cells Function to Read and Write to Cells There is yet another way to read and write to cells. The Cells ftmction lets you address cells using a numerical row and column numbeling scheme. Here is an example illustrating how Cells works: Sub
CellsExamplet)
1 Make l'Sheetz'' Worksheetsl'lsheetzttl.Activate '
The
first
entry
the
active
sheet '
is
the
row,
the
second
is
the
column
''
'
D.5 USING EXCEL FUNCTIONS FROM WITHIN VBA VBA permits you to use most Excel functions within your own custom functions. Since Excel has a large number of bllt-in functions, this is a powerful feature.
894
%.AN
To VlsuAL
INTRooucTloN
Usl NG
FoR APPLICATIONS
BAslc
FUNCTIONS
There is only one complicated piece of the Black-scholes calculation: Computing the cumulative normal distribution (the &&N)''function in the formula). Based on the example appendix, we would like to do something like the following: at the start of .this
Function
dl (12 BS End
-k*E&(-r*t)*NomSDist((Ln(s/k)+(r-d-v-2/2)*t)/(v*tG0.S))
BS(s, k, v, r, t, d) 2 d + v (Log ( s / k) + (r * dl 0 S t v s*Eol-d*el*WorksheetFuctiqn.NonsDistldl)
/
'*
=
Unfortunately, this doesn't wrk. The reason it doesn't work is that VBAdoes not understand either itLn'' or ttNormsDist.'' Though these are f'uctions in Excel they are we can not functions in VBA, even though VBA is part of Excel. lnstead of using of However, there is VBA version the EELog,'' w hich VBA f'unction. is the same no use version of NormsDist. Fortunately, there is a way for you to tell VBA that NormsDist is located inside of Excel. The following example will show you the error you get if you fail to call ttn,''
NormsDist correctly:
-
2
)
t ) /
*
(v
'N
t
*
2. Enter thefolloving: Ftmction
di (:12 Bs End
= =
=
Bs (Log
=
dl
-
s''Ex.p
v,
,
*
v
(
-d*t)
+
Sqrt
d)
t,
r,
(r (t )
=
*nom-msdist
d
+
''
v
(dl)
2
-k*Exp
/
2)
(
-r*t)
*
t )
/ (v
*
A'No:rmsDist
Sqr
(t ) )
(d2)
Ftmction
Comment'. To save a little typing and to make the function pore readable, we are and separately. You will also notice that defning the Black-scholes ttog,'' Which-as we noted abpve-is built instead of entering tn,'' we entered /t. into N&B G'dl''
'd2''
3. Enter ilrt? the spreadsheet Hit <Ellterw.
lblf vill get
f/2c error
5
)
Function
function will now evaluate correctly to 2.78.
The Object Browser The previous example illustrates an extremely powerful feamre of VBA: lt can access the f'unctionsbuilt into Excel if you tell it where to find them. way you locate other functions is to use the Object Smwzr, which is part of VBA. Here is how to use it: 'l'he
1. F,r/n within a ?nt-lcrtp module, press #2ctitle f'Olp-jcclBrowsel;
f/7d F2 key. F/l
vill
''
pp
ltp
a dialog l'ltzvvith
pnd'uke
Hsttb
sr/ncribn
not #c/?2W''.
This erroroccurs becausethereis no version, howeverspelled, of E
''
''kcdf''
''
(s k, ( s / k)
.
box /'/7t7l says Libraries. 2. In re top /e# )?(?lfwill see a #?-t/p-tftpwn Click (7/l the listwith, at a ?nf???nIf?n, #t/w?larrow at fe right ofthis Pne. Ftxl will see a #?-t?#-#ow?7 ''7SA and as lwt:l entries. lhere ???tzy be t/l/ycr entries, #epc?z#fn,g upon /lt?wyou have set 1Ip Excel-)
tab.
''
0
-
f'z4#
T'4btflffe./
895
''
=
The Black-scholes
Function
t??1 the
%
-k*Eot-r*tl*WorksheetFuction.NomsDisttdz)
Fpnction BS(s, k, v, r, t, d) Bs=s*Ee(-d*t)*NormSDist((Ln(s/k)+(r-d+v-2/2)*t)/(v*te0.5))
1. Click
VBA
FROM WITH I N
With a correctly referenced NormsDist, the function becomes
Using 'VBA to Compute the Black-scholes Formula
End
ExcEl-
3. Click tpn FSA. 4. 1lt /'/?d
'.classes
5. To /'/7c rights that llpzc/rib?74.Ln
''
''
list click 'f.'4???!7t??-.
n the t-I?-d available ''
t'Nlth.
''
of ,#.ft'I#7 box, i'?7.F'AA.Note that ''
Ifyottright-click
t??- 4.NortnuDist. /'/?t7l ''Log '' ?zlf//-??.
will see
t??7
the
?7t7/zf?-cIf
t?n
7. Click
''
..l.nk
''
logaritlun.
6. Re-htrn to #7c top /tr./i'box, vp/7i'c/7,7tpw says right ofthis ff?lc.
'f
have a list of all /'/7c lnath inclttded is in l/y.- list, but /7//' tllelp, Jn# c lick t?n #?d?l you
?7t)w
,',:/1/
Tff-o.g
''
P'AA. Click
''
t?n
the lt/w?7 arrobv at the
t??7 Excel.
8. In #?d ''Classes'' list, 9. To the
?-f,g/7t
in #7d
click t??, HWorksheetbhmction. ''elnbers
of
''
nsw have a list ofExcel !7IIf//=n hmctionsthat ???J.'y be ct7pc#-//tl?/ a ?/yttc?-t) module /J,)) specvking tlWorksheeTunction-lnctionname-''6 and ''A'(??-??l5'.!%-/.'' kote J'/ycf both are 'klbr/cv/7cd/r.#-fl7crf/?i
''
Tffw
box,
.,/1/
''
5If
you are curious about this, do the following-.Select Vewlobject Browser or press F2. Click on the then select ''Excel-'' Under 'Tobjects/Modules'' click drop-down arrow under ''Lbraries/Workbooks''; on tApplication''; then under ''Methods/properties'' scroll down to KWormsDist-'' You have now just located TNormsDist'' as a method available from the application. lf you seroll around a biq you will see that there is an enormous and ovenvhelming numberof functions available to be called from VBA.
6By the way, you should not make the mistake of thinking that you can call any Excel functon simply and it won't work. While most functions by prefacing it with ttWorksheetFunction.'' Try it with are accessible from VBA, the only way to know for sure which functions you can and cannot call is by using tlte object browser. ttsqrt''
% AN
896
To VlsuAl-
Ihl-rlloouc'rlotq
BAslc
ARRAYS
Appl-lcvlohls
FoR
iltcluded iWthis Iist. Note also that &&Logis ncftdtf in this list, ilfl be t7I$7t7?-' t'f#./-c?-d??/' is base 10 /.7ydefattlt base), that Excel ''Log can spec' a e.1 F#A l is base 1$)/?e?-w,5' ''
rl/(?lf
''hmction
'
If you create any VBAfunctions that are even moderately ambitious, you will need to use the object browser. lt is the hart and soul of VBA.
D.6 CHECKING
.
FOR CONDITIONS
Frequently, you want to perform a calculation only if certain conditions are met. For example, you would not want to calculate an option prie with a pegative volatility. before pfceeding with the It makes sense to check to see if your inputs make ses with an en'or message-if they do not. calculation and aborting-possibly Then The easy way to check if a conditin exists is to use the construct JJ Else.s Hre is an ekample of its use in checldng for a negative volatility in the . Black-scholes formula:g .
.
.
.
.
Function If
BS 2(s,
v BS
2
k,
0 Then
> =
BS(s,
k,
v,
BS 2 End End
volatilityl'o
(uNegative =
CvErrtxlErrvalue)
If Function
This function checks to see if volatility is greater than 0,'if it is, the function computes the Black-scholes formula using the BS f'unction we created eadier. If volatility is not greater than zero, then two tlzings happen: (i) a message box pops up to inform e' you o f the mistake and (ii) the f'unction rettlrns a value indicating that there is an n'or. In general you should be cautious about putting message boxs into a function
(as
opposed to a subroutine), sinc evely time the spreadsheet is recalculated the message box will pop up. Because error-checldng is often critically important (youwould not want to quote a client a price on a deal for which you had accidentally entered a negative volatility), it is worth expanding a bit on the use of the CVEIT function. .
Xscroll
down to the tt:Log'' entry and then click on the button at the bottom left. lf you use tog'' in a spreadsheet or if you use A4WorkslieetFunction-log'' in a f'unction,you wll get the base 10 in a function, you will get the base e logarithm. Note also that, logarithm. However, if you use eisqrt'' is not included and, hence, is not available in VBA. as mentioned earlieq **''
ltog''
S'There is also a Case
hle.
.
.
.
897
lfthe user enters a negative volatility, you cottld just have Excel ret'urn a nonsense -99.
This would be a bad idea, however. Suppos yott value for the option, such as complicated with worksheet have a many option ciculations. If you failed to notce the the would be treated as a true option value and propagated throughout your error, calculations. Alterflatively, you could have the function rett:nz a string such as you entered a negative volatility.'' Entering a string in cell when you should have a number could liave unpredictable effects on calculations that depend on the cell. It is obvious that an addition between a string and a number will fail. However, suppose you are performing a frequency count. Are you sure what will happen to the calculation if you introduce a string among the numbers in your data? Excel has built-in error codes that are documented in VBA'S online help. For example, XIEUNA remrns :t#N/A,'' xlEn-lkef rettlrns 'WREF! and xlErrvalue returns -99
'toops,
.
''
Kt#VALIJEI''. By using CVEIT along with one of the built-in error codes, you guarantee tat your function will retunz a result that Excel recognizes as a numerical error. Excel programmers have already thought through the issues of how subsequent calculations should respond to a recognized enor, and Excel usually does something reasonable in those circumstances.
D.7 ARRAYS
Else MsgBox
X'
%
Select construct that we will not use but that is documented in y'
VBA'S
help
9You need to be aware that VBA will expect the Then,'' $Else,''and iEnd 1F' pieces to be on separate Atlf Then,'' for example, the code will fil. lines. lf you write e'Else'' on te same line as 4:lf
Often you will wish to use a single variable to store many numbers. For example, in a binomial option calculatiop, you have a stok prie tree. After 11 periods, you have ?7 + 1 Possible stock prices. lt can be usef'ul to write te lowest stock price as ,S'(0), the next as S(1), and the highest ms Sl4. The variable S is then called an arraysrt is a single valiable that stores more thap one ntmber. Each item in the array is catled an element.
Thirtk of an array as a table of numbers. You access a specifc element of the array by specifying a row and column number. Figure D.1 provides an example of an array.
De6ning Arrays When you create an array, it is necessary to tell VBA how big the an'ay is going to be. Y u do this by using the Dim statement (EDim'' is short for size of the array). Here are some examples of how to use Dim to create a one-dimensional array: ltdimension''-the
Dim
p
(2 )
As
Double
This creates an array of three double-precision real numbers, with the array index running from 0 to 2. (By default, the first subsclipt in an an'ay in VBA ls 0.) If you had written Dim
p
(3 to
s)
As
ootzble
you would have created a three-element array with the index running from 3 to 5. In this example we told Excel that the valiable is type 'Double.'' This was not necessary-we could have left the type unspecised and permitted Excel to determine the type automatically. It is faster and easier to detect mistakes, however, if we specify the type.
898 k. AN INTRODUCTION ;)' t')' ;':' ;'' ((' k-f S''F''qq' j'k))p)r't))q'. )' r' j5' 1* (f 'jjjjkj-jjjj;' tjf '.(' .y'(' i ('' (.'' (''' r' ( )--.----' yt)f .tjyj'i. y' ..' ' j't,'(t'tjjjtrrrijij,r j'( jtf yt'. )' i'. '.' 7* 1,114*. ' ,))* jjgjjrjjj'kf )'. tk'lttt).:f rsrf rr'-(t')' (' r' yt'. ' ! (i jj!qir ;!.ifi(. y i.'.' 1. . ' ttf y' t'. . .)'y) 'j't'jjjjjjjytyi jjjjjtjrrjrjj' ! ! . !(. y (.).j...r. :i''. jji. !: ..l. . . . )jq. $g),.q q qjq j :.r.prti; yyty ..y yj y.gy . j jjjjj:;jjjrj. r . . . y y g .... .... . y..;j. .y.jtj j . y ' t.':. .. '.(...)y;...);..-... ...;!y . ;)j.. ;) .;-.ri-.. '.'. '.. @' )' Fy'tl $' !' 'j'1* (
.'.' ''. .'' .jj-jjjjj'. '71* ' -'' ''-'.-'..,'-' -'. -' '(S' 5'77''11,7751* 1* ' ' ' '. E ' :.! Ei
(''ji.
--
: *1'1. E... . -
q
' :
.
-
.
..
:
' :
:' E'
'
'
.
--
-
.l.
-
.
.
-.
...
,.. (E , ( -. - -; . - i ,- . . . .-
.
...
-
..
.
.
...
.
....
. . . . . . . . . ..
. . .
.
.
. . .
. -.-. . . .. . .
.
.
-)i
.
.
.
E. E
.
'. . .. .. .. Example of an array with 3 rows and 5 columns. By default, VBAnumbers rows and columns start with 0. If yh array is named X, th number ,,8,, is retrieyxedas X(1#' 3).
. .. . .:
''
... . .
. ,jjj.
.
.
ITERATION
BASIC FOR APPLICATIONS
EE q'rqqE':' 5 E.'. E(EE . E.
''
jltEiEE'' E'Ei'E' E:' E.E ' E E :.. !.EE iE'i:iE.' (: E''.'.: .E'. E E(' .'. irrr)l !iC/IIjE ( . ' ' . . ...)...t . . j)kp. t1l. .ilfE . . . . - : : -i E; .. ;E E.f - i.! ..i ! ! ; : .. !-y . . -- -. . . E . E . -- .-.. . . - -. .
. .ii;!i
TO VISUAL
-- .
.
.
.
.. .. .
columnNumber
'
Row Number
3
1 2
3.91
12
0 -19.8
a
1
0 -82.5
44
4
3
23
-5
1
8
6
17.2
-33.183
24 7
'
%
899
The difference between UseArray and UseArrayz is the way arrays are declared. In UseArray, there is a dimension statement, and then array elements are created one by one. ln UseArrayz, there is no dimension statement; and the An'ay fnction (builtinto VBA) is used to set the initial values of the array elements (this is called litializing the array). UseArray will fail without the Dim statement, and UseArrayz will fail vith the Dim statement. Finally, notice the repetition in these examples. The statements that put numbers into t he array are essen tially repeated three times (albeitmore compactly in UseArrayz) and the statements that read numbers out of the array are repeated three times. If the array had 100 elements, it would take a long time to write the subroutine in this way. Fortunately, we can perfonn repetitive calculations by iteration. ,
'
D.8 ITERATION You can also create arrays with multiple dimensions. For example, the following Itr' e valid Dim statements: 8)
Dim
X(3,
Dim
Y(l
to
4,
-5
Dim
Z(i
to
4,
-5
to 3) to 3,
columns-a The irst statement creates a two-dimensional array that has 4 rows and 9 and 9 columns. 4 x 9 array. The second also crates a tv-dimensional ari'ay with 4 rows 25. 9 x 25 = 900, 4 4 Since which il 9 x tlukt-dimsional x x array The third creates a this lastan'ay has 900 spates, or elements. numbers Here is aroutine thatdefnes athree-element one-dimensional array, reads oui dialog boks: into the writes then array into the array, and UseArrayl)
Sub
X(2)
Dim
=
0
k(i)
=
1
X(2)
=
2
MsgBox MsgBox MsgBox
Double
x
=
UseArrayzl) Arrayto, (X(0))
MsgBox
(X(i))
MsgBox
(x(2))
Sub
UseArrayLoopo
Dim
X(2)
For
i
=
=
End
Double
As
0 To i
2
0 To
2
i
Next
=
(Str(X(i)))
i
Next
(X(0)) (X(1)) (X(2))
MsgBox
End
Sub
MsgBox Sub
The following translates the syntax in the Nrst loop above:
subroutine UseArray You should enter this code and execute it to see what happens. The can also be wlitten as follows: Sub
Here is an example of afor loop. This subroutine does exactly the same thing as the UseArray subroutine:
For.i
Sub
End
A Simple JorLoop
X(i) As
X(0)
Many option calculations are repetitive. For example, when we compute a binomial stock price price, option we generate a tree and then traverse the tree, calculating the when price each node. Similarly, option at we compute an implied volatility, we need to calculation repeatedly until an-ive perform a at the correci volatility. VBA provides we with the ability write of code that can be repeated as many times to one or more lines us like. as we
i,
For
i
=
0
to
2
Repeat the following statements three times, the first = 0, the next time i = 1, and nally
time setting i i 2. =
Set the th value of X equal to i.
Go back and repeat the statement for the next of i.
value
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Creating a Binomial Tree In order to create a binomial tree, we need the following information:
* The initial stock plice.
-
up and down by wlzich the stok
ReDim
Stocklz)
Dim
i
'
As
Integer
D j.m t As
Integer
n u
( Range (
=
P0
1
=
ReDim
/
''
1
u
pyovide
default
'
number
'
move up t initial down
( P0 )
Range
=
d
llu''
''
move
Stocktn
of
2
if
steps
specified
steps
no
+
1)
'
binomial
of
any
price
stock of
array
stock
* Do ,Jnl# prices
@ While
calculations
previous
1)
P0
=
' ' ' ' t '
adopt the that the column holds the convention The point in time. prices for stock row holds a given For prices example, stock the first time. row over prices resulting stock from a11 up moves, holds the prices resulting from a single second stock row holds down move, etc.
'
The
We will
first t
For
2
=
loop To
t)
Cellstl,
second For i 2 To Cellsli, t) Next i
'
The
=
Cellsll, is loop
t
-
1)
*
u prices
stock
across
at
a given
time
t =
Cellsli
.
.
Loop and Do
.
.
.
Loop and Do
.
.
.
Loop Until
.
.
.
Loop While
Wnd
.
If you ever think you need them, you can look these up in the onlih also In Next constnlct, which we discuss below. is a Fol f'tzc .
.
.
.
.
AND WRITING
D.9 READING
help. There
.
ARRAYS
A powerful feamre of VBA is the ability to read arrays as inputs to a ftmction and also to write f'unctions tat rettlnz arrays as output.
time
over
n =
.
.
-
1,
t
-
i)
*
d
Arrays as Output Suppose you would like to create a single f'unction that rettll'ns two numbers: The BlackScholes price of a call option and the option delta. Let's call this f'unction BS-3 and create it by modifying the f'unction BS from Section D.5.1.
t
Next End
is
.
* Do While
Worksheetst'toutputlll.cells.clearcontents
Cellstl,
Other Kinds of Loops Although we will not discuss them, there are other looping constructs available in VBA. The following kinds of loops are available:
steps
Worksheetslttoutputt'l.Activate
' Erase
Note that this subroutine uses the ReDim command to speclf'y' a iexible array size. Sometimes you do not know in advance how big your array is going to be. In this example you are unsure how many binomial periods the subroutine must handle. If you are going to use an array to store the f'ull set of plices at each point in time, tlzis presents a problem-how large do you make the array? You could specify the array to have a any user is ever likely to use, but this lind of practice very large size, one larger t.11a11 could get you into trouble if memol'y is limited. Forttlnately, with the ReDim statement
E
) )
''n''
Range
=
* The use of the Cells f'unction means that you can perform the calculation exactly as you would if you were writing it down, using subscripts to denote which price you are dealing with. Think about how much more complicated it would be to use tradiiional row and column notation (e.g.,EW1'') to perform the same function.
VBA permits you to specify the size of an array using a variable.
()
DrawBihomialTree
901
tree.
moves.
Suppose we wish to draw atfe Wherethe initial price is $100,wehave 10 binomial periods,and the moves up and down are tt = 1.25 and d = 0.8t Here is a subroutine, complete with comments explaiping the code, that will create this tree. Youfrst need to nameaworksheet rutput'' and then wewillwrit the treto this worksheet. Thenumber of binomial steps and the magnitude.of the moves are read from named cells, which can be in any worksheet. I have placed those named cells in Sheetl in VBA-examples.xls.
sttb
%
* This subroutine does not price an option; it merely creates a binomial stock price
* The number of time periods. @The magnimdes
AN D WRITI Nc ARRAYS
Sub
Function
dl d2 nd1 nd2 delta
Several comments'. * There is a simple command to clear an entire worksheet, namely: Worksheets (It//?-kJ/7ddr?7c???d).Cells.CleKontents.
3(p,
Bs
/
(Log(s
=
di
=
-
v
*
k, k) -
t
(r
-
WorksheetFuction.NomsDisttdl)
=
WorksheetFuction.NomsDisttdz) =
Expt-d
*
d
0.5
=
t)
*
d)
t,
r,
v,
+
nd1
+
0.5
*
v
a
2)
*
t)
/ (v
*
t
a
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price BS End
s
=
3
*
delta
Arraytprice,
=
To VISUAL
-
READI NG AN D WRITI NG ARRAYS
BASIC FOR APPLICATIONS
k * Expl-r delta)
*
$
903
Arrays as lnputs
t)
Function
The key section is the line
We may wish to write a function tat processes many inputs, where we do not know in advance how many inputs there will be. Excel s built-in t'unctions sum and average examples of this. They both take of cells as input. For familiar two range can are lt turns out that it is easy to example, you could enter in a worksheet write functions that accept ranges as input. Once in tlle function, the array of numbers from the range can be manipulated in at least two ways: As a collectiolt, or as an actual array with'the same dimensions as the range. 1%um(a1:b8).''
BS
3
Arraytprice,
=
delta)
We assign an array as the function output, using the array function introduced in Section D7 If youjust enter the f'unctionBS-3 in your worksheet in the normal way, in a single cell, k will remrn a single numben In this case, that single number will be the option price, which is the Erst element f the array If you want to see both numbers as output from the function, you have to enter BS-3 as an an'ay fanction spanning multiple cells; Select a range of two cells; enter the formula in the first, and then press Ctrl-shift-Enter (instead of just Enter). There is a 50% probability you just discovered a catch. The way we have written BS-3, the array ouqmt is horizolltal. If you enter the rrfayfunction in cells A1:A2, for example, you will see only the option price. If you entef the funtion in A1; t, #ou will see the price and the delta. Wllat happens if we want vertical output? The answer is that we eanspose the array using the Excel function of that name, modifying the last line to
read BS
First, here are two examples of how to use collection. Excel has built-in functions c/led Sumsq and Sumprod, which (as the names suggest) sum the square d e lements of a range and sum the product of the corresponding elements of two or more arrays. We will see how to implement similar f'unctionsin VBA. Sumsq takes a set of numbers, squares each one, and adds them up:
The array
Function
=
WorksheetFuction.TransposetMraytprice,
.a
Sumsqtx)
Sum
=
For
Each
Sum
0
y In Sum
=
+
x y
a
2
Next
Sumsq
Sum
=
Functibn
End
3
as a collection
deltal)
rhis will make the output vertical.
There is also a way to make the putppt both horizontal and vertical. We just have to return a 2 x 2 array. Here is an illpstrattop of how to do that:
The function Sumsq can take a range (e.g.,'W1:A10'') as its argument. The For fac/? construct in VBA loops through each element of a collection without our having to know in advance how many elements the collection has. There is another way to loop through the elements of a collection. The f'unction Sunrod takes two equally sized arrays, multiplies them elementby element, andretnzs the sum of the multiplied elements. ln this example, becuse we are working Fith two collections, we need to use a more standard looping construct. To do ihls,we need to first count the number of elements in each array. This is done using th Count property of a collection. If there is a different number of elements in each of the two arrays, we exit and ret'urn an error code. Sumprodtxl,
Function
xl.count xz.count If nl <> n2 Then if arrays equally not 'exit CYErrtxlErrNum) Sumprod
nl n2
=
=
=
Now it does not matter whether you select cells A1:A2 or A1:B 1', either way, you will see both the price and the delta.lo
End
If
Sum
=
0
For
i
=
Sum
=
1
To
Sum
+
=
Sum
i Sumprod
Next
lowhatdo you see if you select cells A1-.B2? What about A1 :D49.
End
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We can also eat the numbers in the range as an array. The as an array trick doing that is that we need to lnow the dimensions of the array. i.e., how only to and columns has. The f'unction Rangerfest illustrates how to do this. it many rows
The array
'
:.2 9j5
For example, here is the macro code Excel generates if you use the chart Wizard set up a chart using data in the range A2:C4. You can see, among other things, tht to the selected graph jtyle was the fourth line paph in the graph gallery, and that the chart ttl-lere is the title.'' Also each data s/j is in a olumn and the frsi column was t itled x-axis used the as was ,
Function
RangeTesttx)
(Efategorytvabels:uizl'').
1 prod x.Rows.count r x.columns.count c =
=
t Macrol ' Macro
=
i
For For
l
=
j
1
=
prod
To
To
prod
=
r
1
c *
#'
xti,
McDonald
Robert
1
Macrolo
Activesheet.chartobjects.Addllg6.s,
RangeTest E n d Ftmction
=
WorksheetFunctiop
.Transpose
(Array
(prod,
r
,
Application.cutcopyMode Activechart.chartWizard
This function again multiplies together the cells in the range. lt remrns not only product, but also the number of rows and columns. the When is read into the function, it is considered by VBA to be an array.'l Rows and Columns are properties of an array. The constrtlct -v
x.Rows.count
tells us the number of rows in the array. With this capability, we could multiply arrays, check to see whether two ranges have the same dimensions, and so on.
Source:=Range('1A2:C4H),
=
End
Gallery:=xlLine, CaEegouLabels:=l,seriesLeels
TiEle:=tHere :=0, HasLegend:=i, ''Y-.Axis l'X-mxis ValueTitle : : 11
i62).Select
252.75,
False
=
is tt
#
the
Titlen,
ExtraTitle
CategoryTitle :
=
=
11 11
Sub
Using Multiple Modules You cap split up your fpctipns and sgbroptines among as rpany podules as you likefunctions from one module can cll another. for example. Usingmultiple modules is oen convenient for clarity.
Recalculation Speed
D.10 MISCELLANY ln this section we discuss
miscllaneous
topics.
Gtting Excel to Generate Macros for You Suppose you want to perform a task and yop don't know how to program it in VBA. F& example, suppose you wnt to create a subroutine to set up a graph. You can set cornmantis that accomplish the up a graph manually and tell Excel to record the VBA same thing. You then examine the result and see how it works. To do this, select ToolslMacrollkecord New Macro. Excel will record al1 your actions in a new module located at the end of your workbook, i.e., following Sheetl6. You stop the recording by clicking the Stop button that should have appeared on your spreadsheet Fhen you how started recording. Macro recording is an extremely usef'ul tool for understading Excel and VBA wqrk and interact.
l l'ou
can verify this by using the VBA functon lsArray. For example, you could write
y
IsArray
(x)
and y will have the value
39, =
PlotBy:=xlcolumns,
Format:mq,
=
by
Range(''A2:C4H).Se1ect
i
Next
2/17/99
.
Sub
j
Next
Macro
recorded
'true''
if
-v
is a range input to the function.
One unfortunate drawback of VBA-and of most macro code in most applications-is that it is slow. When you are using built-in functions, Excel performs clever internal checking to lnow whether sopething requires recalculation (you should be aware that on occasion it appears that this clever checking goes awry and something that should be recalculated isn't). When you write a custom function, however, Excel is not able to perform its checking on your functions, and it therefore tends to recalculate everything. This means that if you have a complicated spreadsheet, you may find vcp'y slow recalculation times. This is a problem with custom functions and not one you can do anything
about. If your calculation writes to the worksheet, you can signiscntly routine by t'urning off Excel's screen updating. You do this by
speed up your
Application.screenupdating=False
If you want to check the progress of your calculations, you can turn Screenupdating off at the beginning of your subroutine. Whenever you would like to see your calculation's progress (for example every l00th iteration), you can turn it on and then immediately turn it off again. This will update the display. Finally, the keystroke Ctrl-Break will (usually)stop a recalculation. Ctrl-Break is reliable if your macro writes output to the screen or spreadsheet. more
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Debugging We will not go into details here, but VBA has sophisticated debugging capabilities. For example, you can set breakpoints (i.e., lines in your routine where Excel will stop calculating to give you a chance to see what is happening) and watches (whichmeans that you can look at the values of variables at different points in the routine). Look up in the online help. Itdebugging''
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an asset and the price of the contract hedging the asset. Bear spread The sale of a call (or put) together with the purchase of an othenvise identical higher-strike call (or ppt). Bermudan option An option that can only be exercised at specihed times during its life. Bid price The price at which a dealer or marketmaker buys a seculity. Bid-ask spread The difference between the bid pricexand the ask price. Binary option An option that has apayoffthatis a discrete amount forexample, $1 or one share; Also called a digital option. Binomial tree A representation of possible asset price movements over time, in which the asset price is modeled as moving up or down by a given amount each peliod. A version of the Black-scholes Black formula fonnula in which the underlying asset is aftltures price and the dividend yield is replaced with the risk-free rate. See equation (12.7)(p. 38 1). equation The partial diffefenBlackcholes tial equation, equation (2l (p. 682), relatlng ' price, delta, gamma, and theta, that must be satis(ied by derivatives. The Black-scholes-npl//s solves the Black-scholes equatit. Blackcholes The fonnula giving the formula price of a European call option as a function of the stock price, strike plice, time to expiration, interest rate, volatility, and dividend yield. See equation (12.1)(p. 377). Bootstrapping This term has two meanings. First, it refers to the procedure where coupon bonds are used to generate the set of zero-coupon bond prices. Second, it means the use of historical returns to create an empirical probability distribution for returns. Boundary condition The value of a derivative claim at a certain time, or at a particular price of the underlying asset. For example, a bundary condition for a zero-coupon bond is that the bond at maturity is worth its promised maturity value. Box spread An option position in which the stock is synthetically purchased (buy call. sell put) at one price and sold (sell call, buy put) at .11)
a different price. When constructed with Euros pean options, the box spread is equivalent to zero-coupon bond. Brownian motion A stochastic process in which the random valiable moves continuously and follows a random walk with nonnally distributed, independent increments. Named after the Scottish botanist Robert Brown, who in 1827 noticed that pollen grains suspended in water exhibited continual movement. Brownian lpotion is also called a Wielterprocess. The purchase of a call tr pt) toBull spread gether with the sale of an othenvise idtical higher-strike call (or put). Aposition crated bybuyifly Butterlly spread call, selling two calls at a highet strike price, and buying a fourth call at still higher strike jfie, wit.h an equal distance between strike plies. The buttey spread can also be creatd sing pts alones or by buying a straddle and ingtlfingit with the purchase of out-of-the-money calls and puts, Or in a variety of other ways. A spread position in Which Calendar spread the bought and sold options or have the same underlying asset but diferent timej to maturity. A contract giving the buyer the Call option right, but not the obligation, to buy the underlying asset at a prespecihed price. A period during which Call protection a callable bond cannot be called. Acontractual feature of a callable Call schedule bond, specifying the price at which the company can buy the bond back from bondholders at different points in time. A bond where the issuer has the Callable bond right to buy the bond back from bondholders by paying a prespecied amount. An options contract thatserves as insurance Cap against a high price. (See also lnterest rate cap.4 A contract that insures a borrower Caplet against a high interest rate on a single date. A collection of caplets is an interest rate cap. An option with a maximum Capped option payoff, where the option is automatically exercised if the underly.ing asset reaches the price at which the maximum payoff is attained.
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Another term for owning an asset, typically used to refer to commodities. (See also Carry market and Cost ofcarlyj. A situation where the folavard Carry market price is such tat the return on a cash-and-carry is the lisk-free rate. A procedure in which the Cash flow mapping cash iows of a given claim are assigned--or mapped-to asetof benchmarkclaims. This provides a way to approximate the claim in terms of the benchmark claims. A procedure where settlement Cash settlement entailsacash paymentfrom one party to the other, insteadof delivery of an asset. The simultanous spot purCash-and-carry and fonvard sale of an asset or commodity. chase The use of a cashCash-and-carry arbitrage and-carryto effect an arbiage. Cash-or-nothing call An opyion tlpt pays a if of cash tlze asjet price exceeds amount fixed the strike price at expiration or zero othenvise. An option thay payj a Cash-onnothing option fixed amount of cash if the option is in-the-money or zero otherwise. An option that pays a Cash-or-nothing put tixed amount of cash if the asset price is less than the slrike price at expiration or zero thenvise. See collateralized debt qbligation. CDO One of the most imporCentral lirnit theorem tant results in statistics, which states that the sum of independent and identically distributed random valiables has a limiting distribution that is normal. When a futures optract Cheapest to deliver gives the seller a choice of asset to deliver to the buyer, the asset that is most protable for the short to deliver. A fonnula used to Cholesky decomposidon construct a set of correlated random variables f rom a se t of uncorrelated random variales. The present value of a bond's fuClean price t'ure cash flows less accrued interest. organiation, typiA snancial Clearinghouse cally associated with one or more exchanges, that matches the buy and sell orders that take place during the day and keeps track of the obligations and payments required of the members of the clearinghouse.
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The purchase of a put and sale of a call at a higher strike price. The difference between the strike Collar Avidth prices of the two options in a collar. Collateralized debt obligation a tinancialstructure that consists of a pool of assets, financed by issuving financial claims that reapportion the return on the asset pool. An option where Collect-on-delivery option the premium is paid only when the option is exercised. Commodit'y spread Offsetting long and short positions in closely related commodities. (See also Crack spread and Crltsh spread-) An option that has an option Compound option underlying the asset. as Shaped like the cross section of an Concave UPS id e- down bowl A tenn in tax law describing Constructive sale the owner of an asset entering into an offsetting position that largely eliminates th risk of holding '
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A fonvard curve in which futures prices are rising with time to expiration. A bond that beContingent convertible bond comes convertible once a contingency (for examp le the share price is greatf than $100for 30 days)has occurred. The value of lejwing an opContinuation value tionunexercised. You make an exercise decision by comparing the continuation value to the value of immediate exercise. Continuously compounded interest rate Xway of quoting an interest rate such that if $1 is invested at a continuously compounded rate of ?-, the payoff in one year is er. A techniqu used in Control variate method Monte Carlo valuation ip which simulated asset prices are used to compute two derivatives prices: The price of the derivative that is being valued, and the plice of a related delivative for wlzich te value is known. The en'or in valuing the derivative with a known plice is used as a control for that with the unknown price. A nonmonetary return to Convenience yield ownership of an asset or commodity. A risk-free position consisting of Conversion purchased put, and a written call. an asset, a
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A bond which, at the option of the bondholder, can be surrendered for a specified number of shares of stock. Shaped like the cross section of abowl. Convex The second derivative of a bond's Convexity price with respect to a change in the interest rate, divided by the bond price. The greater of (i) 65 deCooling degree day grees Farenheit minus the avrage daily temperature, and (ii) zero. The interestcost of owning an asCost of carry dividend paylpents received ak less lease or set, result of ownership-, the net cas flow resulting a f'romborrowing to buy an asset. Covered call A long position in an asst together with a written call on the same asst. A zero-investment Covered interest grbitrage strategy with simultaneous orrowing in one currency, lending in another, and entering into a forward contract to guarantee the exhang rate ' svhen the loans mamre. A long position in an asset couCovered write pled with sale of a call option on the same ajst. The difference between i price Crack spread of crude oi1 futures and tliat of equivalent ' . . . . .. amounts of heating oil and gasoline. payoffdewhere the derivative A claim Credit pends upon the credit rating or default stat'us of a
Convertible bond
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Risk resulting from the possibility that a counterparty will be financially unable to m eet its contractual obligations. The difference between the Credit spread yields on a bond that can defauli and on an otherwise equivalent default-free bond. A bond that has payments Credit-linked note determined at least in pa.rt by credit events (e.g., default) at a different lirm. The use of a derivative on one Cross-hedging underlying asset to hedge the risk of another underlying asset. The difference between the price Crush spread of a quantity of soybeans and that of ihe soybean meal and oil that can be produced by those soybeans. A function Cumulative distribution function giving the probability that a value drawn from a distribution will be less than or equal to some specilied value. Credit risk
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Cumulative normal distribution f'unction The cumulative distribution function for the normal distribution; equation.
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in the Black-scholes
A swap in which the parties make payments based on the difference in debt payments in different currencies.
Currency swap
An investment in Currency-translated index an index denominated in a foreign currency, where the buyer bears both currency and asset risk. Debnture
curedonly
A bond for which payments are seby the general credit of the issuer.
The maximum ainount of debt Debt capacity that can be issued by a firm or secured by a speu cifc asset. between th Default premium The differn yield on a bond and that on an othenvise equivalent default-free bond.
A contract in which the swap buyer pays a regular pfemium; in exchage, if a default in a specitied bond occurs, the swap - se ller pays the buyer the loss due to the default. A defen'ed reDeferred down rebate option bate option for which the current stock price is above the rebate barrier.
Default swap
A clltim that pays $1 at Deferred rebate option expiration if the price of the underlying asset has reached a barrier prior to expiration. Deferred swap A swap with terms speciied today, but for which swap payments begin at a later date than for an ordinary swap.
A defen'ed rebate Deferred up t-ebate option option for which the current stock plice is below the rebate barrier. The act of the seller (e.g.?of a fprlvard contract) supplying the underlying asset io the buyer.
Delivery
Thq ehange in the price of aderivative due Delta to a change in the price of the underlying aasset. Aformula using Delta-gamma approximation the delta and gamma to approximate the change in the derivative price due tp a change in the price of the underlying asset.
approximation forDelta-gamma-theta A mula using the delta, gamma, and theta to approximate the change in the derivative plice due to a change in the plice of the underlying asset and the passage of time. Hedging a derivative position Delta-hedging using the underlying asset, with the amount of the underlying asset determined by the derivative's sensitivity deltnl) to the plice of the underlying
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Effective annual interest rate A way of quoting an interest rate such that the quoted rate is the annual percntage increase in an amount invested at this rate. lf $1 is invested at an effective annual rate of r, the payoff in one year is $ l + r. The percent change in an option price Elasticity for a l % hange in the price of the underlying asset.
fonvard contract the quantity to be bought or sold depends upon the perfonnance of a stock or stock index. An option that can only beexEuropean option ercised at expiration. An option permitting the Exchange option holder to obtain one asset by giving up another. Ctln-ency. Standard calls and puts are exchangk options in Differential equation An equation relating a which one of the two assets is cash. variable to its delivatives and one or iof inThe exchange of the strike price (or Exercise dependent variables. strike asset) for the underlying asset at the tenns continuous Generally, Diffusion process a specified in the option contract. ifajes stochastic process in which uncertai Under the tenns of an option Exercise price with time. Also used to describe the Brwnian the amount that can be exchanged for contract, (random) part of an lt process. the underlying assey. Another name for a binal'y opDigital option Exercise style The circumstances under which option holder has the right to exercise an opan Dirt'y price The present value of a bond's f11tion European'' and 'lAmericgn'' are exercise ture cash flows (this implicitly includes accrued styles. interesi). A derivatives contract in which Exotic option The distance between the Distance to default an ordinary derivative has been altered to change current 11:4.11 asset value and the leyel at which the characterlstics of the derivative in a meaningdeviations. default occurs, measured in stnard 11way. Also called a llolsttntdard option. Risk that is, in the limit, Diversiliable risk Expectations hypothesis A tenn with multiple eliminated by combining a large number of asmeanings, one of which is that the expected 1sets in a portfolio. ture interest rate equals the implied forward rate. Down-and-in A knock-in option for which the The expected loss condiExpected Tail Loss banier is less than the currnt price of the tlndertional upon the VaR loss being exceeded. Anlying asset. other name for Tail VaR. ' Down-and-out A knock-out option for which The date beyond which an unexerExpiration the banier is less than the current price of the cised option is worthless. underlying asset. Drift The expected change per unit time in an Anothername for the theoretical forFair value ward price: Spot price plus interestless the f'uture asset price. value of dividends. Duration Generally, the weighted average life Creating new financial of the bond, which also provides a measure of the Financial ensneering instruments by combining other derivatives, or bond's sensitivity to interest rate changes. Two more generally, by using derivatives pricing techcommon duration measures are ??7(?#(#e# dltraniques. //6,/) and Macalday dltration. asset. Derivative A hnancial instrument that has a value detennined by the price of somethin else. Diff swap A swap in which paymentj are based on the difference in Eoatng interest ratej on a given notional amount denominated in a sinjle
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Floor An option position that guarantees a lninimum plice. Forward contract An agreement tat sets toprice and quantity-at day the tenns-including which you buy or sell an asst or commodity at a specific time in the ftlttlre. The set of forward or futures Forward curve prices with different expiration dates on a given date for a given asset. The annnalized percentage Forward premium difference b.etween the forward price and the spot price. Forward rate agreement A forward contract for an interest rate. Another name for te fonvard Forward strip Ctl
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A position that is paid for in full at the outset. A prepaid forward, for exampl, is a funded position in a stock. See also ulmded. An aprement tat is similar Futures contract conact except that the buyer and to a seller post margin and the contract is markedFutures are typically to-market periodically. exchange-traded. Converting arl investment in Future: overlay assetAinto the economic equivalent of an investment in asset B by entering into a short fures position on assetA and a long futures position on
Fuhded
'forward
asset B. The change in delta when re price o the underlying asset changes by one unit. Gap option An option where the option ownr has the light to exercise the option at strile K3 if the stock price exceeds (or, depending on the option, is less than) the price Kz. For an ordinary option, KL Kz. continuous A Geometric Brownian motion stochastic process, a.(1), in wllich the increments adt + o'dz, where are given as dxtjjxt) 6IZ is the increment to a Brownian process. A result tlat pennits a Girsanov's theorem change in the drift of an It process accompanied by a change in the probability distribution of the Brownian motion driving the process. A term generally refening to delta, Greeks gamma, vega, theta, and rho, all of which measure te change in the price of a delivative when
Gamma
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there is a change in an input to te pricing formula. The collateral, over and above the market value of the security required by the lender when a security is borrowed. with A measure of the efficincy Heat rate which heat can be used to produce electricity. Specihcally, it is the number pf Britis Thermal Units required to produce 1kilowatt/hourof electzicity. The greater of (i) the averHeatlng degree day minus daily 65 degree Farnheit, temperattlre age and (ii) zero. In a hedgipg trasaction. the ratio Hedge ratio of the quantity of the fomard or futures position to the quantity of the underlying asset. An action-such Hedging as ntedng into a reduces ihe risk of derivatives position-that loss. An option pricing mdel in Heston model whichthe instantaneous variapce of t stock ret11:11f() llows a man-reverting sqare root pro-
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Interest rate cap A contract that periodically pays the difference between the market interest rate and a guaranteed rate, if the difference is positive. ln-the-money An option tat would have value if exercised. For an in-the-money call, the stock price exceeds the stlike price. For an in-temoney put, 4he stock price is less than the strike plice. Investment trigger price The price of an investment project (or the price of the good to be produced) at which it is optimal to invest in the project. It process A continuous stochastic process that can be written in the fonn dxtj e(.Y(l), 11 dt +tr(X(?). 11 dzt), where dzt, is the increment to a Browflian process. If x follows an It process, It's It's Lemma Lemma describes the process followed by fx). For example, if is a stock price and fx) an option price, It's Lemma characterizes the behavior of the option price in terms of the process for the stock. '
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Data that is charateiizd -lleteroskedastici that variances not equal, either okr tilpe are by different obselwaions at a point in time. across or
The standard deviation of continuously compounded asset, rettlrfl on the histolical prices. using tlred meas Data that is charactelized by Homoskedasticity variancesthat are equal over time of afoss differentobservations at a point in time. The failure of an effect to reverse itHysteresis self as the underlying cause is reversed.
Historical volatility
Theforwardinterestrate lmplied forward rate between time tt and lime h (h < ?a) that mkes an investor indiferent between, on the one hand, buying a bond maturing at la, and, on the other hand, buying a bond mattlring at 11 and reinvesting the proceeds at this fotavardinterest rate. Tmplied repo rate The rate of ret'urfl on a cashand-cany The volatility for which the lmplied volatilit'y theoretical option price (typicallycomputed using the Black-scholes fprmula) equals the observed market price of the option.
Jensen's inequalit'y and
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A process fr Jump-difhuion model rl asset price in which the asset most of the time follows an It process but can alsojump discretel with occurrence of the jump controlled by a Poisson PIDCeSS.
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Another name for vega. option An option in which there can only be a linal payoff if, during a specised period of dme, the price of the underlying asset has reached a specihed level. An option in which there can Knock-out option only be a linal payoff if, during a specilied period of time, the price of the underlying asset has not reached a specihed level. A Kolmogorov backward equation partial differential equation, (see equation (21.32) (p. 691), that is related to the Black-scholes equation and that is satissed by probability distributions for the underlying asset.
Kappa
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Kurtosis Ameasure of thepeakedness of aprobability distribution. For a random variable with mean /4 and standard deviation c, kurtosis is the fourth central moment divided by the squared Jz)4/(r4. For a nonnal random variance. Ex variable, kurtosis is 3. -'r
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Ladder option If the barrier L > K is reached over the life of the option, a ladderoption at expiration pays maxto, L K, Sv K4. Ifthe banier is not reached, the option pays maxto, Sp K). Another name for vega. E Lambda Lattice A binomial tree in which an up move followed by a down move leads to the same price as a down move followed by an up move. Also c Llle (1 a ?-c c5??2/? ?7illjl rrcc. The assertion that two portfoLaw of one price lios generating exactly the same return must have the same price. Lease rate The annualized payment required to borrow an asset, or equivalently, the annualized payment received in exchange for lending an asset. Arise in the stock price volatilLeverage elect ity when the stock plice declines. London lnterbank Bid Rate. See LILlBm BOR. LIBOR London Interbank Offer Rate. A meaborrowing rate for large ipternational of the sure banks. The British Banker's Association determines LIOR daily for different currencies by surveyingat least 8 banks, asking at what rate they could borrow, dropping the top and bottom quartiles of te responses, and computing an arithmetic average of the remaining quotes. Since LIBOR is an average, there may be no actual transactions at that rate. Confusingly, L1BOR is also sometimes refen'ed to as jt lending rate. This is because a bank serving as a marketmaker in the interbank market will offer to lend money at ahigh interestrate ILIBORIandbon'ow money at a 1owinterest rate (LlB&). (The difference between LIBOR and LIBID is the bid-ask spread in the interbank market.) A bank needing to borrow will thus pay LIBOR, and a bank with excess funds will receive LIBD. A derivative contract that, at maLog contract turity, pays the natural log of an asset price. -
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The allocation of assets between stocks and bonds in an attempt to invest in whichever asset is going to have a higher return. The procedure of revaluing a Mark-to-market portfolio or position to renect current market Prices. A stochastic process for which Martingale Xtj, where *(r) is inEkxt + formation available at time t. in a The percent ehage Modified duration bond's plice for a unit change in the yield. Modified duration is also Macaulay duration divided by one plus the bond's yield per payrent/liod. A procedure fpr pricMonte Carlo valuation discounting expected claims derivative by ing the expected payoff is cmputed where payoffs, usingsimulated prices for the undrlying asset.
A probability distribution in which the natural logarithm of the random valiable is normally distlibuted. Aposition is long with respect to aplice if Long the position profits from an increase in that price. An owner of a stock prohts from an increase in the stock price and, hence, is long the stock. An owner of an option prots from an increase in volatility and. hence, is long volatility. The party to a fonvard contact Long forward who has an obligation to buy the underlying asset. Lookback call See Lookback option. An option that, at matulity Lookback option v) or minipays off based on the maximum of the option. life stock the price over mum (..wl and a Sv A Iookback call has the ppff - Iookbackpu? has the payoff Sv Sv. See Lookback option. Lookback put
Market-timing
The percent change in a bod's price for a given percent change in one plus the bond's yield. This calculation can be interpreted as the weighted avefage life of the bond, with the weights being the percentage of th bond's value due to each payment. The level of margin at Maintenance margin which the contract holder is required to add cash or securities to the margin account. A bond that Mandatorily convertible bond makes payments in shares instead of cash, with the number of shares paid to the bondholder typically dependent upon the share price. A deposit required for both buyers and Margin sellers of a futures conact, which indemnies the counterpart'y against the failure of the buyer or seller to meet the obligations of the contract. The requirement that the owner of Margin call a margined position add funds to the margin account. This can result from a loss on the position or an increase in the margin requirement. Owning a latge percentage of Market corner the available supply of an asset or commodity that is required for delivery under the tenns of a derivatives contract. Atraderin an asset, commodity, Market-maker orderivative who simultaneously offers to buy at one price (thebid price) or to sell at a higher price (the offer price), thereby a market.''
Selling options without an 0ffsetting position in the underlying asset. Another term oprht? Net payoff Non-traded asset ' A cash flow stream that cannot be purchased directly in nanial market. are nonMany corforate investment projt tradedbecause they can only be acquired by buying the entire company. Risk that remains after a Nondiversiliable risk large number of assets are comblned in a portfolio. A binomial tree desribNonrecombining tree ing asset price moves in which an up move followed by a down move yields a diferent price than a down move followed by an up move. See Entic option. Nonstandard option A bell-shaped, symmetNormal distribution ric, continuous probability distribution that assigns positive probability to al1values from t'bell curve-'' (See to +x. Sometimes called the I !l?, it l/?e (2 ?-d??l. ) t1s () :ell 1?-47 The dollar amount used as a Notional amount scale factor in calculating payments for a folavard
Lognormal distribution
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Naked writing
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The same as the askprice. Offer price A forward contract in Off-market forward which the forward price is set so that the value of the contract is not zero. A government bond that is not one of the recently issued bonds. The most recently auctioned govOn-the-run ernment bonds at the government's specifc auction maturiries. Open interest The quantity of a derivatives contract that is outstanding at a point in time. (One long and one short position count as one unit outstanding.) A system of trading in which buyOpen outcry ers and sellers in one physical location convey offers to buy and sell by gestuling and shouting. The percent change in an opOption elasticitjr tion price for a l % change in the price of the underlying asset. Selling Option overwriting a call option against a long position in the underlying asset. The party with a short position Option writer in the option. The 11 draws of a random variOrder statistics able sorted in ascending order. An option that would be exOut-of-the-money ercised at a loss. An out-of-the-money call has the stock price less than the strike price. An outof-the-money put has the stock price greater than the strike price. An option in which Outperformance option the payoff is determined by the extent to which one asset pgiceis greater than another asset price. A term used generOver-the-counter market ally to refer to transactions (e.g., purchases and sales of securities or derivatives contracts) that occur without the involvement of a regulated exchange.
Off-the-run
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contract. futures contract, or swap. The notional amount for an Notional principal interest rate swap. The units. in which a payoff is denominated.
Numeraire
Par bond A bond for which the price at issue equals the maturity value. Par coupon The coupon rate on a par bond. Partial expectation The sum (or integral) of a set of outcomes times the probability of those outcomes. A derivative where the final Path-dependent payof depends upon the path taken by the stock price, instead of just the final stock price.
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Payer swaption A swaption giving the holder the right to be the hxed-rate payer in a swap. Paylater strategy Generally used to refer to option strategies in which the position buyer makes no payments unless the option moves more into the money. The value of a position at a point in time. Payoff The .term often implicitly refers to a payoff at expiration or maturity. A graph in which the value of Payoff diagram a derivative or other claim at a point in time is plotted against the price of the underlying sjet. A characteristic of a derivaPayout protected tive where a change in the dividend payout on the underlying asset does not change the value of the derivative. Perpetual option An option that never expires. A probability distribution Poisson distribution that counts the number of events occurring in an interval of time, assuming that the occurrence of events is independent. An 11 x 11 matrix with elements Positive-dennite 0, i azj is positive-dehnite if, for every i 11 E''f=I j.=j ?f j t7; g > 0. A covari.l ance matrix is positive-detinite. ' Power option An option where the payoff is based on the price of an asset raised to a power. A contract calling Prepaid forward contract for payment today and delivery of the asset or commodity at a time in the future. The price the buyer Prepaid forward price pays today for a prepaid fonvard contract. Prepaid swap A contract calling for paymnt today and delivel'y of the asset or commodity at multiple specied times in the future. Price Iimit In futures markets, the size of a 1tures price move such that trading is halted temporarily. Price participation The extent to which an equity-linked note benehts from an increase in the price of the stock or index to which it is linked. Price value of a basis point The change in a change in the bond price due to a l yield of the bond. Frequently abbreviated PVBP. The payoff less the future value of the Profit original cost to acquire the position. Prolit diagram A graph plotting the proht on a position against a range of prices for the underlying asset. :#
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Proprietary trading Taking positions in an asset or derivative to express a view-for example, that a stock price will rise or tat implied volatility will fall. The change in the price 9f a derivative due Psi to a change in the dividend yield. Purchased call A long position in a call. A long position in a put. Purchased put Acontractgiving the buyerthe right, Put optitm but not the obligation, tt) sell the underlying asset at a prespecified price. A relationship stating that the Put-call paity difference between the premiums of a call and a put wit te same strike price arld time to expiration equals the difference between the present value of the folavard price and .the present value of the strike price. Puttable bond A bond that the investor can sell back to the issuer at a predeternzinedprice schedule. E
The sum of squared increments to a Brownian motion. quantle if Quantile A data point is the of the data lies below that poipt. Quanto A derivatives conact with a payoff in which foreign-currency-denominated quantities are treated as if they were denominated in the domestic currency. Quasi-arbitrage The replacement of one asset or position with another that has equivalent risk and a higher expected rate of returrl.
Quadraticvariation
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.An option that has a payoff based on the maximum or minimum of two (or more) lisky assets and cash. For example, the payoff to a rainbow call is maxts'r, Qv,#), where Sv and Qvare lisky asset prices. A stochastic process, Xt), in Random walk which increments, 6(r), are independent and Xt identically distributed: Xt) /l) + 6(1). Achange in te credit rating Ratings transition of a bond from one value to another. Buying ?n of an option at one Ratio spread strike and selling ?? of an otherwise identical option at a different strike. Real options The applications of delivatives theory to the operation and valuation of real (physical) investment projects.
Rainbow option
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Realized quadratic variation
The sum squared condnuously compounded asset returns, typically measured at a high frequency. Another term for realized Realized volatilit.y quadrac variation. A claim that pays $1 at the time Rebate option the price of the underlying asset reaches a barrier. Receiver swaption A swaption giving the holder tlle light to receive the lixed rate in a swap. A binomial Recombining tree ee desclibing in wlch price asset moves an up move followed by a down move yields tlze same price as a down move followed by an up move. Also called a
Iatdce.
The percentage of par value reRecovery rate bond holder in a bgnkruptcy. by ceived a A market plice or rate used to Reference price detecmine the payoff on a derivatives contract. Mother name for a repttrchase agreeRepo ??ld?11.
The annualized percentage difference between the original sale price and linal repurchase price in a repurchase apeement. Repricing The replacement of an out-f-thecompensation option with an at-themoney money compensation option. The sale of a security Repurchase agreement coupled with an agreement to buy it back at a later date. The simultaneous Reverse cash-and-carry short-sale and forward purchase of an asset or commodity. A short position in an asReverse conversion set coupled with a purchased call and writtetl put, both with the same strike price and time to expiration. The position is equivalent to a short bond. Another name for reverse ?'e#!f?= Reverse repo cltase a,grccplcnf. The purchase Reverse repurchase agreement of a security coupled wit.h an agreement to sell it at a later date. The opposite of a repurchase Repo rate
agreement.
The change in value of a derivative due to change in the interest rate. a A term describing an investor who Risk averse with an expected prefers a' to taking a lisky.bet value equal to x. Rbo
ltisk management The active use of derivatives and other techniques to alterrisk and protect profitability. Atenn describing an investorwho Rislt neutral is indifferent between receiving and taling a lisky bet with an expected value equal to x. The difference between the exRisk prernium pected returfl on an asset and the risk-free rate; the expected return differential that compensates investors for risk. Risk-neutral measure The probability distribution for an asset transfonned so that the expected returfl on the asset is the lisk-free rate. In the (Risk-neutral probability binontial model, the probability of an up move in the asset plice such that the expected return on the asset is the risk-free rate. -
Self-financing portfolio
A portfolio that retains specised characteristics (e.g., it is zeroinvestment and lijk-free) witout the need for additional investments in the portfoli. The time in a tansaction at which Settlement a1l obligations of both the buyer and the seller are fulhlled. The position in shares that Share-equivalent has equivalent dollar risk to a derivative. (See also Delta-j For an asset, the ratio of the risk Sharpe ratio premium to the ret'urn standard deviation. A position is short with respect to a price Short if theposition prots from adecrease in thatprice. A short-sellen of a stock profits from a decrease in te stock price and, hence, is short the stock. A seller of an option profits from a decrease in volatility and, hence, is short volatility. A call that has been sold. Short call The party to a fonvard contract Short forward who has an obligation to sell the underlying asset. A put that has been sold. Short put The rate of returfl paid on collatShort rebate eral when shares are bonowed. The sholt-sale of a Short-against-the-box stock tat the sholl-seller owns. The result of a short-aginst-the-box is that the short-seller has botlz a long and short position and, hence, bears no risk from the stock yet receives the value of te shares from the short sale.
917
Short-sale
A transaction in which an investor borrows a security, sells it, and then returns it at a later date to the lender. If the security makes payments, the short-seller must make the same payments io the lender. A shout call optipn expiring at Shout option time F has the payoff maxto, S #), K, where is the time and S is the price at which the option holder thereby guaranteeing an expiration payoff at least as great as S K. Ameasure of the symmetry of aprobSkewness ability distribution. For a random valiable with mean Jz and standard deviation t)-, skewness is the third central moment ivided by tlie cubed /2)3/,3 stan d ar d deviation E (or a normal variable, skewness is 0. (See also Volatility .ke1'.) The difference between te price Spark spread of electricity and that of the quantity of namral gas required to produce the eleclicity. Spot curve The set of zero-coupon bond prices with different maturities, usually inferred from government bond prices. The current market price of n assett Spot price Spread Simultaneously buying and selling closely related derivatives. Aspread in options is a position in which some options are bought and some are sold, and all options in the position are calls or all are puts. (See also Calendar Jprcf7t'/ and Commodityspread.j An option with a payoff where a Spread option spread (the difference between prices) takes the place the of the underlying asset. A probability distribution Stable distribution for which sums of random variables have the same distribution as the original random variable. The normal distribution is stable because sums of nonnally distributed random variables ' are normally distributed. A hedging strategy in which an Stack and roll existing stack hedge with mamdng f'umres contracts is replaced by a new stack hedge with longer dated futures contracts. Hedging a stream of obligations Stack hedge by entering futures contracts with a single maturity, with the number of contracts selected so that changes in the present Iztz/l/c of the futtlre obligations are offset by changes in the value of this of futures contracts. -
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The net present value of a project at Static NPV point time. ignoring the possibility of postin a
poning adoption of the project. The use of options to hedge options. with the goal pf creating a hedging portfolio that has a delta that nlgurally moves in tandem with the delta of the option being hedged. An equation Stochastic differential equation characterizing the change in a variable in which one or more of the diferential terms are increments to awstochastic PFOCCSS. Stochastic process A mathematical model for a randomprocess as a function of time A process in which the inStochastic volatility stantaneous volatility can val'y randomly, either as a function of the stock price or other variables. An average of the prices of a group Stck index stocks. A stock index can be a simple averof of stock prices, in whch case it is eqttally age is/tzq/l/tW, or it can be a weighted average, with the Nkeights proportional to market capitalization, in wh tch case it is valtte-b,eighte. The purchase of a call and a put with Straddle the same strike price and time to xpiration. Tax reculations controlling the Straddle rules circumstances in which a loss on a claim can be realized when a taxpayercontinues to own related securities or derivatives. The purchase of a put and a higherStrangle strike call with the same time to expiration. A technique used in Monte Stratified sampling Carlo valuation in which random numbers are drawn from each. percentile (or other regular ipterval) of the disibution. Another term for exerdse price. Strike price Hedging a stream of obligations by Strip hedge offsetting each individual obligation with a futures contract matching the maturity and quantity of the obligation. Acronym for Separate F?'t?t'#n.ofRegSTRIPS klcrct llltel-est twl# Priltcipal of Secttl-ities. STRIPS are the interest and plincipal payments from Treasury bonds and notes traded as individual securities. A bond that makes payments Structured note that, at least in part, are contingent on some variable such as a stock price, interest rates, or exchange rates.
Static option replication
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A claim to a portfolio that pays the holder a portion of the pordblio only if a particular event occurs. (An example of an event would be the asset losing between 25% and 26% of its value.) Acontractcallingforthe exchange of paySwap ments over time. Often one payment is sxedin advance and the other is floating, based upon the realization of a price or interest rate. The difference between the sxed Swap spread rate on an interest rate swap and the yield o a Treasury bond with the same mturity The lifetime of a swapt Swap tenor
Supershare
Another name for ienor. An option to enter into a swap.
Swap term Swaption
-lt/tp
Tail VaR The expected loss condltional upon being exceeded. VaR loss the Tailing A reduction ip the quantiiy of an sset held in order to offset future icoine received by th e asse t Time to maturity or expiration of a cppTenor tract,frquently used when refening to swaps. A repurchase agreement lasting for Term repo of time longer than one day. specified period a Theta The change in the value of a derivative due solely to the passage of time. Another term for theta. Time decay A swap in which one party Total remrn swap pays the total return (dividendsplus capital gains) on a reference asset, and the other party pays a floating rate such as LDOR. Traded present value The value an investment project would have once the investment was made; also called tbvill .lclf/-'(J'. See b-aded presellt l?t-//lfc. T'win security .
The asset whose price deterUnderlying asset mines the proitability of a derivative. For example, the underlying asset for a purchased call is the asset that the call owner can buy by paying the strike price. A position that is not paid for at the Unfunded outset, and for which cash inflows and outflows can later occur. A fonvard contract, for example, is an unfunded posiyion in a stock. See also hmded.
Up-and-in A knock-in option for which the barrier exceeds the current plice of the underlying asset. Up-and-out A knock-out option for which the banier exceeds the current price of the underlying asset.
Value at risk
The level of loss that will be exceeded agiven percentage of the time overagiven horizon. Vanilla A standard option or other derivative. Forexampl, ordinal'y puts and calls are options. A fonvard contract that settles Variance swap based on cumulative squared asset returns. Vega The change in the price of a derivative due to a change in volatility Also sometimes called kappa or Iambda. The sale of an option at one Vertical spread strike and purchase of an option of the same type (call or put) at a different strike, both having the same underlying asset and time to expiration. The standard deviation of the continVolatility uously compounded return on an asset. Volatility clustering The tendency of high volatility days to be followed by high volatility days. Generally, implied volatility as Volatility skew a function of the strike price. Volatility skew refers to a difference in premiums as reflected in differences in implied volatility. Skew is sometimes used more precisely to refer to a difference in implied volatilities between in-the-money and out-of-the-money options. Volatility smile A volatility skew in which both in-the-money and out-of-the-money options have a highervolatility than at-the-money options (i.e., when you plot implied volatility against the strike price, the curve looks like a smile). 'vanilla''
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Volatility surface Athree-dimensional graph in whichvolatility is plotted against strike price and timeto mattlrity.
Volatilit'y swap
A fonvard contract that settles deviation
basedon some measure o f the standard of returns on an asset. Warrant
An option issued by a 5rm with its
ownstock as the underlying asset. This term also refersmore generally to an option issued in tixed supply. Weather derivative
A devative contract with based weather-related measurepayment on a a ment,such as heating or cooling degree days. See Brobmtian ???t?lt?n. Wiener process A call that has been sold; a short Written call
call.
Written put
put.
Written straddle
A put that has been sold; a short
The simultaneous
call and sale of a put, and time to expiration.
sale of a with the same strike price
Yield curve
The set of yields to maturity for bonds with different times to maturity. The single discount factor Yield to maturity for which the present value of a bond's payments is equal to the observed bond price.
Zero-cost collar
The purchase of a put and sale of a call where the strikes are chosen so that the premiums of the tavo options are the same. A bond that makes only a Zero-coupon bond single payment, at maturity. Zero-coupon yield curve The set of yields to matulity for zero-coupon bonds with different times to matulity.
Acerbi, C., 2002. Measures of Risk: A Coherent Repreentation of Subjective Risk Aversion,'' Jollrnal &,./Bankilq tplt Finance, 26(7), 1505-1518. Acerbi, C. and Tasche, D., 2002, the Coherence of Expected Shortfallz'' Joltnlal t)/#twlJ?2.j' all Finance, 26(7), 1487-1503. Acharya. V. V., John, K., and Sundaram, R. K., 2000, 4EOnthe Optimality of Resetting Executive Economics, 57(1 ), 65-1 01 Stock Optionsz'' Jottrnal t:?./-F??t7??cD/ Alexander, C., 2O01, Malket Vodels,Wiley, Chichester, England. Allayannis, G., Brown, G., and Klapper, L. F., 2003, Structure and Financial Risk: Evidence from Foreign Debt Use in East Asiav'' Jottnltll ofFinance, 58, 2667-2709. Allayannis, G., Lel, U., and Miller, b., 2004, Governance and the Hedging Premium Around the World,'' Working Paper, Darden School, University of Virginia. Allayanniss G. and Weston, J., 2001, RThe Use of Foreign Currency Derivatives and Firm Market Values'' Revieb' t/-/Fntpzt:i't'# Stltdies, 14( l ), 243-276. Andersen, T., Benzoni, L., and Lund, J., 2002, Empirical Investigation of Continuous-Time Equity Return Models.'' Journal ofFinalce, 57(3), 1239-1284, forthcoming. . Andersen, T. G., Bollerslev, T., Diebold, F. X., and Labysd, P., 2003, ltModeling and Forecasting Realized Volatility,'' Econonletricq, 7 1(2), 579..-625. Andersen, T. Q, Bollrslev, T., Frederiksen, P. H., and Nielsen, M.. 2005, in Financial Markets,'' Unpublished, Kellogg School, Northwestern University. Anderson, R. and Sundaresan, S., 2000, ::A Comparative Study of Structgral Models of Corporate Bond Yields'. An Exploratory lnvestigation,'' Joul-nal ofBanking CInJFillallce, 24(1-2), 255269. Arnason, S. T. and Jagannathan, R., 1994, t'Evaluating Executive Stock Options Using the Binomiaf Option Pricing Model,'' Working Paper, Carlson School of Management, University of ttspectral
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k. INDEX
Ask price, 1l Asset
as investment asset, 688-690 leaserate of, 14-15
Sharpe ratio and, 659-660 underlying,21 valuationof, 558 volatilityof, 5 10 Asset allocation general,151 as index futures use, 150-151 Asset-or-nothing call, 706 Asset-or-nothing opdon, 686 banier options, 7 15-7 16 A sset priCe '1.46..447
averageas, distributionof.. 608-612
Asset swap, 255 Asymmetry
in insurance purchase, 9-97 of zer-cost collar, 77
As-you-like-it option, 465 At-the-money call, 284 At-the-money geometric average price? 447 At-the-mpney optton, 4, 530 At-the-mofley put, 76, 284 At-the-money written staddle, 822-823 Autoregressive, 750 Average plice options, 466 Average strike options, 466-467
Back-to-back transaction, 250 Backwardation, 170 Backward equation, Kolmogorov, 69 1-692
Bakshi, G.&772 Balance sheet CDO, 853 Bankers Trust, Procter & Gamble swap with, 263,
264
Bankruptcy, 841n. See also Default bondratings gnd probpbillty of, 847-850 hedgingand, 103 reducedform models, 851 Banks. capital to cover losses, 495n Barclays, Russian-doll CDO from, 854 Barrier COD option, 737 Barrier option, 7 17-7 l 8 all-or-nothing,710-717 .149-453,
INDEX
asset-or-nothing, 715-716 cash-or-nothing, 710-715 desning barrier for, 450 types of, 450-451 Barrier present values, 403 Bartram. S. M., 107 Bartter, B. J., 313, 793n Basel Committee on Banking Supervision, value at risk and, 8 15 Basis risk, 116, 196-198 in hedge with reji j ua1 risk, 15311 in T-bond and T-note futures, 233 Basket options, 735-736 Bates, D. S., 743n, 77 1n, 772 Bltxter, M., 662n, 674 Bear spread, 72, 300, 301 Below-investment grade bonds, 847 Benchmark, 459 Bernstein, P. L., 592n Best, A. M., 847n Beta, index futures and, 152 Bettis, J. C., 491 Bias convexity, 219-221 - futures price, by risk premium, 140-141 Bid, origins of term, 12 Bid-ask bounce, l2, 756 Bid-ask spread, 1l-l 2 Bid price, 11 Bills. See Treasury bills Binomial fonnula, graphic interpretation of, 3 l9, 320 Binomial interest rate model, 793-798 Binomial model, 355, 528 Black-scholes formula and, 375 for dollar-denominated investor, 72.4-727 Greeks in, lognormality and, 355-358 origins of, 598 path dependence and, zltzln for pricing reload options, 532-534 Binomial option pricing, 313. See also BiomialE .
.441-442
E
tree American options and, 329, 330 early exercise of option, 343-346 one-period binomial tree and, 31 3-323 options on other assets, 330. put options and, 328-329 two or more binomial periods and, 323-328 .
-336
Binomial pricing formula, 315-318
Binomial tree, 314, 553, 555-556 altelmative, 358-359 for American call option, 364 Black-Derman-rfoy model and. 798-808 constfucting, 321-322 Cox-Ross-Rubinstein approach, 359 for dollar and Niltkei index, 720 generalizationto many periods, 326-328 lognormalityand, 351-380 tree, 324 nonrecombining optionGreeks and, 441-442 path through, 356, 357 prepaidfonvard and, 363-365 for pricing of Amelican call option, 331, 335,
350 for pricing of American put option,
330, 333
for pricing of European call option, 323, 324, 327, 349 for pricing of European put optiop, 328 for project value, 555..2556 recombining tree, 324 stock price paths on, 356 with two or more binomial periods, 323-328 valuation with risk-neukal probabilities, 618, 6 19 VBA for, 900-901 Binomial valuation, of callable noncovertible and convertible bonds, 518
Black, Fischer, 375, 376, 503, 798. See also specic formulas and models Black-cox modl, 846 Black-Derman-rlby model, 779, 798-808 constructing tree, 804-805 Black formula, 381-382, 560n bond options, caps, and, 790-792 Black-scholes analysis Delta-hedging in practice and, 432 multivariate, 700-701 option pricing and, 70, 29011,429-436,
679-698
Black-scholes equation, 679-698 equilibrium remrns apd, 686-688 interest rate derivatives, for, 783 jumps and, 696 market-maker hedging claim and, 72311 for pricing options, 679, 681-690 risk-neutral pricing and, 690-691
%
937
Black-scholes formula, 375G05. See also specisc types of options applying to other assets, 379-382 assumptions about, 379, 649-650 binomial model and, 313-314 for bonds, 779 call provision and, 5l7 for computing value of debt, 509 computing with Visual Basic for Applications, 894-895 derivation of, 585 heuristic derivation of, 604-$05 history of, 376 implied volatility and, 400-402 inputs to, 377-378 lognormal distribution of stock price and, 453 market-makers and, 4l3 Monte Carlo valuation and, 625 option Greeks, 382-395 for option premiums, 34n for payoffs to equity and debt, 505 pricing formula for exchange option and, 460 prost diagrams before maturity, 395-399 value at risk and, 836 value of options and, 524n for valuing bond convertible at maturity, 5 1zl-5 16 volatility and, 763-773 Black-scholes Greeks, 441-442 Black-scholj-Merton methodology, 679n Black-scholes partial differential equation (PDE). See Black-scholes equation Bodie, Z., l72n, 299, 528, 814n Bodnlm G. M., 107 Bohn, J., 8z.14n Bollerslev, T., 748, 751 Bondts). See also specihc types basics of, 205-214 ' Black-scholes equation for, 779 boundary condition for, 680-681 box spread alternative tos 74 callable, 516-520 catastrophe, 6 cheapest to deliver, 231 commodity-linked, 478-48 l contingent convertible, 520 convertible. 84, 513-516 coupon, 2 10-2 1l, 475 coupons and yields, 242-244
;
lxoEx
938
% INDEX
currency-linked,481 as debenre, 504n delta-gammaapproximations for, 784-785 duration-hedgingwith, 779 withembedded options, 482-486 equilibriumequation for, 781-784 equity-linked,476-478 featureso, 520-522 high- nd low-coupon. 233 interestrates and. 780 Marshall & llsley, 83-85 off-the-run,206
on-the-nm,206 optionson, 286-287, 335-336 par value of, 496 ir payoff diagram, 61 price conventions for, 241-245 prices,yields, and conversion factors for, 232 recoveryrates for, 850...-85J Treastlrpbond f'utures, 230-233 valueat risk for, 826-830
valuingbond convertible at matutity, velifyingvolatility for, 803-804 verifyingyields for, 802-803 volatilityfor. 828-830
514-516
J'1?# Street Jt/lf?-//tWgovernment bond listing,
207
zero-coupon, 28-29, 6l, 206-208, 474...475, 794,-796 Bond options caps, Black model, and, 790-792 delta-hedging for, ?98 Bond pricing, market-making and, 779-785 Bond ratings bankruptcy probability and, 847-850 default and, 847-852 recovery rates for, 850-851 Bond valuation, based on stock price, 520 Bond volatility, 794-796, 812 Bootstrapping bonds, 211 probability distributions, 831-832 Borrowers, rates by class of, 158n Borrowing, 4l8 arbitrage and, 139 swaps and separation of, 263 Boundary condition, 680-681 of European call opon, 680-681
boundary conditions, 686 Box spread, 72-73, 74 Boyle, P. P., 431, 630n, 63211,633. 724n
for terminal
Brealey, R., 105n Brennan, M. J., 516n, 565n, 574n Broadie, Mark, 622n, 633, 635, 636, 772 Brown, G.W., 107 Brosvnian motion, 649-674. 650 azitbmetic,653-654 continuouspaths in, 672-674 geometric,655-659 Girsarlov's theorem and. 662-.663 andIt processes, 653-659 propees of, 652-653 quadriaticvaridon of, 652-653 risk-neutralprocess and, 660-663 Bull spread, 71-72, 300 Greks for, 392 prost diagram for, 87 Bulow, J., 528, 529 Bulow-shoven expensing proposal, 528, 529, 532 Burghardt, G., 25811 Butterlly spread, 81-82, 300 asymmetric,82-83, 302, 303 proht diagram for, 87 Buyer long as. 23 Iisk management
byN
98-100
Buying of index and put, 69 vs. short-selling,
14
Calendar spread, 397-399 Cilable bonds, 51*520 convertible,519-520 nonconvertible,517-519 Call option, 31-38, 547, 558, 684-685. See also Purchased call option 706 asset-or-nothing, at-the-money,284 binomial ee for pricing European, 323
Black-scholes formula for, 375-378 as cap, 62-63 704-705 cash-or-nothing, in CD strucmre, 485 .
collarand, 73
down-and-out cash call, 7 12-7 13 elasticity for, 393 equitplinked foreign exchange call, 731-732 European call option, 293 exercising prior to dividend, 295-296 fixed exchange rate foreign equity call, 730-731 formula for, 460 fonnulas for banier, 717 gamma for, 384, 385 as high-beta security, 620-621
as insurance, 47-48, 99-100 insuting by selling, 95-96 on nondividend-paying stock, 29+-295 payoff for, 33-37, 320 perpetual, 404 premium of, 60 price apd Greek information for, 415 pricing of, 325-326 protit diagram apd, 60-61 profit diagram of insured house and, 62 psi for, 388, 392 put-call parit'y and, 68-70 as puts, 289-290 selling of, 108-109 strike price properties for, 304 summary of, 52 up-and-in cash call, 7 15 up-and-out cash call, 7 15 written, 37-38 Call payoff, 35 Call prolit, 35 Ca1lprotection, 5.16 Call schedule, 516 Cap, 62-63, 792, 805 bond options, Black model, and, 790-792 interest rate, 792 selling of, 95
Capital
insurance against loss and, 834-835 insurance companies and, 437 regulatoly 815 short-seller needs for, 15 .
Capital Asset Pricing Model ICAPMI,659-660 Capital expenditure, rejearch and development as, 563-565
Capital gains, deferring, 490-495 Capital income, taxation of, 74
$
939
Capital loss, 49011 Capital management, long-term crisis in, 236 Caplets, 805 Cap level, collars and, 113 CAPM. See Capital Asset Pricing Model (CAPM) Capped options, 717 Capped participation, for CD structure, 485-486 Cam P., 433', 772n Carry, 18 1 Carry market, l 8 1-l 84 Cash-and-carry, 137 reverse, 137 eansyctions and cash flows for, l37 Cash-and-carry arbitrage, 137, 177 lease rate and, 180 with lending, l78 Cash call down-and-out, 7 12-7 13 price and delta of, 708 up-and-in, 7l5 up-and-out, 7l5 Cash coupon payments, 477-478 .
Cash flow
for company that borrows at LVOR and swaps to lixed-rate exposure, 255 in credit default swap, 861 distribution through derivatives, 10l for floating-rate borrower using swap, 256 of market-makers hedging with fonvard rate agreements, 256 portfolio and, 418.-419 short-selling and, 14 standard discounted, 371 stocks, puts, and, 283 from swaps, 249 on total returrl swap, 272. 273 unhedged and hedged for dollar-based hrm, 265 valuing derivatives on, 552-554 Cash flow CDO. 853 Cash flow mapping, 829 (8PDRS) Cash index, S&P's Depository Receipts 14811 and, Cash interest, 479-480 Cash-or-nothing call, 647, 704-705 Cash-or-nothing options banier options, 710-715 supersharepayoffs and, 709 Cash-or-nothing puq 705
940
k
llqoEx
lNoEx
Cash put down-and-in, 7 13 up-and-in, 7 1zI-7 15 up-and-out, 715 Cash-settled contract, S&P 500 as, 143 Cash settlement, 30
Catastrophe bonds, 6 CBT. See Chicago Board of Trade (CBT) CDO. Se Collateralized debt obligation (CDO) CDs alternative structures for, 485-486 economics of. 50-51 equitplinked, 48-52, 483-485 payoff on, 49, 50 CDS CDOs and, 861-862 i-ndex, 864-866
CDX, 865 Central Iimit theorem, 592-593 Certainty
single-harrelextraction under, 565-569 valuationunder, 679-68 l
CEV model. See Constant elasticity of variance (CEV) model Cheapest to deliver bond, 231 Chicago oard of Trade (CBT) contracts aded annually at, 9 f'uturescontracts traded on, 10 Onechicago and, l43 quantityuncertainty contract and, 1l6n Chicago Board Options Exchange (CBOE) indexof implied volatility (V1X),743, 757, 759 Onechicago and,. 143 supersharesand, 709 Chicago Mercantile Exchange (CME) contractstraded annually at, 9 futurescontracts traded on, 10 Onechicago and, 143 Chi-squared distribution, 607n Cholesky decomposition, 644, 657 Chooser option, 465 Cisco, 528n Clean price, 243 Clearinghouse, 142, l44:1 Clearing members, l42 CME. See Chicago Mercantile Exchange ICIMEI Co-cos. See Contingent convertible bonds (co-cos) Coherent risk measures, 837n
Collar, 73-75 in acquisitions, 538-542 CD structure as, 485, 420-440, l09 price protection as, 503 proht diagram for. 87 short fomard contract vs., 75 strategies with, 112-1 13 use and pricing and, 76-77 zero-cost, 76, l 10-1 12, 491-492 Collar width, 73
Collateral credit risk and, 30 .repurchase agreements and, 234 Collateralized debt obligation (CDO), 853 CDO index and, 864-866 CDS and, 861-862 with correlated defaults, 85*857 wit.h independent defaults, 853-856 Nth to default basket and, 857-858, 859 types of, 853, 854 Collect-on-delivery call (COD), 737 Collin-Dufresne, P., 25811 Commodities lease rate for, l86 options on, 334 synthetic, 17 l value at lisk calculations and, 8l4 Commodity extraction costs of, 568 optimal, 56*567 as option, 565-572 with shut-down and restart options, 572-579 single barrel under certainty, 565-569 single barrel under uncertainty, 569-570 Commodity fonvards, 169-184 equilibrium pricing of, 171-172 as hedging strategy, 196-200 pricing by arbitrage, 174-178 .
Commodity futures as hedging strategy, 19*200 as synthetic commodities, 17211 Commodity lease rate, 178-1 81 Commodity-linked bonds, 478-481 Commodity spreadsy 195-196 Commodity swaps. 247-254, 268-270 commodity swap price, 268-269 with variable quantity and. price. 269-270
Compensation options. 503, 523-538 end of'?, 526 expensing option grants and, 528-531 at Level 3 Communications, 534-538 repricing of, 531-532 selling of, 295n valuation inputs for, 527-528 valuation of, 525-526 Compound call otion, exercise decisions for, 453 Compounding, continuous, 875-879 Compound option parity, 454 Compound options, 453-456, 467-468 currency hedging with, 456 Concave function, 103 Conditional expected price, 603-604 Condence intervals, lognormal, 600-602 Constant elasticity of variance (CEV) model, 763, 766-767 implied volatility in, 767 pricing formula for, 767 Constantinides, G. M., 68911 Constant matulity easury (CMT) rate, 264 Constructive sale, 58. 49l Contango, 170-171. 298 Contingent convertible bonds (co-cos),520, 521-522 Contingent interest, bonds and, 520 Continuation value, 634-635 Continuous compounding, 875-879 Continuous dividends, 132-133, 134 Continuously compounded rate, 148n, 875 Continuously compounded returns. 353-354 Continuously compounded yields, 213-214 Control variate method, 630-632 Convenience yield, 182-1849 183 Convergence trades, 236 Conversion, 285 Conversion premium, 515 Convertible bond, 84, 482n, 495--498, 513-516 callable, 519-520 contingent, 520 Convexity, 224, 228-230 delta-gamma approximation and, 42311 strike price and, 304 Convexity bias, 219-221 Cooling degree-day, 200 Cooper, L., 48611
%
941
Core, J. E., 524
Corn, futures prices for, l90 Cornell, B., 235 Corn fonvard markeq seasonality and, l 88-191 Corporations derivatives issued by, 503-542 tax deferral for, 492-495 Correlated It processes, 657 Correlated stock prices, simulating, 643-645 Correlation coecient, computing, 154n Correlations. See also Volatility historical, 83l value at risk and, 828, 838-839 Costts) of commodity extraction, 568 of hedging, 1O6 storage, 18 1-l 82 Cost of carry, 141 Coupon bond, 210-21 1, 242-244, 475 options in, 482--483 as perpetuity, 480-48 1 duration of, 225 '
zero-coupon, 474.-475 Coupon bond prices, zero coupon bond prices deduced from, 211-212 Covered call, 64-65 profit from, 69 selling, 63-64 written put and, 65 Covered interest arbitrage, 15*157
Covered put, 65-66 Covered writing, 63-64 Cox, J. C., 292n, 313, 359, 65011,690, 691n, 692n, 763, 7671%846 Cox-lngersoll-lkoss (CIR) model, 787-788 Vasicek model and, 788-790 Cox-lkoss-lkubinstqinbinomial tree, 359, 555-556 fracking'' process, 195-196 Crack spread, 196 Crane, D., 8 l4n Credit default swaps, 858-862 cash ;ows in, 861 single name, 860-861 Credit event, 273, 841 Credit instruments, 853-866. See c//'t? specific WPCS
Credit-linked notes, 863-864
7
942
% IN D Ex
1N DEx
Credit risk, 1l l5. 30-31 841-867. See also ,
,
Default of interest rate swap, 258n, 263 in LIBOR rate, 258n swaps and, 263 value at risk and, 815 Credit spread, 842 Credit spread curve, 860 Credit tranches, 858n Crosbie, P., 8:14n Cross-hedgingyxl 1z1 16 with imperfect correlation, l53 with index fumres, 151-.154 weather derivatives and, 199-200 Crush spread, l95 Cumulative normal distribution'function, 589 Cumblative normal distlibution, inverse, 622-623 Cumulative standard nonnal distribution f'unction, 409-410 Currencies LIBOR quotes in, 160 optionson, 286, 332, 381 valueat risk calculations and, 814 Currency contracts, 154-157 Currency,hedging, 451-453 withcmpound options, 456 Currency-linked bonds, 481-482 Currency-linked options, 727-732 Cul-rency options, 290-292 Cul-rency prepaid forward, 155-156 Currency risk, in Nikkei index, 726 Currency swaps, 264-268 formulasfor, 267 typesof, 267-268 Currency-translated index, 721 Currency translation, 693 Curves. See also specilic types impliedvolatility, 741 swap,258-260, 270 yield,208 .
dividend risk and, 434 Daimlerenz, Davydov, D., 767n Dax index, 735 DCF. See Discounted cash llow (DCF) Debenture, 504n
Debreu, Gerard, 370n Debt connict with equity, 510 defaultable, 506 of Enron, 252 investment incentives harmed by, 510n leverage and expected return on, 506-510 multiple debt issues and, 51 1-512 as options, 503-510 value at maturity, 505 Debt capacity, 105 Debt-to-asset ratio, as function of asset value of lirm, 509-510 Decision trees, 557. See also Binomial tree DECS (Debt Exchangeable for Common Stock), 48211,497 Default, 841 bond ratings and, 847-852 CDOs and, 853-856 concepts and terminology of, 841-843 cumulative rates of, 851 by large companies in 2002, 865-866 Merton model of, 84.3-845 related models of, 845-847 Defaultable debt, 506 Default premium, 221 Default swaps, 273-274 credit, 858-862 pricing of, 862-864 XYZ debt issue and, 860 Deferred capital gains, 490-495 Deferred down rebate option, 712 Deferred swap, 261-262 Deferred up rebate option, 715 DeGroot, M. H., 59211 Delivery, cash settlement and, 30 Delivery value, Treasury bonds, 231 Delta, 382, 383-384, 44l formula for, 410 as measure of exposure, 416-417 .
put, 385 Delta approximation. of portfolio returns, 820-821, 822 Delta-gamma approximation, 423, 424-425, 665 for bonds, 784-78,$ of portfolio returns, 820n Delta-gamma-theta approximqtion, /.142
Delta-hedged bond portfolio, 782 Delta-hedged positions, 435 Delta-hedging, 414, 417.-422, 686n of all-or-nothing options, 707-708 of American options, 430 for bond option, 798 market making and, 413.-438 mathematics of, 422-429 in practice, 432-433 prost calculation and, 418-419 selling en'or in. 433 for several days, 420-422 supershares and, 709 for two days, 417-418 Delta-neutral position, 433 Dependent bootstrap, 831 Derivative, l construction from other products, 4 corporate applications of, 503-542 credit, 858-860 snancial 51414use of, 106-107 growth in trading of, 7-10 nonsnancial tinn use of, 107 perspectives on, 3 uses of, 2-3, 10-1 1 valuing on cash flow, 552-554 verifying formula for, 683-686 weather derivatives as cross-hedging, 199-200 Derivative prices, as discounted expected cash iows, 692-693 Derman, E., 710n Deutsche Bank, econornic delivatives and, 6, 7 Deutsche Terminbrse (DTB) exchange, 434 Diagrams. See specihc types Diebold, F. X., 748n Differential equations, and valuation under certainty, 679-681 Diff swap, 268
Diffusion process, 651 Digital cash option, 686n Digital share option, 68611 Dirtjr price, for bond, 243 Discounted cash flow (DCF),'551-552 standard, 371 techniques for option prices, 350 for valuing claim on 5', 668 Discounted expected cash flows, derivative prices as, 692-693
943
Discounted expected value, option price as, 6 17-621 Discounted present value, l29 Discrete dividends, 131, 134 stocks with, 36 1-365 Discrete dividend tree, 362-363 Discrete probability distribution, Poisson distribution as, 636-638, 639 Disney, Roy, 492 Distance to default, 843 Distress costs, hedging and, 103 Distribution of asset prices, 608-612 chi-squared, 607:1 exponential, 63711 lognonnal, 587-612 normal, 587 of payoffs, 617 Poisson, 636-638, 639 '
rettlrn, 831-832 stable, 592 two-parameter, 587 Diversisable jumps, in Mertonjump model, 697-698 Diversidable risk, 6 Diversilication, in portfolio, 819 Dividend, 510 binomial model and, 361-365 continuous and fonvard price, 132, l34 discrete apd forward price, 131, 134 on index, 28n liquidating, 297 on options, 5*57 option value and, 527:1 pricing prepaid fonvards with, 131-133 Dividend-paying stocks, 68 1 compound option model and, 455-456 Dividend lisk, 434 Dixit, A., 574n, 578-579 Dollar-based investor, 719, 720, 721-724 Dollar-denominated call option, on euros, 290-291 Dollar-denominated investor, binomial model for, 724-727 Dominion Bond Rating Service, 84711 Donaldson, William H., 84711 Down-and-in cash call, 7 12 Down-and-in cash put, 7 13 Down-and-in options, 450, 452
944
k
INDEX
INDEX
Down-and-out cash call, 7 12-7 13
Equilibrium equation, for bonds, 78 1-784
Down-and-out option, 450
Equilibrium pricinp of commodity fonvards.
Down rebate option, deferred, 7 12 Drift geometric Brownian motion and, 656-657 geometlic random walk with, 663 Drugs, development process for, 563-565 Duflie, D., 650n, 674, 7z14n,768:1, 772, 852n. 858n, 8631 86411 Duration, 223-228 delta-gamma approximation and, 423:1 hedging of bond portfolio and, 78 1 Macaulay, 225 as measure of bond price risk, 828 modihed, 225 Duration-hedging, with bonds, 779 Durntion matching, 227-228 Dutch auction, 7
171-172 Equilibrium returns, Black-scholes equation and, 686-688 Equilibrium short-rate bond price models, 785-790 Cox-lngersoll-lkoss model, 787-788 Rendleman-Bartter model, 785-786 Vasicek compared with Cox-lngersoll-lkoss, 788-790 Vasicek model, 786-787 Equit'y con:ict with debt, 5l0 elasticity of, 507 leverage and expected return on, 506-510 as options, 503-510 tax-deductible, 495-498 Equity call, foreign, 728-729 Equity-linked bonds, 476-478 Equity-linked CDs, 48-52 reverse-engineering for, 51 valuing and structuring, 483-485 Equity-linked foreign xchange call, 731-732 Equitplinked forward, 7 19 Equitplinked note, 48 options in, 483 Equity-linked products, 48 Eurex, 9 Euribor (European Banking Federation) lnterbank Offer Rate, l 60 Euro-denominated put option on dollars, equivalence to dollar denominated call option on Euros, 290-29 l Eurodollar contracq interest on, l60n Eurodollar futures, 158-160, 218-223 Eurodollar prices, swap curve with, 258-259 Eurodollar strip, 158 folavard interest rate curve implied by, 260 ten-year swap rate and, 261 Euro interest rate swap, 268 European call, Monte Carlo valuation of, 625-626 European call option, 684-685 binomial tree for pricing, 323, 349 Black-scholes formula for, 375-378 two-period, 323-326 European exchange option, electricity generation and, 560
Early exercise of option, 343-346 Eberharq A. C., 524 Economic derivatives, 6, 7 Economic obsen'er perspective, on derivatives, 3 Effective annual rate, l48n, 875 Efhcient Monte Carlo valuation, 630-633
Mlasticity
for call option, 393 of defaultable bond, 507 of equity, 507 option, 389-395 of portblio, 395, 5 l 1-512 Elasticity of variance', constant, 76*767 Electricity nonstorability of, 172-174 peak-load generation of, 559-563 Emanuel, D. C., 43l 767:1 Embedded options bonds with, 482-486 notes with, 488-489 End-user perspective, on derivatives, 3 Energy markets natural gas fonvard curvq and, 194 oil and, 194-195 Engle, Robert F.. 748, 749, 750 Enron, 847 hidden debt of, 252 ,
European put option binomial tree for, 328 Black-scholes formula for, 378 European-style option, 32, 293. See also specisc tJ'pes Black-scholes formula applied to, 379-380 strike prices for, 298, 300 time to expiration and value of, 297 EWMA. See Ekponentially weighted moving average (EWMA) Excel Monte Carlo valuation and, 622:1 Visual Basic forApplications (VBA), 885-906 Exchange option, 459.-461, 732-733 generalized parit'y and, 287-292 insnitely lived, 468-469 Exchange rate Asian options and, 452 history of changes in, 8 Exchange-traded contracts, credit risk and, 30, 31n Exchange-traded index fund, 709 Exercise, 32 moneyness and, 304 of options, 57 Exercise price, 32 Exercise style, of option, 32 Exotic options, 443--46 1, 703-736 Expectations hypothesis, 213 Expected interest rates, yields and, 796-797 Expected return Black-scholes equation and, 690 on debt an4 equity, 506-510 of lognormally distributed stock, 597 Expected return on equity as function of asset value of firm, 509-510
Expected tail loss, 832 Expensing, of option grants, 524-525, 528-531 Expiration change in, effect on option plice, 297-298 of option, 32 Expiration date, 21 of option, 32 Exponential distribution, 637n Exponential function. 87*878 Exponentially weighted moving average (E 74*747
),
k
945
Exposure, delta and gamma as measures of, 4 16-4 17 External hnancing, hedging and, l03
Fair value, l34
Fama, Vugene.376 FASB. See Financial Accounting Standards Board (FASB) Faulkender, M., l07 Feynman-Kac solution, 692n Financial Accounting Standards Board (FASB) on compensation options, 524 contingent convertibles (co-cos)and, 521 derivatives reporting requirements and, 10 Statement of Financial Accounting Standards (SFAS) 123R0 525 Financial assets buying, 11-12 short-selling, 12-14 Financial engineering, 3.-4. 47 l 473 Modigliani-Miller theorem and, 474 security design and, 473-498 for tax and regulatory considerations, 490-498 Financial firms, derivatives used by, 106-107 Financial forwards. See Fonvard contracts Financial markets impactsof, 4-5 role of, 4-6 Financial options, real options and, 558 Financial policy, Modigliani-Miller theorem and, 473-474 Financial products, routine categories of, l .
,
Financing hedging and, l03 short-selling and, 13 of zero-cost collar, 76-77 Financing cost, 60 Fitch Ratings, 84711 Fixed collar offer, 538 Fixed exchange rate foreign equity call, 730-73 I Fixed-rate bonds, swaps and, 256 Fixed stock offer, 538 Floating collar offer, 538 Floating interest rates, currency swaps and, 268 Floating stock offer. 538 Floors, 59-62
9 46
IN D Ex
w Ilq D E x
Ford Motor Co., hedging for risk management by, l08 Foreign assets. See tz/'t/ Nikkei 225 index options on, 727-732 Foreign equity call, in foreign currency, 728-729 Foreign exchange call equity-linked, 731-732 hxed exchange rate, 730-731 Foreign stock index. See also Nikkei 225 index foreign equity call and, 729-730 risk of, 7 l 8-7 19 Formtllas. See specic formulas Forward commodity, 169-184, 196-200 cun'ency, 154-155, 156 equity-linked (quanto),7l9 ilterest rate, 205-237 prepaid, 727 summary of, 52 synthetic, 66-70, 1l2, 135-138 Forward ontracts, 2 1-31. l 25, 128, 279. See also Futures contracts canceling obligation to buy ot sell, 14211 collateralization and, coverd interest arbitrage for. 15*157 currenc'y prepaid fonvard, 155-156 equating with futures, l66 - futures prices and, 146-147 futures prosts and, l46 graphing payoff on, 25, 26 hedging with, 92-93, 98-99 long and shol-t positions, 43 oflmarket fonvard. and, 69 vs. outright purchase, 26-28 over-the-counter, 142 payoff on, 23-26 as single-payment swkpp,247 on stock, 133-141 synthetic, 135-136 ' as zero-cost collar, l 1l-l 12 ' Forward curve, 169 for natural gas, 193 Forward premium, 134-135 Fonvard price, 561 '
666....64:/
on st' forclaim of coltar and, 77-78 .
'
.
cost futureprice predicted by, of gold,' l 88 leaserate and, 179-18 l
140-141
premiumfor fonvard contract risk-neutralprice as, 553 for S' Q?'670-672
and, 134
7
storage costs and, l 8 1-182 Forward pricing formula, interpretation of, 14l Fonvard rate. implied, 208-210 Forward rate agreement (FRA.), 214-215, 806-808, 826 Eurodollar futures and, 2 19-221 hedging with, 256 settlement at time of borrowing, 2l5 settlement in arrears. 2 15 Swap as, 25811 synthetic, 21*218 Fonvard sale, hedging with, 92 Forward strip, 169 420-440 collar, 109 FIkA. See Fonvard rate agreement (17RA) French, K. R., 14711 Froot, K., l03n Fuller, K. P., 53811 Fully leveraged purchase, of stock, l27 Funded CDO, 862 Funds, exchange-traded, 709 Future contracts, fonvard prolits and, 146 Future interest rates, 221n Future price, predicted by fonvard price, 140-141 Futures asset allocation use in, 150-151 commodity 196-200 currency, 154-1 55 Eurodollar, 218-223 gold, 184-188 index, 150-154 interest rate, 205-237 oil, l94 pptions on, 381-382 quanto contract and, 7 19 stock-index, 22 on terrorism, 24 Treasury-bond, 230-233 Treasurpnote, 230-233 , Futures contracts, 125, 142-150, 279. See also ' Fonvard contracts on CBT, CME, and NYMEX, 10 equating wit.h fonvard contracts, 166 Eurodollar contract as, 25811 Eurodollar futures, 158-160 forward contracts and, 21
forward price and, 146-147 hedge quantity and, l53 index, 22 on individual stocks, 143 margin, marking to market, and, 144-146 mark-to-market proceeds and margin balance in, l46 Nilckei 225. 149-150 options on, 332-334 quantity uncertainty and, l l6n quanto index contacts and, 149-150 S&P 500, 143-144 S&P index arbitrage and, 147-149 X, 10 traded on CBT, CME, and Futures exchange, 9 Futures overlay, 151 Future value fonvard price as, 133 in proht diagrams, 36 of option premium, 43
Gamma, 382, 384-386, 428,
.442
to approximate change in option call, 385 formulafor, 4l0 hedgingand, 420 as measure of exposure, 416
price, 423-424
755
comparison with EWMA volatility, 755 estimating using Excel, 777-778 exponential, 748 maximum likelihood estimation of. 752-753 volatility forecasts in, 753-754 Gastineau, Q L., 106n Gates, Bill, on compensation options, 526 Gczy, C., 107 General collateral repurchase. agreement, 234 Generalized parity, exchange options and, 287-292 Geometric average, formulas forAsian options based on, 466-467 strike calls and puts, /147 ,446..447
947
Geometric Brownian motion, 649. 655-659 Black and Scholes and, 679, 681-682 It's Lemma and, 665 Geometlic random walk, with drift, 663 Geman, H., 695n Girsanov's theorem, 662 Glasserman, P., 63011,633, 635, 636 Gold exaction of, 568-569 fonvard price of, 188 gold-linked notes and, 486-488 investments in, l87 notes with embedded options and, 488-489 prepaid folavard plice of, 188 producer's perspective pn tisk management and, 91-98, 109-1 13 valuing future production of, 187-188 Gold f'utures, 184-188 Goldman Sachs, economic derivatives and, 6, 7 Gorton, G.. 172n Government securities, repos for, 234 Graham, J. R., 107 Granger, Clive, 750 Granovsky, R. J., 728 Graphical interpretation, of binomial formula, 319, 320 Greek alphabet, 873 '
Greeks in binomial model, 441-442 for bull spread, 392 dehned, 382-388 formulas for, 410-412 option, 382-395
Gamma-hedging, 436
Gamma-neutrality, 433-436 Gap option, 457.-459, 536, 68611,706-707 GARCH (Generalized ARCH) model, 751-754,
k.
as portfolio measures, 388-389 Ltd., 74 v. Citibank Grossman, S. J., 130n Guay, W. R., 524 Gupta, A., 258:1 Guth, Robert A., 526 Grn
,rnvc&l??;c?,?x
Haircut, 15, 234 Hakannsony N. H., 709 Hamilton, D., 851 Haug, E- G., 56211 Haushalter, G. D., 107 Heath, D., 793n, 794n, 798
948
% Ilq D Ex
lNDEx
Heath-larrow-Morton model. 8 l 1-812 Heating degree-ay, 200 Heat rate, 560 Hedge basisIisk in, l53 cross-hedgingand, 114-1 16 stack, 197, 198 strip, 197 in swaps, 250-251 Hedged position, 4 Hedged prost, 9/ Hedge funds, repos for, 235 Hedge ratio, l 13-1 l9, l 14, 22811 quantity uncertainty and, l l &-1 19 variance-minimizing, 118 Hedging, 2. See also Cross-hedging', Delta-hedging Asian options nd, 445, 448.-449 barrier options and, 449 of bond portfolio based on duration, 78 l commodity futures, using, 19*200 currency, 451-453, 456 delta-hedging, 417-422 of diff swap, 268 empirical evidence on, 106-107 with furodollar contract, l 60 with forward contract, 92-93, 98-99 with forward rate agreements, 256 frequency of re-hedging and, 431-432 of jet fuel with crude oil, 199 by market-makers, 413 . option risk in absence of, 414-415 reasons for, 100-101, 103-1 06 reasons for not heging, 106 short-selling and, 13 swaps and, 247 value added by, 101-103 volatility and, 74l 757-763, 76911 zero-cost collar and, l I l Heston, S. L., 763 Heston model, 768-77 1 Heteroskedastic, 749, 750 High-beta security, call option as, 620-621 Histogram for assessing lognormality, 608-609 jumps and, 642 for risk-neutral stock price disibution, 628 Historical correlations, 831 Historical returns, of risk arbitrageurs, 541 n ,
Historical volatility, 360, 361, 744-746, 756n Ho, T. S. Y., 793n, 798 Home equity insurance, 6 Homeowner's insurance with house, 6 1-62 as put option, 45-47 Homoskedastic error term, 749 Horowitz, J., 831n Hoskins, W., 25811 Houweling, P., 863 Huddart, S., 527n Hull, J. C.. 768n
Hysteresis, 578-579
IASB. See lnternational Accounting Standards Board (IASB) IBM, volatilities for, 742, 745 lmmediacy, as dealer service, 1ln Implicit borrowing and lendinp in swap, 260-26 l Implicit short position, 62n Tmplied forward rate, 208-2 10 expectations hypothe' sis and, 2 13 Implied repo rate, l35 Tmplied volatility 400-402, 741-744 in CEV model, 768 computing, 400-402 for IBM call options, 742 jump risk and, 764-765 over time, 743 using, 402 Importance sampling, 633 lncome-linked loans, 6 Index. See Jl/w Stock index; specific indexes buying of, 69 currency-translated, 72 l for futures contracts. 22 payoff from shol-t-selling, 63 profit from shol-selling, 63 put and, 59-62 lndex futures. See also Futures contracts cross-hedging with, 151-154 hedge rate dtermination, l52 shorting of, 152 uses of, 150-154 lndividuals, tax defen'al for, 49 1-492 lnsnite investment horizon, ev luating project with, 558
lnlinitely lived exchange option, 468-469 Igersoll, J. E., Jr., 516n, 787-788 Inglis, Martin, l08 lnsurance adjusting by changing strike, 9*98 call option and, 47.-48, 99-100 homeowner's as put option, 45-47 market-maldng as, 436-438 minimum price guaranteed with put option, 93-95 options as, 45-48 portfolio, for long nln, 603 by selling a call, 95-96 selling of, 63-66 strategies for, 59-66 value at risk and, 834-835, 836-837 lntel, peak-load manufacturing at, 559 Interest, 4 l 8-4.19 accrued, 243 cash, 479-480 contingent, 120 on Eurodollar contract proceeds, l60n lnterest in-kind. 478, 480 lnterest rate Black-Derman-lby model and. 798-808 Black fonnula to price options on, 79l continuous compounding and, 875-879 conventions for, 241-245 determining, 15-16 equilibrium short-rate bond price models, 785-790 futures and fonvard prices and, 14*147 reasons fo( swapping, 262- 263 S&P index arbitrage and, 147-148 short-term, 783 yields and expected rates, 796-797 Interest rate cap. 792 Interest rate curve, fonvard cun, 260 Interest rate fonvards and futures, 205-237 lnterest rate models, 779-808 binomial, 793-798 Cox-lngersoll-lkoss, 787-788 Heath-larrow-Morton model for, 81 1-812 Rendleman-Bartter, 785-786 Vasicek, 78*787 Interest rate risk Eurodollar contract for hedging, 160 swaps and, 263 lnterest-rate sensitive claims and VaR, 826-827
k
949
lnterest rate stacks, 223 lnterest rate strips, 223 Interest rat swaps, 254-263 euro swap, 268 simple, 254-255 swaps and, 256 lnterest pte tree Black-Derman-rfby, 800, 801 three-period, example, 793, 797 Internal rate of return, 208 Intenmtional Accounting Standards Board (.lASB) on compensation options, 5.24 derivatives reporting requirements and, 10 hedge accounting, 10*107 In-the-money option, 43, 304 Inverse cumulative normal distribution, 622-623 Investment. See also Net present value (NPV) in gold, l87 in Nikkei 225 index, 728 NPV rule and, 548-55 1 in oil well, 572-578 solving for optimal decision, 556-558 staged, 564 under uncertainty, 55 1-558 when well shutdown is possible, 576-577 Investment asset, as underlying asset, 688-690 lnvestment cost, as strike price, 547 lnvestment grade bonds, 847 Investment horizon, evaluating project with z-year horizon, 554-558 Investment jroject, 547 as call option, 558 Investment trigger price, 551 Iomega Corporationqdividend of, 57 1t process, 655-659 It's Lemma, 663-666 Black and Scholes and, 679 bond pricing model and, 78 1, 782 log contract and, 663-666 multivariate. 665-666 applied to &', 667-668 Jagannathan, R., 532 James, J., 812 Jarrom R., 8 l 2, 852
Jarrow-lkudd binomial tree, 359 103n, 370n, 594-595, 62 1n, 663, 664, 665, 88 1-884 bond pricing and, 784
Jensen's inequality,
950
%.lN D EX
IXDEX
exponentialfunction and, 881-882 priceof a call and, 882-883 proofof, 884
Jet fuel, hedging with crude oil. 199 Jorion, P., l n J. P. Morgan, 8 14 J. P. Morgan Chase, 526 Judd, K. L., 62ln, 630n Jump difksion model, 763 Jump risk, implied volatility and, 764-765 Jumps multiple, 643 option pricing with, 69*698, 772 simulating with Poisson listribution, 639-643 in stock price, 672-674 Junior bonds, 511-512 Junior tranche, 511
Kani, I., 7 1On ' Kapadia, N., 77211 Karatzas. I., 650n, 652n, 662n, 664n, 674 Karlin, S., 691n, 692n Kemna, A. G. Z., 63011 KMV model (Moody's), 8/.1z1m Knock-in'options, 450 Knock-out options, 450 Kolmogorov backward equation, 69 1-692 Kulatilaka, N., 52711 Kurtosis of a distribution, 609, 643
Ladder opdon,740 Land, as option, 567 Latin hypercube sampling, 633 Lattice, 325n Lauder, Estee and Ronald, 49 l Law of one price, 3 15 Lease market, for commodity, 178-179 Lease rate, l4, 479n of asset, 1+-15 cash-and-carry arbitrage and, 180 commodity, 178-18 l forward prices and, 179-18 1 for gold, silver, and other commodities, 568-569 storability and, l 76-178 storage costs and, 182 for valuing claim on Sl, 669
Lee. M., 646, 69611 Lee, S.-B.. 793n, 798 Leland, H. E., 709 Lending. of security, 12 arbitrage and, 139 LEPO. See Low exercise price option (LEPO) Leptokurtosis, 609 Level 3 Communications, compensation options at, 534-538 Leverage, and expected rettlrfl on debt and equity, 506-5 10 Leverage effect, 766 Li, D. X., 858n LIBOR. See London Interbank Offer Rate ILIBORI Linear regression, hedges and, 115n Linetsky, V., 76711 Liquidating dividend, 297 Liquidity, of futures contracts, l42 Liquidity premium, 222 Loan balance, of swap, 260-261 Loans, income-linked, 6 Logarithmic function, 876-878 Log contract, 760-762 ' Lognormal confidence intelwals, 600-602 Lgnormal distribution, 587-612 and the Black Scholes formula, 605 estimating parameters of, 605-607 probability calculations and, 598-605 for value at risk calculations, 814 Lognormality binomial model and, 355-358 binomial tree and, 351-360 comparison with three-period binomial approximation, 357 geometric Brownian motion and, 655-656 histograms for assessing, 608-609 Lognormally distributed variable, 593-595 Lognormal model, of stock prices, 595-598, 744 Lognormal random variables, generating 11 correlated, 644-645 Lognormal stock prices, simulaying, 623-624 Lognormal, binomial, tree, 359 London Interbank Offer Rate (LDOR) computing, 258:1 Eurodollar futures and, 158-160, 218-221 history 3-month LIBOR and 3-montlt T-bill yield, 222-223 ' interest rate and, 147-148 '
.
186,
three-month forward rates, 259 vs. three-month T-bills, 221-223 Long (buyer),23 Long call. proht, 53 Long fonvard, prolit, 53 Long folavm'dcontract, synthetic, 67 Long fonvafd position, 44, 52 Long position, bonds, repos, and, 235 Long positions,' l2, 44, 52 floors and, 59-62 prolit diagrams for, 44 Long put profit, 53 Long nm, portfolio insurance for, 603 Long Term Capital Management, 236 Longstaff, F., 633, 635 Lookback call, European, 739 Lookback put, European, 739 Loops, in (VBA), 901 for loop, 899 Losses capital,490:1 withput and short fonvard, 40 Loss given default, 842 Low-coupon bonds, 233 L OW discrepancy sequences 633 Low exercise price option (LEPO), 133 Lublin, Joann S., 526 Lux, H., 709 ,
Macaulay, Frederick, 22511 llacaulay duration, 225 for zero-coupon bond, 22911 MacBeth, J. D., 767n Madan, D.. 433 Maintenance margin, 145 Managed CDO, 853 Mandatorily convertible bond, 84 Mapping, cash flow, 829 Marcus, A. J., 52711 Margin, 144 maintenance, l45 marldng to market and,.144-146 for writlen options, 57 Marginal utility, declining, 369-370 Margin balance, in futures contracts, 146 Margin call, 145 Marketts). See also specific types in contango, 170-17 l
%
951
economic derivatives, 6 over-the-counter, 10 for risk-sharing, 6 Market corner, 233 Market-maker, 4, 413-414 bid price, offer price, and, 12 delta-hedging all-or-nothing options and, 707-708 dividend risk and, 434 exposure in currency swaps, 266 insurance and, 437-438 overnight proht of, 421 over-the-counter options and, 43311 prot of, 427-429 risk of, 414-417, 437-438 and risk of extreme price moves, 432-433 roles of, 413-414 selling prepaid fonvard and,. 130 Market-maker perspective, on derivatives, 3 Market-maling, 413-438 bond pricing and, 779-785 as insurance, 436-438 synthetic fonvards in, 13*138 Market lisks, value at risk and, 8l5 '
Market-timing, 733-734 Market value, of swaps, 253-254 Market value CDO, 853 Marking-toemarket, 142, 415, 418 margins and. 144-146 proceeds and margin balance, 167 Mark-to-market proceeds, in futures contracts, 146 Manied put, 58n Marshall & llsley security, 83-85 tax-deductible equity and, 495-498 ,
Martingale, 651 'iMatched book'' transaction, 250 Mathematics, of delta-hedging, 422-429
Maturity default at, 843-844 effect on option price, 297-298 payoff for Marshall & Ilsley bond, 84-85 prolit diagrams before, 395-399 yield to, 208 McDonald, R. L., 43311,527n. 565n, 689:1 McMurraya S., 491 Mean return. estimate of, 607 Mean reversion, in alithmetic Brownian process, 654-655 Measurement, of volatility, 744-757
952
k
INDEX
IN DEx
Merger Northrop Grumman-TlkW. 538-542 options in agreements, 538-539 plice protection in, 503 Merrill Lynch, MI'TTS from, 48 Merton, Robert C., 1, 292:1, 376, h.03,639, 641, 650:1, 664:1,673, 674, 679n, 696, 697, 843 jump diffusion model, 763 jump pricing model, 697-698 Merton default model, 843-845, 846 Nletallgesellschrft A. G. (MG), 198 Mezzanine tranche, CDOs and, 854, 856 Microsoft, compensation options and, 524, 526, 527-528 Miller, H. D., 650n, 69111 Miller, Merton, 376, 473. See also. Modigliani-Miller theorem Mispriced option, rbitrage for, 318-323 Mitchell, M., 54111 MITTS (Market lndex Target Term Securities), 48 ' Mixmre of normals model, 609. 647 Modeling, tjf discrete dividends, 361-362 Modified duration, 225, 237 Modigliani, Franco. 473 Modigliani-Miller theorem, 473-474, 507 Moneynes, exercise and, 304 Monotonicity, 837n Monte Carlo valuation, 528, 617-645, 8 14n accuracy of, 626-627 of American options, 633-636 antithetic variate method, 632 arithmetic Asian option and, 627-630 for basket options, 736 computing random' numbers, 621-623 control variate method of, 630-632 efficient, 630-633 of European call, 625-626 importance sampling and, 633 Latin hypercube sampling and, 633 low discrepancy sequences and, 633 naive, 631, 633 for nonlinear portfolios, 822-826 stratilied sampling and, 632-633 for value at lisk of two or.more stocks, 818 of written straddle after 7 days, 826 Mood, A. M., 59211, 88411 Moody's bond rtings, 847
lnvestor Services, 847n KMV model, 814n Moon, M., 563:1, 565 Moore, David, 55911 Moral hazard, 47n Morgan Stanley, TRACERS and, 865 Morgenson, G., 53l n Moving average, exponentially weighted, 74*747 Multi-date swap, 247 Multiple debt issues, 51 1-512 Multivariate Black-scholes analysis, 700-701 Multivariate It's Lemma, 665-666 Multivariate options asket options, 735-736 exchange options as, 732-733 options on best of two assets, 733-734 Myers. S.. 105n Myers, S. C.. 5l0n
Naik, V., 646, 69611 Naive Monte Carlo valuation, 631, 633 Naked writing, 64 Nationally Recognized Statistical Rating - Organizations (NRSROj), 847 Natural gas? seasonality, stfage, and, 191-195 futures contract, 191-194 Natural resources, commodity extraction and, 565-572 Neftci, S. N., 65211,662n, 674 Nelson, D. B., 74811 Net payo 28 Net present value Ih1PVI correct use of, 549-550 investment and, 548-55 l static, 548-549 Netscape, 492-493, 498 Neuberger, A., 760 New York Mercantile Exchange ( X) contracts traded annually at, 9 futures contracts traded on, 10 gold futures contracts on, 185 light oil contracts on, 194 natural gas futures on, 192 Nikkei 225 index, 735 currency risk and, 726 futures contracts and, 149=150, 268 .
investing in, 718-719, 720-724 put warrants and, 728 No-arbitrage bounds, with transaction costs, 138-139 No-arbitrage pricing, 70 Noise term, geometric Brownian motion and, 656-.657, 658 Nonconvertibl bonds, callable, 517-519 Nondiversifiable risk, 6 Nonhnancial hnns, derivatives used by, 107 Nonhnancial risk management, 105-106 Nonlinear portfolios, value at risk for, 819-826 Nonmonetary returfl, convenience yield as, 183 Nonrecombining tree, 325 Nonstandard option formulas, Black-scholes equation and, 683-684 Nonstandard options, 443-461, 703-736 Nonstorablity, of electricity, 172-174 Normal density, 587-588 Normal distribution, 587-593 cumulativ function, 589 cumulative inverse. 622-623 standard, 409-410 Normal probability plots, 609-612, 610 jumps and, 642-643 Normal random variables conversion to standard normal, 590-591 sums of, 591-593 Northrop Gnlmman-x'R.W merger, 538-542 Notes. See also specilic types with embedded options, 488-489 equity-linked, 48, 483 gold-linkqd, 486-488 strucmred, 474 Treasury note futures, 230-233 Notional amount of swap, 249 NPV. See Net present value (NPV) NQLX. See Onechicago
Numeraire, 693-696 NYMEX. See New York Mercantile Exchange (NYMEX)
O'Brien, J., 709 OCC. See Options Clearing Corporation (OCC) Offer price, l l Offer structures. 538
$
953
Off-market forward, 69 Off-the-run bonds, 206, 236 Oil, 194-195 hedgingjet fuel with, l99 Oil extraction, 565-572 with shut-down and restart options, 572-573 valging inlinite oi1 resen, 570-572 Oi1futures. 194 Oil market, 198 Oil prices. See also Swapts) derivatives markets and, 7 producer price index for, 8, Onechicago, 143 One-period binomial tree, 3 13-323 On-the-run bonds, 206, 236 On-the-nm/off-the-run arbitrage, 235, 236 Open interest, 1/14 Open outcry, 142 Operational lisk, value at risk and, 8 15 Optimal investment decision, solving for, 556-558 Optionts). See also Investment', Parity; specitic types all-or-nothing, 685-686, 703-709 American-style, 32, 329, 330, 403, 404 arbitrage for mispriced, 31 8-323 Asian, 48, 9 asset-or-nothing barrier options, 7 15-7 16 on the average, barrier, 449-453, 7 17-7 18 bartier COD, 737 basket, 735-736 on best of two assets, 733-734 Black-scholes analysis in pricing of, 429-436 Black-scholes equation for pricing of, 679-698 on bonds, 286-287, 335-336 bonds with embedded. 482-486 Bulow-shoven proposal and, 529 buying, 34, 56-58 call, 31-38 capped. 7 17 closing prices for S&P 500 Index from CBOT, 33 334 on commodities, commodity extraction as, 565-572 common debt and equity as, 503 compensation, 503, 523-528 compound. 453-456, 467-468 in coupon bonds, 482-483 J
.446/.-447
,
954
l tq DEx
k. INDEX
currency, 286, 290-292, 332, 38 1, 727-732 debt as, 503-510 distribution of returns in portfolio and, 8 19-820 on dividendrpaying stocks, 455-456 embedded, 488-489 equity as, 503-5 10 equity-linked foreign exchange, 731-732 in equity-linked notes, 483 Etlropean-style, 32 European vs. American, 293 exchange, 28/-289, 459..-461 exercise of, 57, 279, 3O4 exotic, 1, 703-736 linancial and real, 558 on futures, 332-334, 38 1-382 gamma to approximate change in, 423-424 gap, 457-459, 706-707 inlinitely Iived exchange options, 468-469 as insurance, 45-48 in-the-money, 43 ladder, long and short positions, 43 maximum and minimum plices of, 293-294 multi-period example, 349-350 multivariate, 732-736 one-priod binomial example, 348-349 outperformance, 459 overpriced arbitrage of, 31 8 path-dependent, zlzlz! payof and prolit diagrams for, 33-43 perpetual, 403-405 power option, 690 pricing using real probabilities, 347-350 put, 38-43. 328-319 rainbom 734-735 real, 547-580 rebate, 7 l 6-7 17 reload, 532-534 repricing of, 531-532 risk premium of, 394 Sharpe ratio of, 394-395 shout, 739 spreads of, 70-73 on stock index, 330-331 on stocks, 283-286 on stocks with discrete dividends, 380-381 style, maturity, and strike of, 292- 304 stlmmal'y of, 52 .443-46
.740
synthetic, 285-286 terminology for, 32-33 underpriced, 319 volatility of, 393-394 warrant and, 512 written, 37-38, 40-42 Option-based model, of debq 51 1 Option elasticity, 389-395, 391 Option grants, expensing of, 528-531 E Option Greeks, 382-395 formulas for, 410-412 Option overwriting, 63-64 Option premium graphs for, 397. 398 Option plice computation with expected value and tnle probabilities, 347-350, 620 computing, in one-period binomial model, 314-315 delta- and delta-gamma approximations of, 426 as discounted expected value, 347-350, 617-621 jumps and, 69*698, 764 taxes and, 34l Option pricing formula, for commodity extraction, 567-568 Option pricing model. See also Black-scholes formula; Binomial model evidence of volatility skew in, 771-773 Option risk, in absence of delta hedging, 414-415 Options Clearing Corporation (OCC), 56-57, 434 Option writer, 37, 38 Order statistics, 610 Ordinal'y options, 706-707 '
Ornstein-uhlenbeck process, 654-655 Vasicek model and, 786 Onvall, Bruce, 492n OTC. See Over-the-counter market Out-of-the-money options, 43 Outperformance feature, valuing, 535-536 Outperformance option, 459 stock option (OSO), 534 Outputs, arrays as in VBA, 901-902 Over-the-counter ontracts credit risk and, 30-31 forward coneacts, 142 Oventhe-counter market, 10 Over-the-counter options. marfet-makers and, 433n
Paddock, J. L., 56511 Palladium, Ford risk management and, 108 Par coupon, 210 Parity, 28 1. See also Optionts) bounds forAmerican options, 310-31 1 of compound options, 454 generalized.and exchange options, 287-292
put-call,281-287
Partial expectadon, 603-604 Par value of bond, 496 zlxlzl Path-dependent optiom
banier options
as, 449
Monte Carlo valuation and, 627-628 Payer swaption, 27l Paylater strategy, 113, 114, 45711 Payoff, 23 Asian option, 683n call, 35 on CD, 49, 50 combinedindex position and put, 59-62 comparisonof long position vs. fonvard conkact,27 distributionot 617 at expiration, 36, 63, 65 23-25 on forward contract. for future values of index, 25 graphingon forward contract, 25, 26
neq 28 for purchased and for purchased and supershare,709
written call option, 33-37 wlitten put option, 39-42
Payoff diagram, 28
for covered call, 66 for covere'd put, 67 for long forward conact, 29 for purchased and written call option, for purchased and written put option, zero-couponbonds in, 28-29
Payoff table, for arbitrage oppo>nity,
33-37 39.-40
288
482n,493-494 Percentage lisk, of option, 391-393 PERCS (Preferred Equity Redeemable for Common Stock), 482n
955
Perpetual options, 403 Perpetual puts, 40,4-405 Pepetuities, 480-481 Petersen, M. A., 107 Pharmaceutical investments, 563-565
Physical measure, 844 PIBOR tparisInterbank Offer Rate), 160 Pindycklk. S., 565 Poisson distribution, 636-638, 639 pricing options with, 694-696 simulatingjumps with, 639-643 Poisson process, 845 Portfolio elasticity of, 395, 51 1-512 gold as asset in. 187
Greek measures for, 388-389 lisk assessment for, 813 risk premim of, 395 value at risk for nonlinelm 8 19-826 Portfolio insurance, for long nm, 299, 603 Positive-definite conelations, 644-645 Positive homogeneity, 837 Positive lease rat, strability and, 176-178 Power option, 690 Premium, 32 for call and put options, 39 default. 22 1 forward, 134-135 future value of option premium, 43 liquidity, 222 option, 284, 300n Premium for fonvard contract, folqvard plice and, 134 '
option. 57n Payout-protected Peak-load electricity generation, 559-563 Peak-load manufacturing, at Intel, 559 Shares), PEPS (Premium Equity PMrticipating
Perpetual calls, 404
$
Prepaid fonvard. 727 binomial tree with, 363-365
currency prepaid, 155-156 Prepaid forward contract (prepay),127, 684 pricing by analogy, 128=129 pricing by arbitrage, 129=130 pricing by discounted present value, 129 pricing with dividends, 131-133 on stock, 128-133 Prepaid folavard price for claim on S'', 666-667 of gold, l 88 Prepaid swap, 248, 27 1 Enron and, 252
956
'w
INDEX
Ilq D Ex
Present value. See also Net present value (NPV) banier values, 403 calculations of, 684 of cap payments, 805 of future stock price, 179:1 pricing by discounted, 129 of project, 547 traded, 550 Price ask, 11 bid, 11 clean, 243 dirty, 243 futures and forward, 146-147 guaranteeing with put option, 93-95 offer, 11 stdke, 32, 97-98 Price bonds, denoninated in stocks, commodities, and currencies, 47 1 Price limit, l42 ' Price options, average, 466 Price risk, derivatives and, 7-8 Price variabilily derivatives markets and, 7-8 Probability of bankruptcy, 847-850 distribtion, to stock price, 593-603 in high and low states of economy, 372 1og normal, 599-600 normal plots, 609-612 risk-neutral, 32 l 618 in value at risk assessment, 813 Probability calculations, for lognormal distribution, 598-605 Probability distribution' for VaR and tail VaR, 833 Probability measure, 8.414 Procter & Gamble, swap with Bankers Trust, 263, 264 Producer, risk management by, 91-98 Producer price index, for oil (1947-2004),8 Production, seasonality in, 188-191 Proht, 28 call, 35 daily calculation for market-maker, 422 of delta-hedged market-maker, 428 diagrams of, 53 at expiration, 36, 63, 65 for gold prices, l09 ,
hedged, 93 from insurance on house, 46 for long positions, overnight market-maker, 42 l for purchased call option, 33-37 'l4
put, 39 on short fonvard position, 92-93 spark spread and, 560 unhedged, 92 for written put option, 40-42 from written straddle, 80 Protit from delta hedging, intelpreting, 418-419 Prolit diagram, 28 for bull spread, 72, 87 for butterfly, 87 for calendar spread, 399 for collar, 87 for covered call, 66 for covered put, 67 from holding call option, 398, 399 of insured house, 62 before maturity, 395-399 for no arbitrage, 70n. for ratio spread (2:1),87 - for straddle, 79, 87 for strangle, 87 for unhedged buyer, long fonvard, and buyer hedged with long forward, 99 zero-coupon bonds in. 28-29 13811 Pro forma arbitrage calculation, Proprietary trading, 4l4 Psi, 383, 388, 411.-412 for call options, 392 Pulvino, T., 54111 Purchase, of stock, 127 Purchased call, gamma for, 384 Purchased call option, 396 payof and prolit for, 33-37 proht diagram for, z.!.z:l Purchased option, Greek for, 383-388 Purchased put, gamma for, 384 Purchased put option, 44-45 payoff and prolit for, 39-40 Purchase of shares, 418 Purchasing Manger's Index, 7 Put-call parity 68-70, 7 1 28 1-287. 530 versions of, 305 ,
Ratio spread, 73
adjusting insurance with, 96-98 Black-scholes formula for, 376, 378 buying of, 69 calls as, 289-290 cash-or-nothing, 705 in CD stnlcture, 485 collar and. 73 covered, 65-66 delta for, 385 down-and-in cash put, 7l3 early exercise and, 296-297, 345-346 gamma for, 384 homeowner's insurance as, 45-47 as insurance, 59 payoff and prost for purchased. 39-40 perpetual, 404-405 prenum for, 39 risk of, 47 strike price properties for, 304 summary f, 52 theta for, 390 up-and-in cash put, 7 1z1-715 up-and-out cash put, 7 15 Put premium, for gold options, 109 Put sales, as hedge, for shar repurchase, 522 Put strikes, proft using various, 97 Puttable bonds, 520 Put warrants, 522-523, 728
652-653
realized,255-757
Quantile,61 1 Quantityu-certainty, l l 6-1 19, 694 Quanto,1l6n, 149-150, 718-727, 7l9 Quantooption, equitplinked foreign exchange as,
731-732
Quasi-arbitrage,139-140 Rainbow option, 734-735 Random numbers, computqing, 621-623 Random walk model, 351-352 stockprices and, 352-353 Rate of return, 12 Ratings transition, 848-850 Ratio, hedge, l 13-114
95g
Rational option pricing theory, 292n
Put option, 38-43, 328-329
Quadraticvariation,
%
call
proht diagram for, 87 Real assets, 547 Realized quadratic variation, 755-757 Realized returns, Sharpe ratio and, 395n Realizqd volatility, 756n for IBM, 757 Real options, 547-580 Rebate deferred up, 7 15 short, l 6 Rebate option, 450, 7 16-7 17 Rebonato, R., 8 12 Receiver swaption, 27 1 Recombining tree, 324-325 Recovery rate, 842 for bonds, 850-851 Reduced form bankruptcy models, 852 Reference asset (obligation),860 Regressionts). hedges and, 1l5n Regression beta, hedging and, l 15n, 154 Regulation, hnancial engineering for, 490-498 Regulatory arbitrage, 2-3, 4 Regulatory capital, value at risk and, 8 15 Re-hedging, frequency of, 431-432 Reiner, E., 7 l0n, 72711 Reinsurance, insurance companies and, 437 Reinvestment dividend, l32 Reload option, 532-534 Rendleman, R. J., Jr.. 313, 793n Rendleman-Bartter model, 785-786 Rennie, A., 662:1, 674 Repo rate, 16 implied, 135 Repos (repurchaseagreements), 233-235 haircuts and, 234 Long--ferm Capital Management (LTCM) crisis ( 1998) and, 236 Repricing, 531-532 Repurchases, put sales as hedge against, 522 Research and development, as capital expenditure, 563-565 Restart options, for oil production, 572-578 Returnts) continuously compounded, 353-354 standard deviation of, 354-355 596 variance of continuously compounded,
%.Ilq D Ex
958
Return distributions, bootstrapping of, 831-832 Reverse cash-an-carry, 137 Reverse cash-and-carry arbitrage, l77 apparent,175-176 leaserate and, 180 Reverse conversion, 285 Reverse repo, 234 Revlon stock, constructive sale of, 491 Rho, 383, 387-388, 411 Risk, l See also specific types basis, 116, 196-198 bondprice, 818 of coupon bond, 225 crediq 11, 15, 30-31, 8 15, 841-867 .
diversihable,6 dividend,434 durationas measure of. 237 of extreme price moves, 432.-433 in foreign stock index, 718-7 19 insurancepurchases arld, 97 jump, 764-765 market,815 of market-maker, 414-417 nondiversisable,6 operational,815 of in insurance. 436 poolinwi of put options, 47
in short-selling, 15 swapsand, 263 valueat risk and, 813-839 volatility,741 Risk arbitrageurs, 541 ltisk-averse investor, 105. 346 decliningmarginal u'tility and, 369-370 risk-neutral process and, 661 Risk-free bond, valuing, 372 Risk-free rate of return, Monte Carlo valuation and, 6 l 7-618 Itisk management, 2, 91-1 19 buyer's perspective on, 98-100 cash-and-carry as, 137 nonfinancial, 105-106 producer's perspective on, 91-98 reasons for, 100-107 for stock-pickers, 154 Risk measures, subadditive, 837-838 Risk-neutral distribution, value at risk and, 835-837
-
IN D Ex Risk-neutral
investor, 34*347
ltisk-neutral measure, 662-663
Risk-neutral pricing, 320-321, 343, 34*350,
369-374,690-693 as forward price, 553 reasonsfor success of, 373-374 for valuing claim on S', 668
Risk-neutral probability,
321, 346, 347, 7 13, 725,
726
Risk-neutral process, 660-663
for dS' Qb,670-672
Risk-neutral valuation, of stock, 373 Risk premium fo'nvardconlact earning of, 140 fonvardprice bias by, 140-141 of option, 394 of portfolio, 395 Sharpe ratio and, 659 valueat risk and, 83*837 Risk-sharing, 5-6 Risky stock, valuing with real probabilities, 372 Rogers, D. A., 107 Roosevelt, D., 858n Rosansky, V. I., 172n Ross, S. A., 359, 555, 690, 787-788 Rouwenhorst, K. G., 172n Rubinstein, M,, 292n, 359, 555, 709, 710n, 72411, 727, 73511 Russian doll CDOs, 854 Ryan, M. D., 728
Sales, constructive, 49l Saly, P. J., 532, 53411 S&P 500 futures contract, 143-144 S&P 500 index arbitrage and, 147-149 volatilit'y estimates for, 745 S&P Depository Receipts (SPDRS), 148:1 Scarcity in short-selling, 15-16 Scholes, Myron, 375, 376, 503. See also Black-scholes formula Schwartz, E. S., 516n, 56311,565, 565:1,574n, 633, 635 Scott, L. O., 76811 Seasonality corn folavard market and, 188-19 1
in dividend payments, 13211 natural gas and, 191-195 SEC. See Securities and Exchange Commission
(SEC) Securities. See also specic types Treasury, 20511 Securities and Exchange Commission (SEC), 847 derivatives reporting requirements and, 10 Seculity design, 4 snancial engineering and, 473-498 Semlinancing, 419 portfolio and, 422 Seller, short as, 23 Seniorities, of debt-holders, 51 1 Senior tranche, 511 CDOs and, 854, 856 SFAS. See Statement of Financial Accounting
Standards (SFAS) Shao, J., 83111 Shapiro, A. C., 235 Sharets), convrtible bond exchanged for, 513 Share-equivalent of option, delt as, 383-384 Share repurchases, 510, 522
Sharpe, W. F., 313 Sharpe ratio, 39511, 659-660, 663, 688 bond plicing model and, 782-783 of option, 394-395 Shiller, Robel't, 6 Shimko, David, 86n Short (seller),23 Short call proft, 53 Short fonvard contracq collar and, 75 Short folavard.position,44-45 Short forward prolit, 53 Short position, 12, 44-45, 52 bonds, repos, and, 235 insuring with cap, 62-63 Short put proht, 53 Short rebate, 16 Short-sale, 12-14 cash flows and, 14 risk and scarcity in, 15-16 of wine, 13-14 . Short-term interest rate, 783 Shout option, European, 739 Shoven, J. B., 528, 529 Shreve, S. E., 650n, 652n, 662:1.664n, 674 Shut-down, of oil production, 572-578
%. 959
Siegel. D. R., 115, 56511, 689n
Siegel, J. J., 299 Simulatiops of lognonnal stock prices, 623-624
Single-barrel extraction of oil under crtainty, 565-569 under uncertainty, 569-570 Single Ipme credit default swaps, 860-861, 865 Single-payment swap, 247 Single stock f'utures, l43 Singleton. K. J., 85211 Smith, C. W., 103n, 107n Smith, D., 243n Smith, Randall, 49211 Solnik, B., 258n Soutlnvest Airlines, jet fuel hedging by, 199n Spark spread, 560, 561 SPDRS. See S&P Depository Receipts (SPDRS) Special collateral repurchase agreement, 234 Speculation, 2 hnancing wit repos, 235 on foreign index, 728-729 short-selling and, 13 on volatility, using options, 78-85 Speculative grade bonds, 847 Spot price, 23 Spread, 70-73, 7 1 bear, 72 bid-ask, 11-12 box, 72-73 bull, 7 l commodity, 195-196 crack, 196 crush, l95 ratio, 73 vertical, 7 1 Spread option, 562 Stable distribution, 592 Stack and roll, 198 Stack hedge, 197, l98 Staged investment, 564 Standard and Poor's, 84711.See also S&P 500 entlies Standard deviation estimate of, 607 of returns, 354-355 Standard normal density, 587-588 Standard normal distribution, 409-410 Standard normal probability density function, 409
k. Ix oEx
960
Standard normal variable, conversion of normal random variable to, 590-59 1 Statement of Financial Accounting Standards (SFASIIZ3R, 525, 527, 532, (SFAS) 133, 106 Static NPV. 548-549, 555 Static option replication, 433 Steiner, R., 23311 Stiglitz, J. E., l30n Stochastic diferential equations, 649 Stochastic process, 650, 768n Stochastic volatility 744, 763, 772
Stock alternative ways to buy, 127-128 calls on nondividend-paying, 294-295 cash flows for, 283 with discrete dividends, 36 1-365 forward contrats on, 133-141 options to exchnge, 288-289 prepaid fonvard contract (prepay)on, 128-133 risk management for picking, 154 risk-neutral valuation of, 373 short-selling. 14 synthetic, 285 value at risk for, 8 15-8 19 Stock index, 22-23 fonvafd contract vs. option and, 37 option on, 330-331 Stock index futures, 22 Stock options. See Compensation options',
Optionts) Stock prices bond valuation based on, 520 conditional expected price, 603-604 current price as prsent value of future price, 129, 179n jumps in, 672-674 lognormal model of, 595-598 portfolio value as function of, 827 as random walk, 352-353 simulating correlated, 643-645 simulatingjumps with Poisson distribution, 639-643 simulating sequence of, 623-624 standard deviation correspondence and, 602 Stock price trees, option price trees and, 442 Stock purchase contract, 495-496 Storage as carry, 18 l
IXDEX
of corn, 189-190 costs,fonvard plices. and, 18 1-1 82 costsand lease rate, 182 of electricity, l 72-174 of natural gas, 19 1-195 positivelease rate and, 17*178
Straddle, 78-80
at-the-moneywritten, risk profitdiagram for, 87 on single stock, 823-824 strangleand. 78-79, 80 written,79-80
of, 822-823
Straddle rules, 58 Stragle, 78-79, 80, 87
Strategic options, 558 futureinvestment options as, 558 Stratified sampling, 632
Strike
foreignequity call struck in domestic 729-730 pricingoptions for, 769
currency,
Strike price, 32, 299-304, 547, 560
averageas, 446.-447 convexityand, 304 ' effecton option price,
299-304 risk management and, 97-98 Strike price cpnvexity, 30 l Strip, Eurodollar, 158 Strip hedge, 197 Strip, interest rate, 223 STRDS, 206 Structured note, 474-482 Stulz, R., 8 l4n Style, of options effect on price, 293 Subadditivity, 837-838 Subordinated tranche, CDOs and, 854, 856 Subrahmanyam, M. G., 258n Supershares, 709 Supply, of corn, 190 Swapts), 125, 247, 279, 826 accreting, 263 amortizing, 263 asset, 255 cash flows from, 249 collateralization and, 486:1 commodity, 247-254, 268-270 computing rate, 257-258 currency, 264-268
default, 273-274, 858-863 deferred, 261-262 financial settlement of, 248-249 as fonvard rate agreements, 258n implicit loan balance of, 260-261 interest rate, 254-263 market value of, 253-254 physical settlement of, 248 prepaid. 248 total rate of return, 864 total return, 272-274 with variable quantity and price, 269-270 variance, 758-759 volatility, 759, 828-830 Swap counterparty, 250-251, 255-257 Swap curves, 258-260, 270 Swap price, 250 Swap-rate calculation, 274 Swap spread, 259-260 Swap tenor, 254 Swap term, 214
Swaptions, 27 1 Synthetic commodity, 171, 172n Synthetic fonvards, 66-70, 1l2, 135-1 36, 16 l box spreads and, 72-73 in market-maldng and arbitrage, 136-138 Synthetic FlkAs, 2 16-2 18 Synthetic Nikkei investment, 719 Synthetic options, 285-286 Synthetic stock, parit'y and, 285 Synthetic T-bills, 285
Tailing, 132 Tail VaR, 832, 83*837 Tax-deductible equity, 495-498 Txes box spreads and, 74 corporate deferral, 492-495 for derivatives, 58 on employee options, 527n on equity-linked CD, 48414 hnancial engineeling for, 490-498 hedging and, 104. 106 individual deferral. 491-492 option prices and, 34l Taylor, H. M., 69 1n, 69211 Taylor series expansion, of bond price, 22811
k
961
T-bills. See Treasury bills Term repo, 234 Terrolism, futures on, 24 Theta, 383, 387, 428, +42 delta-hedging and, 425-426, 427 formula for, 4 10-41 1 hedging and, 420 for put options, 390 Thiagarajan, S. R., l07 TIBOR (Tokyo Interbank Offer Rate), l60 Times Mirror, 492-495 Time-varying volatility (ARCH), 747-751 T-note. See Treasury-note futures Total rate of return swaps, 864 Total returfl payer. 272 Total return swap, 272-274 TRACERS, 865 TRAC-X, 865 Traded present value, 550 Trades, arbitrage and, l39 Trading of derivatives, 7-10 proprietary, 4l4 Tranched CDO claims, 853 Tranched CDX, 866 Tranches, 5 l l ratings of, 85+-856 Transaction costs, 2 in bonds, 233 future overlays and, l 51 hedging and, 106 no-arbitrage bounds with, 138-139 Translation invariance, 83711 Treasury bill rate. monthly change in ( 1947-2004), 9 Treasury bills, 244-245 LIBOR vs. 3-month T-bills, 22 1-223 quotations for, 2/14 stocks and, 150 synthetic, 172n, 285 yield on. 147 Treasury bondts). See Bonds; Treasury-bond futures Treasury-bond futures. 230-233 Treasury-note futures, 230-233 Treasury securities. See t7/'tp Treasury bills issuance of, 205n Trottman, Melanie, 199n
%.INDEX
962
lxoEx
of inhnite oil reserve, 570-572 of 1og contract, 761-762
True probabilities pricingoptions with, 369
Monte Carlo, 617-645 of oi1 producing hrm, 57 1 of option for electricity and gas production,
valuationwith, 619-621
Trust, tax-deductible equity and, 495-498 TRW. See Northrop Grumman-TlkW merger Tu. D., 831n Tufano, P., l07 Turnbull, S. M., 852n Twin security 550 Two-parameter distribution, 587 Two-period Europxean call, 323-326
560-563 of option to invest in oil, 57 1-572 investment horizon, of project with z-year 554-558 of project with inhnite investment horizon, 558 with risk-neutral probabilities, 618, 619 with true probabilities, 619-621 utilitpbaed, 369-37 1
Uncertainty discounted cash llow and, 551-552 investment under, 551-558 quantity, 116-1 19 single-barrel extraction under, 569-570 Upderlying asset, 21 Unfunded CDs. 862 Uniformly istributed random variables, sums of, 622 U.S. TM Code, on capital income, 74 Unit of denomination, nureraire as. 693 Up-and-in cash call, 7 15 Up-and-in cash put, 71/1-7 15 Up-and-in options, 450 Upand-out cash call, 7l5 Up-and-out cash put, 715 Up-and-out-currency put options, 452 Up-and-out option, 450, 452 Upper DECS, 497 Upward-sloping yield curve, 107n Utility-based valuation, 369-371 Utility weights, in high and low states of economy, '
'
372
Valuation. See also Monte Carlo valuation of American options, 633-636 bond,520 undercertainty, 679-681 of claim on S'', 66*672 of claim on S Qd>670-672 of compensation options, 524-525, 525-526 of derivatives on cash flow, 552-554 equation of, 680
Valu
at risk (VaR.), 813-839
alternativerisk measures and, 832-835 for bonds, 82*830 estimatingvolatility in, 830-831
Monte Carlo simulation for, 822-826 for nonlinear portfolios, 8 19-826 for one stock, 8 15-8 17 regulatorycapital and, 8l5 lisk-neutraldistribution and, 835-837 subadditivelisk measures arld, 837-838 for two or more stocks' 8 17-8 19 uses of, 814-815 VaR. See Value at Iisk (Valk) Valiable prepaid fonvard (VPF) contract, 492 Variance, constant elasticity of, 763, 766-767 Variance estimate, variability of, 607n ,
Variance swap, 758-759 Vasicek, O., 779, 781n Vasicek model, 786-787
Cox-lngersoll-lkoss model and, 788-790 Vassalou, M., 846 VBA. See Visual Basic forApplications IVBAI Vega, 382, 386, 41 l for at-the-money 4p-strike options, 386 Vertical spread, 71 Visual Basic for Applications (VBA), 885-906 arrays and, 897-899 Black-scholes formula computation with, 894-895 checking for conditions and, 896-897 creating button to invoke subroutine, 888-889 differences between functions and subroutines, 890 functions can call functions, 889 illegal function names, 889-890
iteration and, 899-90 1 object browser and, 895-896 storing and retrieving variables in worksheet, 890-893 subroutine in, 888 using Excel functions from within, 893-896 VIX volatility index, 743, 757, 762-763 Volatility, 741-773 asymmetric' buttey spread and, 82-83 averaging and, 447-448 Black-scholes model and, 763-773 in bond pricing, 794-796 for bonds and swaps, 828-830 butterny spreads and, 8 1-82 detenninistic changes over time, 772-773 early exercise of option and, 343-346 for electricity and natural gas, 560-562 equitpholders and, 510 estimating, 360, 361, 830-831 Health-larrow-Morton model and, 8 12 hedging an pricing, 757-763 historical, 744-746 for IBM and S&P 500 index, 745 implied, 400-402, 741-7/14 measurement and behavior of, 744-.757 of option, 393-394, 553n pricing of, 759-763 speculating on, 78-85 stochastic, 744 straddles and, 78-80 of various positions, 87 verifying for bonds, 803-804 zero, 322 Volatilit'y clustering, 748 Volatility forecasts, in GARCH model, 753-754 Volatility frowns, 742 Volatilit'y skem 402 in option pricing model, 77 1-773 Volatility srnile, 742 Volatility smirk, 742 Volatilit'y surface, 742 Volatility swap, 759 Vorst, A. C. F., 630n Vorsq T., 863 .
5Varrant,512 put,522-523
k
963
Watkinson, L., 85811 Weather derivatives, 199-200 Webber, N., 8 12 Weston, J., 107 White, A., 76811 White, Gregory L., l08 Wiggins, J. B., 768n Wilmott, P., 674 Writing of covered call, 6,4-65 selling insurance and, 63-64 Written call option, 44-4s, 96 payoff and protit for, 37-38 with different underlying stocks, 824-826 Written options. See also Written call option; Written put Greek for, 383-388 margins for, 57 Written put covered call and, 65 with different underlying stocks, 82+-826 payof'f and proht for, 40-42 profit diagram for, 42, with same underlying stock, 823-824 Written straddle, 79-80 at-the-money, 822-823 butterlly spread and, 8 1-82 Wu, L., 77211 Wyden, Ron, 24 '
.lzi
Xing, Y., 846 XYZ debt issue, default swap and, 860
Yen-based investor, 7 l9, 720-721 Yen fonvard conact, synthetic creation of, 157 Yield bond, 242-2,44 continuously compounded, 213-214 convenience, 182-1 84 dividend, 132 effective annual rate, continuously compounded rate, and, 14811 expected interest rates and, 796-797 verifying bond, 802-803 Yield curve, 208 Black-Derman-Toy model and, 798
964
k. INDEX
bond pricing lrnodel and, 780-78 l upward-sloping. 107n Vasicek and CIR models and, 789 Yield to maturity, 208, 241-245 Yurday, E. C., 858n, 864n
Zero-cost collar, 76-78, l 10-1 l 1, 49 1-492 financing of, 76-77 forward contract as, l l 1-l 12 Zero-coupon bopd, 206-208, 474-475, 779 commodity-linked, 479 default and, 841-843
equitplinked, 476 infening price of, 2 l l-2 12 Macaulay duration and, 229n movement of, 829 payoff, prot, and, 28-29, 61 price of, 684 STRIPS as, 206 valuation of, 783 value at risk for, 827-828 Zero-coupon bond prices, 794-796 Zero-coupon debt, 51 1 Zero pfemium, of fonvard contract, 69 Zero volatility, 322
