P
PREFACE TO THE SECOND EDITION xxiii INTRODUGION
CHAPT'ER
.
1
xxi
Financial Derivatives A Bvief Introductien
1 1 lntroduction 2 2 De6nitions of Types Derivatives 3 3,l Cash-and-carry Markecs 3.2 Priee-l-iscovenr Markecs 3.3 Expiratiun Datc 4 and Yutttres Fonvards 4 1..l Ftltutes 5 Optlons 5.1 6 Swaps
5
7
Some Notation
7
9
6.1 A Simpte Interesr Rate Swap 7 Concluslons 11 References 11 8 11 9 Exercises
10
X
Contents
viii
C
PTER
*
Contents
A Primer on the Arbitrage Theorem
2
1 lntroduction 2 Notatlon
14
2.1 Asset Ptices
15
2.2 States
of the
2.3 Retums
c
PTER
1.1 lnforrnation
'
E
16
17
3.1 A Filst Glance at she Arbitrage Theorem 3.2 Reevance of the Arbicrnge Theorcm of Synrhecic Probabilities
21
and Submartingales
3.5 Normalization 3.6
24
3.7 The No-Arbitrage Contlition 4 A Numerlcal Example PossibiliLies
4.l Cttse 1) Arbitrage 4.2 Case 2) Arbitmge-Frcc
Prices
IncleLerminacy
with
26
27 27 28
29
Lattice Models
29 32
;
CHAPTER
Dividends
6.2 The Case with Foreign Currencies 36 7 Some Genemlizations
34
8 Conclusions: A Methodology for Pricing 37 Assets
Theorem Exercises 11
4
72
Pricing Detivatives 77
2 Pricing Functions 78 78 2.1 Forwards 80 2.2 Options 3 Applicatlon: Another Pricing Metlzod 85 3.1 Example 86 4 The Problem
37
38 9 References 10 Appendix: GeneMization of the Arbitrage
.
1 Inttoduction
of the World
7.3 Discounting
47
Modib und Notatien
i
')
'
7.1 Time Intlex 7.2 States
66 5 Partial Derlvatives 67 5.1 Example Differentials 67 Total 5.2 68 5.3 Taylor Series Expansion 5.4 ordinaryDifferential Equatiorus 73 6 Conclusions 74 z References 8 Exetcises 74
25
6 Payouts and Forei> Currencies 6.1 The Case
46
57 4.2 The Chain Rule 59 4 3 The lntegral 65 Intcgration by Parts 4.4
' '
24
Rates of Retum
4..3 An 5 An Application:
45
46 2 Some Tools of Standard Calculus 47 3 Ftmctions 48 3.1 Random Futwtions of Functions Examples 49 3.2 and Limit 4 Convergence 53 4.1 Tle Dcrivative
E
3 A Basic Example of Asset Pricing
()f
Flows
1.2 Modcling Random Behavifar
17
Equalization
Calculus in Deterministic and Stochastic Environments
1 lntroduction
World
2.4 Portfolio
3.4 Martingales
3
13
and Payoffs
3.3 The Use
*
4,1 A Fint Look at Ito's I-emma 88 4.2 Conclusons 88 5 References 89 6 Exercises
E
38
40
'(
..
84
Contents
X
CI4APTER
.
5
Contents
Tools in Probability Theory
1 lntroducdon 2 Probability
4 Relevance of Martingales in Stochastic MMeling
91 91
126 4.1 An Example of Martlngale Trajectories 5 Propeuies of Examples 130 Martingales 6
92 2.l Example 93 2.2 Random Variable 3 Moments 94
3.l First Two Moment.s .3,2 Higher-ordcr Moments
130 6.1 Example l : Brownian Motion 132 6.2 Example 2: A Squared Ptocess 6.3 Exampe 3: An Exponential Proccss 6.4 Example 4: Right Continuous Mattingales l34 7 The Simplest Martingale
94
4 Conditional Expectations
97
97 4.1 Conditiona Probability of (n-ontlitional Expectarions 4.2 Propertics Some Important 100 Models 5
.5.1 Binomial Distributitm in Financial Markcts 101 5.2 Limiting Propertics 102 5.3 Moments 103 5.4 The Normal Distribution 'Fhe l06 Poisson Disttibutitm .5..5 6 Markov Prxesses and Theit Relevance 109 6.l The Relevance I 10 6.2 The Vcctor Case 7 Convergence of Random Variables 7.1 Types of Cxnvergence and Their Uses 7.2 Weak Convergence t 13 116 8 Conclusions References 116 9 Exercises 1 17 10
135 7.1 An Application 136 7.2 An Examplc Martingale 8 Representations
100
137 8.l An Examplc 8.2 Doob-Meyer Decomposition 9 The First Stochastic lntegral
9.1
PTER
*
6
108
2.2 Continuous--fime
112
1l
.
140
to Finance: Tmding Gains
A Summary
.1
12 Concluslons 13 References 14 Exetcises
E
'
143 144
145
C
P
1.52
152 153 154
R 7 Differentiation in Stochastic Environments *
156 1 Introductlon Metivation 157 2 3 A Fmmework for Discussing
119 120 120 Martingalcs
137
.
Representations
2.l Notation
134
147 1l l A Hedgc 147 11.2 Time Dynamics and Normalization Prohabilior Risk-Neutral 11.3
Martingales and Martingale
1 IntrMuction 2 DeVitions
Application
127
133
10 Martingale Methods and Pricing 146 11 A Pdcing Methodology
'
C
124
l11
3 The Use of Martingales in Asset Pricing
122
4 5
Differentiation 161 of The Incremental Errots 167 One lmplication fsize''
164
Conrent 6 Putting the Results Together l 70 6.1 Stochastic Diffcrentials 17 Concluslons 1 7 l 71 8 References 17 1 9 Exerclses
CI'IAPHR
*
8
The Ito Integral
l69
,
2 Two Generic Models
4.l Nonnal Events 4.2 Rare Evenlns
The 1to Integr'al ls
209
a
213
220
Martingale
220
224
Other Properties of the lto lntep-al
4
226
5 lnterals with Respect to JumpProcesses 228 6 Conclvsions References 228 7
184
226
226
4.1 Existence 4.2 Correlation Ptoperries 227 4..3 Adtlition
.
21 1
Rtemann Sums
8 Exercises
227
228
187 189
in Stochastic
lntegration Envitonments The 1to lnfegrul
1 lntroduction
Integral and
3.2 Pathwise lntehyals
175
C
2O4
PTER
.
10
lto's Lemma
1 Introduction E
196
3.2 '
:
First-order
231
232
3 lto's Lemma .3.l The Nttion
199
230
2 Types of Derivatives
2.1 Example
E
9
208
214 2.4 An Expositoty Example lntegral lto he Properties of t 3 E
190 5 A Model for Rare Events 193 6 Moments That Matter 195 Conclusions 7 8 Rate and Normal Events in Practice l96 8.1 Thc Binomial Model 197 8,2 Nonnal Events 198 8.3 Rare Events of The Behavior Accurrtulated Changes 8.4 202 9 Refetences Exercises 203 10
CHAPTER
of the Ito lntegral
2.3 Def nirion: The Ito Integml
174
l76 2.1 The Wierker Process 178 2.2 The Poisson Process 180 2.3 Examples Back Rare Events 182 to 2.4 3 SDE in Discrete lntervals, Again 183 and Normal 4 Characterlzing ltare Events
.
2.1 The Riemann-stieltjes 2.2 Stxhastic Integration
'
173
1.1 Relevance of the Discussion
and SDES
1.2 The Practical Relevance 2 The 1to lntegral
The Wiener Process and Rare Events in Financial Markets
l lntroduction
xiii
Contents
tf
232 ''Size'' in Stochascic Calctllus
Terms
237
3.3 Second-order Terms
238
3.4 Terms lnvolving Cross Product.s 210 3.5 Terms in the Remainder 240 4 The 1to Formula of 241 5 Uses lto's laemma 5.1 Ite's Formula as
a
Chain Rule
241
242 5,2 6 lntegol Form of lto's Lemma 244 Fomuuln Complex Settings 7 lto's in More Ito's Formula as an Integration
Tool
245
K.
Contents
xiv
MuLtivariate Case
7.2 Ito's Formula 8 Conclusions 9 References 10 Exercises
CHAPTER
*
11
and
245
1.l
4 Partial Differential Equations
248
250 251 251
252
on at and
253
tz'r
2 A Geometric Descrlption SDES
of Paths
SDES
255
3.1 Wha: Does a Solutton Mean? 256 3,2 Typcs of Soluriorvs 3.3 Whic Solution Is to Be Prefetzetl? 3.4 A Discussion of Strong Solutions 3.5 Veritication t)f Solutions o SDES l6? 3.6 An Important Example
4 Major Models Lincar Constant
of
SDES
Coefficient
258 2,61
*
12
of
290 292
292 PDES
292
10 Exercises '
13
280
294
The Black-scholes PDE An Applfcltion
2.1 A Geomerric Look
3
PDES
296 PDE
4 Exotic Options Lookback (Jptions
296
Black-scholes Formula
at the
in Asset Pricing
.3.l Constant Dividends
276
293
293 294
8 Cotclusions 9 References
1 lntrMuction 2 The Black-scholes
Eqwafo,xs
1 lntrvuction 275 Riskree Portfollos 2 Forming 3 Accumcy of the Metbod
290
6. 1 (-aircle 6.2 Ellipse 6.3 Pamboa 6.4 Hyperbola
266
Pricing Derivative Products Purtiul Di#erevaial
Equations
CI-IAPTER
265 SDF.S
6
286
Bivariate, Second-Degree 289
A Reminderl
7.l Exampie: Pambolic PDE 258
284
Example 1: Lincar, First-cder PDE 5.2 Example 2: Linear, Second-order PDE
7 Types
255
4.1 267 4.2 Geometric SDL 269 4.3 Square Root Process 270 4.4 Mean Reverting Process 27l 4.5 Omstein-uhlenbeck Process 271 5 Stochastic Volatility 272 6 Concluslons 272 7 References 273 8 Exercises
CHAPTER
lmplied by
254
3 Solution of
282
?83 4.1 Why ls tle PDE art 'sEquarton''? 283 the Btaundary Conditicm? Wha Is t 4.2 PDES 284 5 ClassiEcation of
The Dynamics of Derivative Prices stocetic Dffferenfzl Iqutfffond
Conditions
282
3.1 An Interpretation
Jumps
1 lntrodttction
XV
Ctnntenrs
299 300
301 30l
4.1 301 4.2 Ladder Opions Trigger Knock-in Options or 4.3 302 4.4 Knock-out Options 302 Exocks 4.5 Other 303 4.6 The Relevant PDF-S PDES Practice in 5 Solving 304 5.l Closed-Fonn Solutians
302
304
298
xvi
Contents 5.2
Numerical Solutions
6 Concluslons 7 References 8 Execises
CHAPTER
*
14
Contents
5.1Detennining ?
306
309
3.2 The lmplied
.
310 310
4.1 Cakulation
as
'sMeasure''
5.1 Equivalence of rhe Two Approaches 5.2 Crtitral Sreps of the Derivation 5.3 Inregml Form of the ito Formula
312
2 Changing Means
517 321
5.1 A Normally Distributcd Random Variable 325 3.2 A Normally Distributed Vcctor 327 3.3 'I'hc Radon-Nikolym Derivative Measures Equivalent 328 3,4 4 Statement of the Girsanov Theorem 5 A Discussion of the Glrsanov Tlzeorem
5.1Appliation
to
Probabilides
C
PTER
.
15
CHAPTER
353
*
16
331
5 Conclusions 6 References 7 Exercises
337 340
364
New Results and Tools for lnterest-sensitive Secutities
4.l Drift Adjustment 4,2 Term Stnlcture
334 Equlvalent
36$
365
368 369 Interest Rate Derivatives 375 Complications
3 4
329
359
366 366
1 lntrMuction 2 A Summary
332
SDES
6 Wlch Probabilities? 7 A Method for Generating 7.1 An Example 8 Conclttsions 9 References 10 Exercises
6 Conclusions 2 References 8 Exercises
316
2.1 Merlzoti 1: Operating on Possible Values 2.2 Method 2: Omnating on Prohabilities 3 The Girsanov Theorem 322
Formula
356
and PDE 5 Comparing Mlrtingale 358 Approaches
Pricing Derivative Products
Probablity
352
4 Applicadonk Tbe Black-scholes
Eqtdvtzlcnt Mdzungtge Metsures 312 1 Translations of Probabilities 1l
350
SDF-S
37 1
.376 377
377 378 378
342
CHAPTER
342 343
Arbitrage Theorem in a New Setting
379 1 lntroduction 2 A Model for New lns%ments
Aplicctfrnx
2.1 The Momenr-oenerating
17
Normulfwfon und Rundom lnteredt Rates
Equivalent Martingale Measures
345 1 lntrMuction 2 A Martlngale Measure
.
2.1 'T'he New Environment 346
Function
2.2 Nonnalication
Pnxesses 2.2 Conditiorkal Expectation 3 Converting Asset Prices into Martingales of Geomerric
2.3 Some Undcsirable
346
389 Prommies
2.4 A New Normalization
348
349
2.5 Some Jmplicarions
381
383
392 395
399
F Contents
?(hpiii
3 Conclusions 4 References 5 Exercises
Contnts
404 404 404
440
3.3 Intemretation
3.4 The rt in rhe HJM Approac'h 441 3.5 Another Advantage of the HJM Afproach
3.6 Marker Pracriue
CHAPR'ER
*
18
3.4
Moving to Continuotls Xme Yiclds antl Spor Rates 418
4l2
414
CHAPTER
l9 4.: Discrete 'Tlme 4.? Moving tg Contintlokls Xme
Relationships
6 References 7 Exercises
419
of the
'
423 424 424
5.1 Case 1: A Deterministic
'
CI-IAPTER
19
426 1 Introduction 2 The Classieal Approach
427
3. l Vfhich Forward Rate?
436
.3.2 Arlnicrage-Free Dynamics in HJM
rt
6 Concluslons 7 References 8 Exercises
.
C
'
PTER
*
21
464
465 465 465
Relating Conditional Expectations
to
PDES
1 Introduction 467 2 From Conditional Expectations to
' E
2.1 Case l : Constan.t Discount Factors
434
2.2 Case 2: Bond Pricing
435
472
2.3 Case 3: A Generalizacion
:'
2.4 Some Clatifkations
437 :
460
461
rt Complex 3: More Forms Case 5.3
;
2.1 Example 1 428 429 2.2 Exampe 2 2.3 The General Case 429 the Modcl Race Spfat 2.4 Using 432 2.5 Comparison wkth the Black-scholes World 3 The HJM Approach to Temn Structure
455
zto
5+2 Casc 2: A Mean-Revcrcing
Classical and HJM Approaches to Fixed lncome
444
Classical PDE Analysis for lnterest Rate Derivatives
5 Closed-Form Solutions of the PDE
: .
20
451 1 lntroduction The Fmmework 2 454 3 Market Price of lnterest Rate Risk 457 4 Derlutlon of the PDE 459 4.1 A Comparison
420
5 Conclusions) Relevance
*
'
117
4 Forward Rates and Bond Prices
441
4 How to Fit rt to lnitial Term Structure 445 4.1 Monte Carlo 446 4.2 Tree Models 447 4.3 Closed-Fonn Solutions 447 5 Conclusions Refetences 447 6 448 7 Exetcises
Modeling Term Structure and Related Concepts
1 lntroduction 407 2 Main Concepts 408 nree Curves 409 2.2 Movements on the Yield Cunre 3 A Bond Pricing Equation 4l4 (zonsrantSpot Rate 416 3.2 Scochastic Spot Rates
443
475
475
PDES
469
Contents
XX
W'hich Drifc?
476
2.6 Atlorher Kmd Price Formula 479 2.7 Wlch Forrrtula? PDES 479 to Conditional Expectations 3 From and Other 4 Generators, Feynman-lfac Formula, Tools 482 482 4.l Ito Diffusions 483 4,2 Markov Property 483 4.3 Generattlr of an Ito Diffusion A 484 4.4 A Reprcsentation for 485 4.5 Kolmogorov's Backward Equation 487 5 Feynman-lac Formula
6 Conclusions 7 References 8 Fxercises
CHAPTER
.
22
487
487 487 'rimes
Stopping Securities
and American-Type
489 1 lntroduction 2 Wiw Study Stopping Times? 2.1 American-style Securicics
492 3 Stopping Times of Times 493 Stopping Uses 4 494 5 A Simpliied Setting The Model 494 5.l 499 6 A Slmple Example and Martingales Stopping Times 7 7.I Mmingales
7.7 Dynkin's Fonuula
8 Colulusions 9 References 10 Exercises
BIBLiOGRAPHY 513 JNDEX
504 504
491
492
5O4
nis edition is divided into hvo parts, The lirst part is essentially the revised and expauded vrsiotl of the rst edition and consists of 15 chapters. The sccond part is entirely new and is made of 7 chapters on more recent and more complex material. Overall, the additions amount to nearly doubling thc content of the first edition, The srst 15 chapters are revised for typos and othcr errors and are supplemented by several new sections. The major novelty, howevcr, is in thc 7 chapters contained in the second part (f the book. These chapters tzse a similar approach adopted in the Erst part and deal with mathematical tools for fixed-income sector and interest rate products. The last chapter is a brief introduction to stopping timcs and Arnerican-style instruments. The other major addition to this edition arc thc Exercises added at the ends of thc chapters. Solutions will appear in a separatc solutions manual. Several pcople provided comments and helped during thc proccss of revising the first part and with writing the seven new chapters. I thank Don Chance, Xiangrtmg Jin, Christina Yulzzal, and the four anonymous referees Who provided very useful comments. The comments that I received from numerous readers during the past threc years are also greatly appreciated.
505 505 505 509
Kx
This book is intended as background reading for modern asset pricing thcory as outlined by Jarrow (1996),Huli (1999),Dufie (1996),Ingersoll and other excellent sources. (1987), Musiela and Rutkowski (1997), require, by their very nature, models hnancial derivatives Pricing for utilization of continuous-time stochastic processes. A good understanding of the tools of stochastic calculus and of some deep theorems in the theory of stochastic processes is necessary for practical asset valuation, ncre are several excellent technical sources dealing with this mathematical thcory, Karatzas and Shreve (1991),Karatzas and Shreve (1999), and Revuz and Yor (1994) are the first that come to mind. Others are discussed in the references. Yct cvcn to a mathematically well-trained reader, these sources are not easy to follow. Sometimes, the material discussed has no direct applications in Iinance. At otlner times, the practical relevance of the assumptions is difficult to understand. The purpose of this text is to prtwide an introduction to the mathematics utilizcd in the pricing models of derivative instnzments. ne text approacbes the mathematies behind continuous-time finance informally. Examples are given and relevance to hnancial markets is provided. Such arl approach may be found imprecisc by a tcchnical reader. We Simply hope that the informal treatment provides enough intuition about Some of these dicult concepts to compensate for this shortcoming. Untflrtunately, by providing a descriptive treatment of these concepts, it is diflicult to emphasize technicalitics. This would defeat thc purpose of the book. Further, there ac excellent sources at a tcchnical level. What sccms to be missing is a text that explains the assumptions and concepts behind XX11l
lntroduction
xxiv these mathematicai tbeory.
Introduction
Trading became cheaper. ne deregulation of the hnancial scrvices that 1980s was also an important factor here. Bathcred steam during the nree major steps in the theoretical revolution led to the use of advanced mathematicalmethods that we discuss in this book:
tools and then relates thcm to dynamic asset plicing
1
Audience
xKarbi-
'inew''
*
Black-scholes pzee/ (Black and Scholes, 1973) used the method of arbitrage-free pricing. But the paper was alm inquential because of the techriic.al steps introduced in obtaining a closed-form formlzla for options prices. For an approach that used abstract notions such ms Ito calculus, the formula was accurate enough to win tlze attention of market participants.
@
ne methodology of using equivalent martingale measures was developed later. nis method dramatically simplised and generalized tbe original approach of Black and Scholes. With these tools, a generat method could be used to price any derivative produd. Hence. arbitrage-free prices under more realistic conditions could be obtained,
'
fluring the past two decades, some major developmcnts have occurred in the theoretical understanding of how derivative asset prices are determined and how these priceg move over tne. There were also some rccent institutional changes that indirectly made the methods discussed in the following pages popular. ne past two decades saw the freeing of exchange and capital controls. This made the exchange rates significantly more variable. In the meantime, made the elimination of currency risk world trade p'ew sigafcantly, a much bigher priority. During this time, interest rate controls were eliminated. This coincided with increases in the government budget descits, which in turn 1ed to large new issues of government debt in all industrialized nations. For this reason (among others), the need to eliminate the interest-rate risk became more urgent. lnterest-rate derivatives became very popular. It is mainly the need to hedge interest-rate and currency risks that is at the origin of the reecent prolc increase in markets for derivative products. This need was partially met by :nancial markets. New prducts were developed and offered, but the conceptuai understanding of tbe strucure, functioning, and pricing of these derivative products also played an imporapplicable to tant role, Because theoretical valuation modeis were directly able price to these new products, nancial intermediaries were understanding the of clear and successfully market them. Without such a developsimilar what extent a conceptual framework, it is not evident to ment nlight have occurred. As a resuit of these needs, new exchanges and marketplaces came into of new products became easicr and iess costly. elstence- lntroduction
f/lctprczzzl
gives the formal conditions under which trage'' prohts can or cannot exist. lt is shown that if asset prices satisfy a smple condition, then arbitrage cannot exist. This was a major development that eventually permitted the calctklaticm of the arbitrage-free derivative product. Arbitrage pricing must bc conpricc of any trasted with equillbrium pricing, which takes into consideration conditions other than arbitrage that are imposed by general equilibrium.
. The arbitrage
ne text is directed toward a reader
|
2 New Developments
xxv
: '
'
rrhis
:
' .
ne
Finally, derivative products have a property that makes them especially suitable for a mathematical approach. Despite their apparent complexity, derivativeproducts are in fact extremely simple instruments. Often their value depends only on the underlying asset, some interest rates, and a few parameters to be calculatcd. lt is signiEcantly easier to model such an inStrumcnt mathematicallyz than say, to model stocks. The lattcr are titles On private companies, and in general, hundreds of factors inquence the p erformamce of a comparly and, hence, of the stock itself.
Objectlves
Rcorrectly''
We have the following plan for learning the mathematics products. '
lnls
is sometimes
called
'sthe
Fundamental neorem of Finan.'' zTllis is espedaliy true if one is armed wilh the arbltrage theorem.
of derivative
Introduction
Introdtzction
xxvii
Theorem 3.7 The Qfr-stz''ntx/
3.1 The Arbitruge Thcotem rlahe meaning and the relevance of thc arbitroge theorcm will be introduced lirst, Tbis is a major result of the theory of inance. Without a good understanding of thc conditions under which arbitrage, and henee insnite mathematics that prolits, is ruled out, it would be difficult to motivate the we intend to discuss. 3.2 Rtsk-Nxfl'ul Pxobabilifieg the The arbitrage thcorem, by itsclf, s sufhcient to introduce some of arbithe In particular, later. disctlss main mathematical concepts that we mathematical f'rtzvlxrwrand, morc important, trage theorcm provides a of risk-neutzal prfbabilitics. The latter utilization antl justifies the cxistence valuing assets, utilized in probabilities ney make it posslble are risk related premiums. to to bypass issues usynthctic''
3.3 Wfcnemand Poisson Procenses A1l of these require an introductory discussion of Wiener processes from assumptions'' the a practical poimt of vicw, which means learning and differencalculus. stochastic behind notions such ag Wicner processes, Reconomic
tial cquations.
3.4 Ncw Czctflzu In doing this, some familiarity with the ncw calculus necds to be develsimple oped. Hence, we go ovcr somc of thc basic results and discuss some
examples.
3.5 Mttingules valuation At this point, the notion of martingales and their uses in asset utilized in should be introduccd. Martingale measures and tbe way they are valuing asset prices are discussed with examplcs.
3.6 Pcrtiul Differerttfal Equufitnz.s obtainpcrtic/ Derivative asset valuation utilizcs the nttion tf arbitragc to the pries of these dzfferential equtions (.PDEs) tbat must be satisfied by and equations differential products. We present the mathematics of partial their numerical estimation.
ne Girsanov theorem permits chan/ng means of random processes by varying the underlying probability distrlbution. The theorem is in the background of some of the most important pricing methods.
3.8
ne
Fcynpum-Ktzc Fonnulu
The Feynman-lac formula and its simpler versions give a correspondence between classes of partial differential equations and certain conditional expectations. nese expedations are in the form of discounted future correspondence is useasset prices, where the discount rate is random. ful in pricing interest-rate derivatives. rntis
3.9 Extmple: The text gives as many examples as possible. Some of these examples have relevance to financial markets; others simply illustrate the mathematical concept under study.
'
.
-
-v*
.=...=X=%
-
< .
.-
-mt
::, =
.
,
=
=
a
,
1-nancial
.
*-
,
.
erivatives
A Bricj lntroduction
r'
'
'
1 lntroduction This book is an introduction to quantitative tools uscd in pricing fmancial dcrivativeg. Hence, it is mainly about mathematics. It is a simple and heuristic ltroduction to mathematical concepts that have practical use in hnancial markets. Such an introduction requires a discussion of the logic behind asset pridng. In addition, at various points we provide examples that also require an understuding of format assct pricing mcthods. A1I hcse necessitate a brief discussion of the securities under consideration. This introductory chapter has that aim. Readcrs can consult other books to obtain morc background on derivatives. Hull (2000)is an excellcnt source for derivatives. Jarrow and Turnbull (1996)gives another approach, The more advanced books by Ingcrsoll ( 1987) and Dufie (1996)provide strong lin kA to the underlying theory. manual by Das (1994) PTovides a summary of tlze practical issues associated with derivative contracts. A comprehensive new source i Wilmott (1998). This chapter first deals wit.h the two basic building blocks of llnancial derivatives: options and folwards (futures),Next, wc introduce thc more complicated class of derivatives known as swaps. nc chapter concludes by showing that a complicated map can be decomposed into a number of forwardsand options, This dccomposition is very practical. If one succeeds in pricing fomards and options, one can then reconstitute any and obtainits price. nis chapter also introduces some formal notation swap that will be used tbroughout the book. rrhc
.
1
F
C H A PT ER
2
*
1
.3 Trpes
Financial Derivatives ' '
2 Deinitlons In the words of practitioners,
:
Tderive'
uderivative
DEHNITION: A fmancial contract is a derivative Jcclfzy., or a continby the determined oactly date T is value at expiration gent clfpz, if its (Ingesoll, time F at cash instrument of the underlying market price 1987), Hence, at the time of the expiration of the derivative contrad, denoted by F, the price F( F) of a derivative asset is completely determined by Sv, aunderlying asset.'' After that date, the security ceases to the value of the exist. nis simple characteristic of derivative assets plays a very important role in their valuation. In the rest of this book, the symbols F(Mand Fst, tj will be used alterlmderlying nately to denote the price of a derivative product written on the assumed to yield asset S: at time t. ne inancial derivative is sometimes will always denote the is F times, the other At payout zero, dt. payout a expiration date.
3 Types of DeHvatives We ean group derivative securities under three general headinp; 1, Futures and fomards 2, Options 3. Swaps Fomards and options are considered basic hufltfrlp bloc. Swaps and wbich some other complicated structures are considered hybrid securities, and options. forwards basic of decomposed into eventually sets be can We let St denote the price of the relevant casb instrument. which we call the underlying We can list live main groups of underlying assets; returns gencrated in the production 1. Stocks: nese are claims to and senices, sector for goods 2. Currencies: nese are liabilities of governments or, sometimes, banlcs. 'rhey are not direct claims on real assets.
' i J.
: ; :
(1994).
3. Interest rates: In fact, intcrcst rates are not assets. Hence, a notional asset nceds to be devised so that one can take a position on the direction of future intcrest rates. Futures on Eurodollars is one example. I11this category, we can also include derivatives on bonds, notes, and 'Ilbills, which arc government debt instruments. ney are promises by governments to make certain payments on set dates. By dealing with dcrivativcs on bonds, notes and T-bills, one takes positions on the direction of various interest rates. ln most casesyz these derivative instruments arc not notionals and can result in actual delivery of the underlying asset. 4. Indexes; ne S&P-500 and the FTLSEIOOare tw'o examples of stock indexes. The CRB commodity index is an index of commodity prices. Again, themselves. But derivative contracts can be written these are not on notional amounts and a position taken with respect to the direction of the underlying indcx. main classes are j. Commoditics: 'rhc
Soft commodities'. cocoa, coffce, sugar and oilseeds'. barley, com, cotton, oats, palm oil, potato, soyGrains . bean. winter wheat. spring wheat, and othcrs Metals: copper, nickel, tin, and others metals: gold, platinum, silver Prccious . Livestock: cattle, hogs, pork bellies, and others * Energr. Crude oil, fucl oil, and others .
These underlying commodities are not hnancialasscts. Thcy are goods in kind, Hence, in most cases, they can be physically purchased and stored.
There is another method tor our purposes,
j mptrtant
'
;
of classlfy' ing the underlying
asset, which is
.
3.1 Caxlz-urecan'y
d:real''
2-3, Klcin and Ixderman
3
ddassets''
'eccr/y.
'see pagcs
Derivatives
of
Murkets
Some dcrivative instruments are written on products of cash-and-carry markets. Gold, silver, curencies. and T-bonds arc some examples of cash-and-carly products. ln these markets, one can borrow at risk-free rates (bycollatcralizing the underlying physical assct), buy and storc the product, and insure it until thc expiration datc of any derivative contract. One can therefore easily build an alternative to holding a fonvard or futures contract on these commodities. For example, one can borrow at risk-free rateg, buy a T-bond, and hold it until the delivcry date of a futures contract on T-bonds. nis is equivalent z-fhereis a
signilitlant
amount
of trading on
<noticmal''
French government bonds irl Paris.
CHAPTER'
4
and Futures 4 Fonvards
Financial Derivacives
both represent the same thing at time F. So, in the case of contract. o1d fmuresa wc can indeed say that the equality in (1)holds at expiration. At f < F, F(f) may not cqual S3. Yet we can determine a hmctionthat ties St to F(f ). 'rhey
accepting the deiiveryof tEe underlyng into buying a futures contract and similar examples with currencies, strument at expiration- One can comtruct etc.3 gold, silver, crude oil, lnformation about Ptzre cash-and-carry markets have t'mc more propcrty. should not influunderlying instntment fmure demand and supplies of thc After all, this prices. between cash and futures (forward) ence the storage, interest rates, of risk-free spread will depend mostly on the level supplies future concerning information and insurance costs. Arly relevant expccted to make the cash and demands of the underlying instrument is amount. price and the fumrc price changc iyy the same Gspread''
3.2 Price-Didcxeo
Mazkets
marprice tictzvery The second type of underlying asget comes from for instrument underlying the kets, Here, it is phpically impossible to buy either are goods Such date. expiration cash and store it until some future cash market at the time the too perishable to be stored or may not have a spring wheat. When the derivative is trading, One example is a contract on exchange, the corretraded in tic futures contract for this commodity is exist. sponding cash market may not yct asset until some later ne stzategy of borrowing, buying, and storing the Under these markets. pricc-discovery expiration date is not applicable to of the demand supply and lformation the about future conditionsa any Such price. cash corresponding the underlying commodity cannot influcnce terminolor. the hence market. futures information can be dcovered in thc
3.3 Exleuton
4 Forwards and Futures Futures and fomards are linear instruments. This section will discuss forwards; their differences from fmures will be brielly indicated at the end. OEFINITION: A forward contract is an obligation to buy (sel1)an underlying asset at a spccihed forwardpiceon a known date.
,
,
E
.
I
The expiration date of the contract and the fomard price are written when the contract is entered into. lf a forward purchase is made, thc holdcr of such a contract is said to be long in thc undcrlying asset. lf at expiration the cash price is higher than the forward price, the long position makes a proht; otherwise thcre is a loss, The payoff diagram for a simplifed long position is shown in Figure 1. ne contract is purchased for F(/) at time 1. lt is assumed that the contract expires at time I + 1. The upward-sloping line indicates the proft or loss of the purchaser at expiration. ne slope of the line is one. 100
Dutc
derivative, and St, the ne relationship between F(r), the prke of the only deterministically), cxactly kriown (or value of the underlying asset, is naturally of forwards or fmures, we at the expiration date T. In the case expect
F(T)
=
&;
(1)
should be equal to its that is, at expiration the value of thc futures contract
cash equivalent. futures contrad jromisingthe deFor example, the (exchange-traded) value dlfferent from the livery of 100 troy ounces of gold cannot have a the opiration date of the actual market value of 1tX)troy ounces of gold on klp being ve'ry costly. may end sHowever, as in the case of crude oil, tize storagc press crude oil. storc Bnvizonmental and other efects makr it very expensive to
Pmfit
()r
loss
50
AB
' .
.
(;
50 BC
.
price of the
-
15O
F(t) Puzchc plice of futuros contracq =
:
-5Q
. : .E
:
.100
y Ic
uRE
j
200 uaderlying wsset
E
CHAPTER'
6
l
Financial Derivatives
5 Options or loss during holdcr.
Pr0st ur loss
1
thc day is rccorded accordingly in the account of the contract
5
K
'Z''
0
K
't
sale pricc
shou
.
piceof ,I,c 200
150
underlyiug
asset
-K
-100
F IG U R E 2
.
prolit.4 Given exceeds F(r), then the long position ends up with a lf vertical line BC. that the line has unitary slopc, the segment AB equals the the vcrtical being read dircctly be as At time / + 1 the gain or loss can linc and the horizontal axis. distance betwcen this diagram of a n'hort position under similar payoff the Figure 2 displays circumstances, the mechanics of Such payoff diagrams are useful in undemtanding reader can derivative products. In this book we treat them brie:y. consult Hull (1993)for an extensive discussion. u+.1
<tpayoff''
:
rfhe
4.1 Fvtugc.s
.
; major differences can
Futures and fotwards are similar instrumcnts. ne be stated brie as follows. exchange designs a Futures are traded in fonnalized exchangcs. expiration dates. Fonvards are standard contract and sets some specifie custom-made and are tzaded over-the-eounter. clearing bouses, and Fumres exchanges are cleared through exchange default risk, the reduce thcre is an intrjcate mechanism designed to r=rkel. isy every day the That marked to Finaliy, futures contracts are written. Any prolh is contract is settled and simultaneously a new contract rrlle
4Note that becaase tlle contract
cxpirt)s at J + 1,
Sv+L
will cqual F(t + 1).
Options
options constitute the second basic building block of asset pridng. In later chapters we often use pricing models for standard call options as a major example to introduce concepts of stochastic calculus, Forwards and futures obligate the contract holder to deliver or accept the delivcry of the underl/ng instrument at expiration, Options, on the other hand, give the owner the right, but not tbe obligation, to purchase or sell an asset. nere are two types of options. DEFINITION: A European-type call option on a security St is the right to buy the security at a preset strike prcc K. nis right may be exercised at the cxlrt7lftpn date T of the option. The cali option can be purchased for a price of Ct dollars, called the premium, at time l < F. /tpn is similar, but gives the owner the right to sell an A European put speciNed price at expiration. asset at a ln contrast to European options, American options can be exercised any time between the writing and the expiration of the contract. nere are several reasons that traders and westors may want to calculate the arbitrage-free pzice, G, of a call option. Before the option is lirst written at time t, Ct is not known. A trader may want to obtain some estimate of what t.141price will be if the option is written, If the option is an exchange-traded security, it will start trading and a market price will it may also trade heavily and emerge. If the option trades over-the-counter, a price can be observed. However. the option may be traded irtfrequently. Then a trader may want to know the daily value of Ct in order to evaluate its risks. Another trader may think that the market is mispricing the call option, and the extent of thq mispricing may be of interest. Again, the arbitrage-free value of Ct needsto be determined.
:.
5.1 '
.some
Notutoa
'I'he most desirable way of pricing call option is to flnd closed-form a a a function of the underlling
formulafor G that expresses the latter as asset'sprice and tlze relevant parameters.
At time 1, the only known concerning Ct is tlle one that determiues its value at the time of expiration denoted by F. In fact, tdformulan
!
C HA P T ER
1
.
Firlancial Derivatives
SWaPS
lionvalue 60
commissions and/or fees, and . if thcre are no spreads on St and Ct are zero, . if the bid-ask
50
then at expiration, Cz can assume only two possible values. lf the option is expiring out-of-mony, that is, if at expiration the option holder faces Sv
<
C#ion
<
K
=y
Cz
=
< t
30 '
then the option will have no value. ne underlying asset can bc purchased in the market for ,V, and this is lcss than the strike price K. Nt option holder will exercise his or her right to buy the underlying asset at K. Thus, Sv
valuc
40
(2)
K,
Oplion N'altltt at t+l
ga
.:
/*---
.
10
0
0,
optsaavaluo at expiration
(
40
20
60
100
80
FIGU RE
120
140
Sy
4
But. if the, option expircs in-the-money, that is, if at time T,
(4)
5'z. > K, the option will have some value. One One can buy the underlying security at Sv. Since there are no commissions or K. Market participants, being be K usz on thc option, and wc have uy
-
should clcarly exercise the option. price K and sell it at a higher price bid-ask spreads, te net prct will aware of this, will place a valuc of
We can use a shorthand notation wrjtisg Cv
Value ef
Jd
v
K
>
==>
f,'F
=
uy
-
/
J
a.
40 39
/
Oplion'q value expiration
brorc
6
20 10 / i
0
20
-
;
40
/
/ ./
,
optitm's
FlG UR1
IX trike Price
120
IK
.
(6)
swaps
Swaps and swoptions are among some of the most common types of deriva-
value at expiration
tives.But this s not why we are interested in them. It turns out that one method for pricing swaps and swoptions is to decompose them into forWards
Sk
K
60
K, 01
zsz
C.w1Itlplion 50
-
This means that the Cz will equal the greater of the tw'o values inside the brackets. In later chapters, thig notation will be used frequently. Equation (6), which gives the relation between Sr and Cz, can be graphed easily. Figure 3 shows this rclationship. Note that for Sp :GK, the C1. is zero. For values of such that K < &, the CT increases at te SV. Hence, for this range of values, the graph of Eq. (6) is a same rate as straight line with unitary slope. Options are nottlirtear instruments. Figure 4 displap the value of a call option at various times before exPiration. Note that for t < F the value of tlle function can be represented by a smooth continuous cuwe, Only at expiration does the option value become a piewise Iinear function with a UI'I.k at the strike price.
(5)
K.
max (&
.'
-
.
=
to express both of these possibilities by
and options,
'ritis
illustrates the special role played by forwards and basic building blocks and justifies the special emphasis put on as titemin following chapters.
'
options
3
'L
; ..
C H A P T ER
10
*
1
Financial Derivatives
9 Exercises
DEFINITION: A swap is the simultaneous selling and purchasing of cltsh qows involving various currencies, interest rates, and a number assets. of oter snancial
Tls very basic intcrest rate swap consists of exchanges of interest paycounterpatics borrow in sectors where they have an advantage m cnts.Thc then exchange the interest payments, At the end both counterparties and W gl secure lower rates and the nvap dealer will earn a fee. It is always possible to decompose simple swap deals into a basket of simplerforward contracts, The baskct will replicate the swap. The forwardscan then be priced separately, and the corresptnding value of the from these numbers. This decomposition into swap cam be determled of buildingblocks fonvards will signilicantly facilitate the valuation of the
Even a brief summary of swap instruments is outside the scope of tMs book. As mentioned earliez, our intention is to prtwide a heuzistic inlroduction of the mathematics behind derivative asset pricing, and not to discuss the derivative products themselves. We limit our discussion to a typical example that illustrates the main points.
swapcontract.
6.1 A
smplelnfcresf
Rlfe Jtt'up
'g
a swap into its constituent components is a potent example engineering and derivative asset pricing. lt also illustrates the
Decomposing
of snancial special role played by simple fomards and options, We discuss an interest rate swap in detail, Das (1994)can be consulted for more advanced swap structuresJ ln its simplest form, an interest rate swap between two counterpardes and B is created as a result of the following steps:
.4
needs a $1 million tloating-rateloan. Counterparty 1, Counterparty flxed-rate loan, But because of market conditions and B needs a $1 million with relationships various banks, B has a comparative advantage in their rate/ :oating bon-owing at a and B decide to exploit this comparative advantage. Each counter2. party borrows at the market where he had a comparative advantage, and then decides to exchange the interest payments. 3. Counterparty borrows $1 million at a flxed rate. ne interest payments will be received 9om counterparty B and paid back to the lending bank. 4. Counterparty B borrows $1 nlillion at the lloating rate. Interest payments will be received from counterparty W and will be repaid to the lending bank. 5. Note that the initial sums, each being $1 million, are identical. Hence, are called notmal pzntn/f-. they do not have to be exchanged. the also in 'rhe interest payments are same currency Hence, the counterconcludes the interest dlfferentiala. only the interest parties exchange rate swap.
conclusions
'
we have rcvicwed some basic derivative instruments. Our first, to give a brief treatment of the basic derivative twofold: purposewas securitiesso we can use them in examples; and second, to discuss some notation in derivative asset pricing, where one first develops pricing formulasfor smplc btlding blocks, such as options and forwards, and then decomposesmore complicated structures into baskets of forwards and opway, pricing formulas for simpler structurcs can be used to value tions. complicated stnzctured products. more
!
Hull (21400) is an excellent source on derivatives that is uniquc in many ways. Practitioncrs use it as a manual; begirming graduatc students utilize it as a textbook. It has a practical approach and is meticulously written. Jarrow and Turnbull (1996)is a welcome addition to books on derivatives. Duffie (1996) excellent is an sourcc on dmamic asset pricing theory. Howeverpit is not a source on the details of actual instruments traded in the markets.Yet, practitioners with a very strong math background may fmd it useful Das (1994)is a uscful reference on the practical aspects of derivative instruments.
In this chapter,
.
rrhis
..4
I
.4
..4
rrhey
rrhis
sother recent sources on practical appiiraations of s'waps are Dattatreya et aI. Kapnef and Mnmilall (1992). 6nis means that has a comparalive advantage i.n borrowing at a EXGIrae, .z4
(1994)and
,
9 Exercises 1. Consider the following investments:
;
'.
* An investor short sells a stock at a price and wites an at-the-money call option on the same stock with a strike price oj K ,,
Financial Derivatives
CHAPTER'I
.i>.
j'A
* An investor buys onc put with a strike price of K3 and one call option at a strike pricc of Kz with KL S Kz. price A'1, * Am investor buys one put and writes one call with strike and buys one call and writcs onc put with strike price KZLKL :s #2).
,
(a) Plot the cxpiration payoff diagrams in each case. (b) How would these diagrams look some time before expiration?
*
rl *
2. Consider a fixed-payer, plain vanilla, intercst rate swap paid in arrears with the followlg characteristics'.
r ltra
* ne start date is in 12 months, the maturity is 24 months, * Floating rate is 6 month USD Libor. * The swap rate is x 5%, (a) Represent the cash flows generatcd by this swap on a graph. (b) Creatc a synthetic equivalent of this swap using two Fomard Rate Avcements (FlkA) contracts. Describe the parameters of the selected FRAS in detail, (c) Coeld you generate a synthetic swap using appropriate interest rate taptions?
a,
.
r
&>
.
.w.. .
:*
er on t e e eore
=
3. Let the arbitrage-frec 3-month futures price for wheat be denoted by Ft. Suppose it costs c$ to store 1 ton of wheat for 12 months and per year intcrest rate applicable to traders to insure the srtme quantity. The (simple) of spot wheat is r%. Finally assume that the wheat has no convcnience yield. .$
(a) Obtain a formula for Ft. 1tX)$,c 150$ and the spot price 1500, r 5%, J (b) Let the Ft of wheat be St 1470. ls this Ft arbitrage-free? How would you form an arbitrage portfolio? (c) Assuming that all the parameters of tlze problem remain the same, what would be the profit or loss of an arbitrage portfolio at expiration? =
=
=
=
=
1* 4. An at-the-money call written on a stock wit.h current price St trades at 3, ne corresponding at-tbe-money put trades at 3.5. There are no transaction costs and thc stock does not pay any dividends. Traders can brrow and lend at a rate of 5% per year and all markets are liquid. (a) A trader writes a forward contract on the dclivery of this stock. ne delivery will be within 12 months and the price is Ft. What is the value of F/? 101 for this con(b) Suppose the market starts quoting a pri Ft arbitrage portfolios. Iw/ tract. Form =
=
1
Introductlon
A11 current methods of pricing derivativc assets utilize the notion of arbifrtzpc.ln arbitrage prcz;g methods this utilization is direct. Asset prices are obtitled from conditions that preclude arbitrage oppornmities. ln equiIibrium pricing methods, lack of arbitrage opportunities is part of gcneral equilibrium conditions. ln its simplest form, arbitrage means taking simultaneous positions in different assets so that one guarantees a riskless profit Mgher than the risklcss return given by U.S. Treasury bills- lf such prosts exist, we say that there is an arbitrage opportunity. Arbitrage opportunities can arise in two different fashions. ln the firqt way. one can make a series of investments with no current net comntment. yet expect to make a positive proht. For example, one can short-sell a stock and use the proceeds to buy call options writtcn on the same securi. ln this portfolio, one snancesa long position in call options with short Pitions in the underlying stock. lf this is done properly, unpredictable mtwements in the short kmd long positioms wili cancel out, and the portfolio Will be riskless. Once commissions and fees are deductcd, such investment Opportunities should not yield any exccss prolits. Othenvise, we say that tkere are arbitrage opportunities of the fil'st kind, In arbitrage opportunities of the second kind, a portfolio can ensure a negative net commitment today. while yielding nonnegative profits in tlie
future.
13
14
C H A P T ER
+
2
A Primer on
the
Arbitrage Theorem
Wc use these concepts to obtain a practical dehnition of a price'' for a linancial asset. We say that the price of a security is at a level, or that the security is correctlkpriced, if there are no arbitage opportunities of the Erst or second kind at tlaose prices. Such arbitrage-free asset prices will be utilized as benchmarks. Deviations from thesc indicate opportunities for excess prohts. In practice. arbitrage opportunities may exist. nis, however, would not prices. In fact, determining arbitragereduce our interest in aarbitrage-free'' valuing derivative assets. We can imagine at free prices is at the center of least four possible utilizations of arbitrage-frec prices, One case may be when a derivatives housc decides to engineer a new fmancial product. Because the product is new, thc price at which it should be sold cannot be obtained by obsening actual trading in Iinanci::tl markets. Under these conditions. calculating the arbitragc-free price will be very helpful in deterrnining a market price for this product, A sccond example is from tisk management. Often, risk managers would like to mcasure the risks associated with their portfolios by running some case'' sccnarios. These snulations are repeated periodically. Each time some benchmark price needs to be utilized, given that what is in qucstion is a hypothetical cvcnt that has not been observed.l A third example is marking lTp market of assets held in portfolios. A treasurer may want to know the current market value of a nonliquid asset for which no trades have been obselved lately. Calcuiatirig the corresponding arbitrage-free price may provide a solution, Finally, arbitrage-free benchmark prices can be compared with prices obsen'ed in actual trading. Signcant differences between obscwcd and arbitrage-free valucs might indicate excess profit opportunities, r'Iqhfsway arbitrage-free prices can be used to detect mispricings that may occur during short intenrals. If the arbitragc-free price is above the obsenred price, the derivative is cheap. A long position may be called for. When the opposite occurs, the derivative instrument is ovcrvalued. The mathematical environment providcd by the no-arbitrage theorem is the major tool used to calculate such benchmark priccs.
15
7 Nota tion jng of tbe notation is sometimes as important as an understanding mathematical Iogic. underlying
'ifair
d
2.1 A-sdefPrice. ne index t will represent time. Securitics such as options, futures, forwards, and stocks will be represented by a vector of asset prices denoted by st.This array groups all securities in hnancial markets under one symbol:
st
:
=
may be riskless borrowing or lending, Sglt) may denote a particular stock, S(f) may be a call option written on this stock, may f subscnpt in St represcnt the corresponding put option, and so on. meansthat prices belong to timc rcpresented by the value of /. ln discrete St, St-vj However, time,securities prices can be expressed as &, in continuous timc, the f subscript can assume any value between zero and intinity. We formally write this as Herc,
.1(1)
N(/)
'rhe
.%,
.
! . '.
t
(E
g(Jj(x))
.
t
.
.
.
,
,
.
.
.
.
j2)
.
ln general, 0 denotes the initial point, and write
f
rcpresents the present. If we
(3)
x,
then s is mcant to be a futuredate.
2.2 statesoj the World '
To proceed with the rest of this chapter, we necd one more concept-a that, at the outset. may appear to be very abstract, yet has significant practical relcvance. We let the vector H?'denote all possible states of the world, Ctmcept
E
2 Notation
For cvnmplc, it is no clear lNote that devising such scenarios s not at all straightforward. that markcts will have the nessary liquidity to secure no-arbilrage conditions if they are Mt by some extreme shock.
(1)
.
5'x(/)
Gtworst
We begin with some formalism and start developing the notation that is an integral part of evcry mathematical approach. A correct understand-
of the
1
JF
. Where
=
:'
,
(4)
each zt;j representg a distinct outcome that may occur. These states are mutualb exc/uvc, and at lcast one o them is guaranteed to occur.
16
C H A P T ER
>
2
the
A Primer on
In general, fnancial assets will have different values and give diferent payouts at different states of the world wi. It is assumed that there are a finite number K of such possible states. It is not very dimcult to Wsualizc this concept, Suppose that from a instant. Clearly, trader's point czfview, the only time of interest is the securities prices may change, and we do not necessarily know how. Yet, iri a adowntick'' small time interval, securities priceg may have an or a or may not show any movement at all. Hence, we may act as if there are a total of three possible states of the world. dxnext''
5
i'uptick''
2.3 Rehms
ne states of the world wi matter because in different states of the world returns to securities would be different. We let #fj denote the number of units of account paid by one unit of security i in state j. These payoffs will have two components. ne first component is capital gains or losses. Asset valucs appreciate or in the asset, an appreciation leads depreciate. For an investor who is capital and leads depreciation gain to a capital loss. For somebody to a a who is in the asset, capital gains and losses pill be reversed.z The second component of the dq is payouts, such as dividends or coupon Some assets, though, do not have such payouts, call and interest payments.S fptions and bonds among these. discount put The existence of severai assets, along with the assumption of many states of the world, means that for each asset there are several possible kj. Mafrce. are used to represent such arrays. the payoffs dij can be Thus, for the N assets under consideration, grouped in a matrix D:
columnwise. Each column of D represents payoffs to different assets in a given S tate of the world. If current prices of all assets are norlzero, then one can divide the /th the corresponding Sitj and obtain the gross rc/lgrn,&in different row of D by will have a I subscript in thc general case when state s of te world. The D time. depend on Payoffs
2.4 Portfolo :
uzzd Payogs
17
Example of Asset Pricing 3 A Basic
Arbirrage Theorern
A portfolio is a particular combination of assets in question. To form portfolio, one needs to know the positions taken in each asset under a considcration.The s'ymbol 0i represcnts the commitment with respect to the fth asset. Idcntifying all .(pf,i 1 N) specilies the portfolio. A positive % implies a long position in that asset, while a negative #;. impliesa short position. lf an asset is not included in the portfolio, its correspondingei is zero. If a potfolit delivers thc same payoff in all states of the world, then its value is known exactly and the portfolio is tiskless. =
Itlong''
Rshort''
' ' :
.
.
3 A Basic Example of Asset Pricing
' '
,
We use a simplc modcl to explain most of the important results in pricing derivative assets. Witb this example, we hrst intend to illustrate the logic used in derivative asset pricing. Second, we hope to introduce the mathematical tools needed to carry out this logic in practical applications. The model s kept simple on purpose. A more general case is discussed at the end of the chapter. We assume that time consists of and a period'' and that tbis tlzesetwo periods are separated by an inten'al of length book will reprcsent a but noninfinitesimal inten'al. We consider a casc where the market participant is intcrcsted only in tnow''
Wnext
'rhroughout
#ll
#1x.
.
(5)
D=
ds
'
three assets'
dNK
One can look There are hvo different ways one call visualize such a mat. of a given unit payoffs matrix D as if each row represepts to one at the world. of Conversely, the security in different states one can look at D
dismall''
.
1. A risk-free asset such as a Treasury bill, whose gross return until next Period is (1+r).4 in that it is constant rcgardless nis return is of the realized state of the world. 2. A.n underlying asset, for example, a stock St). We assume that during the small intewal Stj can assume one of only fw't:) possible values. This miltimum Iw'o states of the world. of a is risky because its payoff means is different in each of the two states. IGrisk-free,''
'
5
2To realize a capital gain, one must unwind the postion. 'Another example, bGsides dividend-payng stocks and coumn bonds, is nvestmcnt in practice of to market'' lcads to daily paytauta to a contract holder. futures. Howeverv in lhe case of futures, these payouts may be negative or Nsitivc. 'l''he
'<markng
'r
1We maat multipjy tlae risk-free rate,
r,
by the time that elapses, A, tty get the proper rcturn.
C H A PT ER
.
2
A Primer on
Arbitrage Theorem
the
3. A derivative asset, a call option with premium C(t) and a strike price period. Givcn that the tmderlying asset has Co. The option expires two possible values, the call option will assume two possible vales as well. (Tnex-t''
3), and two states nis setup is fairly simple. nere are three assets (N of the world CK 2). The lirst asset is risk-free borrowing and lending, the second s the underlying seeurity, and tbe third is thc option. The example is not altogether unrealistic. A trader operating in real (continuous) time may contemplate taking a (covered)position in a parprices of ticular option, If the time interval under consideration is these assets may not change by more than an up- or a downtick. Hencc, the assumption of two states of thc world may be a reasonable approximation.s We summarize this information in terms of thc formal notation discussed earlier, Mset prices will form a vector St of only three elements, =
=
19
of h Basic Example Asset Pricing
3.1 A Fiy'sf Qlfmceat the Arbitrcge Tlseozem we are now ready to introduce a fundamental be used in calculating fair market ory that can first we will simpllfy' the notation even scts. But borrowing and lending is selected by the risk-free alwayslet
#(/)
rcselt in fmancial thevalues of derivative as-
of furter, The amount investor. Hence, wc can
(8)
1,
=
sarlier, the time that elapses was called A. In tltis partkular example we 1et
(9)
= I The arbitrage theorem can now be stated: .
THEOREM; Given the Sf Dt desned in (6)and (7),and given tbat the two states have positive probabilities of occurrence. ,
B(t4 ,T,
st )
=
(6)
,
Cbt)
1. if positive constant,s satisfy
which these prices apply, Payoffs will be grouped in a matrix Dt, as discussed earlier. There are three assets, which means that matrix Dt will have three rows. Also, there are two states of the world; the Dt matrix will thus have two columns, #(f ) is riskless borrowing or lending. 1ts payof will be the sarne, regardless instant.'' The Sltj is risky of the state of the world that applies in the A) down to Sgt + A), Finally, and its value may go either up to &tf + or will Ctf change in line witb movements value of thc call option thc market ) will given be by: in the underlying asset price Stj. Thus, Dt rrhe
Rncxt
(1 + rA)#(l) Dt
k%(f
=
)
+
c1(/ +
.)
(1+
rjBtj
&(/ + ) C'ztf +
,
(7)
)
wheze r is the annual riskless rate of return. 5In fact, we show later tha a ntinuoas-time Wiener proceess, or Brownian moton, can s we let ll1c A go toward zero, be approximated arbitzarily well by such two-state presses,
Sftj C(1)
r)
(1 +
1 where Bt) is riskless borrowing or lending, Jtf ) is a stock, and C(t) is the value of a call option written on this stock. The t indicates the time for
41, z can be found suc that
x$j(?
=
(1 +
+ 1)
r)
+ 1)
katl
G (f + 1) C2(f + 1)
asset prices
*1 1/32
,
(10)
then there are no arbitrage possibilitiesi6 and 2. if there are no arbitragc opportunities, then positive constants #L #z satisfying (10)can be found. The relationship in (10)is called a rcpresentation. It is not a relation that apossible'' + 1) are can be obsen'cd in reality. In fact, 5'!(1+ 1) and SzLt futurevalues of the underlying asset. Only one of them-namely, the tme that belongs to the state that is realized-will be obsewtd. What do the constants represent? According to the second row la c of the representation implied by the arbitrage theorem, if a security pays 1 in state 1, and t) in statc 2, thcn ,
(11)
-$(f) (1)41 =
,
Thus, investors arc willing to pay 'I'j pol(current)units fcr an offcrs one unit of account in state 1 and nothins in state 2. Simiindicates how much investors would like to pay for an ainsurance tdinsurance
icy'' that larly,tl,z
fhote that if 1 + r frst row of tlle matrix
w
1, we 'need to have
equation.
l#I
+
bpz
m 1 as well.
This is obtained from the
20
C H A P T ER
A Primer on
2
.
the
Arbitrage Theorem
Example of Asset Pricing 3 s%Basic
policf' that pays 1 in state 2 and nething in state 1, Clearly, by spending 4//1+ h, one can guarantee 1 unit of account in the future, regardless of which state is realized. nis is conlirmed by the first row of representation (10). Consistent with this interpretation, 4f i 1, 2 are called state prices. 7 At this point there are several other issues that may not be clear. Onc can in fact ask the following questions: ,
'
.E
E
:.
For the moment, Iet us put the first two questions aside and answ'er the third question: What types of practical results (ifany) does one obtain from the existence of #a? It turns out that the representation given by the ls very important for pradical asset pricing. arbitrage theorem t,
'
Thcorem
ne arbitrage thcorem provides a very elegant and general method for pricing derivative assets. Consider again the represcntation;
1 st) c(f)
(1+ sLt
=
(1+ r) szt + 1) czt/ + 1)
r)
+ 1)
c,(/ + 1)
=
(1 +
rll
+
(1+
'c
(1+ (1+
=
=
j j, /,,,1
rla.
<
P;
r)4!
P1 + Pz
=
Considcr the equality implied by the arbitrage tbeorcm again, Note that the representation (10)implies three separate equalities'. 1 Stj
(12) '
(1+
=
=
l
Now multiply the right-hand
,5'(f) C(/)
1
=
( l + r) 1
=
-
(1 + r)
'
(15) (16) (17)
zvht + 1 )
side of the last two equations
by
g(1+
rjnkyt
g(.1 +
r)
j
+. 1) +
(
(ja)
(1 +
rjzyt
1
1
.
.( liarry
t
'
($
j/j j%
=
-(y .y,
p
;
Cz(t + 1)j (t +. 1) + ( 1 + r) (.$z
(j
+. J ) ..y jh Sg(j
'l-hisis tlae case
fllrtberconditions 9As long
(19)
+ 1)j
Rtlt, we can replace (1 + r)#j, i 1, 2 with the corresponding This means that the two cquations bectme k
hence, they should
rlllz
=
(13),
1
7Note that, in general, statc prices will be subsc-ripts. nis is omitted here for notationnl simplicity.
(1+
C1(f + 1) + I/zzC2(f+ 1).
to obta? .
+
1+ r 1+ r
:
time-dependent;
rjlij
4).$'j(r+ 1) +
=
C(l)
(14)
rl/a
<
Risk-adjusted probabilities exist if there are no arbitrage opportunities. i'mispriced assets,'' we are guaranteed to find ln other words, if there are no i#zl. Multiplyiny these by the riskless gross ret'un'l positive constants (1, 1 + r guarantees th e e xistence of (/ 1 , # c 1..8 'rhe importance of risk-adjustcd probabilities for asset pricing stems from the following: Expectations calculatcd with them, once discounted by the risk-free ratc r, equal the current value of the asset.
E
(13)
Because of the positivity of state prices, and because of 0
3.3 TW Usc of Synthetic Pmbttbilifies
;
Define!
h
Y.
:
Multiplying the Erst row of the dividend matzix Dt by the vector of /l, z, we gct 1
A
,
,
oj the Arbiftage
/j'S are positive numbers, and they sum to one. As such, they jjence, interpreted as tw'o probabilities associated with the two states uncan be consideration. We say because the tnle probabilities thas der of of the the world will in general be difstates two the occurrence govern and the These dehned by Equation (14)and provide from are ferent tlirect information concerning the true probabilities associated with the no tisk-adjusted two s tates of the world. For this reason, (#1 6J are called Probabilities. thetic syn dinterpreted''
=
* How does one obtain this theorem? 1): tl'z have to do with no arbitrage? . What does the eZstence of this result relevant for asset pricing? * Why is
3.2 Rclaunce
21
.y.
j
1)
Yi ,
(20)
.
=
1, 2.
GJ )
with jnuv states of thc world. witlauncountabl.v mauy statcs one needs for the existence of risk-adjusted probabilites.
as r is not equal to -1, we c.arl always do this.
r
!
C H A P T ER
22
.
2
A Primer
on tle
Arbitrage Theorem
1)1 1 + 1) + ccctf + r) ct) (1+ gz-tqlf be interpreted. Now consider how these expressions -
Exarnple of Asset Pricing 3 A Basic and assuming that asset prices are nonzero,
Rea rranging,
(22)
.
23
rrjle
expression can which 1/(1+r), by multiplies right-hand the in the brackets side term on the discount factor, On the otber hand, the term inside is a riskless tme-period the brackets can be interpreted as some sort of expected value. It is the sum of possible future values of 5(/) or C(/) weighted by the /1 &. Hence, the terms in the brackets are expectations calculated using the risk-adjusted probabilities. expected As such, the equalities in (21)and (22)do not represent values. Yet as long as there is no arbitrage, these equalities are valid, and they can be used in practical calculations. We can use them in asset pricing, speced, as long as the underlying probabilities are explicitly Ialt/cr Wit this interpretation of #:, /a, the cunmtt prces of all assets Further, thc cottsideration become equal to their diacounted opected JItzytpF.. discounting is done using the risk-free rate, although the assets themselves
(1 +
r)
=
(1 +
r)
=
Elrue (5'(f+ 1)j Etrue
E
(1 +. r +
risk prernium for
ln order to emphasize tbe important role played by risk-adjusted probaprobabilities dicbilities, consider what happens when one uses the tated by their nature. expected values by using the tnze probabilities First, we obtain the dcnoted by PL, #z:
St)
<
c(I)
<
''true''
Etruefct
+
+ 1)1
=
(23) (24)
.
assets, when discounted by the risk-free rate, Because these are these expectations will in generallo satisfy <
(1
1 E trur kgj + + r) 1
c(l) < tl +
r)
E true
jgjj
(25)
+ j. jj
(26)
.
To see why one obtains such inequalities, mssume othemise:
CBWesay with the bcta'' asseys,
'Tmarket.''
tin
u(1)
=
C(1)
=
1 (1 + r) 1
(1 + r)
E
true
E true
gy(/ ..).
E
(27)
,j)j
jct.r +. j)j
.
1
Etrue Ikvf+ 1)1
(1+ rj 1 .E'true ( ct (1+ r)
.
(32)
+ ((lj
(33) .
(a4)
,
that satisfy
1
:
(1 + r) 1 r)E j ( +
. i
E '!3(st + 1)j (C(l + 1)1
=
=
stj
(35)
C(f).
(36)
'Fhese equations are very convenient to use, and they internalize any risk Premiums. lndeed, one does not need to calculate the risk premiums if one tlses synthetic expectations. corresponding discounting is done using the risk-free rate, which is easily observable.
(28)
.
Etrtgejctj + 1)j z x C/7
,
:
jjj
=
(31)
ne importance of the no-arbitrage assumption in asset pricing should become clear at this point, lf no-arbitrage implies the existence of positive constants such as (Jj always obtain from these constants then y, 31we#acan the risk-adjusted probabtlities and work w1f.11asynthetic'' expectations
dirisky''
Stj
Etrue gyjf+ 1)j X(l)
This implies. in general, the following inequalities for risky assets:ll
ditrue''
+ 1)j
=
k'41))
premium for C(/))
( 1 + r + risk
are risky,
=
(30)
.
ttnzel
Gtrue''
+ 1) + P2&(t 1)J LPLSLIt 1)j (#jG t + 1) + Pzhlt
.
expected returns from the risky assets equal riskBut this means that however, is a contradiction, because in general risky assets less return. nis, will command a positive risk premium. If there is no such compensation for risk, no investor would hold tllem. 'Thus, for risky assets we generally have
,
+
+ 1)) rct/ C(f) .
dprobabilities''
Etruefslt
(29)
.5(/)
'rhe
general'' becausc one can imagine risk'y assets that are negativcly correlated Such assets may have negative risk premillrns and are called Gnegative
''For nugative beta asscts i : '...,
the inequanties are reversed.
24
C H A PT ER
3.4 Murtiagules
un,d Su
.
2
on the
A Primer
Arbitrage Theorem
of 3A Basic Example Asset Pricing
rt-ing'zvlw;
E
martingale with respect to the risk-adjusted probability #. Second, was a it is not the S that is a martingale, but rather the St divided, or tbat te normalized, by the ( 1 + rP. latter is the earninp of 1$ over s periods if investcd and rolled-over in the risk-free investment. What is a martingale i9 the ratio. A.fj interesting question that we investigate in the second half of this book is thcn the folloving. Suppose we divide the St by some other asset's would the new ratio, Prjcc, say Ct; $ * A-z+, (42)
This is tlze right time to introduce a concept that is at the foundation of pricing fmancial asgets. We give a simple definition of the terms an (j jeave technicalities for later chapters. Suppose at time f onc has information slzmmarized by lt. A random variable Xt that satishes the equality EP
g/wlfl) =
Xt
for all
u
>
11O
(37)
0,
i9 called a martingale with respect to the probability #.12 1 instead we have
f
:
K
for aIl s > 0, (38) submartingale with rcspect to probability Q. thenA-, is called a Here is why these concepts are fundamental, According to the discussion inthe prcvious sedion, asset prices discounted by the risk-fwe rate will be under the true probabilities. but become martingales under submartingales the risk-adjusted probabilities. nus, as long as we utilize the latter. the market toolsavailable to martingale theory become applicable, and of obtained exploiting under consideration by the can be values'' the assets
where .
>
(), and where
E # (A-,+x1f,)
=
,
=
1 s-t-vs. (1+ r)
'
.g.6 Eqxulfzutfon of Rates oj Refum By using risk-adjusted probabilities, we can derive anotber important resultuscful in asset pricing. In the arbitrage-free representation given in (10),divide both sides of tlze equality by thc current price of the assct and multiply both sides by (1+ r), the gross ratc of riskless remrn. Assuming nonzero asset prices, we O btain
(39) (40)
ktf .!31
and r are the secuzity price and risk-free return, respectively. # is Here stws probability. According to this, utilization of risk-adjusted risk-adjusted thc w ill alI (discounted) asset pr icesinto martingales. convert probabilitics
s xo
.a lt
alaie,x
a-;j
s-irstnote
1)
&(/ + J-sz
-
+ I)
y(f)
C1(rY 1) + Cjtj
Cz(l
.>o
1)
+
=
=
c(/)
-
(1 +
r)
(43)
(1+
r).
(44)
that ratios suck as
yt
+ j)
- 5'tf )
szt
-
'
,5'(1)
..j.
j
)
(45)
are the gross rates of ret'urn of in states 1 and 2, respectively. The (43) and (u) imply that if one uscs , /2 in calculating the expected values. a1I assets would have the same expected return, According to this new result, utmder all expected returns equal the Hsk-frcc , #a,7* rturn r.13 nis is anothcr widely uscd result in pricing linancial assets. st)
tqlmlities
'
Y
I-L
=
'znertz arc other ccmditions that a martingale must satisfy. In later chaplers, we discmqq hem in demil. Ir1 tlne meantime, we assume impdtly that thse condilional expectations t e xist-that is, they are flnite.
+
ytfl
,
is important to realize that, in linance, the notion of martingale is always associated with two concepts, First, a martinga le is always defned with respect to a certain probabiliT.Hence, in Scction 3.4 the discounted stock price. gj X St.vs., (41) t+s (j +. r)x
,
c /+d
with respect to some other probability, say #*? The answer question is positive and is quite useful in pzicing interest sensitivc this to derivative instruments, Essentially, it gives us the flexibility to work with with an asset of our choice. a more convenient probability by normalling But these issues have to wait until Chapter 17.
xt-vsis denned by Xt-vs
I+S
t)c a martingale
Wfair
-Y,
'l'he
=
EQ gXf+.vlfllAE Xt
equalit.y martingale
25
, '
lln
a'Ild
.
'h
.
probability theory, the phrase
#z.''
Ttunder
#I,
6''
means
d
one uses the probabilities
.151
26
C H A P TE R
3.7
ne
No-Arbit-e
Arbitrage Theorem
on the
A Primer
2
.
4 A Numerical Example
4 A Numerical Example
Cendite
Within this simple setup we can also see explicitly the cpnnection between the no-arbitragc condition and the existence of 1. #z. Let the gross remrns in states 1 and 2 be given by Rjt + 1) and Rc(l + 1) respectively: R1(f + 1)
Rlt + 1)
=
.%(/ + 1) 5'(r)
=
Szt + 1) St)
(12)using 1 (1+ r)# + (1+ =
1 R1/1 =
A sjmple example needs to bc discussed. Let the current value of a stock jx gjvcn by St
(46)
-
Now write the first tw'o rows of
-I- 1) k$'1(/
(47)
Szt + 1)
r4#2
equation
+ Rznz. E ,
will be satished if and only if:
Rj
(1 + r)
<
<
(48)
1
and, at the same
'
R2.
Rl
<
<
E
,
R)
<
Rz
<
(51)
1.1
0
5()
vujtyjyjugthe uons:
)
dividcnd matrix with the vector of
1 ( l )j + (1.1)4c 1(j() ltltj/j +. 15:4, .1
=
=
) and
irwest the proceeds
C
in the risk-free investment to realize insnite gains. Again Equation (48) ! will not be satissed with positive 4y, z, because the right-hand side will always be p ositive under these condltions. I)z the
=
0#1 + 50/2
'
Now suppose a premium C Iast equation
'
gelds
.
Thus, we see that the existenc.e of positive
1,
is closely tied to
;
W
implies, in tlus simple laich
<
(1 +
r)
<
R,,
=
.
(52)
,
z's yields three equa-
(53) (54) (55)
25 is observed in hnancial markets, Then the 50#z
condition
Rb
#1 h
4. 1 Cse 1: Arbitmge Possibilities
(1+ r).
If this was the case. then one could short the St
150.
100 150
=
C
!
R2.
This means that by borrowing infinite sums at rate r, and going long in Sltj, we can guarantee positive returns. So therc is an arbitrage opportunity. But then, the right-hand side of (48)will be negative and the equality will not be satis:ed with positive l a. Hence no 0 < 1, 0 < #z will exist. A similarargument can be made if we have
(50)
Note that the numerical value of the call prcmium C is leR unspecihed. Using this as a variable, we intend to show the role played by #i in the arbitrage theorem.
z
r)
=
1.1
100
For example, suppose we have
(1 +
100
Hcnce, there are only 11$,5states of the world. Tjjere exists a call option with premium C, and strike pricc 100. ne option expircs next period. Finally, it is assumed that 1 unit of account is invested in the risk-free assetwit ja a return of 10%. We Obtain the following represcntation under no arbitrage!
:
=
=
and
these new symbols:
t) ((1+ r) - R1)1 + ((1+ r) - Rzl#z, j. 42 to bepositive. nis will be the case
(49)
100.
=
stock can assume only two possible values in the neM instant:
rrhe
subtract the Nrst equation from the second to obtain: wherewe want time, the above
27
=
25
(56)
or
,
*,
seuing, tlmt there are no arbitrage possibilities. ':
=
1
,.
(57)
'
28
C H A PT E R
Substitming this in
*
on the
A Primer
2
j yu&Application: Lattice Models
Arbitragc Theorem
4.3 An lndetenrkinao
(54)gives
#j
and
#2, the hrst equation is not
1,1(.25)
+ 1
.1
ne same method of determining the unique arbitrage-fzee value of the call option wou jd not work if thcre were more than two states of the world. Consider the systcm For cxample,
(58)
.25.
l= But at these values of
29
(1,5)#
1
satisied:
1 1*
(59)
,
=
C
25, it is Clcarly, at the obsen'ed value for tbe call premium, C impossible to hnd #:, k that satisEes al1 three equations given by the arbitrage-free representatlon. Arbitrage opportunities thereftre exist. =
1.1
1.1
1.1
/1
100
50
150
c
0
0
50
43
(63)
,
.
Hcre, thc first two equations cannot be used to detcrmine a unique set of into the third equation to obtain a C. Therc yj > 0 that can be plugged , such of j s. sets are many I.n order to determine the arbitrage-free value of the call premium C, ucorrect'' j. ln principlc, this can be done onc would need to select the usjng the underlying economic eqtlibrium.
,
E
4.2 Csc 2: A&'bitrtlge-lk-rcc Prices Consider the same system as before
1 100
1.1
5 An Applicatlon: Lattice Models
1.1
/1
100 150
=
C
0
(60)
.
/2
50
Simple as it is, the eumple just discussed gives the logic behind one of the ?'rl)&l.14 Common asset pricing metbods, namelyj the so-called lattice The binornial modcl is the simplest example. We brictly show how this pricing methodology uses the results of the arbitrage theorem. call considera call option Ct written on the underlying asset St. option has strike price and time < f cxpires T; F. It is known that at at G expiration, the value of the option is given by most
E, E
But now, instead of starting with an observed valuc of C, solve thc Erst two equatitms for 1)1 #a. ncsc form a systcm of tw'o equations in two unknowns. Rnheuniquc solution gives
.
'rhe
,
.7273,
l Now use the
SO Iution:
(Jz
=
third equation to calculate a value of
c
consistent
cz
mx
=
gkz c(),oj
(64)
-
.
We rst divide tlae time interval (r f) into n smaller intewals, each of size a. choose in the sense that the variations of s, during a we A Can be approximated reasonably well by an up or down movement orlly. According to tis we hope that for small enough the underlying asset Pnce St cannot wander too far from the currently observed price Thus we assume that during k the only possible changes in S J are an up moveme'nt by =o or a down movement by -o-X&
with this
-
''small''
'
j
C
=
9.09.
this price, arbitrage prolit.s do not exist.
At
(61)
.1818.
=
(62)
Note that, using the constants #j, #z, we derived the arbitrage-free prico C 9.09. ln this sense, we used the arbitrage theorem as an asset-p ri cing tool. lt turns 0ut that in this paticular case, t-he reprcgentat ion given by the with positive and unique #i. nis may not arbitrage theorcm is gatished be always true.
=
:
,
.
%%.
'
''
=
Sl+. ''Also called tree zyltiel.
'
i.
=
st +
g.y
St -
G'U'X
.
(65)
r
C H A P T ER
A Primer
2
.
on the
5 An Application: Lattice Models
Arbitrage Theorem '
su4
values of the call option at any time l + A to 1he aties's two (arbitrage-free) rbitrage-free) the option as of time /. The lhp is known at this value of (a order equation make the usable, we need the two values Ctlh In to PO int. Cdovm Given these, calculate the value of the call option Ct at we can an d /+
Su3 Su2
su2
'
timc f
Xu
su S
yigure 2 shows thc multiplicative lattice for the option price Ct. ne arbitrage-fzee values of C3 are at this point indeterminate, except fo the ln fact, Sventhe lattice for St, we can determine the expiration using the boundary condition expiration the of C3 at values
s
a
''nodes.''
Stl
Sl
E
Sd2
-
Cp
sd2 sd3
=
max (&
01 - G,
Once this is done, one can go backward using sd*
G
F 1G U R E
Clearly, the size of the parameter (r determines how far St-vhcan wander during a time inten'al of length k&. For that reason it is called the volatilit. parameter. The (m is known. Note that regardless of tr, in smaller intervals, St Will change less. The dynamics described by Equation (65)represent a lattice or a binomial frcc. Figure 1 displays these dynamics in the case of multiplicative up and down movements. risk-free rate r for the Suppose now that we are givcn tbe (ct-mstant) probabilitiesyls risk-adjusted the detezwne pcriod Can we We know from the arbitrage theorem that the risk-adjusted probabilities and l'bovmmust satisty'
1 (1+ r)
=
Uasfahownwhj up
'
.
.:r4''il
+
hownst-
c'vr-all .
f cuS
Cu
=
1
(j
.j.
rj
downj
g/urfri+a up
+
dovln.u
l5ln the second half of the bOOk. we will relax the assumpton now we maintain this assumption 'ERemember hat Pgov.. 1 #up. =
that r is constant.
K)
i'.
(Su2
- K)
(s -
K)
cu
rrhtls,
Ct
(su4-
cu2
(66)
ln this equation, r, St, fJ-, and k are known, ne til'stthree are observed in the markets, while A is seleded by us. the only unknown is tlle /.s, which can be determined easily. 1 t$ Once this is donc, the lhp can be used to calculate the current arbitrage-free value of the call option. In fact, the equation
(69)
.
'
#up
+ st 1 +1r gzurt-.
doum
+
Repeating this several times, one eventually reaches the initii node that gives the cun-ent value of the option. Hence, the proccdure is to u:e the dynamics of St to go forwardand determine the expiration date values of the call option. Then, using the risk-adjusted probabilities and the boundary condition, one works backward with the lattice for the call option to dete=ine the current value Ct. It is the arbitrage theorem and the implied martingale equalities that make it possible to calculate the risk-adjusted probabitities Pupand gaown.
::.
.
-
(68)
.
C Cd
cd
E
cd2
f6y)
(*d2-
K)
cd'
But for E
(Sd*
F 1G U R E
. :
.
.
2
- K)
'
'
.
32
C H A PT E R
*
A Primer on
7
the
.
:
Arbitrage Theorem
'
payouts
6
cln this procedure Figure 1 gives an approximation of all the possible paths that may take during the period F - 2. The tree in Figure 2 gives approximation of all possible paths that can be taken by the price of an option written is small, then the lattices will be close the call on St. lf approximations to the tnle paths that can be foilowed by St and Ct.
Foreign Curencies
and
1
=
ut
33
(1+ r)
gsv:nu
,
-U'
'
LW
r'd ga-l U *-
gj)
1 '
where # is the risk-neutral probability. and where we ignored the time SUbscripts. Note t jaat the qrst equation is now different from the case with no-dividends, but that the second equation is the same. According to this, an asset has some known percentage payout d during the period e ach time risk-neutral discounting of the dividend Jyyjn: asset has to be done the a using thc factor (1+ J)/(1 + r) instead of multlplying by 1/(1 + r) only. It is also worth emphasing that the discounting of the derivative itself did not changc. Now consider tbe following transformation!
:
*-'
6 Payouts
and
.
Foreign Curtencles
In this section we modify thc simple two-state mo del introduced in this chapter to introduce hvo complications that are more often the casc in practicalsituations. The 6rst is the payment of interim payouts such as dividendsand coupons. Many sccurities make such payments before the cxpiration date of the derivative under consideration. These payouts do change tlle pricing formulas in a simple, yet at ftrst sight, counterintuitive fashion. The second comp licationis the case of foreign currency denominated assets.Here also thc prking fonnulas changes slightly.
6.1 The C-e u'itlv Dztrleads ne setup of section3 is first
k
(1+ r) (1+. d)
s
,
wliich means that the expectcd return under the risk-free
.
given by;
Stu st
E
r
(1 +
rL) (1-f- dtl
=
Qearly, as a first-order approximation,
'
ycar,and are small:
modilied by addzg a dividend equal to Note tw'o points. First, the dividends are not lump-sum, J, percent of the dividend : are paid as a percentage of the price at time t+,&. second, subscrpt r instead of f + a, According to this, the d is rate f known as of time f Hence, it is not a random variable given the informatlon : set at 1t. ne simple model in (10)now becomes: ,
ut-vh.
sdl'sd
Supa +
=
-
.
1+
rh
2 1+
1+ d
but paymenthas
Using this in the previous equation; S.
iq
.
E'-
'*''
.
s
or
measure is now
,
if d, r are defned over, say, a
(r -
dlh.
.
1 +. (r
l
-
djh,
.sp
u u
#J
Bt..h
St
'hn
cl
=
d
Bt-vh +
.$d
d t su ,+a
+ dt
/+a
cd/4-?k
cu/->k
.d
r+a
p r+a j a s
1/32 d y,
6
,
equations:
'
p
where B, 5', C denote the savings account, the stock, and a call option, as usual, Note that the notation has now changed slightly to reNect the discussion of Section 5. Can we proceed the same way as in Secticm 3? The answer is positive. With minor modilications, we can apply the same steps and obtain *0
According to lf we were term for Jxi, Price, will be written as; :7
!
S
=
tl (y
+ d'' ..4.
yj
j
-.,-
+
s
,+a
o st +. (,
uncse .
dlsth +
a,yrapus.
-
ds l
(7t))
-
tz.-s/api.f-a:
this last equation, we can state the following, to let go to zero and switch to continuous time, the dnT wllich represents expected change in the underlying asset's given by (r djstdt and the corresponding dynamics can be
.'
Wbere us,/'sj
f +. r - ajsth, unpredictable component,
or again, after adding a random,
dt represents stooassc
.js
.dlstt
+
(ryapy,
an innnitesimal time period.
disozential equations
wsl,c
stufsetIwi,,l more detail irl later chaptem.
34
C H A P T ER
2
*
the
A Primer on
Arbitrage Theorem
nere is a second interesting point to be made with the introduction of payouts. supposenow we try to go over similar steps using, this time, the equation for Ct shown in (7i); c
=
1
(j
End
paycuts
E
opportunities
gc upu
cldj
+
1
c
E
/
Ct
C
',
ut
C'+A
)
z
(,
-
dlh (;
c;
4 ru #/,,
,
cs,q. 2k
w-
=
=
(1+ rTl (j .j. pj 1 (1+. r)
(e upu
d
z
d
s
+ c
(CN/?
+ C
x 1 + rh.
(1+ r,j < 1 + (r (1+ rTA) we again obtained a different ne exqected rate of retum
.
.
probability P:
E p ea et
Forefgn Currencies
by adding an investment opportunity The standard setup is now moded savings foreign account. in a currency In particular, suppose we spend et units of domestic currency to buy one the et is the exchange rate at time t. Assumc unit of foreigncun-ency. U.S. dollars (USD) is the domestic currency. supposealso that the foreign savings interest rate is known and s given by rT. raus
(1+ r y )
f+.
/
j
1 .
,
1+
modiscations in the formulas, but in practice they may These are make a signihcant difference in pricing calculations. ne case of foreign currencies below yields similar results.
6.2 TRe Ctzse
ed
(1+ r y )
r)
Again, note that the ftrst equation is different but the second equation is the same. Thus, each time we deal with a foreign currency denominated the risk-neutral discounting of the forcign asset that has payout rl during asset has to be done using the factor ( 1 + r)/(1 + rT). Notc the first-order approximation if rf is small:
i
glight
lvifl
(1 +
w-i4.zs
'
and Ct during a period A are . The expected rate of returns of the risk-free /: dkprent under probability the now rA) dLj
r)
wllere the Ct denotes a call option on price et of one unit of foreign c'ur#.1S renc'y. The strike pricc is we proceed in a sinlar fashion to the case of dividends and obtain the following pricing equations:lg
Thus, we see that even though there is a divided payout made by the underlying stock, the risk-neutral expected retllrn and the r isk-free discounting remains the same for the call option written on this stock. Hence, in a risk-neutral world fumre returns to Ct have to be discounted exadly by the same factor as in the case of no-dividends. In other words:
(1 + (1 +
o, cu
ct
Ct-vh = 1 + r. ct
=
C/+. =
.
We would obtain
Stn St
u
.
.
E>
(1 +
1 '
+ r)
35
in investment and the yields of these investments ovcr using the following setup; summarized be a can now
ne
'
Foreign Currencies
'
!:
y
)a.
result.
of the et and C are dkyerent under the 2 1+
s> G+a. z t
:
r
-
(g
-
r
y
)
1 + za.
Here the K is a strike price on tilc exchange rate ';. If the cxchange ratc exceeds the X' at time t + the tmyerof thc caE will reccive te difference e,+ - K times a notional amount N. t9As usual we omt the time subscrips for convenience. ,
,
k
k
cHh
36
PT E R
-
2
A Primer on
t
heArbitrage Theorem
8
A Melodology conclusions:
for Pricing Assets
2.3 Didcotmting
switch According to the iastremark, if we were to iet go to zero an d given by rctdt. But the drifl to SDE's, the drift tenns for dCt w iil be denominated asset, det, will ow bave to term for the foreign currenc.y rflcfttf. be (r .'
models usingcontinuous-time fact, j.f is continuous,
Ieads to a change in the way discounting then the discount factor for an intenral t In js done, w'ill be given by the exponentialjnctm of lcngth
-
e 7
general, such simple U P to this point, the setup has been very simple. In rcal-life financial assets. Ixt us brielly examp les cannot be used to Price consider some genera lizations that are necded to do st.
8 Conclusions: A Methodology for Pricing Assets
ln 1, 2, 3, with f Up to this point we cons idercd discrete time continuous-t ime asset pricing models, this will change. We have to assume that f is continuous:
l e (0,x) ter, we
.
The arbitrage theorem provides a powerful methodolor for determining values of major jr market fmancial in practice. assets steps of this fa ne mcthodology as applied to inancial dcrivatives can be summarized as follows:
.
Consider
,
to track the dynamics of the underlying obtaina model (approximatc) price, asset's 2. calculate how the derivative asset price relates to the price of the 1
(72)
.
'
.'
with this chapto t-he asmall'' time interval A dcalt Symbolin Jl. the injnitesima l intcrvals denoted by
This way, in addition Ca:II
.
underlying asset at opiration or at other boundaries. 3 obtainrisk-adjusted probabilities. 4 calculateexpected payoffs of derivatives at cvrtlfoa risk-adjusted probabilities. 5. Discount this expectation using the risk-free return. .
,
.
.
7 2 Stutes of fh,e World
.
limited In continuous timc, the valucs that an asset can assume are not of continuum and uncountably many POSS ibilities a to two. There may be states of the world. stochastic zs./.erTo capture such generalizations, we nee d to introduce example, as ment ioned above increments ill security ential equations. For using Prices S: may be modeled dSt
=
p'tvh
dt +
o-t
ul
,
change in the price of where t he symbol dSt represents an insnites imal inlinitesimal dt is thc predicted movemen t during an t lae security, the 1 uuuvztn cyaaxavintcrval #/, and gtst JIFI is an unpre dicable,. inlmneslmal stochastic differdefming It is obvious that most of t he concepts used in Step. ential cquations nee d to be developed step by ut
.
-
,.
=.
-,
.--aw.m
'
'jau.d
7
ojaeaov
E .
,
using
these
In order to be able to apply this pricing methodology, one needs familiarity with the following types of mathematical tools. First, the notion of time needs to be defined carefully. Tools for handlingchanges in asset prices during time periods must be developed. requires continuoua-time analysia. nis Second, we need to handle the notion of randomness'' during such jusaktesjmal periods. conceptssuch as probability, expectation, average value, and volatility during inlinitesimal peliods need to be carefully deEned Tllis requires the study of the so-called stochastic calculus. we try to discuss the intuition bchind tlae assumptions that lead to major results in sthastic calculus. we laeed to understand how to obtain risk-adjusted probabilities and how to determine the correct discounting factor. ne Girsanov theorem states the conditions under which such risk-adjusted probabilities can bc used. ne theorem also gives the form of these probability distributions. ''inlinitesimal''
(73)
#W$
(74)
,
.
7.1 Tlme l
.
-rA
ne r becomes the continuously compounded interest rate. lf tere exist dividends or foreign currencies, the r needs to be modied as explained in Section 6.
Some Genetalizatlons
=
37
r
C H A P T ER
38
.
2
A Primer
on the
ArbitrageTheorem
lppendix: Generalization
jc
Further, the notion of martingales is esscntial to Girsanov theorem, and, world. consequcntly,to te understanding of the of various movements Rnally, there is the question of how to relate the this is standard done using calculus, quantitiesto one another over time. In equivalent concept is differentialequations, In a random environment, the equatkm (SDE). a stochastic lz//rezc/ Needless to say, in order to attack these topics in t'urn, one must have calculus. concepts amd results of somenotion of the wcll-known thc notion of derivative, of notion the (2) There are basically three: (1) integrala and (3) thc Taylor series cxpansion.
xowdcfine
.
''risk-neutral''
of the
a portfolio, :,
39
s the vector of commitments
to each asset:
0L
'
#
=
(,20
.
oxv ln dealer's termfnology, 0 gives the positions takcn at a certain time. Multiplyill# the 0 by Stt we obtain the value of portfolio 0:
d'standard''
'
./# t
x =
j=l
'
9
ArbitrageTheorem
References
sit )tj.
qy
This is total investment in portfolio : at timc /. TN 2t) In matr . payoff to portfolio 0 in state j is izzz1 d ij f3i expressed as
,
*
lngcrsoll ln tis chapter, arbitrage theorem wa,s treated in a simple way. accessible, quite is that detailed treatrncnt (1987) provides a much more with a strong quantitative background may Readers beginncr. to a even The original article by Harrison and Kregs ( 1979) prcfer Dufse (1996). material can be found in Harrison may a lso be consulted. Other related RutkowRki Musiela and in chapter (1997)is , and Pliska (1981).The first read afler this chapter, cxcellent and very easy to
jqj
.
o'j?
.
DEFINITION:
.
either
:
Accordirig to the arbitrage theorem, if there are no arbitrage possibilities, therl there are state prices, jsij., such that each assct's price today etjuals a linear combination of possible future values. Thc theorem state prices then there js also true in reverse. lf there are such (supporting) are no arbitrage opportunitics. ln tis section, wc state the gencral form of thc arbitzage teorem. First we brielly dehne thtt underlying symbols. * Desne
a matlix
o, N' js thc total number thc.
Wol1d.
Of
=
securities
: is an arbitrage portfolio, or simply an arbitrage,
one of the following conditions is satisqed:
if
.z
According to this, the portfolio /? guarantees some positive retttrn irl all states, yet it costs nothing to purchase. Or it guarantees a nonncgative teturn whil having a negative cost today.
q ,
'l'he
'
dcussed
following theorem is the generalization of the arbitragc conditions earlier.
THEOkRM: dbK
'
.
dx
deline an arbitrage porolio..
'
payoffs, D1
fll
()x
s'v ...s () and D'o > () a s,o (j ajalj p,o u (j. '
'
,
xvr
1.
.
'Isupporting''
faf
we can now
(78)
'
dl K .
:.,
oj
.
:
,
10 Appendix: Geneulizatlon of the Abltrage Theorem
#xj
form tls is
(75)
Jow and K is the total number of statcs
j Skch
'
@
:
opportunities,
s
! E
jj tjwre are no arbjtrage that
wNotc the differenc.c Ixtwzen summaton
to j.
=
thcn there exists a
D*. with respect
>
0
(,79) o
and summation
with respect
.:
C H A PT ER
40
*
A Primer
2
on the
2 If the condition in (77)is true, then there are tunities. *
Arbitrage Theorem no azbitrage
world there cxist
This means that in an arbitrage-free #11 5) =
dNK
dsj
oppor-
:suc ja that .
(b)
*K
D
#2r .
(d)
and
1)1
d:
-1-
.
.
.
+
.
#r,
=
f+
.
1
=
y
..4.
r;
E zaxgct.v aj.
use the normalization by St and Iind a new measure # under wlajch the normalized variablc is a martingale. whatis the martingale equality that corresponds to nonmalization
,
2, In an economy there are two states of the world and four asscts. You are givcn the following prices for three of these securities in different states Of the world:
:
: ,
'
(82) : Price
-
g
)-q
j
=
(83)
(Jv.
A security secmrity s
2=1
Security (
The 40 is the dixount in riakless mrrtlwnp t:
1 1 Exerclses nondividend
1' You are given the price of a world where t here ropean call option Ct in a
'
state1 jx
8() q)
Dividend State 2
State 1
State 2
yj
1
($:
j
ls()
1()
rp
current prices for A, B, C are 100, 70, and 180, respectively. (a) Are the prices of the three securities arbitrage-free? ''current''
pay ing stock St and a Euare only two possible states:
320
if u ocmzrs
260
if d occurs.
=
=
risk-neutral martingale measure P* using the normalization by risk-free borrowing and lending. Calculate the value of the option under the risk-neutral martin#itle measure using
define
Sr
.5,
=
hy s:? Calculate the option's fair market value using the #. (e) that the option's fair market value is independent state Can we (t) of the choice of martingale measure? (g) How can it be that we obtain the same arbitrage-free price although we are using two different probability measures? (h) Finally, what is the risk premium incorporated in the option s p rjce? can we calculate this value in the real world? Why not?
, , 7.
,
=
(#'T
(c) Now
'
(81)
Jn this matr ix D the hrst row is constant and equals 1. This implies that State of the world the rcturn for the flrst asset is the same no matter which security riskless. is is realized. So, the tirst Of D with the Using the ar bitrage thcorem, and multiplying the hrst r0w Obtain I), we state pricc Vector S1
at time
=
c,
' Jxi
ne
'
if eac h statc under considcration has a nonzero probability of occurrence. Now supptse we c onsider a special type of return matrix where 1 #21
280.
with A
E
=
#d .5J,The annual interest rate is constant at r 5%. 3 months. ne option has a strike price of
of the two states are given by
(a) Fin d the
(80)
all i
i/f > 0 for
=
=:
have
Note that according to the theorem we must
probabiiities is St current timc is discrete, ne K 280 and expires
vjwtrue prife
.
*1
dLK
*
SN
/
41
11 yxercises '
;
:
(b) lf not, what type of arbitrage portfolio should one form? (c) Determine a set of arbitrage-free prices for securities A, C.
(d) supposewe
B, and
introduce a fourth security, which is a one-period ftltures contract written on B. What is its price?
7
c HA PT E R
.
2
the
A Primer on
Arbitmge Theorem
43
1j Eercises
5. You arc given the folloeng information concerrting a stock denoted
(e) Suppose a put option with strike price K 125 is written on C. ne option expires i.lzperiod 2. What is its arbitragc-frce price? =
by st.
value 102. . currentvolatility 30%. =
3. Considcr a stock and a plain vanilla, at-the-money. put option written on this stock. ne option expires at time /+, where denotes a small intenal. At time f, there are only two possible ways the St can move. lt can either go up to 5 lu+ or go Jtlwa to Sd/+.a. Also available to traders is borrowing and lending at annual rate r. risk-free sb
. '
xrmual
=
5%, which is known to be you arc also given the spot rate r months. during the 3 next stant con lt is hoped that the dynamic behavior of Sf can be approximated reasonably observation intervals of Iength 1 wellby a binomial process if one assumes =
@
'
,
(a) Using the arbitrage theorem. write down a three-equation system . withlw'p states that givs the arbitrage-free values of s, and E (b) Now plot a hvo-step binomial tree for St. supposeat every node the markets are arbitrage-free. How many thrceof the treesptems similar the preceding could thcn be to case equation entire for the tree? written (c) can you find a three-equation sptcm with 4 states that corredsto the same trcc? spon How do wc know that all the implied state prices are internally (d) consistent?
month.
.
a European call option written on ne call has a 120 and an expiration of 3 months. Using the price K and thc risk-free borrowing and lending. Bt, construct a portfolio thatreplicates the option. the replicating portfolio price this call. Using (b) 100 such calls to your cusSuppose you sell, over-the-counter, (c) tomers. How would you hedge this position? Be precise. (d) Suppose the market price of tMs call is 5. How would you form arbitrage portfolio? Considcr (a) Strike
.
.
,
4, h four-step binomial trce for the price of a stock usingthe up and down ticks given as followsz u
=
#
1.15
=
nese up and down movements apply to one-month = 1. We have the following dynamics for St, k up St-a
=
uSt
down
s'aa
=
-
.%
is to be calculated
6.
supposeyou are
at
-k-free
5%.
QJrest
=
6%,
*
u periods denoted by
ds t,
u$j
.
wsy..j
;
,
+1
.
Jwn
zzz
ySt +
whcre the e is a serially uncorrclatcd following values! 6
up and dtaw'rldescribe the two states of the world at cach node. Assume that time is measured in months antl that t - 4 is the expiration for a suropeancall option c, written on st. nc stock does not pay '-markct oarticioants''j to erow at dividends and its orice is expec-ted (bv r-is c-onstant of ris kn 15 Tlie to rate rate -%.
given tlle following data:
* Risk-free yearly interest rate is r The stock price follows:
.
where
date anv an-annual
./
an
'
1
vt.
=
=
cglt s ( y
binomial process assuming the
with probability p sjth probability 1 p. -
-
-be
-
ne 0 < p < 1 is a paramcter. Volatiiity is 12% a year. The stock pays no dividends and the current stock price is 100.
Now consider the following questions.
E
(a) According to the data given above, what is the (approximate) annual volatilty of St if this process is known to have a log-normal distribution ? ('b) calculate the four-step binomial trees for the St and the Ct. (c) Calculatc the arbitrage-free price C, of the option at time t 0.
(a) supposeJt is equal to thc
risk-free interest rate:
:
,
an d oat the st is arbitrage-free, What is the value of p? (b) Would a p 1/3 bc consistent with arbitrage-free St3
=
=
..L
'(.
7 cHh (c) Now suppose
J,t
P T ER
-
2
A Prinwr on the Arbitcage Tlaeorem
''
!
k7.
is givcn by: Jt
=
r + risk
'N
oywwygpwztyyfyswzw.
xwpmww..m -%
,m
-
4--x
premium '
(d)
What do the p and 6 repesent un der these conditions? ls it possible to determine the value of p?
alcxllxlsln .
7 Using the data in the previous question, you aTe now asked to approximate the current va lue of a European call option on the stock ,St. ne option has a strike price of 100, and a maturity of 200 dap. an appropriate time interval /, such that the binomial has 5 steps. tree What would be the implied u amd t? probability? What is the implied the tree for the stock price S:. Deternne Determine the tree for the call premium Cf.
RR
eter *
lnlstlc *
a:tjc
.
nV1rOn
.
ents
'
(a) Determine (b) (c) (d) (e)
tOC
.
: '
Wup''
,
1.
1 Introduction
:
The mathematics of dcrivative assets assumes that time passes continuously. As a reslzlt, new information is revealed continuously, and decision-makers may face instantaneous changes in random news. Hence. technical tools for pricing derivative products require ways of handling random variables over ienitesimal time intcrvals. The mathematics of such rarldom variables is known s stochastic calculus. Stochastic calculus is an intcrnally consistent set of tperational rules that are different from the tools tf calculus in some fundamental
.
Istandard''
,
wap.
2
At the outset ? stochastic calculus may appear too abstract to be of any use to a practitioner. first impression is not correct. continuous time nis is both simpler and ticher. market fmance participant gets some praca once ticeit is easer to work with continuous-time tools than their discrete-time equivalents. In fact. sometimes tbere are no equivalent results in discrete time. In thissense stochastic calculus offers a wider variety of tools to the fmancial
.
E
,
For example, continuous time aaalyst. mrtfolioweights. This way, reyicating
permits inlinitesimal adjustments in assets with Prtfolios becomes possible. In order to replicate the underlying option, an
q . i
! 4
:
' .. '
.
5
''nonlinear''
''simple''
C
: .c'
46
C H A PT ER
*
3
Deterministic and Stochastic Calculus
asset and risk-free borrowing may be used. Such an eact be impossible in discrete time.l
1
1 Infov'nwtion
replication
j yunctons
VII
@
.
.
s
the response of one variable to a (random) is, we would likc to be able to differentiate
'rhat
cbange in
of interest. variotlsj'unctions wtAu'd
like to calculate sums of random increments that arc of We jjyterest to uS. This leads to the notitan of (stochastic)tegral. arbitrary function by using simplcr wc would likc to approximate an Taylor scries approximations, ftmctions. vjjjsIeads us to (stochastic) We model wOu ld like thc Finally, dynamic behavior of to * z-nntinuolls-tj me rasdom variables. This leads to stochastic differential eqlt *
It may be argued that the manner in which information llows in linancial markets is more consistent with stochastic calculus than with caiculus.'' interval'' may be diffcrent on diffcrcnt For example, the relevam trading days, During some days an analyst may face morc volatile markets, . the basic obscrvain othel's less. Changing volatility may require chanpng tion period,'' i,e., the of the previous chapter. Also, numcrical methods used in pricing secqlrities are costly in terms of computer time. Hence, thc pace of adivity may makc the analyst choose coarser or finer time intervals depending on the level of volatility. Such approximationg can best be accomplished using random variables defned will be needed to over continuous time. The tools of stochastic calculus defme these models.
;
( dstandard
Wtime
*'
'-----sns.
if
1.2 M.odeing Rl
like to calculate wcwould anothcr variablc,
':
z someTools of
:'
:
E
.
Bfltwitn'
A more teclmical advantage of stochastic calculus is t.hat a complicated E randomvariable can have a very simple structure in contitmous time, once the attention is focused on infinitesimal intervals. For example, if the time Perjod under consideration is denoted by dt, and if Jl is then asset prices may safcly be assumed to have two likely movements: ': ! tmtick or downtick. strudure may be a good apUnder some conditions, such a interval #l, but not necessarily inflnitesimal dering an proximationto re ality l.2 denoted by time'' inten'al large in a thc 1to integralFinally, the main tool of stochastic calculus-namely, the Riemann linancial markets than appropriate in be to use may more caiculus. used standard in integral nese are somc reasons behind developing a new calcu1us Before doing this, however, a review of standard calculus will be he 1p11 After all, althoughthe rulcs of stochastic calculus itre diferent, the reasons ftr devcloping such rules are the same as in standard calculus:
Standard Calculus
calJn tus section wc review the maior concepts of standard (deterntinistic) xlus. Even if the reader is familiar with elementary conccpts of standard cajculus discussed here, it may still be worthwhile to go ovcr the examples in this section. The examples are dcvised to highlight exactly those points at which standard calculus will fail to be a good approximation when underlying variablcs are stochastic. .
,
i'infinitesirnaly''
: '
ibinomial''
''discrete
:
'
3 Functions
suppose and B are two sets, and let be a rule which associates to every elemeut x of ,4, exactly one elemcnt y in B.3 Such a rule is called kfunction or a mappiag. In mathematical analysis, functions are denoted by .d
f
.
:
-->
B
or by
,'
:
.,4 '.
y
=
f (A),
(2)
If the sct B is made of real numbers, functionand write
then we say that
f
is a real-valued
.
Would be thlz case Will the undcrlyillg State Space s itself discrete. nis values in the fut-ure. number possibie of ' ther underlying asset pricc can assumc only a Iinite values, and it may be ' ptlksible c)f the variable two random assume binomial can one 2A easicr te wor k with than, say, a rimdom variable that may assumc any onc of an signiqcantly uncountablenumber of Nssible values.
IVIIICSS, Of Course.
.'
f
..4
R, If the sets and B are themselves collcctions os functions, then f transforms a function into another function, and is called an operator. Most readers will be familiar with the standard notion of f'unctions. Pewer readers may have had exposure to random functions. ',
.->
.,4
'.
aThe
set
ad
is called the domain, and the set B is called the ranne of
)'.
C H A PT E R
48
3
Deterministic
Stoclastic Calculus
and
Flznctm.s
3.1 Ru
.
In the function y
E
(4)
,d,
flxt,
=
x.
Often y is assumed to
y. once the value of x is given, we get the element signi:cant following number. consider the Now be a rea f nere is a set p: wherc w c Wz-denotes a state of the Jzi(s depends on w e JF; f on x c R /
or
y
=
fxb
:
R x l#r
u?),
->
alteration. world. The function
(5)
R
x s R, w
q
''plug
wherc the notation R x JF implies that one has to variables,one from the1,) set H,', and the other from R.
in'' to
f.j
'
''
two
has the following property: Given a w QE I,F-.the 1J?) of x only. nus, for different values of w G HJ function becomesa f', different functions of x. Two such cases are show'n in Figure 1. we get tql and ftxa fx, tzlzl are two functions of x that differ because the second elementw is different. Wlien x represents time, we can interprct f @,t&ll and f (-r.u7z) as two differcnttrajectories tha t depend on differcnt states of the world. vJ) Hence, if u? rcpresents the underlying randomness, the function flx, Anotber name for random functions is be called a random hmction. . The function
stochastic processes. With stochastic proccsses. x will represent tne, and limit our attention to the set x k 0. we otken fun damen taI pout. Randomness of a stochastic process is in this Note trajector.y of the as a whole, rather than a particular vftlue at a specsc tejxns words. other the random drawing is done from a collection ln time. Pointin Of trajcdtries. Choosing the state of thc wozld, 1:), determines the complete trajcdoly
.;
(6)
I'I'C
49
3 ytmctions
2 Emmple: of Fuxcfimzs
There are some important mctions that play special roles in our discussjon. We will brie:y review them.
. ExponentiGl 3.2,1 Z71& The insnite sum
f (x,
Function
7
(7)
.,
) ''
.
,
.
.:
can
Converges to an irrational number between 2 and 3 as n --> x. nis number is denoted by the letter e. ne exponential functionis obtained by raising c to a powcr of .r:
y
..
ffkx.w)
E
;
.ew
y
!'
s p
(yj
.
'I'liis function is generally used in discounting asset pricts in continuous
ime. ne exponential function has a number of important properties. It is infmitelydlfferentiable. is, begirming with y elx) the followng opy' CF36On rnjat
;
'
=
Carl
be repcated
intinitely by recunively letting d
)' =
Jx f
tx-p'Ll
e%ez =
:
x 1
dfxt dx
(g)
.
We exponentjal function also has the interestng multiplicative
:
.'
F l Gu R E
(?'f(x)
be the right-hand
property:
eA+Z
(10)
,
.
Bhally if js random variable x a ,
.
7.
then
.p
=
ex will
be random as well.
50
C H A PT ER
3
*
Detenninistic
Stochastic Calculus
and
51
t, jutwtions
3.2.2 F/zcLogarithmic Function The logarithmic function is delincd as the inverse of the exponential function. Given
Thus, fundions of bounded variation are not excessively function will be of bounded variation/ fact any
'
udsmooth''
In
,
.'
Fvvmple ?.2.// Consider the function xzl'l
y the natural logarithm of
.y
=
e'T
(11)
EER,
.'r
'
-
is given by
ln(y)
(12)
0.
x,
=
T(/)
A practititaner may sometimes work with the logarithm of asget prices. Note that while y is always positive, there is no such restrictitn on x. Hence, the Iogarithm of an asset price may extend from minus to plus infinity.
0
=
11).'.s
tL .'.s
.
.
-
.
.
,S tn
=
represents the length of the fth subintewal, consider Now a function of time /(f ), dclined on the interval
'rhe
(/j
-
Wc form the sum f
=
l
(t),F1!
0.2
(15)
It/f ) - f (ff..l)I
This is the sum of the absolutc valucs the next.
of a11changes in
.f(.)
o .oxa
-Q.4
.'
from one q to
Clearly, for cach partition of the inten'al (0,T1, we can form such a Given that uncountably many partitions are possible, the sum can assume uncountably many values. If these sums are boundcd from above, the ftmction f (.) is said to be tf bounded variation. Thus, bounded variatbn
-.e m.:Q
'
Sum,
implies
V=
max
If(fj)- f (fj-1)I < ,
j=1
x,
(17)
()
=
0.4
(14) j .
when t
1
c.6
.
;
'.
n
()
s
fttl
Ij-lj
f.. (0,F1 -.->. R.
wjjen () < (
'
,
(13)
:J:
f
u,jyagujaro,,
,
.
N
5 It can be shown that /(f) is not of bounded variation, That this is the case is shown in Figure 2. Note that as I --> 0, f becomes excessively ne concept of bounded variation will play an important role in our dismlssions jater, one reason is thc following: asset prices in continuous
3.2.3 Funcffon' of Bounded Pzo/ft?rl The following ccmstruction will be used several times in later chapters. Supposc a time interval is given by (0,F1. We pardtion this inten'al into n, as n subintcrvals by selecting the ti, i 1, =
j
(.Tr
t ajn =
0.2
0.4
FIGU :
(16) '
5To show
lhis
formally, choose tilc partition 2 < 0< 2n + 1 ln - .l tyyir ljjk, pagjtjorj js .2,
of thc intewal , herethc maximum is taken over all possible partitionsvariations (0, F1, ln this in f (.), sense, L$ is the maximum of al1 possible Speaking, tf On and it i9 finite V is the tottll . g0,F1. Rougitly V measures the ltmgth t'f the trajectory followe d by (') 11S / 80e9 from 0 to F, W
yjjeu
oo
vauatjoa
I-)yytjj)
sAlhlf#lrl
.
-o
:
Ve
'
.
:: '
:. .
right-hand
- yt, ) j
side of this equality
I
=
4
<
1
2
4It can be shown that if funcion :as derivative ewewlwre a a is of bounded variatijan.
)'
.
o.8
0.6
RE
.
.
.
jp1wl + L3 n .j.
becomcs arbikarily
<
c
a 5
<
on
j.
.,c
.1
7
..).
.
.
.
..).
((),r1, tlacn tlte funuion
(1m 1
ln + 1
large as rl
-->
j
cx).
.
jjo
C 1-IA P T E R
52
.
Deterministic
3
and
Stochastic Calculus
4 Con:
E
time will have some unpredictable part. No matter how linely we slice the time interval, they will still be partially unpredictable. But this means that trajectories of asset prices will have to be very irrcgu jar. As will bc seen later, continuous-timc processes that we use to represent asset priccs have trajectories with unbounded variat jon.
,
.
' )
.
-
.
!
Xn
,
.
.
.
(z()j
,
-fx n
PEFINITION:
'P
)
.
''eventual''
wecall . the .r
n
>
'
:
'''
XX)
$221
y(x + a) ytxl -
.
jx
%
':
where
nc
(21)
limit of xn.
lim
a-.(j
(23)
,
is an increment in x.
variablc
x can represent any real-life phenomenon. Supposc it reptime.l Then would correspond to a linite time intewal. (x) would be thc value of y at time x, and the + ) would represent the fx Valuc of y at time x + /, Hcnce, the numerator in (23)is the change in during .F ratio itself becomes the rate of change in a time interval &.
.
'
ne
For example, if y is the price of a certain asreprcscnt the rate at which the price
(23)would
;
changes during an intewal 1. Why is a limit being taken in (23)?In defining the derivative, the limit has a practical use. It s taken to make the ratio in (23)independent of the size of f , the time intewal that passcs. Fbr making the ratio independent of the size tf , one pays a price. The derivative is dclined for inhnitesimal intervals. For larger intemals thc large/and delKative becomes an approximation that deteriorates as A gets
.
,
'
?
.
Wclosenessz''
.
5t
larger.
.
Ene
.
reader
should not conltse
dentivewith ti,e term
'the
::
ZTHt)C
L .
. r.
j
'rhc
the same intewal. ysetduring at timc x, the ratio in
In words, a:n convcrges to x' if a;?l stay: arbitrarily close to the point x* E after a Iinite number of stem. Two important questions can be asked. Can we deal with convergen of xn if these Were random variables in; stea d of deterministic numbers? This questim is relevant, since a random number A:s can Gmce iva bl assume an extrcme value and suddenly may fall y Vel'y far f'rom any x* even if n > N : We Secmdly, since one can define differtmt measures o f shotlld in principle be ablc to define convergence in differcnt ways as well. .g Are these dehnitions all cquivalent? We will aoswer thesc questions later, However, convergence is clearly j ing a qllantity that does nOt easily a very important concept in approximat lend itself to direct calculation. For example, we may want to defme notion of integral as the limit of a gequence. .
.&
=
rescnts
'j
N.
=
be a ftmction of x e R. Then the derivative of fxj with respect to x, if it exists, is formally denoted by the symbol fx and is given by
of rea 1numbers xn converges DEFINITION: We say that a Sequence > 0, there exists a N < x guch that (x) < arbitrary if for x* c to
for all
Lct
.
Thc notion of conveqence of a sequence has to do with the tx). In the case where xn represents rcal numbers, we V aluc of x n as n -can state this mtre formally;
e
chain r//d.
,
ne
Ixu- x*I<
53
changc, But it is a rate of change for irnitesne derivative is a rate of jormaj dctinitjon flrgt. yycjgjvr imal movements. a
''object''
an object that changes as n is increased. This of functiong, or a sequence of E an be a scquence of numbers, a sequence essential point is that we are obscrking successive vers ion: ooerations.
O
usingthc
.E
.
wherexp rcpresents
Limit
(Ttotl
Suppose we are given a sequence X2 r
and
(fsmoothness''
4 Convergence an d tmit
,
e rgence
4. j 7-he negivutivc The rlotion of the derivativcs can be Iooked at in (at least) tw'o differof with the ent way s. First, the dcrivative is a way of dealing of of variables under considdefming of change rates functions. lt is a way particular, if trajcctories of asset prices are irregulary'' then Ic tion. era with exist. time not respect to may their derivativc second,the derivative is a way of calculating how one variable remonds to a change in another variable. For example, given a change in the price of the underlying asset, wc may want to know how the market value of an option written on it may movc. These types of derivatives are usually taken
!
'
Y() , X1
,
..
.'
--derivative
lhe mathematical operation of differentiation securilies'' used in fmance,
is one of he few dcterministic variables one can imagine.
or taking a
,;d
.;.
C H A PT ER
54
.
3
Deterministic
4.1.1 Ewample: Tc Faponcntial Function
arld
As an exnmple of derivatives, consider the exponential fx)
=
Aerx,
x c R.
Stochastic Calcults
function: :
*
=
=
=
FL
g
.,4##m.Wr?zI/foa # ).2 Example.. T/zcDerivative 4: an To see an example of how derivatives can be used in approximations, COn sider the following argument. Let N be a fnite interval. nen, using the defmition of derivative in (23) we can write approximately and if is
,
(24),
zsmall,''
(x + ) y(x) + yx a,
equality means that the value assumcd by /(.) at point + vhisapproximated value of f.4 at point x, plua the derivative by tbe be can by 1, Note that when one does not know the aact value jx multiplied ?), the knowledge of (x), A, and k is sucient to obtain an of fx + s approximation. result is shown in Figure 4, where the ratio Tbis
rxj
.x
,
(25)
4
,'
The quantity fx is the rate of change of f (x) at point x. Note that as ets largcr, the term er* increascs. nis can be seen in Figure 3 from the increasinggrowth the f (.) exhibits. ne ratio .r
fxs =
(zg)
.
r/'@).
X
and
convergence Limit
4
A graph of this function w ith r > 0 is shown in Figure 3. T,a'king the derivativewith resped to x formally: vAdfxj dx
'
'
f(x
(26)
r
(28)
becomes smaller represc nts the slope of the sepuent denoted by AB. As and smaller, with A ftxed, the sepncnt AB converges toward the tangent at tlle point A, Hence, the derivative fx is te slope of this tangent. when we add the product fxh to we obtain the point C, This point approximation Whether B. be taken of this will be a as an can or and on the shape of the a bad approximation depends on the size of function f (.). l
.
is the percentage rate of change. ln particular, we see that an exponential functionhas a constant percentage rate of change with rcspect to x.
) - y(x)
+
.f(x)
i'good''
f (x)
'
rj.j
:
4
'
.'.
'.
.j,(x. A; .j(x)
. I I
.'
I I
.:
A
I I
.
I
Slope
=
I l
r (Aer)
.
I I
j
'
j
'
'
'F I G U R E
j7 j G kr R E B1f
'
I I
x+
.
I I
x.
time, and if is the tspresents'' thcn /'(x + 1) will belong lo !he Ksfuture.'' and a1l quantities that relae to the o j'x y /-(x) x. are In tlus sensc, they can a 'e c'prediction'' used for obtauing of x + a) in real time. nis prcdction requires a crude IMViD: a numerical value fer h, thc value of the derivative at the point x.
'
3
.
.
r
x tepresents
However
'
'
. :. .
.h ..
,
.x
'xpresent,''
56
CH APT ER
*
Deterministic and Stochastic Calculus
3
j
and
convergence Limit
:(x)+fxl
57
f (x)
i
ftx)
eawr
.
i
:
X
a
f (x+A)
......
A
( j l ( L i
.
i g
i
..
FIG U RE
i i
.
xa
,
X+A
X
i s i !
@ g i i 1
a+A
F I G tJ R E
$
' Two suple examples will illustrate these points. First, considcr Figure 5. Here, is largc. As expected, the approximation f (x)+ X A is not very .: near fx + A). Figurc 6 illestrates a more relevant examply, We consider a function /'(.) that is not vely smooth. The approimating fx + ) obtained from '
.
4. l.3 Example: High Wtiation Consider Fiplre 7, where the function extreme variations even in small intervals
1
/x) .
is continuous, but exhibits
Here, not only is the prediction
:
y(x +
c
,
k)
(x) + fx
(30)
,
likely to fail, but even a satisfactory definition of fx may not be obtained. Take, for examplc, the point What is the rate of change of the function the point at It is dificult to answer. lndeed, one can draw many /11) with differing slopcs to flx) at that particular point. It appears tangents that the function (.r) is not diffcrcntiable, .ra.
fx
.x()?
+
)J
(29)'r
.f(.x)
+
.
x
/:
' tion to the truc fx +). mayend up being a vezy unsatisfactory approxlrna the function becomes, thc more such ; clearly, the morc approximationsare likely to fail. considcr an extreme case in the next example. .f(.)
''irregular''
4.2
.
..
f(K)
'
E
' i '
j : Xo
F 1G U R E
6
x
Chuin Rxlc
The secoad use of the derivative is thc chain I'ule. ln the examples discussedearlier, fx) was a function of x, and x was assumed to represent time.The derivative was introduced as the responsc of a variable to a variationin timc. In pricing derivative securities, we face a somewhat different problem. Tbe Price of a derivative asset e.g., a call option will depend on thc price of the undcrlying asset, and the pricc of the underlying sset depends on time-g Hence, there is a chain effcct. Tue passcs, new (small)events occur, tbe price of the underlying asset changes, and tMs affects the derivative asset'sprice, In standard calculus, the tool used to analyze these sorts of nIle.'' cbaineffects is known as the ,
,
''chain
9As time passes, tlalzexpiration date of a contract comes closer, and evcn if he underlying price remains constant, thc price of the ca11 opton will fall.
Get's
.
T!U
:
58
c H A PT
ER
.
3
Dcterministic
and
Stochastic Calculus
and Limit
Supposc in tbe example just given x was not itself the time, but a deterministicfunction of time, denoted by the symbol f k 01
nen the function
=
jvalellt
4.3 The lntegrtzl
(32)
yt /(#(r)). =
Thc jyytegral js the mathematical tool used for calculating sums. In contrast to the S operator, which is used for sums of a countable number of objects, intcgrals denote sums of uncountabk injnite objects. Since it is not objects that are not even countable, a formal clear how one eould integral has be dcrived. of to definition The gcner al approach in dehning integrals is, iri a sensea obvious. one would begjn with an approxmation wolving a countable number of objects, and thcn take some limit and move into uncountable objects. given that dfferent types of limits may be taken, the integral can be delined in various ways. In standard calculus the most common form is the Riemann integral. A somewhat more gencral integral dehned simiintegral. In this scction we will review these lazly is the Riemann-stieltes delinitions.
.
The qucstion is how to obtain a formula that gives the ultimate effcct of ' change in f on the y,. a In standard calculus the chain I'ule is desned as follows. DEFINITION:
For
f
Gsum''
and g defined as above, we have
#.f(#(/)) dt) dgtt dt
Jy
-=
dt
;
(g3)
.
-
i
According to this, the chain rule is the product of two derivatives, First, the derivativeof (g(/)) is taken with respect to g(l). Second, thc dcrivative of E. ,(r) is taken with respect to 1. tinal effect of I on .y/ is then equal to tbe product of these two cxpressions. The chain rule is a useful tool in approximating the responses of one variable to changes in other variables. Take the case of derivative asset prices, A trader obsen'es the price of the undcrlying asset continuously and wants to know how the valuation of ' the complex derivative products written on this asset would change, If the ) product, these changes can be observed . derivative is an exchange-traded l0 from tlle markets directly. However, if the derivativc is a structured product, its valuation needs to be caiculated in-house, using theoretical pricingmodels. nese pricing models will use some tool such as the S rule'' shown in (33). In the example just given, fx) was a function of xt, and x, was a deterministicvariable. There was no randomness assoc iatedwith xt. whatwould happen if x, is randoma or if the function f (.) depends on some random : z, as well? ln other words, .
'rhe
)
4.3.1 The Riemann lntegral we are given a detenninistic function /(/) of time l G (0,F1. Suppose we are interested in integrating this function over an intcrval (0,F) r y(x)ds g4l a which corresponds to the area shown in Figure 8. In ordcr to calculate the Riemann integral, we partition the interval g0,rj into n disjoint subintervals
:
,;
'
'>
I
,
''
,
dtchain
variable
chain rule formula? formula change in stochastic environments?
The answer to the Ergt question is no. The chain rule formula given in (33) carmot bc used in a continuous-time stochastic environment, In fad, dfstochastic calculusj'' we mean a set of methods that yicld the formulas by course, 10(,')f
t
/() 0 =
then consider thc approximating n
here is always the quegtion
security at that iustant.
of whether
...
J=1
DEyqxlrrlox:
<
<
.
.
tn
<
.
=
T)
(35)
).
(36)
sum t i + j..j
f
.
1. Can we still use the same 2 How does the Chain rule
same as that of standard calcu-
Thc puose rules, though, are diferent. jus,The
is called a composite functionand is expressed as
.f(.)
the chain rule and that approximate the laws of motitm of
to equ variables in continuous time. random of stochastic calculus is the
(31)
,g(r).
xt
59
4 convergencc
(t
2
-
f..-:
oiventhat mpx Itj
/?-j
-
,
I-->
0,
te Riemann integral will be defined by the limit '
n
j''q
''
tile markets are corrcctly pricing t.*:. ).
f=1
wherethe
'.
.E
.
t i +. -.
jj..j
2
j tf
.-.
jj..y
)
v ()
limit is taken in a standard fashion.
ts j #s
j
(37)
.'
(
60
C H A P T'E R
*
3
Deterministic
and
.
Stochastic Calculus
'
4
:
and
convergence Limi f(t)
: :
..
l
:.
'
..
.
... : ..7. .'
i'
8
i
$)
t!
tz
: 1
$3
: : : : : ; .1
,
: l : : : : : : l : 1
: : l : : : : : : 1 : 1
qn-1
tazzRn
.1
E
: : 1: : 1
.1
':
: t4
t,
.
' : The tcrm on the left-hand side of (37) wolves adding thc areas o j n fj-l)/2) 1) and rectangks constructed using (fj fj. as as the base ffti + the hcigbt. Figure 8 displays this construction. Note that thc small area A is appro ximately equal to the area B. nis is especially true if the base of is, docs not thc rectangles is small and if the function f (/) is smooth-that vary heavily in small intewals. In case the sum of the rectangles fails to approximate the area under the CUWC. We m ay be able to corrcct this by ccmsidcring a hner partition. As the Iti - fj-j I>s get smaller. the base of the rectangles will get smaller. More rectangles will be available, and the area can be better approximated. Obviously, the condition that ftj should bc smooth plays an important path followed by /41)r role durlg this process, ln fact, a very method. Using the ter- ' may be much more diflicult to approximate by this k minology discussed before, in order for this method to wor the fundion (f ) must be Riemann-intcgrble. A counterexample is shown in Figurc 9. Here, the function /(f ) sho< j gteep variations. If such variations do not smooth out as the base of the L. rectanglcs ge ts smaller, the approximation by redangles may fail. with the in dcaling will that be important EE We have one more comment Ito integral later in the text, The rectangles used to approximate tlie area
: : :
: : : : : I 1
'
:
,
under the curve were constructed in a particular way. To do this, we used thc value of fltj evaluated at the midpoint of the inteaals (. 1/-: Would the same approximation bc valid if the rectanglcs werc dclined in a different fashion? For example, if one detined the rectangles either by
E
T'l G U R b
I
: : : : 1
:
:
: :
:
: ; :.
2
: : : l : :
'
ytillti
-
,
Or
.
(38)
/f-l),
(39)
by
fti-jjti
-
would the intcgral be diffcrcnt? To answer this question, consider Figure 10. Note that as the partitions gct liner and finer, rectangles defmed either way would cventually approximate thc s'ame area. Hence, at the limit, the approxnation by rectangles would not givc a different integral evcn when One uses different heigbts for deining the rectangles. lt tums out that a similar conclusion cannot be reachcd in stochastic environments. Suppose f p)) is a function of a random variable Wz)and that we are interegted in calculating
.
'irrcgular''
:
'
,
'
v
'
ln
ytpl,llt
(4p)
.
Dnlike the deterministic case, the choice of rectangles
.
/(p;,)(p;,
.
,
)
j..j
-
)...
-
p;,-,
)
delined by
(41)
('
:
' u
:
62
C H A PT E R
ftt) ca
Deterministic
3
.
and
..
Stochastic Calculus
4
upwr reaangle
0.25
0
.k,)
i
i
(
i
i
0.15
Roxaugleusing
i
j
'
j
I-ower Rectangle
()
O.4
0.6
FIGIJRE
1o
0.2
the where
:
0.95
hxj
E
0.8
1
'
t
Aq
.
T
n
.
0
l'Ntate that
(14$ i -
Wzi and f 1)
#K are i
=
A(x)dx.
f=1
g
fi + / i -
a
1
fti) -
.f(J2-1)).
(49)
time. Hence, it may appcar that the Riemann-stieltjes integral is a more appropriate tool for dealing with derivative asset prices. However, before coming to such a conclusion, note that alI the discussion thus far involved deterministic functions of time. Would the same dcfinitionsbe valid in a stochstic environment? Can we use the same rectangles to approxnate illtegrals in random environments? Would the choice of the rectangle make a difference? e answer to these questions is, in general, no. lt turns out that in Stochastic environments the functions to be integrated may vary too much for a straightforward extension of the Riemann integral to the stochastic A new definition of integral will be nceded.
: '
':
.
(43)
.
E
rjwjj
(44)'
case
correlated.
-
.
i.( E
#(-) #.f(.)
Because of these similarities, the iirnit as maxj ti I - ff-j I --> 0 of tbe right-hand side is known as the Riemann-stieltjes integral. ne Riemann-stieltjes interal is uschzl whcn the integration is with respect to incements in /(x) rather than the x itself. Clearly, in dealing with Iinancial derivatives. this is often the case. ne price of the derivative asset depends on the underlying asset's price, which in turn depends on
We have alrcady discussed the equality
dfx)
(48)
.
.
f (47).
yzlx)dx.
=
This definition is not very different from that of the Riemann integral. In fact, similar apprtmmating sums are uscd in both cases, lf x represents tlrrle r, the Stieltjcs integral over a partitioned interval, y, wj,is given by
.
+
(47)
Jy(x)
(42)',
4 3 2 The Stieltjes Integral integral is a differcnt definition of the intcgral. Define the 'I'he stieltjes diyerentialdf as a small variation in the function (x) duc to an infmitesimal variation in a':
g(x) df (.t),
With
;
To see the reason behind this fundamental point, consider the case where . 1#) is a martingale. nen the expectation of thc term in (42),conditional on information at time fj.-l, will vanish. nis will be the case becausc, by dchnition, future increments of a martingale will be unrelated to the current : information set. on the other hand, the same conditional expedation of the term in (41) will, in general, be nonzcro. 11 Clearly, in stochstic calculus, express ions that utilize different deEnitions of approximating rectangles may lead to . different results. result. Note that when Finally, wc would like to emphasize an important /(.) depcnds on a random variable, thc resulting integral itself will be a 5 random variable. In this sense, wc will be dealing wit.h random integrals.
fx
(46)
xa
E
H6,-1). f ('r'1k1)(11$ -
dfxj
gxjfxx).
'Ihc.n the Sticltjes integral is defined as
(
-
=
:'
will in gcneral result in a different cxpressicm from the rectangles:
dA)
(45)
function /?(.v) is given by
'
=
hxtdxxg
Inipoilt
i
.
.1
Limit
(Note that according to thc notation used hcre, the derivative fxxj is a of x as well,) Now supposc wc want to integrate a ftmction /l(.r) orjctjon rCSPCIX to 3:: W ith
.
o.2
and
ccnvergence
'
.E
.
64
C H A P T ER
.
'
:
Detenninistic and Stochastic Calcolus
3
*
:
4
cxmvergenceLimit Elnd
65
J.J, J Evnmple In this scction, wc would like to discuss an example of a RiemannStieltjes integral. Wc do this by using a simplc function. Wc 1ct g(8)
;
.lit)
avt
=
(50) '
,
where a is a constant, This makes g(.) a lincar function of value of- the integral
.5/.
'
12
w'hatis the
t1
s /
A
(51)
J.s(/)
gus.
-
,
V-
'!
.,
2*
.
0
a
=
s
-'r-
,
kv.r
IY-Y
if the Riemann-stieltles' dehnition is used? Directly zttaking''the integral gives
auhd'St )
W
1 2 -St
.a:'''
*
'tYr=
.
(52)
a
ko.v.F2-Il>rzs-zsuxasprovs: .,
!
()
--
VS
,
2
'
So T
t
0
dvbjt)
=
1 1 a -S2 2 z'- -&2
z
(53)
xjujxnt (sooss/a
:
F l G U R lJ
Because g(.) is linear, in this particular case the approximation by redangles works well. This is especially true if we evaluate the height of tlle redangle at the midpoint of thc basc. Figurc 11 shows this sctup, with 4, a Due to thc lincarity of. g(.), a single rcctangle whose heigbt is measured the midpoint of the intewal & 5'wis sufficient to replicate the sladed at area.In fact, the area of the rectangle 5'().,dS,$z is
4 4 Iwtcgvution !r-yPuvjs ln standard calculus there is a useful rcsult known as integration by parts. lt can bc used to transform some integrals into form more convenient to a deal wit. A smilar restgt is also very useful in stochastic calculus, even tlmugh the resultng formuia is different. Consider two differentiable functions (f) and (/), where / (E (0,F1 represents time. Then it can be shown that
:
=
,
-
1
2
1
l5'z- s'lpj a -sv 2 - 2.-s =
!
T ./',(1)(1)
g,
ne Riemann-stieltj 'es approximating sums measure tangle exactly, with no need to augment the number rec rectangles.
,
(54)
thtz area tlnder the of approximating
.
()
dt
=
E./'(F)(F)
-
/'(())(0)j
I (f) and ftt) where With
r -
(j
/7/(f)y(r) dt,
(55)
ftmctions are the derivatives of the corresponding respect to time. 'They are themselves functions of time /. ln the notation of the Stieltjes inlegral, this transformation means that 2.11expression that involves an integral
'
1
.
I 25'
11
by
rectangles.
.5l
Sv
'
.
Now, let us see if we can get the same result using approximation
+ vb'v a 2
E
z-vcwx,
ji--kka.'
.
r
r
! i
r
1'
G
i
Equal eiangLes
is a f'unction of time.
f.l .
;
l'
'
'
htj
dft)
U'
66 Can
c HA PT ER
Deterministic and Stochastic Calcultls
3
.
j parrial Derivatives
flnancial markets. However, partial derivativcs are very useful as inPrjce in termediary too ls They arc useful in taking a total changc and then splitting from different sources, and they are useful jt into components that comc in tos/ diycyentiations. on seforedealing with total diffetentiation, we have one last comment changes, ytjal derivatives. Because the latter do not represent pa stochastic or dctcrministic envitjwj.e is no difference between thcir use in yosments. We do not have to devclop a new theory of partial differentiation cnvironments. stochastic ja j'sake this clearer, considcr the following example.
now be transformed so that it ends up containing the integral
z n
.
.
'
f f) dh($).
(57) .
The stochastic version of this transformation is very useful in evaluating , 2 Ito integrals. In fact, imagine that is random while (.) is (conditionusing integration by parts. we ally)a detelnninistic function of timc, stochastic intevals as a function of integrals with respect to can exprcss variable. ln stochastic calculus, this important role will be deterministic a playedby Ito's formula. .f(.)
Gobscrved''
'lnhen,
'
.
5 Partlal
vo
5 j sxampje -
Derivatives
Consider a call option, Timc to expiration affects the price (premium) of call in dfferent First, time tlw two expiration date will as ways. passes, the h and the remaining lifc of the option gets shorter. This lowers the approac But at the same time, as time passes, the pricc of the underlying premium. will change. This will also affcct the premium. Hence, the price of a asset callis a function of tw'o variables. It is more appropriate to write
Considcr a ftmction of tw'o variables ys / f)
'
,
.
=
FCS:
,
f
),
+. t2
(61)
,
St is the (random)price of a tinancial asset and / is time. involves simply differentiating F(.) Taking the partial with respect to With respect to OFCSf,t) (62) = PS? Here cut is an abstract incrcment in St and does not imply a similar acttzal change in rcality, In fact, the partial derivative Fs is simply how much thc function F(.) would have changed if we changed the ; by one unit. nc Fs is just a multiplier. Whcre
,
G
,yyy
=
.t
.
.%:
.3.
(58) '
where Ct is the call premium, St is the price of the underlying asset, and f is time :E ' Now suppose we the time variable f and differentiate Fxt /) with rcspect to St. The resulting partial derivative,
.
iiftx''
,
r
oFst,
f)
0'% -
(59)
F,
s.2 Total Digeventias we obsewe a small suppose
,
change in tlic price of a call option at time Let this total change be denoted by thc differential dh. How much of this valiation is due to a changc in thc underlying asset's price? How much of the variation is the result of the expiration date getting nearer as time Passes? Totai differentiation is used to answer such questions. Let f (x;,r) be a function of the two variables. Then the total differential is detined as
'
effect of a change in the price of the unwould represent the (theoretical) derlying asset when time is kept ftxed. nis effect is an abstraction, because 1 in pradice one needs some time to pass before can change. ) The partial derivative with respect to time variable can be dehned simi: larly as , fFst, tj Ft (60) = pf
f
ct
,
.
N O te that even though St is a hmction of time, we are acting as if it does not change.Again, this shows the abstract charader of the partial derivative. As t changes, St will change as well. But in taking partial derivatives, we behaveas if it is a constant. Because of this abstract nature of partial derivatives, this type of differentiationcannot be used diredly in rcpresenting actual changes of asset
dj-
yst
,
/)
ofst,
tj
dt. (63) 0St ln other words, we take the total chfmge in St and multiply it by the partial derivative Js. we take the total change in tne Jf and multiply it by tbe partial derivative j;. The total change in f(.) is thc sum of these two Poducts. According to this, total diffcrcntation is calculated by splitting an obsewed chan ge into different abstract components
'
' ,
=
dSt +.
t?f
.
.
y...
'.. (
' .
g'
68
C H A PT ER
Deterministc
3
*
Stochastic Calcolus
and
69
j partial Derivatives
Tlae convcntion in calculus is that, in gcneral, tcrms t:f order ((;r)2or negligible if is a determiniaic variable,l4 Thus, if hi#her are assumed to be t jaat x is detenninistic, and lct (x xa) be small, then we could we assume Taylor scrics approximation:fjrst-order use the
5.3 Taylfr Sevies Ea'panxsitm
.x
bc an infinitelydifferentiable function of arbitraryvaluc of x; call this x(). Let
fxj
The Taylor scries expansion of
DEHNITION:
deuedas flxt
fxj
=
.'z(.r)(.K
+
1
-i'Tx=(Ab)(,Y
=
f=0
-(f
1
-
fxj
R, and pick an
:
around
irtl
:
R is
fxt
'
1 2
-/.(.b)(,Y
+
a+
.t())
+ 3. x
Ab)
-
(E
.x
*b) 2
-
nis
'
'
(.x
-
i.
Under thcse conditions.
:.
i
I0)
)jx)
becomes an equa lity if the
(64)E
'
E
i (.ra)(Jr
-
,
Jx g)g(x
respect
to x
one js
and the
'
Jf(xl
:
d.f(A)
Ixre,
1lZ
wc surely have
fxx
=
>
I-ri-
>
.
'
'
5.3.1 The equation kcctpritf-tlrer
(x) flxoj +
.Jf(-),the
latter become constaats,
xtxoltx
-
x()) +
1
jxxtxeltx -
,
..
j,l
'
'.
lf so the terms (J.x)5 '
'
.
: '
(J2)
x(j)
w())2.
(73)
point is quitc relevant for the later discussion of stochastic calculus. ln fact in order to prepare the groundwork for Itffs Lemma, we would like to consider a speczc example.
,
oncc xa is plugge,d in
-
'rhis
:7
lhat
LLnj,
j cajud a frst-order Taylor series approximation. often,a better approxis mato,i can be obtained by including tlae second-order term;
;
the result even smaller,) Under these conditions, we may want to drop some of the terms tm tlle ' right-hand side of (64)(we c,an argue that they are negligible. To do this, E We mllst adopt a dConven tion'' for smallness and then eliminate al1 terms Small When that are <(negligible is a trm aerd ing to this criterion. But Cnolgh tfl be negligible? mplies
xtx
'
(66)
a higher power, we multiply it by a small
(71)
Approximations
tx)1 tx(jl+
E
.
dA''
.x
E
xol3
in terms of the differentials dx
yxj
')
(Each time we numberand make
las point 13,1711: indcpcndentof x.
tjao
(70)
.x
(65).. E..
'Lsmall''.
(69)
-
js written as a function of instead of the usual tyecause we arc considering the limit when approaches xa,
!
.x()l
is denoted by
fxt - /(x0).
.x
Ix,- wnl > Ix: raise 1x1 to -
-
.x())
As a result we obtain the familiar notation arld df
,: granted, We will, however, discuss some of its implications. First, note that at this point the expression in (64)is not an approxl 'mx- ! tion, ne right-hand side involves an inhnite series. Each element involves ''simple''powers of orily, but there are an inhnite number of such ele-? ments.Because of this, Taylor series opansion is not very useful in practico. Yet the expansion in (64)can be used to obtain useful approximations.' > we considcr Equation Suppose (64)and only look at thosc x near xu. That is, suppose
xlz
(a
y(.) ja
:
-
variation
.ra)
'
are not going to elaborate on why the expansion in (64)is valid if t we fx) is continuous and smooth enouglz. Taylor series cxpansion is taken for
'rhcn,
(68)
0.
.
where fftxtl) is the jth order derivative of f(x) with evaluatedat the point &.13
(.x -
.-..>
the ivnitesimal
.
.x())
has a derivative at xtj and if we 1et
x))
-
'
(67)
T(*k,)+ X(3:(,)(;r abl.
A
g. .( .
' k..
,
(:x)4
,
.
.
.
,
will be smaller
than
(l'fA)2.
70
c H A PT ER
5..3,2Example: Dumron
Consider the exponential and t G (0,F1:
and
.
Deterministic
3
and
Stochastic Calculus
cbave-ly
function where t denotes time, F is ftxed, r B
t
=
looc-rt
r-r)
>
5 paxial
vjoure 13 plots the exponential cun'e with the second-order
pP
This fundion begins at t 0 with a value of Bij 100c-rT. Then it increases T, the value of Bt approaches 100. at a constant percentage rate r. As t visuaiized value. Hence, Bt could be as the as of time 1, of 100 to be paid value of at time F. lt is the present a default-free zero-coupon bond that matures at time F, and r is the corresponding continuously compounding =
sl u
i
=
:
We are interested in the Taylor series approximation of Bt with respect to f assuming that r, F remairl constant. A hrst-order Taylor series expansion around / t will be given by
,
.
'
=
(r)l00e-r(F-:)(f
-
j()),
t
(u
'
(75)
((),F),
where the fil'st term
:
/o. The on the right-hand side is #/ cvaluatcd at f ' second term on the right-hand side is the hrst derivative of BI with respect to f, evaluated at /a, times the increment t /(). : Figure 12 displays this approx'lmation. The equation is represented by a F T. The Erst-order Taylor series approxconvcx curve that increases as I a4. imation is shown as a straight line tangent to the cun'e at point Note that as we go away from fll il either directitm, the line becomes a worse approximation of the exponential cuvc. At t near l0, on the other hand, the approximation is qlzite close. =
'
.p < /
-
dBt Bf
j g
j(;, sj.
=
1
-(
1. -( r-/)2(r-o)2 J,
,)(r-rj))+
r-
,
f
s p, wj,r
,
-(T N
-
tjtr '
'
1
- aj + -4F 2
-
*'
/) c
(r
150 .
-
r(,) a ,
90
'
8c
;.
t
e
((),T1, r
'
A
Parabola
s0
..
:
50 40 20
5
1(1
15
''-''
FlGU RE
25
:
30
j
30
:.
2:
A
ztn
. L(
i: O
Time )
l2
5
10
$
IS
FlGURE (
j
70
q,1.
60
s
.
t ''
().
(10()e-rn(F-?))
Rr
BI
c
>
>
0,
This expression provides a second-order Taylor series expansion for the Percentage rate of change in the value of a zero coupon bond as r changcs
'
1O0
(76)
.jyjz,
(r )1.f)()g
-l)j
jj()(je-ro(T
by oividing
--y
c
fa)
0,
5
+
.,jw./*)(j
-
-
.
100c-dF-la)
1. z
(r)1(J()g-r(r-/o)jf
..j.
The r;ghsjaand side of this equation is a parabola that touches the exponent jaj curve at point A. Because of the curvature of the parabola near 10, we expect this cun'e to be nearer the exponcntial function. Taylor Note that the difference between the first-order and second-order hinges on the size of the term (.1 /(J)2. As / nears Sef jes approximations Smaller. More importantly, it becomes smaller faster f this terms becomes f()). t han the term (f - approximations show how the valuation of a discount Taylor series xesc bond changes as thc maturity dase approaches, A second set of Taylor scties approximations can be obtained by expan djyjg s, wjth respect to r, keeping /. T flxed. Consider a second-order approimationaround the ratc rt);
.
yieldto p'tzfzrf/A'.
t
1(j(u-r(r-/o.) +.
--/
B
Taylor series
-rmoximation'.
0,
(74)
-
Derivatives
'
:.
20
25
30
Trilth=
s.'
E
C H A P T ER
*
Deterministic
3
imd
Stochastic Calcultts
6 Ccn
side measures the percentage fatkt of change infinitesimally. The Iight-hand in the bond price as r changcs by r - r(T, Where rlj can be interpreted as the current rate. We see two terms containing r - ra on the right-hand side. In hnancial markets the coefficicnt of thc s1'St term is called the modilied duration. The second tem is positivc and has a coefficient of 1/2(F - f )2. It represents the so-called convexity of the bond. (verall, the second-order Taylor series expansion of Bt with respect to r shows that, as interest rates the value of the bond decreases (increases). increase(decrease), nc of the that the bigger these changes, the smallcr their implics bond vexity'' relative effects,
clusiom
ju an ordinary dterential
equation,
dxt .-dt
.
wherc tjw unksown
s xt, a
axt + b,
=
function.More
'.
.x/
In tbe case of the ODE,
:
CCViCW
thC
dB3
Sf
Of
Concept
dB t
=
dt
-r,s,
Bv, rt
with known
>
changes ' This exprcssion states that Bt is a quantity that varics with /-i.e., in Bt arc a function of t and of Bt. The equation is called an ordinary dfferential equation. Here, the percentage vaiation in Bt is proportional to r: some factor rt times dt UV (78)E t -% dt. ,
'
'
s
Now, we say that the fundion #,, dchned by
Bt
e-
=
..(J'
ro du
(3
.
6
Here
t he solution
,
b
=
,
,
,4
.4
.
xt'
(85)
.,
E
E .
k. .
l ,
conclusions
equivalents. 'Ihe other jmportant concept of the chapter asmallwas the notion of ResS.'' ln Particular we need a ctmvention to decide when an increment is Small enough to be ignored.
(S1)
0,
thc unknown element is a vector. Under appropriate cond it ions thc solunlultiplied by the vector b. -1 y--.i e the inverse of tion would bc a' =
=
chapter reviewed basic notions in calculus. Most of these concepts elementary. While the notions of derivative, integral, and Tayior series may all be well known, it is important to review them for later purposes. Stochastic calculus is an attempt to perform similar operations when the utjderlyjug pjjenomena are continuous-time random processes. It turns out that in such an environment, the usual dehnitions of derivativc, integral, and Taylor scries approximations do not apply. In order to understand stochastic Vtrsions of such concepts, one first has to understand their deterministic
is '''
=
-
an + btds
Wee
(80)
x.
to be determined.
,1,Y
(64)
.
'l'his
'
the unknown is x, a number .x -1/2, ln a matr equation,
sta
(79)..
,
rrhat
=
FD
where t h e un known xt is again a function of r.
,
.:
solves the ODE in (77)in that plugging it into (79)satisfies the equality (77). is, Thus, an ordinary diferential cquation is hrst of all an equation. it is an equality where there cxist one or more unknowns that nced to be ' determined. A very suple analogy may be useful. ln a simple equation, 3.):+ 1
jy
''E
(
t 1
/
C-
1, was
Readers wjjj recognize this as the valuation function for a zero-coupon bond. This example sbows that the pricing functions for hxed income scctzritics can be characterized as solutions of some appropriate differential equations. ln stochastic settings, we will obtain more complex versions of tjlis result. Finally, wc need to defme the integml equation
,!
=
=
(83)
'
(77)
0.
=
E
Calculus
dt,
-r/f(
=
tjw solution, with the condition Sz
,
Standafd
third major notioll
is
.
''
that we would like to from equation differential (ODE). For exrdinay an ample, consider the expression WC
preciscly, it is a ftmction of t !
(f).
=
'icon-
5.4 Ordinur.y Differertial Equation.s
(8a)
:
.
.
..
cHAP
4
'r
ER
Deterministic
3
.
and
calculus stochastic
7 References
4.
(a) Xn
(b) (c) Xn Xn
=
an,
=
(1+ l/njn
=
(-1)''-'1/n!.
n
=
5.
1, 2, 3, where
with s'j t
(-LJ, given abovc,
'
convergent?
(c) xs
(d) xu
=
sin
/lW')
.
converge as N
( g$
series h'
.x?
-->.
7. Suppose x
=
1, 2, 3
.
xu.v
g.
considerthe
functon; /(X)
.
%
=
.j
'
,
()
:
@) Now consider
. ,
Is this sequence bounded?
(; < .x:
V-n )/ V-n
1im -->
(x)
.u
.<
.v4
4
ns/n
a
<
(9,11 into .!,
4 pieces,
xg a, whcre you choose the xj. ney may or may not be equally spaced. Calculatc the following sums numerically: .
3. Determine the following limits: N-#X
jjxlax.
splitting the intewal atj
:'
lim (3 +
5
(a) Take the integral and calculate
i.
(-1)n/a
.y j
N-1
n
with A'() gven. Write X,' as a partial sum. Whcn does tMs partial sum converge?
.
+.
ooesthe
:.
.i,
3 (/'-,.)
n(-14''
1, converges to 3. Use mathematical induction.
.
(-1)*
=
=
g,yn r
=
1/n
E
1.
=
2. I f it exists, h'nd the limit of the following sequences for n
(b)x,,
defincd y the recursion formula:
xh
u=1
(.
sin
the partial sum
!
:. (i) What is the gross return to 1$ invested during ? (ii) Now suppose 5% is the annual yield on a T-bill with maturity Ap what is the compound return during one year?
-
showthat
sn +1 6
nt
=
1/k!
is conv ergent.
Suppose t h e yearly interest rate is 5%. Let A be a timc intervai that repeats: n times during 1 year, such that we have:
(a) xn
N =
p=1
,
(d) Are the sequences
the partial sum y.
Exercises
1, writethe scquences (A%) for
sbowthat
.
ne reader may at this point prefer to skim through an elementary calcu1us textbook. A review of basic differentiation and integration rules may especially help, along with solving some pradicc exercises.
8
75
g yxercjses
fxijxi j.:
-
xf-j)
.
4
E : . '
j
j : .: .
';
. '.
=
j
fxi-ltxi -
xf-1).
'!.
76
C H A P T ER
3
*
Deterministic
Sttxhastic Calcultts
and
(c) What are the difercnccs between these two sums and how well do they approximate thc true value of the integral?
flxj
9. Now consider the function
fxj
discussed in this chapter:
x(sin(1)))
=
0
*
:
.r
0.
=
(a) Take the intcgral and calculatc
fxtdx, 0 (b) Again, split the interval g0,11into 4 pieces, 0
=
<
A:l <
.%2
<
<
.173
3:4
=
4 i uuc 1
(c) How do thesc (d) Avhy?
':
sums approximate
'
the true integral?
=
f (.r,z, y)
=
(y .j.
.x;(j
i
,' l
x+y+z
(j
.j.
x)(j + z)(j
'
.t :
x+ y+ z + zjtj-.j- yj .j.
y)
'
.
!
Take the partials uith respect to x, y, z, zrspectively.
E
' .
'rhis
: '
E
.F
.. '' .
some aspects of pricing derivative instruments that set them apart thc gcneral theory of asset valuation. Under simplifying assumptions, from one can express tht arbitrage-free price of a dcrivative as a function of securities, and then obtain a set of formulasthat can be used some to price the asset without having to consider any Iinkages to other financial markets or to the real side of tlle economy, There exist spccific ways to obtain such formula. One method was discusstd in Cbapter 2. The notion of arbitrage can be used to determine a probability measure under which fnancial assets behave as nmrtingales, oncediscounted properly. The tools of martingale arithmetic become availble, and one can easily calculate arbitrage-free prices, by evaluating the imPlied expedations. This approach to pricing derivatives is called the method fh.fequivalent martingale measures. The second pricing method that utilizes arbitrage takes somewhat more a directapproach. One first constructs a risk-free portfolio, and then obtaias a PRrtitll dlverential equation (PDE) that is implied by the lack of arbitrage opPortunities. PDE is either solved aaalytically or evaluated numerically. In either case, the prouem of pricing derivatives is to find a function Fs 1* r) tliat relates thc price of the derivative product to 5,, the price of te underlying asset, and possibly to some other market risk factors. Wllen atoclea-form formula is impossible to determinc, one finds numerical wap describe the dynamics of Fst, f). nis chapter provides examples of how to determinc such pricing fulctions Fst, t) for linear and nonlinear derivatives. These concepts are '
4f-1)-
1t). Consider the following functions! flx, z, y)
nere are
':
-
Notution
l Introduction
i
xJ-1)
)(;t' f (.r2-1
erlvatlves
.::
1,
E
f (x)(..rf i= 1
*
.1
by choosing the xi numcrically. Calculate the following sums! 4
:
Mool.s a
'j
xn
*
rlcln
E
.
C H A P T ER
78
*
4 Pricing Derivatives
j pricing Functions
:
clariliedand an cxample of partial differcntial equation methods is given. nis discussion provides some motivation for the hmdamental tools of stochasticcalculus that we introduce later.
parameters'?S We of such a contract at time I in terms of the underlying arbitrage argument. use an one bup one unit of physical gold at time t for St dollars using suppose ftlnds borrowcd at the continuously compounding risk-free rate r?. The r; during the contract period. Let the inserance and is a ssumedto be flxed unit time be c dollars and let them be paid at tne ?-. costs per storage of holding this gold during a period of length F t will be total cost ne given by
E
2 Pricing Functions
,
The unknown of a derivative pricing problem is a functionFtk/ /), where is the price of the underiying asset and f is time. Ideally, the linancial anut will to obtain closed-form formula for F /). The Black-scholes a alyst try option of the underlying asset call in of tbat pricc gives the tcrms a formula the best-known relevant is perhaps other parameters case, There and some simpler. considerably examples, other some are, however, many In cases in which a closed-form formula does not exista the analyst tries to obtain an equation that governs the dynamics of Fst, r),1 In this section, we show examples of how to determine such F(&, f). The discussionis intendcd to introduce new mathematical tools and concepts that have common use in pricing dezivative products.
79
,
d
.
r 'A(v-tj
't
o
+
,
r
:
E
.
'rh
: '
' . ?
consider thc class of cash-and-cany goods.z Here we show how a pricng, functionFlst, /), where st is the underlying asset, can be o ytajned for forward contracts. In particulaz, we consider a forward contract with the followingprovisions:
thisgvestlae equality
,
F(.
.
'
t,
f)
=
ertr-ljs
t
+
(w
-
tjc.
g)
nus
At some future date F, wherc
we used the possibility of exploiting any arbitrage opportunities and obtained an equality that expresses the price of a forward contract F ( ,, 1) as function of uh, t and other
(1) :.
s a parameters. In fact, we determined ah---,-a 'jz, /) tl--t gi-estl--au--ftl,e orwa-dc--t--ct at anyti-,- t. Of the azguments in Fvt f), f and are variables. They may change duing the life of the contract, Onstthe other hand, c, r/, and F al'e parameters. It is assumed that tey will remain constant during F t. The fundion Flst, tj jn is linear in st. nus. forward contracts are (3) called linearproaucts. Jwater we will derive the Black-scholes formula which
..
,
win b- p-id or o-e '--it -f gou. F til * The contract is signed at time f. but no payment changes hands un time F.
dolla-
,
,
Hence, wc have a contract that imposes an obligation on both counterpar-E one that delivers the gold, and the onc that accepts the delivery. How can one determine functionFtu, /) that gives the fair market value : ttes-thc
,
xx
ote thc sense iri which this is a derivajivv contract. Ont!e thc contract is signcd, it becomes securit.y and can be (raded l:s own. To tradc the Jonvard on contzact, onc nced nor kavr in possession arjy pjyysiealgold. In fact, such instruments can be derived from Mderlyng assels that do not even cxist concretely. Derivatives written on equity indices are one such ejass.
:.
a
of a closed-form formula does not necessarily imply the nozlexiasttmf. tlf a pridng funclin. It may simply mcan that we are not ablc to apre's the pricing funditm ftmctions can be' in terms Of a simple formula. For example, a11 continuous and expandcd as an inllnite Taylor series expansion. At the same time, truncating Taylor seres .in ' error. order to obtain a closed-form formula would in genezal lead to an approxlmntion
lrfhe nonexistencc
Seperate
Tnotional''
Msmoclth''
2Sec Chapler 1 for defmitien,
(zj
where the hrst term is the principal and interest to be returned to the bank at time T, and the second represents total storage and insurancc costs paid at tirne T. This is one method of securing one unit of physical gold at time F. One borrows the necessary funds, buys the underlying comrnodity, and stores it until time z'. e forward contract is another way of obtaining a unit of gold at fime F. One signs a contract now for delivcry of one unit of gold at time F, with the understanding that all payments will be made at expiration. Hence, te outcomes of the two sets of transactions are identicall nis means tat they must cost tlw same; othemise, there will be arbitrage opPorttmities. An astute player will enter two separate contracts, buying the cheaper gold and selling the oxpensjve one simultaneously. Mathematically,
:
Fozvzrds
(w.f)c,
4Behind tllis staement xutgjzt.
'
r
;
':
*: ' . .
there are assumptons,
such as zero default risk of the folward
'
.z
CHAPT ER
80
*
4
.
Pricing Derivatives
3 pcicing
provides a pricing function FSf, t ) for call options. This formula will be nonlinear in St. lnstruments that have optionlike characteristics are called nonlinear products.
affeding thc option's price. Hence, unpredictable movemcnts in St can be property imoflsct j)y opposite positions taken simultaneously in CL. change the Flut SO me conditions r) time way on can once the ovcr poses Of S is given. time Path To Sce jaow this properl can be made more explicit, consider Figure 1. The jower part of tMs ligure displays a payof diagram for a short position in s/ A unjt of the underlying asset, St, is borrowed and sold at price S. 'rhis
,
!
2.1.l Boundary Conditions what a boundar.y conditm Here we have to mention brie is. Su the notion formally that the date want to express pose we ge nearen'' To do this. we use the concept of Iimits. We let
FunctiAns
E
(Texpiration
.
:'
(4 Note that as this happcns, lirzzc
;-> F
c(r-,)
uu
(5 '
1
.
One question here is the presence of rt. In reality, this and S$ are rando variables, and one may ask if the use of a standard limit concept is vali lgnoring this and applying the limit to the left-hand side of the expressio in (3),sve obtain Sv
Fl.V, F).
=
:
(6
According to this, at cxpiration, the cash price of the underlying asset an the price of the forward contract will be equal. nis is an example of a boundary condition. At the expiration date-i-es pricing functien Fst f) assllm at the boundary for time variable f-the a special value, Sp. The boundary condition is known at time t, althou the value that St will assume at F is unknown.
'
r
,
2.2
tzns
i ':
E .
Determining the pricing fundion F(St f) for nonlincar assets is not easy as iri the case of fomard contracts, This will bc done in later chapte At this point, we only introduce an important property that the F(St, t should satisfy in the case of nonlinear products. This will prepare th gmundwork for further mathematical tools. Suppose Ct is a call option written tm the stock Ss. Let r be the czonst Iisk-free rate. K is thc strike price, and F, t < F, is the expiraticm dat nen thc pricc of the call option can be expressed as s ,
Ct
=
Fvh
,
f
(
),
L'
'
'
.1 ..
The pricing function F(St /) for options will bave a fundamental property Under simplifying conditions, the St will be the only sourcc of randomne ,
5ne
intercst
rate r is constanl and, hence, is droppcd as an argument
of +-.).
:
.
..
.
:
pI ou R .E
i... E
. y
.$.
.
C H A P T ER
*
E
Pricing Derivatives
!
.
83
? pjcing Functions
The hrst panel of Figure 1 displays the price F(St /) of a call option written on St. At this point, we leave aside how the formula for F(St /) is obtained and graphed/ Suppose, originally, the underlying asset's price is 5. nat is, initially we . on the Fst /) cunre. lf the stock price increases by dvb ac at point thc short position will lose cxactly tlle amount dS, But the option position gains However, we see a critical point. According to Figure 1, when St incrcases by ds the price of the call option will increase only by dct; this latter change ls smaller bccause the slope of the curve is less than one, i.e.,
DssjxlrjajoNi Offsetting changes in C( by taking the opposite posiof the underlying asset ls called delta hedging. Such a tjou jn yv unjts . is dclta neutral and the parameter F is called the delta. joljo port
,
,
.,zl
,
'
.
'
.
.
dCt
<
(8)
d5'J .
Hence, if we owned one call option and sold one stock, a price increax equal to dSt would Iead to a net loss. But this reasoning suggests that with careful adjustments of positionm such losscs could be eliminated, Consider the slope of the tangent tolE Fst f) at point A. This slope is given by :
,
.
alut t) (9) = Fx, aut Now, suppose we are short by not one, but by F units of the underlyin stock. Then, as SI increases by dut, the total loss on the short position be Fsdst. But according to Figure 1, this amount is very close to dct. It indicated by JG ) Clearly, if dSt is a small incremental change, then the 0C: will be a ve good approximation of the actual change dct. As a result, the gain in th offset the loss in the short position. option position will (approximately) unpredictably. will Such a portfolio not move should be rclated Thus, incremental movements in Fvh f ) and equation such as some .'
,
'
'
.
'
.%
,
'
7
dlFk5'll + dljut,
l)1
=
gtj,
,
where gtj is a completely predictable function of time t. 7 If we learn how to calculate such differentials, the equation above can used in finding a closed-form formula for F(S,, tj. When such closed-fo Iormulas do not exist, numerical methods can bc used to trace the trajectfollowed by Fst 1) ) Wo following dtzmititn formalizes some tf the concepts discussed Zis SCCtiOfl. :
,
:
ries
,
J.S1
,
'
'
'
formula that wc provt later. 7.4nd of other possible parameters Of thc problem.
6It comes frum thc Black-scholes
F I c L1R'E
g q.
:: .
&
,
.
J,
84
CHA PT ER
that when dSt is
It is important to reale
.
4
Another Pricing Method J AppLication:
Pricing Derivatives
(10)
PQ x dC' 'ihedge''
may be less satisfactoly an cxtrcme movement, the TMs can be scen in Figure 2. If the change in St is equal to dvh, then the dc, would far exceed the loss -F, dst. corresponding thc clearly, assumption of continuous time plays implicitly a fundamentaI role in assct pricing. In fact, we were able to replicate the movements in the option position by infinitesimally adjusting our short position in the assct, The ability to make such infinitesimal adjustments in the underlying hinges on the assumption of continuous time and the abclcarly Portfolio transaction increments, such of costs, As shown earlier, with sencc will deteriorate quickly. approximations
willfail, With
3. Equation (11), called the total differential of F(,), gives the change in prtduct's price in terms of changes in its deterrninants. Hence, aerjvatgve might think of an analyst who lirst obtains estimates of dSt and then one equation for the total differential to evaluate the dFut, I). Equauses the tion (11) can be used oncc thc partial derivatives F,, Ft are evaluated numerjcally.This, on the othcr hand, requires that the functional form of Flst, /) be known. However, a1l thcsc depend on our ability to take total differentials as in (11).Can this be done in a straightfonvard fashion if underlying variables
the approximation
''large,''
85
i
'
stochastic processcs? The answer is no. Yct, with the new tools of stochastic calculus, it can
are continuous-tmc q
Glarge''
be done. 4. Once the stochastic version of Eq. (11)is determined, one can comfor valuing a dcrivative asset in the following way. pletethc Using dclta-hedging and risk-frcc portfolios, one cari obtain additonal relationships among dFst, t), g5',, and dt. nese can be used to eliminate a11diferentials from (11). 5, One would then obtain a relationsbip that ties only thc partial derivatives of F( to each other. Such equations are called partial diffcrential equations and can be solved for F(.r, /) if one has enough boundary conditions, and if a closed-form soluticm exists. ''program''
3 Application: Another Pricing Method
';
.
rrhis book dealg with the matlwmatics of derivative assct pricing. lt is not a text on asset pricing per se. However, a discvussionof gcncral methods of Pricing derivative assets is unavoidable. nis is necdcd to illustrate the tym of mathematics that we intend to discuss and to provide examples. We use the discussion of the previous section to summarze the pricing method that uses partial differential equations (PDEs).
.)
, '
Thus, we arc led to a problem where the unknown is a function.nis argument shows that partial differential equations and their solutions are topics that need to be studied, An eumple miglit be belpful at this point.
1. Assume that an analpt obsen'es thc currcnt price of a derivative roduct FSt, t) and the underlying assct price St in real time. SupposcE P the analyst would like to calculatc the change in the derivative asset's price r /), given a changc in tbe price of tbe underlying asset dvh. dF't 2. Here the notions that we introduced in Chapter 3 start to bccome useful. Remcmber that the concept of differentiation is a tool that one can E use to approximate small changes in a function. ln this particular case, we If indccd have a function F(.) that depends on St, t. we can use the 4 L standard calcelus, we could write '
,
aj
.
,)
sxamjo
Suppose you know that the partial derivative of F(x) with respcct to (0,A'l is a known constant, b:
* EE
'rhus,
'
dFut
,
1)
Fs dSI + Ft dt,
=
where the Fj are partial derivatives,8 pF F
'
=
as
,
Fx
Ft
7
(12)
g
=
l
,
t
.
and wherc dlljut, tj d eno tes the total change.
F@)
F(&, /) which denotcs the price of the derivatie difference be-een which denotes the partial Jtlrfvsl/f'p'eOf Ie-IS, 1) Witb respect to t.
tNote the imporlant at time t, and
FJ,
,
b.
This equation is a trivial PDE, It is an expression involving patial derivaa tjv; oj s(x;, a term wjth unknown functional form. Using tMs PDE can we tell the formof the function F(x)? answcr i9yes. onlylincar relationships have a property such as (13).ne Thus, F(x) must be given by
(11)( '&
c?-
=
=
a + bx.
.
tr
y
j(
(14)
The form of Fx) is pinned down. However, the parameter a is still unknown. It is found by using the so-called conditions.''
'' E ' ',r .
.'
C H A PT ER
86
For cxample, if we kllew that at the boundary x FX)
then a
can be
*
=
4 Pricing Derivatives
'rhe
(15)
,
E
determinedby a
=
10
(16)
bX.
-
87
change in F(.) is given by the relation (n one uses total differentiation. according right-hand side of But to the nzles of calculus, this equa(19), the jaolds exactly only during infinitesimal intervals. jnhnite time intervals, tion Will hold only as an approximation, Eq. (19) again the univariate Taylor series expansion. Let /x) be an consider itznitely differentiable function of x (E R. One can then write the Taylor series expansitm of f (x) around xo (E R as 1 2
X,
10,
=
j The Problem
'
Remember that in the case of dcrivative products, one generally has ); some information about the form of F(.) at the expiration date. Such in- ' formation can sometimes be used to determine the function F(.) explicitly, given a PDE. '
ylx) flxoj +. fglxo)(x =
+
1 3.
yfxxxlx)
CK)
4 The Problem
=
'-
=0
The program discussed carlier may appear quite technical at the outset : but in fact is a straightforward approach. Howcver. there is a fundnmental 9 . prl blcm Financial market data are not deterministic. In fact, all the variables under consideration, witb the exception of the time variable /, are likely to be random. Since time is continuous, wc obsen'e uncountably many random variables as timc passes. Hence, Fst f ), St, and possibly the risk-free rate (' rt are all continuous-time stochastic processes. . can wc then apply the same reasoning and usc the same tools as in standard calculus to write i
fixoj
where
is the
1 '
f i (;r0)(.v-
jth-order
ev aluated at we can re jnterpret
.
.
In standard calculus, variabks Hence, to get a relation such as
aljtj
.i2.
9In fact, at this point, thert are dFtj
dW'/ =
f
k
:$
6
zr; +
F( JJ,
used to climlnatc . we still do not know how arbitrage can bc #rJ We leave this aside for the time being. .
'
fxvj
(21)
z (w-
=
p
t/yj +
J'
x())
(a2)
yg grj + Ft Jf
(%)
*
*.
..
linis would make the expreision Taylor series ppwximationa ''For lhat attcr, it may no be true for thc cross-product term,
(17) E r
111n
''
ittjinitesimal
(uterlyljytjstic
! . Q: .
-
j
(19)
terms suciz as dl, #F(r), dS,' arld''
(I)
,
.
;
d,S( +
to x,
:!
problems. For one, given thc equation FJ
with respect
(lsmall''
'
:
under considerat ion are dcterministic.
fx)
depends on the assumption that the terms dtlz (dstjl, and drtln, and those of higher order, are enough that they can be omitted from a multivariate Taylor series expansion.lo Because of such an approximation, higher powers of the diffcrentials dst, dt, or drt do not show up on the right-hand sde of (23). Now, dt is a small deterministic change in 1. So to say that dt )2, (tf/;3 are with respect to dt is an internally consistent statement. However, the same argument cannot be used for dS 42 and t possibly. for (#r;)2.1l F'st it is maintained that dSt)2 and (#r/)2are random during small intezvals.lz nus, they have nonzero variances during dt.
'
Fg:.$', + Fr drt + F, #/,
=
dF(t)
'j
4.1 A Firsf Look zl lto': Lenzm.l,
,
dx Thus, an expression such as
(18) i
(2a)
.
and dx as
,
The answer to this question is no. It turns out that one needs a ca lculus and a differcnt ftrmula when thc variables under consideration are random processes. 'T'he following is a hrst look at some of these difficulties.
.
x(j)
using the approximation dflxj 2
,
Fs dSt + Fr #rj + Ft dt ?
&)
.
partial derivative of
dfx)
'
=
&) a +
.ra.
.
dFt)
(;r-
yxxoj (x -
x(j) +
-
k. .
i '' .
jntewals,
aud prowujoual
we will sec that the mean
to dt.
(dutdrtj,either.
square limits of these
erms are
,
C H A PT E R
.
4
Pricing Derivatives
89
j yxercises
formulas. cox and Rubinstein (1985)is a vezy good example. of the valuation theory can be found in the excellent collection most Finally, Merton in of papers (1990).There arc also some recent sources that give tf valuation theory. Bjrk (1999).Nielsen (1999),and broad summary a such books, three Kwok (1998)are
This poses a problem. On one hand, we want to use continuous-time variances during dt. So, we use positive randomprocesses with nonzero for the average values of d-tjl and (Jr,)2. But under these connumbers with rebe inconsistent to call dSf42 and drtjl aitions,it would that and then equate them to zero. a step can be taken if the spect to Jf,in question standard calculus. of are deterntinistic, as in thc case variablcs stochastic with environmcnt a continuous flow of random- E Hencc, in a ' writc the relevant total difercntials as: ness,wc laave to 1 1 ds; + y/zrrJr,2 +. Fsr dSt drt. (24) #, dst + Fv d% + F, dt + dFt)
valuation
4dsmall''
6 Exercises
-#,,
=
2
''stochastic
of why we need to stu dy nis is an cxample exploit the chain rule in a learn how to to want understand what a dterential means in such and the resulting expressions would aboveshows thatdeterministic calculus, in o btained ones If the notion of differcntial needs to be changed,
stochastic
calculus.'' we environment
A; payoffs
a sctting. The cxample
j(j()()$
:
be different from the
R
then that notion of thc should also be reformulated. In fact, in such a stochastic environintegral we define dtkgrentials by using a new delirtition of integral. Otherwise. ! ment, continuous-time stochastic environments, a formal deEnition of derivative
in doesnot 4
supposeyou ean bet on an American presidcntial dection in which of the candidates is an incumbent. The market offcrs you the following
1. One
.,
E
-1500$
lf incumbent loses
You can take ehcr side of the bet. Let tl'le true probability of thc incumbent winning be dcnotcd by p, () < p < 1.
(a) What is the (b) Is the value
cxist.
bet?
expected gain if p = of p impoztant foz you to make a decisitn on this .6?
@)Would
hvo people taking this bct agree on their assessment of p'? Which one would be correct'? Can you tell? (d) wouldstatistical or econometric theory hclp in deternning the
2 Ctmclwsien.s
market valuc'' of derivative securities One approach used to lind the ; summarized Zforma 11 may at this point be y. Using arbitrage, dctermine lm quatifm that ties various partial derimtjves of an (unknown) function Ftx/a t) to fach other. Then, solvc this s (patial equation for the form of #(.). Using the boundary ) differential) cmdititns. determine the parameters of this function. ) chapter also introduced the fundamental mathematical problem' This finance. Standard formulas from calculus arc faced in continuous-time variables under consideration are continuous-time. the when not applicable of t-hesc processes have nonzero varianceg.: stochastic proccsses. Increments of th e secon d-order terms such as dfl size will makc the nis average nonnegligiblc. '
d'fair
R
#?
(e) What weight
would you put on the word of a statistician in making your decision about this bet? (f) How much would you pay for this bet?
2. Now place yoursclf exactly in the samc sctting as before, where the market quotes the above R. It just happens that you have a close friend who offers you thc following scparate bet, R* ',
''
S
.
.
1500$
lf the incumbcnt wins
1()(9(6
Ij tjje incumbent loses
= -
.:
Kote that the random event bchind this bct is the same as in R. Now the following:
)
Consider
5 References valuation. Ingerso g' an excellcnt source on dynamic asset however, several' good are, treatment. also nere (1987) provides a very understanding of simple asset: lesscomp licatedbooks to consider for an
Duffic
Ij jjlcumbtmt wjns
=
(a) usingthe
R and the A,, construct a portfolio of bets such that you get a gaaranteed risk-free return (assumngthat your friend or the market does not default).
(1996)is
.
L. 's
90
C HA P TER
*
zl
Pricing Derivatives
:
(b) Is the value of the probability p important in selecting this portfolio? Do you care what the p is? Suppose you are Sven the R, but thc payoff of R* when the incumbent wins is an unknown to be determined. Can the above portfolio help you determine this unknown value? (c) What role would a statistician or econometrician play in making all these decisions? Why?
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1 lntroductlon In this chapter, we review some basic notions in probability theoly. Our ftrst purpose here is to prepare the groundwork for a discussion of martingales and martingalc-rclatcd tools, ln doing this, we discuss properties of random variabies and stochastic processes. A reader with a good background in probability theory may want to skip these sections. The second purpose of this chapter is to introduce the binomial process, which plays an important role in derivative asset valuation. Pricing models for derivative assets are formulated in continuous time but will be applied in discrete time intenrals. Pradical methods of asset pricing usin difference methods'' or lattice methods fall within this category. Prices of underlying assets are assumcd to be obsen'ed at time periods separatedby smttl.l Iinite intewals of length In such small intcrvals, it is further assumed,prices can have only a limited numbcr of possible movcments.i Th ese methods a1l rely on the idea that a continuous-time stochastic proCPSS representing the pricc of the underlying asset can be approximated arbitrarily well by a binomial process. This chapter introduces the mechanicsof Justifying such approximations.
. '
p' '
5
' :'
(<small',
)
.
''finite
:
.
,
. !
.
i
.
2 larobabillty
'
o erivative products are contracts written on underlying assets whose prices fluctuate randomly. A mathematical model of randomness is thus needed.
'lotexample, prices can move
$.
'
.
up and down by some preset
.
p ..
f'
.
j
amounts.
:
c H A PT
ER
.
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Tools in Probabilit'yTheory
5
:
'
E
Somc clementary models of probability theory are especially well suited to pricingdcrivative assets. This can be a bit surprising, given that many investorj appcar to be notions of probabilities rather than by an axiomatic drivenby and formal probabilistic modcl. However, the discussion in Chapter 2 1n- ; probabilities area if therc are no dicated that no matter what the the fair market value of linancial represent opportunities, one can arbitrage asynthetically.'' Hence, rcgardassetsusing probability measures constnlcted mathematmarkct participants, chances perceived by of subjective any less products. pricing derivative modeis natural in probability have use a ical ) In working with random variables, one first deines a probability space. the notion of chan E That is, onc explicitly lays out the framework where witlnout falling into be dehned rcsulting probability the some incan and
pepending on what is in the report, we can call it either favorable or unfavorable, nis constitutes an example of an evertt. Note that there may rs's that may lead us to call the hawest report It is ye sevcral ts's. collections of that events are sense thjs jyj report.'' ylence, we may want to know the probability of a given by is nis Jyharvest report favorable), (3)
;
''favorable.''
''intuitive''
Ifavorable
''true''
,
'
consistcncies.
=
Finally, notc that in this particular example the f' is the set of al1 possible reports that the USDA may make public.
In general, there is no reason for a probability to be representable by a simple mathematical fonuula. However, some convenient and snple mathematical models are found to be acceptable approximations for representing probabilities associated with Enancial data,z A random variable X is a function, a mapping, defined on the set G. G 3, a random variable will assume a particular numcrical Given an event value, Thus, we have X 2 -->. B, (4)
2
'
.
.
./,1
,
Where
B is the set made of all possible subsets of the real numbers R. In tel'ms of the eumple just discussed, note that a harvest report'' may contain several judgmentalstatements besides some accompanying numbers. Lct X be the value of the numerical estimate provided by the USDA ald let 100 bc some nnimum desirable harvest, Then mappings
,
J7(.,4) k: (),
any
Wfavorable
(1).
CE3,
E
dPA)
=
(2/
1.
=4G
vuruuc
2. : Ra
To deline probability models formally, onc needs a set of basic states of r tlle world, A particular state of the world is denoted by the symbol 'The symbolI represents all possible states of the world. The outcome tf an, cxpcrimentis determincd by the choice of an fs. ne intuitive notion of an event corrcsponds to a set of elementary %.: ne set of a1l possible events is represcnted by the symbol 3. To each event' .,4e one assigns a probability J'(.,4). These probabilities must be consistently defmed. 'Bvo conditions of consistencyarc the followinp -4
93
2 pobability
,
such as
The firgt of these conditions implies that probabilities of events are either, should sum to onc. zeroor positive. The second says that the probabilities L Here, note the notation dPA). This is a measure theoretic notation an dl be read as the incremental probability associated with an event a E may triplet According to this, aE P3 is called a pro bability ;. space. ne pointoa of l is chosen randomly. rtz1lawhere z4. (E t1, represents fhs probability that the chosen point beltmgs to the sct
favorablc rcport
.(fl,
,
azl
.
G@)
tS
Note tat
:
commodity future during a suppose the price of an exchange-traded he report t U S Department of Agrigiven day depends only on a harvcst that public day. will make during Cu lture (USDA) The specihcs of the report written by the USDA are equivalent to a,n .
2ne
:
Pttant
'
.
3l-lere,
u
.
.
<
X
detine the random variable X. clearly,the values assumed by X are real numbers. A mathematical model for the probabilities associated with a random variable x is given by the distribution jnction G(.z):
.
2.1 Exumplc
100
=y
'( 7. .
G(.) is a function of
=
P(X
:q
sense in which a formula lxcomes a golad approximation question that wc will discuss below.
.Y represens tbresbold.',
a random
(6)
x),
.v.3
variahle,
whereas
to a prtybability is an im-
the lowcr-case
.v
represents
a ceuain
1
.
c HA
94
P T ER
5 Toob in Probabilit Theory
.
J
uoluents IE)en:i'3' Witb
When the function G@) is smoot b and has a dcrivative, we can de- : function is denoted by g(x) and is.. tlie densit.vfunctionof x. obtained by dGxj glxj (X (x E It can be shown that under some technical conditions there always exists a distribution function Gxj. However, whet her this function Gx) can be written as a convenient formula is a diferent question. It turns out that.' there arc some well-known modcls where this is possible. We review three . used in pricing derivative prod-'. probability modas that are trequently
o.4
r c <
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tine
95
'
sxdard
Ne ,..aj
.
'3
:
=
.
o.2
('
basic
Density with
o, ''''
.
These examples are specially constructed so as to facilitatc understanding of more complicatcd asset pricing methods to be discussed later. Btzt first conditional expectations. E we need to review the notions of expectat ions an d
ca>
'
'
.6
-4
Moments i'moments.''
r
t '.
3.1 First T't,tloMtmkcnt.
'
Thc expected value E (-Y1of a random variable X, with density called the hrst moment. lt is defned by x
A('Y)
fxj,
isr '
d.Y,
z.ly .sjxjj:
-c.c
ne vazonco t y(x) is uaecorresponuingprobabilidensiwfunction,. momear first tiw second arouna moment is the mean. vhc - sl-jl while theE
here a w of
Gnsider the nonsymmekic
J'x
d.r by dF(x$.
density shown in Figure 2. lf the mean is the of gravity and standard deviation is a rough measure of the width of the distribution, then one would necd another parameter to charaderize the skewness of the distribution. Third moments are indeed informadve about such asymmctries. Ceter
,
;If the densty does not exi%t, we replace
.
3.2 I-jzgjeg-tlrtlzx Momeats
2
-
()
Cal
i'center
-
=:
In fact, an hiyaer-oraer moments of mormally distrouted ranaom variables be expressed as functions of J,t and fw2. In other words, given the lirst two moments , hkher-ordcr moments of normally distributed variables do nOt provide any additional information.
Fg.' of gravity'' of the distribution, the a random variab le is'nformation about the way the distribution is sprea ds secoodmoment gives . out. ne squarc root of the second moment is the standard deviation. It is a.: t#' obsenmtions X/'n the mean. ln hnancial.' measure o f thc average deviation of markets, the standard deviation a price change is called the volatility. distributed random variable X, of normally the For example, in a case formula ll-known thc density function is given by thc we ' -:54(.v-Jz)2 1 (8) fxj e 2*t7-2 =
8'
G2 is the second moment around the mean where the variance parameter and the parameter Jz, is the first moment. Figure 1 shows exa mples of normal distributions. Integrals of this formula determine the probabilities associated with various values that the random variable x can assume. Note that fx) depends g on only two parmeters, c- and m.Hcnce, the probabilities auociated with a nonnally distributcd random variable can be inferred if one has the sample estimates of these two moments. A rjojaually distributed random variable X would also have highcrorder moments, For example, the centered third moment of any normally distrjhuted random variable X will be given by
There are different ways one can classify models of distribution funcSome random .,bL. tions. One classification uses the notitm of Others neeo. characterized momentsvariables theirhrst by can be fully m'fz characterization. higher-order moments for a full
=
6
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96
c H A f, T E R
.
5
Tools in lrobability Theory
4
Ija other words, a casual obsen'er is more likely to be observations in the case of heavptailed random variables. extremc
tsurprised''
. (
'
dcnsity A Tlonsyrrzmetric (An F(5.3) distribution)
.
4 Conditional Expectations of taking expectations of random variables is the formal hcuristic notion of (Torecastinps' To forecast a random variablc, one utilizcs some information denoted by thc symbol 1t. Expectations calculated using such information are called conditional expectations. expectation ne corresponding mathematical operation is thc utilized be, and in general is, difoperator.''s Since the infonnation could another, conditional expectation the operator is fcrent from one time tt
ne operation equivalcnt of the
. Itormal
by
'
A Neavyaailou l.tlistlibution of fmedom with 5 dchzrrecs
A Ktndfird
97
Expectations conditional
E
densy
KGcondititmal
0 y?I o v Rz
E
z
itsclf indexed by the time index. In general, thc information used by dccision makers will increase as timc passes. If we also assume that the decision maker never forgets past data, the information sets must be incrcasing over time!
S :.
ln financial markets, a morc important notion is the phenomenon of hcav. taib'. Figure 2 displays a symmetric densit which has another chazu acteristictbat differentiates it from normal distributions. The tails of this iL' distributionare heaver relatwe ' to the middle part of the tails. Such dcnsities and called heavptaikd are fairly common witb financial data. Again, are would need a parameter other than variance and the mcan to char- i one acterizethe heavy-tailed distributions. Fourth moments arc used for that ) end. i
/.J
.
.
,
gt,
y.,
g
.
.
.
,
tjij
i = 0, 1, are times when the information set becomes available. ln the mathematical analysis, such information sets arc called an increasof slkma hel. When such infonuation sets become available 111:Seqlence continuously, a different term is used, and thc family It satisfying (9) s Where
.
!,
.
calleda hltration.
' i
The conditional stcps.
J.2. l Nef;py Tails What is the mcaning of heavy tails? A distribution that has heavier tails than the normal cuwe means a point should be carehigherprobability of exlreme obscrvations. alsoBut thistails the normal density that extend to plus madc, has that Note fully and minus inhnities. Thus, a normally distlibuted random variablc could also assume extreme values from time to time. However, in the case of a E heavy-tailed didribut-ion, thcsc extreme observations bave, relatively speak- E ing, a bigllcr fretFenc. nGrmal But thcfc is more to hcau-tailed distributifms than that, In a around distributim, tbsenrations Of would naturally be occtirring the most t he the occurrence t7f extremcs is gradual, Z ccnter. More irnprtantly, Obstrrvations that to large, and then to extreme from frdinaly the PaGage in OCCUI'S a gradtlal fashion. In case of a heavy-tailed distribution. on the hCr Ot to extrcme observations is mofe hand, the passage from Suddcn. The middlc tail region of the distribution contains relatively less i . weight than in the normal density. Ckmpared to the normai density, one s likely to get many extreme obselwations.'' ,
:i
.
,
expedation
can then be defined in scveral
opcrator
#-1 CtmditzontzlProbeility
,
'
srst, the probability density functions need to be discussed furthcr, Is is a random variable with density fttnction /(x). and if a;tl is one possible value of this random variable, then for small dx, we have x
p
'
jx -
wjjj s
dx 2
2
jjxoj dy.
jjt)l
'I'IIiS
is the probability that the x will fall in a small neighborhood of ;r1). neighborhood is charactered by the #x. These quantities are shown in Figure 3. Note that although fxj is a Rtmlinear curve in tlzis ligure, for small dx it can be approximated rcaSonably well by a straight line. Then, the rectangle in Figurc 3 would be
'
'l'he
Gtdistancel'
ttordinary''
.
5./%n
'
:
operator is a function that maps funuions into funclions. That is, it akes as irlput a as output another function.
anu produces fuucion
'r. .
..
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.h,
.
t
98
c HA
P T ER
.
.:
5 Tools in Probability Theory
4 .
99
Expcctation conditicnai
the right-hand side should be read as follows; for a Iry tjs eeression, of possible values that St might assume are weighted by aIl f, tbe sum Vjvencorresponding probab ilities (/'(5',1Ju) fzll and summed. ne averaging the way, any information js done by using probabilities conditional on Iu. incorporated in forecast. the that one has gets
.
'rhis
f(x)
dx
) '(
E
.....:
:.':
l
!
;
f
':
: '.
.
,
.j '. E. . I ,E
ftxol
4-2 Propevties of C
,:,
.$
1 ) j '.
First note a convenient notation. Often, the expectation jormation set It, is writtcn compactly as an i'n
'
,
,
.
:.
t
..
.:
.
t
.,...1 ,.....
-
xo
lcI G IJ RE
Tbe t subscript in E: indicates that in the averaging operation one uses a1l inlormation available up to time /. Thc conditional cxpectation operator Et bas the following properties.
3
.
1. The ctnditional expectation of thc sum of t-w0 random variables is the sum of conditional expectations:
.x
s ngz;+. Fjflj
.
Expectation Operator 4.1.1 cont?kcwtz/ second defining a conditional in step ne
''averaging'
expectation is the operator. ln fact, every forecast is an average of possible future values. values that the random variable can assume in tlle future are weighted by the probabilities associated with these values, and an average is obtained. Hence, the operation of conditional expectation involves calculating a weighted sum. Since the possible outcomes are likely to be not only infinite, but also uncountably many, this sum is reprcsented by an integral. given of some random variable ne conditional expectation (forecast)
)
'.
(.
, '
'
EEl+7'(k%+r+u1l
'
=
ll-s-/lzsl =
-x
stfst6lu
5,,
u
<
t.
'(
F(F(.I.f/IIJs1
: '
wjjee 1q4is contained
<
/.
(13)
E',E'%+T+u1.
(14)
'
''uncortditional''
',
..
'j
: !
=
FI.I&1,
(l 5)
in 1t.
Finally, if the conditioning information set It is empty, then one obtains the expectation operator E. nis means that E has properties similar to thc conditional expectation operator.
'J
(11)
u
According to this, recursive application of conditional cxpectation operators always equals the conditional expectation with respect to the smaller itlformation set:
j
gt
X
+ lsg'tf )j, sugusrj
.+.w+u:
rnw
at timc u is given by
=
According to this, onc can form separate forccasts of random variables and then add thcsc to get a forecast of their total. 2. Suppose the most recent information set is 1t, but one is interested in T > 0. u > 0. That is, one would forecasting the expectation .Ql.z,gz%+z+ul, of Iikc to say something about the forecast a possible forecast. Since the information sct f,+z is unavailable at time f, the conditional expectation ft+w(&+r+,;! is unknown. In other words, F/+r(,%+.w+uI is itself a random Variable. expectations A property of conditional is that the expectation of this future expectation equais the present forccast of
'
Consider a simple example. The odds of a stock market crash will bo tdds of a crash given that an example of unconditional probability. ne one has entered a severe reccssion can be represented by a conditional probability.In tMs particular case the information is the knowledge that a severerecession has begun, ne use of such information may certainly lead probability of a crash, to a revision of the (unconditional)
(12.)
=
!
as /(x).
conditional on
s(.I.I',(I .,E.,(.1.
will fall within a small neighborhood of xa close tt the probability that represented by the quantity dx. Jf these probabilities are based on some information set h, then f @) is called a conditional density. The depcn- .J . is not based on any denceon It is formally denoted by .f(.rIJ,). If the /(x) kmd the density is written ; particularinformation, the It term is dropped
the information available
Expectctiems
't'
:
.. .
J
100
C HA P T ER
(
Tools in Probability Theol'y
5
.
j
Some Important Models
5
;
j.
1. Therc
2.
'
=
F(f +
)
F(l)
-
+aXS,
=
a
>
..
t i.i
k
1
?
:
'
=
F( t
)-
+
Where
#(
I
)
=
a U- ,
-
=
,
(16)
0.
J
f
Or, there is a downtick and prices decrease by F( f )
Arl important element of the discussion involving the binomial process aF(/) is that the two possible values assumcd by each Fjtj depend on This dependence permits a discussion of the limiting betjle parameter haviar of the binomial process. We can ask a number of questions that will evcnmally relate to pricing derivative products. one important question is the following: what does a typical path followcd by the LF f) look like? Ctearly, such a trajectory will be made of a se+J Z and -I'Z's. lf the Probability of each of these outcomes qllence of is exadly Cqual to 1/2, then a realization of jil'lt ), f lo fo + A, .J will, a: getg smaller, convergc t an extremely erratic trajectoy that Cuctuates beeen b-af''s and -JU-'A In fact, as gets smaller, two things happen. First, the obsewation points come nearcr, and seiond, gets Smaller. The F(l) is the increment in the price pcess. What kind of a path is foliowed by F(/) itselo First, note that if F(f) represents the price of a dcrivative product at time 1, then it will equal the sum of all up- and downticks since /p. As -->. 0, F(/) will be given by
:'
is either an uptick, and Plices increase according to ntj
r :
.,
=
+IJ'v''XI #,
Pit)
=
-aX-j
ne time index f starts from k
f
=
fo, fp +
k.
=
(1
and increases by multiples of -
.
.
,
/() +
nL,
.
-
,
.
(20)
stochastic
process s a sequence
of random variables indexed by time.
t
F(f
' .'.'
('small''
: J ;
:
..q .
psee,
.' '.
: j
)
=
F(fo)
dFsj.
+ Io
nat is, beginning from an initial Jprce F(fn), we obtain the price at time f by simply adding a1l subsequent inhnitesimal changes. Clearly, in continuOuS time, therc are an uncountable numbcr of such inhnitesimal changeshence the use of integral notation. Also, at the lintit. the notation for iucremental changes F(f) is replaccd by #F(l), which represents iafiltitesimal changes. Finally, consider thc following question. At the limit, the inlinitesimal Changes dFt) are still vcry erratic. Would the trajectoriey of Ftj be of btwnded variation'?l The qucstion is important, becausc otherwise, the Riemann-stieltjes way of constructing integrals calmot be exploited and a TW dehnition of integral would be needed.
L'. ,.
E
,:
At each time pojnt a new F(f) is observed, Each increment F(f) will equal A-avIf the LFt) are independent of eacb other, the or ,-awCS. either tf incremcnts rFlfl will be called a binomia I stochastic pmcel-, sequence or simply a binomial rmccsat' that a eltemember
E
(19)
p).
-
.
l
!:
(18)
=
.
If/x-I
..
ZF(f) represents the change in the observed price during the time interval A11other outcomes that may very well occur in reality arc assumed for the time being to have negligible probability. Then for fixed f, /, the AF(/) becomcs a binomial random variable. 1n particular, Ll't ) can assume only two possible values with the probabilities
,
,
l
(17) Wsmall''
#(AF(f)
markets.
5.2 Limtfng Propc'rties
'
Considcr a trader who follows the price of an exchange-traded derivative asset F(f) in real time, using a service such as Reuters, Telerate, oT Bloomberg. 'rhe price F(f) changes continuously over time, but the trader is assumed to have limited scope of attention and checks the market price every A seconds. We assume that A is a small time interval. More importantly. we assume that at any time l there a<e two possibitities:
adual
turnover, therc even very oa a giventime periods where #(/) change. Or, in does not some special arc many change by However, it than downtick. may mtre an up or c jrcumstagces, will with complications be dealt later. the being, For time we considcr sucja sjmpier case of binomial processes. oe
'
Disfrirlztio'a in Finfmcnl Murkets
Models
these assumptions are somewhat articial for xotctbat trading day, in markets with high
:
Tilis section disctzsses some important models ior random mriables.These models are useful not only in thcory, but also in practical applications of asset pricing. tn this section we also extend the notion of a random variable to a random process. 5.1 Binemll
someImportant
:. '..
claapterg.
!
C H A P T ER
102
.
5
Tools in Probahility Theory
.
:f
5
Another important point is the following; the integral in (21)is taken with respect to a random process, and not with respect to a deterministic variable as is the case in standard calculus. Clearly. tMs intdgral is itse a random variable. Can such an integral be successfully dehned? Can we lzse the Riemann-stieltjes procedure of approximations by appropriate redangles to construct am integral with respect to a random process? questions Iead to the 1to integral and will be an-ered in Chapter 9,
Important somlt
Models
1
.
o'ie'' a 0'8
'.
0.7
'rhese
o.6
:
Nuauce
0.5
.
o.4
5.3 Merrmts
var(A#(l))
=
pa
proponional toA2 variance
o.2 01 .
c
'
(1
s
0.3
One question that we answer here concerns the moments of a binomial : . proccss. variance of fXF(1) value the and are si Lct t be flxed. nen the expected defmed as follows: plavsj pj-axj, FIAF(r)! + (22) =
9.2
-
XS)l+ (1
-
pl-axjl
-
gFtlFtfjjjz
(z3j'
y?l(; g
'y:)
(24)
e jourth-order
moment
'
wsa
itgelf.
(+(z,Zi)4 asl
(%)
=
=
>
obseaations imply that for small intervals higher-order moments of a binomial random variable that assumes values proportional to vq' Ol'l be ignored.
-t)
d
,',
,
.
,&
5.# The No
j
Distdbafzon
Now consider the following experiment with the random variable F(I) djscussed in the pyevious section. We ask the computer to calculate many reajjzauons of Then, begnrjirig from the same initial point F(0), we yt ). pjot these trajectories. Reginnillg from /: (), in the immediate futurc, F(/) has only two possibje vajues:
,
,
r
=
.
.5
=
E =
4
nese
,r
had ipstead Euctuaed behvecn, sayy +z; and -4A, 2 For sma 117, the value of A2', wouldbe much smaller. When -->. 0, the variance would go to zero sig-. nihcantly faster. Under such conditions, it can be maintained without at itself is not. contradiction that the variance cf x F(/) is negligible, whilc Heuristically speaking, a random variable with a variance proportional. gj , to 12 wgl be approximately constant Jn infmitesimal time xntervazs, i between the difference ance 4 illustrates propo rticmal to Fkurc a var 1. negligible foz' latter becomes Thc proportional and lne) (the 45@ to onc small uj. This last point is also relevant for higher-order moments of the binom valu for simplicity. 'Fhen the expectcd process. Again assume that p be will third moment the is zero and Svenby Wit.h #
jk:
1
() tjje jouujsoykjer moment will become negligible. It is prothat goes to zero faster than the timc inten'al to a power of
...s
portional
.,
.5.
o.8
is obtained as
7(F(/)14
expccted value will equal 0 while the variance is given by alt. t. It is important to rcalize that the variance of the binomial process iigi As A approaches zero, a variance that is proportional: proportional to think';;C will to ,'X go toward zero with the same speed. nis means that if we quantity, then the variance will also V. small but ttonnegligible of
and the
would be proportional to
0.6
a
.:j
=
the variance
0.4
F
.
lf wre have a 50-50 chance of an uptick at any time t, then 1 #
as a nonnegligible. Jn contrast, if
preponiosal toa
p-avzl' = fk.F'(z)J3
the third moment
+
(1 -
pl-at-s.
; (2.5),
F(0 +
''
blence pt)
',y
equals zcro.
'
,
j:. L
jfxey
)
F(0) + ax:-
with probability p
n
with probability 1
=
F(0)
-
a
s unomialat t
-
() +
a.
- p.
(gy)
T r'
.
)04
c H A P T ER
.
5
Tools in Probability
:
'l-heory
5
But if wc Ict some more time pass, and then look at F(r) at, say, f 2, F(f) will assumc one of three possible values. More precisely, we have the , following possibilitics'.
,&
:
F(2A)
a'Vs+ axS
F(0) - aF(0) - av-
=
with probability pl
+ a-
.->
with probability 2/7(1 - p) with probability (1 #)2.
av-
(28) E
.
)
-
.
That is, F(2A) may equal F(0) + 2aXS,F(0) - za-, or F(0), Of these, tlielast outcome is most likely if there is a 50-50 chance of an uptick. E Now consider possible values of F(/) oncc some more time elapses. Sew eralmore combinations of upticks and downticks become possible. For ex-' ample,by the time t 5, one possible but extreme outcome may be
IEconvergence
convelyence.
=
F(5)
-
Another
F(0) + avF(()) +
+ a-
+ a-
+ awq + a'-i
sa-.
(29) . (3t))(.
=
F(0) - a4-
More likely arc combinations F5h)
=
80)
-
-
a-
-
av-
-
a-x
-
av-.
of upticks and downticks. For eumple, a-
+ a
- a,
+ axs
According to the central limit theorema the distribution of Fnhj proaches the normal distribution as n, --> x. and that Assume that p
ap-
F(0)
(34)
.5
=
)
in a row: extreme may be to get five aownticks F(5a)
105
---> x, the time period under consideration in wsch was constant and n indehnitely, sed and looked at a limiting F(f) projected toward a we incrca sdjstant'' future. one question is what happens to the distribution of the random variable remains flxed'? A somewhat different question is #(a) as gy x and of F(aA) as --.> 0 while ni is fixed.g the disttibution to what bappens remembcr Ftt was binomial, but a little tarther otigin tlzat the at Now, number of possible the outcomes grew and it became multinomial. away probability distribution also changed accordingly. How does the fonn ne of tbe distribution change as n --> x? W'hat would it look like at the limit? of random Questions such as these fall in the domain of variables.'' Therc arc two different ways one can investigate this issue. ne jirst approach is that of thc central limit theorem. The second is called uzdt?/
=
F(0) +
somelmporranr Models
+ as/-
E
and Then, for fixed arge n, t hc distribution of FnL) can be approximatcd by a normal distribution with mean 0 and variance alnh. The apProximating density function will be givcn by
(31)
d(l
It
2:
c
?$
(32).
g(F(n,)
''
=
x)
1
=
l'rralni
::E
or F(5a)
=
F4()) - axi- + a-
+ av-h
-
a-
+ a-
are two different scqucnces of price changes, each resulting in the mqme. E Price at timc t 5Z. rlnhere are sevcral other possibilities. In fact, we can considcr the generalt fmkez case and tl'.yto sfldthe total number of possible values Fnt) can nume! Obviously, as n may take any of a possibly inEnite x, Fnt) of values. A similar conclusion can be reached if 0 and n --> t:xz while the product tn remains constant. ln this case, we are considering a time interval and subdividing it into Ener and finer part itions.8 For the case =
?
.-->
:.
->
gucre too
(
fact, this latter type tlf convcrgcnce is ef inlerest lo us. nese types of experimenl wfll in the dumain of weai convergence and give us an apl3rox imatt distribution for a '. Of random variablcs obsen'ed during an ntewul. sequence . 81n
fall
:
E
e-
za
xa
(x)c .
(35)
ne corresponding distribution hanction does not have a closed-form formula. It can only bc representcd as an integral. The convergence in distribution is illustrated in Figure 5. It is important for practical asset pricing to realize the meaning of this convergence in distribution. We observe a sequence of random variables indexed by n.10 As n increases, thc distribution function of the nth random variable starts to resemble thc normal distribution.ll It is the notion of wc;k convelgence that describes the way distributions of whole sequenccs of random variables converge.
!
(33)
' '
0.
-
5
lar.r.jj
.
:
n
-..
x.
at is, we have a stochastic process.
'Again we emphasize tbe whole tFt()j,
.'.
,
'y. .:
svquence
. .: :'.
. .
.. .
that we arc dcaling with the distribution of F(z1), Ffnhj Fqhj, c(:zl), .j.
.
.
.
.
.
and not with
'
r
h S--i-F,R
cu
lz .s
Toou
5
.
lvobabiliw
in
'riwory.
j
inaplies that the Claussian rn odcl is useful Vzhen nexv infornnatio n tvjving during inhnitesimal periods is itself infinitesmal. As illustrated for a .-->. 0, the values assumed by Ft) tje binomial apprtximation, with jj bccom e smaller and sma jj er an tjje variance of new information givcn by
;
at n= 1
Distributiop
!
:
.
,a>'h-
vartastfl)
aa/-
0
zs
...
a
=
..-
-
-
'
-
--
-
-
..
..-
-
-
-
-
-
--
-
.-
-
..-
-
--
-
-
-
-
-
-
-
.-
-
-
-
-
-
--
-
-
-
-
a1h
-
(36)
to Zefo. That is, in inhnitcsimal intenrals, the F(/) cannot Jump. cjyayjgos are incremcntal, and at the limit they converge to zert. Continuous-time vcnions of the normal distribution arc vcrjr useftzl in assct pricing. Under some conditions, however, they may not be sufficient to approvimate trajedoics Of asset Prices obsen'ed in some financial markets, we may need a modcl for prices that show as well, Thcrc were during the October 1987 crash of stock many cxamples of such markets around thc world. How can we represent such phcnomena? The Poisson distributitm is thc second building block. A Poissondistributed random process consists of jumps at unpredictable occurtence The jump times arc assumed to be independent of 1, 2, tnes (, one anothcr, and each jump is assumed to be of the same size.12 Further, durhlg a small time inten'al the probability of obscrking more than one jump ls negllglble. Ll'he total number ol Jt1mP9 Observed up to time f is called a Poiason counting process and is denoted by N,. Fb a Poisson proceF, thc probability of a jump during a small inteal wll be approximated by g oes
Distributson at n
1o7
aqAjs
--.----------------------------------------..--------.-
Models
sonwlmportant
.
.
uu.
,,
tjumps''
Kjumps''
a-.?-
,'-
Diszbution
at
$'l
E!'l
'
e>' rl
,'
g
'
.
. .
---
=
*
s ',
;
.
.
t
,
,
l
Lilniting normal distributien
?'
t p :
',
.
.:f
'
.,
,'
. .
h
?
.,
,
l
.
.
.
.
.
.
. .
.
.......
.
..
.
.
.
1
'
l
1.
l
(. E
A
.
,
,
'.
eer
,
z
,'
ye
.
2 avf
o
-2
,'
f
.$
.
s
j
F 1G U RE
5
t ':.
G<small
''
'
approximately
. .T
.
!
PiNt
5.5 Thc Poisson Distdbution
(37)
,
is a positive constant called the intensity. Notc the contrast with normal distribution. For a normally distributed variable, the probability of obtaining a valuc cxactly equal to zero is nil Yet with the Poisson distribution, if is this probability is
7
%
j----..-.
-
1)
where
.,
----
=
.
s,
,-
Phhj
7,
k
f builde '
0)
=
1
-
A.
(38)
Yence during a small interval there is a p robability that nt jump Will occur. Thus, the trajectoly of a Poisson proccss will consist of a cont.u uous path broken by occasional Jumps, Ihigh's
,
: In dcaling with continuous-time stocbast ic pr ocesses, we need 1wta ing blocks. One is the continuous- timc equivalcnt of the normal distribtltio flm. known as Brown ian motion or, cquivalently. as the Wiencr process. M Of these processes trajectorie: indicates. vious disctlsgion in the pre section continuous. be likely to are
,
.
u BO1h lu :
C: E
..
of tlwse assumptions can be altered. need to l)e independent.
jump times
However, to keep the Poisson characteristc,
:
108
c HA PT ER
probabilty tlmt during a finite intewal by
ne givcn
PiNt
n)
=
v
(39)
,
Fl!
+
=
,
,
w hich
is tbe correspondirig distlibution.
6 Markov Processes and Thelr Relevance
'
-
'''-'
.
The discussion thus far has dealt basically with random variables. Yeta this although, it does constitute a too simple to be useful in nnance. block for more complex models. ln finance, what we really need' are': a model of a sequence o random variables, and often those that
6 1
E
tic proccsses. distribution function, F(x1
,
,
.
'rhc
.
,
xr)
'.
=
@
Probxj
.S
.Yj,
.
.
Xt S x/),
,
.
as / --+ x. In case the indcx f is continuous, ontt is dealing with uncountab many random val-iables and clearly the joint distribution fundion of such procesg should be carefully as will be illustratcd lor Wiene
r J.+
! :,
Process.
'
'? .
gectiony
We diSCu55
:
.l,
.
.
.
.
,
.
.
i
.
,
I J;j
=
o-(1t, t ) = ajrt,
,
(41)
,
grt
,
flk
(42)
-->
dr /
.
14
S
uws.x.-
'
:
.
...$.
' .
=
jxtrt,
j)#/
+ ato,
t l#jj?;.
(44)
We are not talking about the dependence of means or varances of zwg only. ne morc t past shoultl no! innltence statemznls concerttirlg thc whole probabilislic behavior or a -*%'''Y Pttss.
djstan .
:'
E
(43)
steps will bc discussed in more dctail when we develtp the notion Of stochastic differential equations in Chapter 11. There, letting A 0, we u Outain a standard stochastic differential equation for r/ and write it as
=
13Itis quite important that the prtxetks one is modcling in Enance is a Markov process. ) valid oa# for such procG Feyuman-lac theorem tllat we will see in latcr chapters wll be -' not Markmr,Yet it can be shown that shorl-lerm interest tat processcs are, in seneral, fOr short ratc procelsesimposes limitatons orl tbe numerical methods that can be applie d
t).
'rhese
-r:)
-
fljj;
,
'
-
,
a'lid
'
,
+ slst
.
E ((G+s - rt)
with joint X:, DEFINITION: A discrete time process, (XI, j'. xtj, said be F(x: is distribution hmction, to a Markov probability process if the implied conditional probabilities satisfy ;r/5), P-vs PX:+s :!! xt-vs Ix:, S xt-vs l (40 :. '' irdbTwhere 0 < s and P' I 1t) is the probab ility conditional on the mation 9et It i .
j Itj
tjaqyuyj; js somc unpredictable random variable with variance Tlwn, tlie clt , t) ;& wgl be the standard deviation of interest rate incremcnts. ne lirst term on thc right-hand side will represent expected chnnge in interest rate mtwements, and the second term will rcpresent the part tlat is unprcdictablc given It. It turns out that if rt is a Markov process, and if It contains only the current and past values of rt, then the conditional mean and variance will be ftmctions of rt only and we can write:
Z dctail a c1aSS Of stochastic prtcesses thaf lays an important role in derivative asset Pricing; namely, the Marko P uL, :jj : Whicb Vll be in discrete time, wlll try to mtzvaw Processes, Otlf dinlssionj will also clarify some some important aspects of stochastic processes and with continuous time models fo#; notions that will be used later in dealing Catc defivatives.l3 Ztercst In this
rtj jtry-ju -
..s
-
W jwa
j,
'dconstructed''
'or
-re
::
.
Rejevance
SUPPOSe
.
process
.
..
How do these notions help a market practitioner? the Xt represents a variable such as instantancous spot rate r,. assuming that rt is Markov means that the (expected) Then, futu behavi of rt-s depends orlly on tlae Iatest obsewation nnd tilat a condition such as (40)will be valid. We can then proceed as follows. we split changes in interest ratcs into expectcd and unexpected compoyjej.j ts:
is concept building f is continuous time. ovcr observed of random variables (.-,J indexed by an index t, where t ik sequences 0, 1, or continuous, t iE y, x), are called stoch as:': eitherdiscrete,A f stochastic is assumed to have a well-defned Joint ,
.
-,
@
.
.
:
E .
=
109
The assumption of Markovness has more than Just theoretical rele1, 2 vancc in asset pricing. In heuristic terms, and in discretc time t (A-r), is a sequence of random variables such that Markov process, a knowlcdge of its past is totally irreievant for any statement concerning the 3L 0 < % giVell the last obsen'ed value, xt. ln other words, any probabiljty sutement about some future Xt-s, 0 < s, will depend only on the latest obsenratifm xt and on nothing obsen'ed earlier.l4
,
'
'
-
and Their Relevance tj Markov Processes
:
, is there will be n Jumps
.&
a( aln
c=
'
Tools in Probability Theory
5
*
'
4.
C HA PT E R
1 J0
5
.
'(
Tools in Probability Theo
y
5
x
But, if interest rates were not Markov, thcse steps cannot be followe since the conditional mean and variance of the spot rate could potential dcpend on observations other than (he immcdiate past. ls Hence, thc assumption of Markovness appears to be quite rckvant pricing derivatives, at lcast in case of interest ate derivatives.
.
-
'.
rp.j.
qL
TRe Vector Ce
-
for the Rt substitute
; rrhere is another relcvant issue coneerning multivariate Markov pr alld u eesses. We prcfer tta discuss it again in discretc time, f, t + variables. motivating intcrest rates as t'mr Below we will show that, although fw'o processes can be jointl. Mark . when we model one of these processes in a univariate setting, it will, t gcneral, cease to be a Marktw process. ftxed-income. t)f discussed in this can best be ( rrhe relcvance elsewherc) a central concept is the yicld clfzw. The so-called classical a proach, attempts to model yicld cun'e using a single interest rate pr : such as the rt discussed above, On the other hand, the morc recent Hea (HJM) approach, consistent with Blackcholes philosop , Jarrow-Merton models it using k separatc fonvard rates. wltich are assumed to be Mar ' . jointy. But as we will sce below, the univariate dynamics of one elem of a k-duensional Markov process will, in gcneral, not be Markov. Hen Markovness can be maintained ill HJM methtadology, but may fail Z; short-rate based approach. Suppose we have a bivariate process, Irl,S21, whcre the r; represents rate. Suppose also that jointlyt.h rate and the Rt is the are Markov: '
,
.
.
Reevance
'
rt-n
rtn
+ pzpt-x + pb gtnrp-
alrt +
=
=
pj azrt-h
ajr; +
m
R r4-,:
a 2r / +
pzR(
lfl-h
g.c
/2 :- a
.y
,
R tn
,
,
V+A
-
rt
such owjously,
ISAIs o if interest ratcs are not Markov, a very imporlant correslxmdence l'etwee,ll a equations (PDE) and a class of conditionai expecmtions cannot te f yoao p artial differenial PDE'S j tls common.y tablishe d uootccarlomcthods cease to bccome equvalcnt to te tile field of interest rate derivatives.
.
JLE .
l6u
.: ..
..
.j
jjsx
.-ha-utstj
.!
r
:
.
.
..
<2 +
+
pgpt-u
+
azh-z,s
=
+
=
avrt +
4:
+.
.
tzq
jjjl.juj .
twajjrrz-u.
(4y) (zs)
(5:)
.
,
+
rt-s + azrt-u
/()H(+a + hj W?;c + bgp;.aa j
r lW'
.-
r / would
+
.j .
.
(51)
'
j
t
esa tjytj ;,p tjo uoy xjlyusost wtauer processes. rasdom varjabjss with no dopendence on
zyjeyare
a past.
i.
''
rjjl.
O.L
'ntls
.
,
.
,
+
an r; process cannot be Markov. Fo one, a forecast of depelld on %. < t jn addition t the last bserved rt. Hencey uyjjpusujg tjmamjc yjlat assumes Markovian behavior for the a shou rate , r t, wjjj uot reprtsent interest rate dynamic.s correctly, although the joint behavor or the short and long rates is Markov by assumption. , even though jyj a bivariate world the r was Markov, when modeled jyy itqelf, fMarkovness. does t satisfy not the assumption of - Obviously 1 the reverse is also true. Any non-Markov process in uniRriate a world can be converted nto a Markov process by increasing the
Ze
,
.(
tzzl.p;zj
as:
.
;..
(47)
pr ocee (jjug ojs way, and assuming that the coefhcients of Rt-kh become negligible as k ncreases, we will obtain an equatjon for h tllat can be wriueu
(4F
' . Of of each other, and +2,+,t. are two crrtr terms independcnt W jwre +1/+a, &;.j Coccients. l Constant Accor are past H(. ,44 s y The fpi, aja
jytj
x.
-
1
t7l
+
gp:fy.a the right-hand side,
pj pcAi-.a
+
Rlong''
V J'lLRt
jjjl
Now, there is another Rt-z on but this can also be substituted out by using the second row written for time t k:
'o
C 1 Xl
tn
term mplied by the second row of the system, Rt aurj-a + pzRt..t + twzH.'/,
Rearranging:
.
f/+'A
+
to get
t
'rhere
gj Rt
ajr; +
=
=
.,
''
Ktshort''
111
'yhis
;.
..
; 6.2
enjeir
and
current short and long rates depend only on the latest to system(45), obserations of rt and #/.16 Cjearly, this is a special case. But, it is sufhcient to ma'ke the point. We dcrive a univariate model for r/ implied l)y the Marktavian system in (45), derivation is of interest itself, because the recursive method utilized lwre is a standar d too j jn solving differcnce equations in other contexts as wen. In order to obtain a univariate modcl, consider the equation implied by the srstrOW:
Wjth such a model, one can then proceed to paramcterize the yjrt, tj a.l txtrr, /) and hence obtain a model that capteres thc dynamics of inter yates.
Processes uarkov
.i
.2.
any intlepcndent,
icurltically
(2 H A P T E R
.
Tots in Probability Theory
5
) convergelwe
dimensionality of the problem. Ths suggests that one can assume that forward rates are Markov, yet at the same timc assumirig Markovness for spot rates could. in general, be inaccurate. This point will play an important role in modeling intcrest-sensitive securities. Within the contcxt of yield curve dynamics, this point suggests working with k-dimensional Markov procesu :;E rather than non-Markovian univariate modcls,
of
Random Variables
'z J.1 Relevance of Mean ska/rcConveence Mean square (m,s.)convergence is important because the lto intcgral is dcfined as the mean square limit of a certailt sum. If otle uses other dejinitions Of convergence, this lintit may not exist. we would like to dscuss this important point further. Consider a more ddnatural''extension of the notion of Iimit uscd in standard calculus.
:
DEFINITION: A random variable (a.s.) 1, fOr arbitrary 8 > 0,
7 Convergence of Random Variables
etr-ltnz-vaa) >
:.
,
.:
.
aad Tlxeir Uses
7 1.2 Eample Let St be an asset price obsezved at equidistant timc points:
*
.
of three diffcrent convergen In pricing Iinancial securities, a milmum used. criteria are is a criterioo utilized to desn The lirst is mean square conveence. stochastic differenti utilized cbaracterizing in the lto integral. The lattcr is equations (SDEs). As a result, mean square convergcmce plays a fundrlmo tal role in numerical calculations involving SDES.
ftl <
E .
,
,
.
limE rA-,,
a-elxl
.
=
x12 =
0.
According to tMs definition, the random approximation
/() +
<
en xn x -
<
.
.
.
<
;
+ n
=
7:
5k+4(5',a+(5+l)a 5luifzl. -
=
:2
(55)
(56)
Herc Iqs'ta.j-l/+jla ky.jajrepresents tlze increment in the asset price at time rc + i&. The observations begin at tme /() and are recorded every k -
!
mmtes.
Note that .Y is similar to a Riemann-stieltjes sum. lt is as if an ittterval(r0,F) is partitioned into n subintewals and the Xn is defined as an to approximason
(5
n
'
:
error en defme,d
r
j' =
21
d-a
.
=
-
''
Xn
.,t
be a sequence of random variXn, DEHNITION: Let A%,A X 'T-hen in mean square if said Xn is to converge to ablcs.
f() + k
Define the random variable X,,, indexed by n:
rrhis
.
(54)
nis defmition is a natural extension of the limiting operation used in standard calculus. lt says that as n goes to inhnity, the difference between the two random variables becomes negligibly small. In the case of mean square convergence, it was the variance that convergcd to zero. Now, it is the diffcrcnce between Xn and X. In the limit, the two random variables are almost the same.
xs
.
().
-
'.
rrhe
,
to X almost surely
--vj
Thc notion of conveqence has scveral uses in asset pricing. Some tf thes binomial example of the previous se are theoretical, others practical, tion introduced the notion of convergence as a way of approximating --> 0, the appro complicated random variable with a simpler model. imation improved. ln this section, we provide a more systcmatic treatmenh of these issues, Again. the discussion here should be considered a bzief an heuristic introduction. j7 .2.!: itf 7.1 Types Cwcrgence
Xn converges
(5
St #,2.
(57)
/(,
But there is fkmdamental a
will have a smaller and smaller variance as n goes to infinity. Note that for fnite n, the variance of eu may be small, but not necess In doing numcrical calculatio . zero. Tltis has an important implication. imation errors into account explid one may have to take such approx Onc way of doing this is to use the standard deviation of e,, as an estimat '
difference. The
A
sum now involves ranOm processes, Hcnce, in taking limit of a a new type of convergence (56), zriterion
.
should be used.
not applicable.
,1
.
.'
Which (random)convergence
'!l (. ;k. '
..
: '
.
.
ne
L
standard dennition of limit from calculus is criterion
should be used?
.
.
;'
.'
..
'
(2 H A P T E R
114
*
5
Tools in Probabilit'y Theotj
)
It turns out tbat if Sf is a Wiener process, thcn Xn will not converg almost surelyjl? but a mean square limit will exist. Hence, the type of a proximation one uses will make a difference. This important point is take
up during the discussion of the Ito integral in later chapters.
''
r
? '
.(
'rhe notion of m.s. convergence is used to hnd approximations to vl assumedby random variables. As some parameter n goes to infmity, valu assumedby some random variable Xn can be approximated by values variable X. somelimiting random In t he case of weak conveqence (the third kind of convergence), what beingapproximated is not the value of a random variable Xn, but the pr p abilityassociated with a sequence A%, , Xn. Weak convergence is used 1 'matingthe diatribution znc/fon of families of random variables.
q3f
11 5
ylcnce, in dening an lto integral, values assumed by a random variable of fulldamental interest, and mcan square convergence needs to be aj.e uscd. At other times, such specisc values may not be rclevant. Instead, onc conccrned only with opectationa-i.e., some sort of average--of may be
'
7.2 Weuk Cwergence
convergenceRandom Variables
Vantlorl
HOCCD
.v)
(, j;
'
For exarlzple, y( y1 w) may deuote the random price of a derivativc product at expiration time F. ne derivative is written on the underlying asset SI.. We know that if there are no arbitrage oppolunities, then therc exists a d'r isk-neutral'' probability # such that under some simplifying assumptions, the value of the derivative at time t is given by x
,
,
,
'
,
-
,
.
apprtm
=
E p (.(Ak)) n
--->
E
J'
uy
$
(5
lim Pn fl u--,cxk where P is the probability distribution of X if
r
E
.
:.
(5
,
cw
.
.
'
(/'()J
Wus, ingtead of bcing concerned with the exact future value of Sv, wc need to calculate thc emectation of some function F(.) of Sv. Using the concept of uzetz/cconveqence, an approxmation S. of can be utilized. nis may be dcsirable if S). is more convenient to work with than tlle actual random variable Si For example, may be a continuous-time random process, random wbccas Sv may be a sequence deqned over small intervals that depend on some pararneter a. If thc work is done on computers, Sn will z be easier to work with than Sv. nis idea was utilized earlier in obtaming a binomial approximation to a continuous normally distributed process.
'
DEFINITION: Let Xn be a random variable indexed by n with probability distribution Pn. We say that Xg converges to X Breakly and
r,
t
,
: 7
.:;
where f') is any bounded, continuous, real-valued hmdioni EPn t.'l..Yklj is the expectation of a function of Xn under proba@ P is the expectation of a ftmction of X bility distribution #s; E under probability distribution P.
7.J,2 An Example
coasjdera time interval g0,11 and let t c g0,11 represent supposewe are givcn n observations ei, i 1, 2, jyom the uniform distribution 1./(0, 1 ).19 udepeutjently
M(AR)1
According to this dehnition, a random variable Xn converges to X >4 if functions of the tw'o random variables have cpectations that are cl values that are v enoug h 'Thus X n and X do not necessarily assume probabilities cx). arbitrarily close governed by as n close, yet they are .
,
Iar time.q'
=
Next define the random variables Xiltl, i
lpfzk Conveence often interested in values assumcd by a random variable as so We are Paramctcr n gOCS to infinity, F0r example, to defme an ltf) integral, a
YKs FZZUOIX Utjm VZSZFIC Wlt 0. Si&PIC SCIICOFC is fst COIRXXXCU. able wili depend on some parameter zl. In the sccond step, one shows the Ito integrai in the as n - x this simple variable converges to
x (,)
,
z'
.,
YWe
displays occas
ionaljumm.
Rmesents :
hxs
'! '. :. :
to this, Xit)
valueassumed by ei.
:.
SCnSC.
vt
accOr du g
j
,
17The same rvsult applies if, n addilion,
1,
.
.
,
,
.
.
a particun drawn ,
n by
,
...+
72.1 Relevance of
=
.
;. ....
for
. N
(
. '. .' '*'
';z
mayj for examplc, the present. meaus
I
if e; :G f
()
otjjejwjse
.
'
is either 0 or 1, depending on the t and on the
!e( the exyiratitm
2 S t :E 1.
time of some dcrivative ccmtract br 1, while 0
hat probtej
*'.8
(6z)
s
1)
=
j,
jfijl
f2 H A P T E R
116
0
: l
:
:
: 1 l
l
: '
: : ;
,
07
C5' S..q
W
.('
..) ' ?
j
.,'.
l ylou Usillg #/
(f), i
=
1y
.
.
.
p
&, WC
Snt)
=
dcllne the random
1 y-
*
n
n f-l
(-V(l)-
Variablc
.$)
il
(1) 7
..'
..
'
t (63.
f).
example
In the remainder of this book, we do not require lmy further results on pyohability than what is reviewed here. However, a fmancial market paticstudent will always benelit from a good understanding of yant or a Nnance Stocha8tic Of WOCCSSeS. zn excellent introduction ig Ross (1993). the theoy IJIVSCF all tj sjusayev(1977)js an cxccllcnt advanced ltroductitm. cnlar another (19Jg) is source for the intermediate level. 'T'he book by Brzezis a good source for introductory stochastic nuk and Zastawniak (1999) also See the new book by Ross (1999), Processes.
! '
RE
proccss. Tls
9 Ref erences
;
2
I
discussed an impoztant binonal
used to introduce the important notion of convergence of stochaswas exeetmple discussed here also happens to have tic PEOCC sses, The binomial since it is very similar to the binomial tree-models ractical implications, P used in pricing fmancial assets. rotltksejy
'
:
83
second,we
:
s l
117
jtl Exercises
.: :
:
k :
: l0.2
P1 L
Tcols in Probabilicy 'Tleory/
:
:
'n
5
.
.!.
7. Notc that Snt) is a rfec r Figure 6 displays this construdion for n wfaecontinuoua functionwith jumps at %. and the As n x, the jump points become more frequcnt morc pronounced. Thc sizes of the jumps,however, w 111diminK of normally distribut At the limit n x, Sntj will be very close to a will be continuous variable for each t. I'nterestingly, the process rall 7.0 equal to zero. ! the limit, the initial and thc endpoints being identically Clearly, what happens here is that as n --> x, the Snt ) starts to beha normally distributed process. For large n. we m more and more like a with than find a limiting Gaussian process more convenient to work example, as n increases, lt should also be emphasized that in this applications whe ln num ber of points at which Snlt) changes will incrcase. analyss, t tinuous-time we go from small discrete intervals towar d con would often be the case, =
,'
10 Exerclses
''oscillatio
-->
vbtt
1. You are givcn two discrete random variablcs X, F that assume thc Possible values (), 1 according to thc followinglbnf distribution:
'
--y
dom
;
#(Y
1)
=
P F
=
0)
'
astz
plxc
1)
.c
P(=0)
.4
.15
.25
'
(a) What are the marginal distributions of X and F? (b) Are X and J'v independent? (C) Cakulate SIA'Iand FEFJ. Calculatc 11. (d) the conditional distribution 'I-YIF the F (e) obtain conditional expectation FLA-I 11 and the conditional variance FlrlAAlF 11.
;
',
8 Conclusions
=
This chapter brieqy revicwed some basic concepts of probability theo. of pro We spent a minimal amount of time ori t he standard definitions i pojnts, important rnade number of However, a we ability, variables an First, we characterized normally distributed random Nocksbuilding Poisson processes as tw'o basic
=
=
.
Ksuch
bridge. a process is called a Brownian
2. We let the random variable Xn be binomial a process,
.
xn
. . ;
m
i: . '.
: .
J..
!
.
,
a
Bi
=
2=l
,
.
C H A PT E R
118
5
*
TYIS
j'
Probability Theo
in
%*47-
:
W herc each B; is indepcndent and is distributed according to 1 with probability p Vi 0 witll probability 1 p. 0, 1, 2, 4 and pl Calculate the probabilities #(-Y4 > kj for k the distr ouuonsncuom b) Calculate the expected value and the variance of x s or n =
#(Z
<
z)
1
=
e
-
P
kj rz
for k
=
0, 1, 2,
,
.
,
.
.
1 lntroductlon Martingales are one of the central tools in the modern theory of fmanee. In this chapter we introduce the basics of martingale theory. However, tltis t1.2 eory is vast and we only emphasize those aspects that are directly relevant to pricing rmancial dcrivatives. We begin with a comment on notation. In this chapter, we use the notation kl.rzi or hv%tto represent $'Smal1'' changes in F; or &. Occasionally. we may also use their incremental versions J##;, dst, wbich represent stochas6* Changes during innitesimal intewals. For the timc being, the reader Can interpret these differentiats as stochastic cbanges obsen'ed over a continuous time axi's. will be formally desned concepts nese itj chapter9. AJI denote a small interval, wc use the symbols h or infmitesimal interval, on the other hand, is denoted by dt. In latcr chapters, we show thatthese notations are not equivalent, An operation such as
'
:
.
,
q
,
g
,3 1 + A+ + --j- + 2. 3.
;
Az
=
.-'y
.
.
.
X,( F 1
. =
1
.
:........mVariance
the mean SIZI alld the
'jao
,
rr
late
Llinnitesimal''
L
show that
(b) Calcu
and
L,
(a) Use the expansion
to
'
'E
,
eh
#
'
k4
<
l;!
'
.
ke-k =
.
)
random variablc Z has Poisson distr ibutionif =
,
y'
the meafl and the variance of S.
(e) Calculate 4, A
artln ales an ale e resentatlons art1n @
'
z 1 + zc.
=
0
.
:
-z
(a) Determ ine and plot the density function of Z. (b) Calculate the F(Z). (c) Obtain the variance of Z. (d) Suppose Z 1 and Zz are both distributed as exponential independent, Calculate the distribution of their sum: S
>
.-
-.
2
.a.
3. We say that Z is exponentially distribute d w ith paramcter the distribution function of Z is given by:
1
.
.
=
t
'V
MM
.
5
-
ta'
Lv.
)
*M
i
Pr(Z).
5',1 .E'Eq%+ -
('
where
:
a js a
<usmalln
=
0,
zterval, is well dehned, Yet, writing
ELQS;4 g ::::u
:
'
'
'
:
is irtformal since ds is only a symbolic expression, as we will see in tl'le delirtition o'f the Ito
.
E
Jntcgral.
.. :
t
',1(.
E.
1l 9 ';
:
' ..
:
.
C H A PTE R
12O
@
.. '
Martingaltts and Martingale Representatio
6
2 Defnltions
J.2 C'orztxxtklu-rllme Murfugtlcs
; E.
gsing diferent inftrmation sets, one can conceivably genezate different Of a Process (&).nese forecasts are expressed using conditiomal expcctdtit-ms. ln particular,
Martingale theory classilies observcd time series according to the way th . mariingale if its trajccteries tren d A stochastic process behaves like a A periodicities. process that, on the average Play no discernible trends or supcrmartingale repesents pro submartingale. The term increases is calied a desnitions o formal gives section This decline. cesscsthat, on thc average, notation. ; these concepts. First some
Ttforecasts''
'
''
.
E J jysj =E
!
Is
lt ; Iv
.
-
.
f
'
j y js known, given 1t. 2 Unconditional
?
(fforecasts''
(1:
jltn t
(E
(0,F))
(
'
E
is calkd a hltration. martingale theory (and throughout the rest of this bk ln discussing assumed by some stochlzstic pr we occasionally need to consider values accomplishcd by selecting tmen This is cess at pmicular points in time. such that . sequence (t) The set
(,)
y;
DEFIMTION: We say that a process LSt, t e (0,x)J is a martingale witb respect to the family of infonnation sets It and with respect to the probability P, if, for a11t > 0,
'
m
-
<
tations.
with continuous-bm. denoted by (.%, t information sets
as time passes,l
t
,
.
2.1 Notctfon indexed by time i
j5yjyyj
is thf formal way of denoting the forecast of a future value, Sy of xi, usingthc inlormation available as of time t. Eu (.wj, u < t, would denote tjle forecast of the same variable using a smaller information set as of or earlier than time u. The defining property of a martingale relates to thesc conditional expec-
,
Suppose we obsen'e a family of random variables dex t. We assume that time is continuotls and deal stoehasticprocesses. Let the obsen'ed process be of (),oaj).. Let fh, t s I0,xlj represent a family maker decision the availablc continuously to become will satisfy s < t < Fj this family of information sets
121
y Ixfinitions
,
j'/ l
(4)
x.
<
3. And if
'
s t jyT j st =
..
0
=
fjl
<
jj <
.
.
,
<
/k- 1
<
tk
=
Herej a11cxpectatitms to the grobabzty P.
timc PCl'iOY OVW a COTRZUOUS timc ZtCFVZI(0,F1. N Value and the endpoint of the inten'al are handled ' way thC initial Whereas fk SWWbO1 ThC this /() is assigned to the iflitial Poillt, notatim. (x), and (-:) the Fmbol ftr F. ln this notation, a: k (lj finer and Iiner pieces. he interval into partitioned would F1 be t (0, . Now conside the random price process St during the fmite inte will (0, 'T'I.At some particular b'mc lj. thc value of thc pricc process each t (.n . lt at set .% If the valuc of St is included in the information js t f it is said that then f.%,t q (0,FIJ is adapted to fh,It.t e (0,FJ)..That value St will bc known, given the infonnation set matingales. We Can 11OW define Continutlv-time t he
-'
-'+
-
Et
.
UVu- Y1
=
f7/kVriul
-
.#L,%I. ,
,
(6) 'dre-
v. uest
;.
:he
'
''?
J
. '?
,
.
,5'
t
is
gxas- Xj change in S
forecast
tf the
=
(),
(g)
over an arbitrary intcnul u > 0 is zzero ju otjjer words, the directiolls of the future movemcrlts in martin' al es are mpossible to forecast, This is the fundmnental characteristic of -
'
7 .; .! ( '.I
.
k
.r.
Et
j.s ,ua j'v'p Luu
-
Z-jg.j are assumcd to be taken with respect
-adaptedj.
:
'
infonnati problejx a! hand, thc L will represent different types of obtain from infonnatson onc, can the will lye to represent ne most natural use of Jr markets time up to reaiized prices in fnanciai
,
Bllt f J. : j s a forecast of a zandom variable whose value is alrcady vealed', gsace Hence it equals If slt) is by dennition 1t a martingale .Et g,jv j would also cqual This gives t+u
,
lpeylcnljjj)g on
'k.j
x
p
,
(5)
rding to this defmition, martingales are random variables whosc Yttkre vahtion. are completely unpredidable given the current information Set. For examplc, supposc St is a martingale and consider the forecast of the change in t over an interval of length u > 0)
.
Wnew's
w,
nat is, the best forecast of unobsenred futurc values is the last obsewation on St.
.
I-CWCSCIX Vzfiotls
.c:
vith probability 1.
(
T
jor aIj t
,
'!
.
('
. '
. '.
. .
a'
C H A P T ER
Martingales and Martingale Representation
6
.
F'
'
j
.
'f'hu
t.jse
Martingales
of
in
Asset Pricing
123
L
thc trajectories of a process displ processes that bebave as martingales. lf atrcndsx'' thcn the process is not ak clearly recognizable long- or short-run ' martingale. 2 i Before closing this section, we reemphasize a very important property o .i' the dehnition of martingales. A martingale is always dehned with respect t probability If with measure, respect to some w some information set, and . change the information content and/or the probabilities associated with tlz l martingale. Process, the process under consideration may cease to be a rrhe opposite is also true. Oiven a process Xt that does not behave probability measure a martingale, we may be able to modify the relevant i martingale. and convert Xt into a :
Ij asset prices are more likely to be sub- or supermartingales, then why martingales? an interest in lt mrns out that although most linancial assets arc not martingales, one them into martingalcs. For example, one can Iind a probability yy cz convert 73 such that bond or stock prices discountcd by thc risk-free sstrjyutjoxmartingales. If this is done, cqualitics such as ra te become Sl7Cjj
'.
f tjl
'
y
Et
(
,.
According to the dehnition above, a process St is a martingale if s ftl movementsare completely unpredictable given a family of information sct Now z we know that stock prices or bond prices are not completely unpre The price of a discount bond is expeded to increase over time. dictable, the are expected to increase . generai, same is trteBt for stock prices. They of ice represents if Hence, t e a discount bond matu li pr theaverage. < time T t F, '
':
(8
(#sJ
Et
<
,
'
not move like a martinga 1e. ( clearly,the price of a discount bond does expected zetu positive will have zisky stock general, , in a a st similazly, martingale. wzite small interval For will can be not we a a and ; 5',j g, E: (9k ,
x
g5',+, -
return. a wherepz is a positive rate of expected A similar statement can bc made about futures or options. For examp passes, the price of European-s optionshave 'itime value, an d as time is a supermarting 1:.4 will decline ceteris paribus. Such a press options
Where
apprnximation hcre is Z tbe Krle S Tayior series expansion of Ef
Utw.a1,
dropping lher-order
Of
lerms
invthring
=
gt
T ay lor series expanslon. term' of tlle corres&nding Valtlc. Rqative 4 time Deep ill the moneyy Amenkmn Puts m2y have ()()
represents
ali
<
T
-
r,
(10)
=
st
,
(j
<
u,
(1z)
fequivalent''
1.
-rus s/t gtr
'
L' :
.
:
F, '
:. '
6
!'
. :..
.j'.
: :
,m!z
j s =
t,
(1a)
and th () e S becomes a martirlgale. Probability distributions that convert equations such as (12)into equalities such as ()g) are called equivalent r/cr/frfu/e measures. Tjwy wjll be treated in Chapter 14. lf this second methodolor is selected to convert arbitrary processes into DvutiRgale: then the transformatitn is done using the Girsanov theorem. In
j.
if
+ tp(),
u
,
: .$,j
g.%+ higher-ordvr EJ
<
, (.) is the conditional expectation calculated using a probability distribution P we may t.ry to tind probability /3, such that an te yiew expectations satisfy
-
where
EP
,-rt
3ne
j-j-gj
'
'
.
,
character.
gc-rus
ws: sz. y-ruy '+u j
,
'>
2A.sample path of a martingale nmy still contain paterns that lmk like short-lived tre However. these up or down trends aze completeiy random and do not havc any s'yste.m
0
,
i'transformed''
',
Bt
Bt
,
?
at
=
foy stock priccs, can be very useful in pricing derivative securities. One important question that we study in later chapters is how to obtain this conversion. There are im fact fww wap of converting submartingales into mmingales. The first method should bc obvious, We can subtract an opected lrentf from e'-*S l tr e-''B l nis would make the deviation.s around the trend Hence, the variablcs would be completely unprcdictable, martingales. This methodology is equivalent to using the so-called representation reSults for martingales. In facq Doo>Meyer decomposition implies that, ungeneral conditions, an arbitrary continuous-time process can be some der decomposed into a martngale and an increasfng (or decreasing) process. Elimination of the lattcr lcavcs the martingale to work with. Doob-Meyer decomposition is handled in this chaptcr. We Second method is more complex and, surprisingly, more useful. Instead of transforming the submartingale directly, we can transform itslzmhabi?ity dawwjzw. lxat is, if one lmd
!
'
.ru
jor bosds, or
.',
The Use of Martingales in Asset Pricing
(& #/+.j
' '. '
.;;
..
:
?. CHAPTER
i 24
Hartingales
*
and
Martingale Representati,xkl'.t :
...
.' '
.
4 Rejevance
of
Martingales in Stochastic Modeling dw,
financial sset pricing, this mcthod is morc promising than the Doob-Meyef, k decompositions. i.
4 Relevance
of
125
L. ''
Martingales in Stochastic Modeling
xc
..:
In the absencc of arbitrage possibilities, market equilibrium suggests tha such that all properl# we can fmd a synthetic probability distribution / martingales: discounted asset prices X behavc as j
=
;()
N
W
E
Ec -' ''U
&+.uIJ,1
.5r,*
u
=
>
'
(14
0.
Because of this, martingales havc a f'undnmental rolc to play in practic assct pricing. But this is not the only reason why martingales are useful tools. Mai is very rich and provides a fertile environment for discuss' tingaletheory variables im continuous time. In this section, we discuss tbe stochastic : usefulteclmical aspccts of martingale theory. Let -Y, represent an asset price that has tlae martingale property wi fzf).and with resped to the probability >, respedto the rtltration
'T-m'e
17,ouaE
i
Figure 2 digplays an example of a zight continuous martingale. Here. the trajectozy is interrupted with occasional jtzmps/ wlaatmakes the trajectory rkht ctntinuous is the way jumps are modeled. At jump times fa. tz, the martngale is continuous rightwards (butnot Ieftwazds.)
.
,
,
E > x ,+a
(1 .
1zt 1 x r,
((
,
=
,
wherc A > 0 represents a small time interval. What type of trajectori would such an X3 lmve in continuotls time? To answer thig question, tirst dctine the martingale /ercncc xt, .-Y/
=
Xtwh
X,
-
'
Xt
:
2
. .
i, 1 tii
b-.
.
(16.J
,
and then note that since Xt is a martingale,
1
i.
.
.
E
p
(1
(Aa'V1610. =
0.S .
k1 '
earlier, this cqtiality implies that increments of a martv should be totally unpredictable, no matter how small the time fnterval consid ' is. But, since we are working with continuous time, we can indced irregular trajectories. vety small A's. Martingales sbould then display very trends lspection, even duri by discenble display should fact, Xt not any If it did, it would become predictabl infinitesimally small time intervals Such irregular trajectores can occur in two different ways. former leads to condnlm J be contittuous, or they can display jumpz. martingales, whereas thc latter are called rkht continuous rzicr/rlpls/e.. . Figare 1 displays an example of a continuous martingale. Note that o. 0. continuous, in the sense that for As mentioned
o
.
s
-n
.
-1
.
-1.5
rrhey
.
'rhe
'
'
-2
c
.E..
trajectoriesare
PLLX:
>
ej
.-->.
0.
for al1 e
>
0.
(18. '
:.1
o.c
a.a
o.4
c.s
F I G U RE
'',
.-..>
SNote that tjw process still does not have a tzend.
t: .
j ':
().*
2
0.7
a,a
().q
1
rIJ
'.
cHAPT ER
126
Martingales and Martingale Representati
6
.
,.
. .1
')
5 poperties
:
This irregular behavior and thc possibility of incorporating jumps in tll trajectoriesis certainly desirable as a theoretical tool for representing assd : prices,especially givcn the arbitragc theorem. But martingales have significance beyond this. ln fact, suppose one is dealing with a continuous martingale Y, that also has a linite secon
Martingale Tmjectories
vosee this, note
AMl
j (.42
<
Et rlN/j
and similarly for E:
Ej
dibad''
pqtj.y tc
''
The
and
NB t
information on
denote the total number of instances of
t ''bad'' news, respectively, tmtil time t. We assume further that arrivesin iinancial markets is totally unrelated to past data, new's are independent. '' good news an d the
Rgood''
l+
-
1
A
.
(24)
(zs) that
j jag,yj u a ..A,a
.))
Ac a
p
s gagyt j u ?
g(j)
.(),
ptasrj a
>
a As a.
.j;
py)
t;a
-
sa
>
(aj;)
().
(1nfact, Mt will be a submartingale.) Hence, changing the underlying probabilites or the information set may alter martingale characteristics of a Process.
'.'
Sibad''
,,
(1
then Mt will cease to be martingalc with respect to It, since
E;
an4'
the way 1) and that th'
.
'good''
..
ATG
(2g)
(Tbad''
'.
of the ncws, but reta in
.
Gibad''
using two ndependent Poisson proces '. we will construct a martingale intervals'' during observed and news. Wi Suppose fmanc ia l mar kets are inquenced by whether it i the
icnore the contcnt dbad.'' 'J ood'' or g
gaxjlj
Hence, jncrements in M, are unpredictable given the family 1t. It can be requirements of matingales, For shown that Mt satishes other (technical) nbad'' example, at time f, we know the zgood7' or news that has already happcned. Hence. Mt is f/-adapted. is Thus, as long as the probability of good'' and news during given by the same expression for botb N1 and NtB, the process Mt will be a martingale with respect to It and these probabilities. However, ifwe assume that news can occur witla a slightly greater Pfobability than news.
)::. 'dgood''
21 0
(a?v,z'j. nis means
.
''small
-
x a,
.v
4.1 An Exfzmple
(22)
'
gyyj
will be
But, approzmatcly,
)t. 0, called frr/e: and is coatinuous variance has finitc square a a proccss martingau. It is signincant tilat one can represent all such martingal motion at a moained time aock.(seeKarat running tlw srownian inteE' words, the class of continuous square other In shreve(1991)1. grable martingales is very close to thc Brownian motion. This suggests th. t' of thc changes and the absence of jumps are two pro theunpredictabilie motioo qt in continuous time. of Brownian ; crties Note what this essentially means. If the continuous square integrabl is appropriate for modeling an asset price, ooe may as we' martingale L assume normality for small increments of thc price process. ') >
such smable by an(:1
u
CNK't -
expectation operator; ApPjy tjje condtional Ev Et ggyjj Et
(19),
(!x)
Ntc
=
=
E
1 27
that the increments of Mt over small intervals
pvc s lv ,
moment
for a1l f
of
Finally, during a small interval , at most one instance of good news x,' and the probablity of this occurren*, one instance of bad ncws can occur, of Thus, the probabilities of incremen for both the news. types same is .
(
&N'G,
changes
ANS
during
I'hNz
t
is assumed to be given approximately by
,&
=
1)
=
PkNjB
1) A
=
l.
(20) .
Mt
The properties of the trajedories of continuous square integrable martingales can be made more prccise. Assume tbat (mj represents a trajectory of a continuous square intelable mmingale. Pick time iltewal a I0,Fj and consider the times (fj)'.
::
..
G -XB
Nt
':
G1)E /
.
, .
.i' ,
wiil be a martingale.
5 Properties of Martingale Trajectories 'E
Then the vaziable Mtb defined by =
'
)
L .
r
( j .
:
i
t
10 = 1,5.
'
k
.:E :
0
<
/1
<
Q<
.
.
.
<
fs-l
<
tn
=
T1
(29)
' ..r
.'j:'
128
c
H A PT ER
.
f MartingaleTrajectories .j popecries
.7'
Martingale Representatioru
and
Martingales
6
'
:
N
prl
.
=
l.A-y-
-',
f
f-1
z
I
1
--.
!.
t
(30)
Heuristically, P'l can be intcrpreted as the length ot. the trajectory followed by X; during the interval (0,F1, The quadratic variation is givcn by
1.--2 =
i=1
.
Izt/j-
Xq
One can similarly deline hkher-order fourth-order variation is defined as U''4 =:!
g,
For example,
*''
)
l--
'*'''Y
*'
%-'
-
x
72
(32
-
r
,
AS
'',
ti
-* ,,
sccutive
x:
,s
t
'
(36)
I ,
Wlarzest'' G-,
X. ''f
IFI,
(37)
all i, the continuity of the martingale implies that will get very near each other. At the limit, max i
S
,
,
.1
?
J .1 fOr
:
When Xt is a continuous martingale, the p-1, P' z P'3 #' 4 happen to 1,av some very important properties. We recall some relevant points, Rcmember that we want Xt to be ctm' tinuous and to have a norizero variance. As mentioned earlier, this mea two things. First, as the partititming of the interval (9, F1 gets fine nn X l get nearer and nearer, for any e > 0 finer,
- X.
*'''''
I.Yj - Xt f mtx j
<
.
changes.
lA) @=1
j
%i-
E
I
Xtt-, I
-
I
.6
,j.
lx t-. - x j4
I'IIMIA)
<
seftause thc rizht-hand side is obtained bv factorinz out the This means that
'
t
1A).- Xq-.,i
i='l
i
the pe' or v 2 arc different measurcs of how much Xt vari obviously, of absolutc changes in z obsewe ne p' ) represents the sum 72 overtimc. The subintewals reprcscnts tbe sums of square ti - lg-:. duringthe
iconsecutive''
.
, (31),.
I2.
,
variations.
-
J-.1
is true. As (0,F1 is partitioned into finer and suhintervals, changes in xt get smaller. But, at the same time, the jyaer of terms in the sum deqning P'1 increases. It turns out that in the pumber continuous-time martingale, the second effect dominates and the case of a ininity. IZ t ne trajectories of continuous martingales have goes toward for variation, cxcept the case when the martingale is a constant. inhnite shown heuristically as follows. We have This can be the opposite surprisinglya
'
we define the variation of the trajectory as
I.',
.
-
A-q-, I
ncon-
0.
.->
(38) 1
This, according
?
to Eq. (37.),means that unless V gets very large, P- will toward in zero some probabilistic sense, But this is not allowcd bccausc go Xt is a stochastic proccss with a nonzero variancc, and consequently 1,,-2> 0 even for very 5ne partititms of g0,F1. nis implies that we must have P'' x. NOw consider the same property ftr higher-order variations. For examPle, consider 1,,-4and apply the same as in (37)-.
,
: ''
.'
q
-->
.
's
'
' 0. #(lA) - z, 1 1> 6) (33) t for all i. Second, as the partitions get finer and lincr, we s -->
.
'
if
/j 'ant %%
-..+
dj-ja
1..,4<
'
,r' jaY2 Xt > O : 1i -l
=
-
.
1.
gnA
s--' .j .
E .
This is tr'uc because Xt is after a1l a random process with nonzero variancct Now consider some propertics of 1z'1and 1z'2, . First, notc that even though Xt is a continuous martingalc, and Xt a t proaches A-r 1 as the subimerval (lj,/j-I 1becomcs smaller and smaller, t jjjsr (jer ' that P' ) also approaches zero. The rea does not mean may find t incremental changes; P'1 of such made of the sum is surprising. After all,
IA) mpx - Xt I
Fl
jV
jn
,r
;.
-
..-.
gj.
j
j
j
3j :
f= 1
Xt .
- i
.
5j
'.
SWC
' :
.
jut
Ilotjjjjtjj maxf jXj - X:(-uI j gjjotjjd rjjsfuyoy ojv,jj yyyoyamoyy jaj yy ,
'
would not p' go towar d z ero as well?
,
' '
,
12-->
9.
(3j)j
(40)
means that 1./-4will tend to zco. The same argument to aIj varjatjons of order meatezthan two.
:
=
j2 4,.,v2
YYS
'
V
:
As long as 72 converges to a well-defined random variable 7 the right-hand sidc of (39)will go to zero. The reason is the same as above. ne x / s a continuous martingale and its increments get smaller as the partition of the inten'al I0,Fj becomes hner. Hence, as fj --> lf-j for all i'.
.'
g
P
As Xt approaches
max IXtj - Xf
3
.
p.x
..
. .
. j :
.
. .E..
: . .
nd doe,s not converge
o infinity.
jx
jxad
ay
cjIOOVIlg
thC
can be applied
SW.YCEECRt lllfgcst OUMCI'VCU
E.
.i
: :'
.J
130
C:H A P T E R
*
6
Martingales
and
:
:
Martingale Representati
Examples
h
:
131
cyartge jndepcndent acr()sg time. lw
if is a small interval, the increments xt <21.8 distribution wit.h mean IJ.Land variance
undcrWthcsehavcconditions, normal
j
during J frlcans 7-1:19
. The variation F 1 will converge to inlinity in some probabilistic se $ and the continuous martingale will behave venr irregularly. P-2will converge to some well-dcsned rando'f quadratic variation The .
variable. means that regardless of how irzegular thc trajectories thc martingale is square integrable and the sums of squares of the incr mcnts over small subperiods converge. This is possiblc because the squ of a sma 11num ber is even smaller. Hence, though thc sum of incremen is 'itoo large'' in some probabilistic sense, the sum of squared increments
Martingales
in Xt is denoted by dxt. lncremental changes in #/ are assumed to
For formal proofs of such arguments. the reader can consult Itaratz Here we summarize the thrcc properties of the traje ' and Shreve (1991).
tories:
Of
ill
a
Xt
tV).
Ny,
'v
(41)
'rhis
ne
fact that increments are uncorrelated
j':'
7g(AA'u
'
not. probabilistic sense. . All higher-order variations will vanish in some higher-order variations of that interpreting this is to say heuristicway no t contain much information beyond those in 1e'l and 1/-2 nese properties have important implications, First, we see that #'1 is n
-
gtjhakrt
/zAll
-
tie,
1+7'
dxu (44) e Assuming that the integral is well defned, we can calculate thc relevant
xtwv
E,
.v()
+.
=
..
a very useful quantity to usc in the calculus of continuous square integrab martingales, whereas the )' can be used in a meanlngful way. Secon liighenorder variations can be ignored if one is certeeun that the underlyi Proccss is a contittuous martingale. nese themes will reappear when we deal with thc differentiation and . tegration operations in sthastic environmcnts. A reader wh0 remembe the desnition of the Riemann-stieltjcs integral can already see that same methodology cannot be uscd for integrals taken with respect to co 1 tinuous square integrablc martingales. nis is the case since the R .jem 'i Stieltjes integral uses the equivalent of Izrl in deterministic calculus an ' consides lincr and finer partitions of the inten'al under consideration. ; stochastic environments such limits do not converge. 2 will calculus discuss Instead, stochastic is forced to use p- We tu s detaillater.
,
expectations.g
considerthe expectation taken with respect to the probability distribu(41),and given the infonnaon on X, obscrved up to time /:
'
tioo given in
,
-
t
.
:.j E / g.v t+
Et
=
l+T
s/ *<
..'
.:.
8
.
. .
;.
Tltis question is more complicated chaplur.
E
... i
)L
9*e
.
e '1
.
lt is not clear why the variancc tyjat
psKsjjjq
.
'
u
=
(46)
gk
(47)
Xt + P.T
=
Clearly, f.m)is not a martingale with respect to the distribution in Eq. and with respect to the irtformation on current and past z.
.
a continuous process whose increments are no Brownian m distributed. Such a process is called a (generalized) mally instant, ivnitesi the of for each f. At obsewe value Xt every a tion. we
suppose Xt represcnts
n
5
Et EA-/+rl
i
6.1 Exumple 1: BraurrkfzmMotion
ay
*'*'
t
so
.
:.:
(45)
.
t
vkmt-vv
'
h) .. r!
.
k&
..
martingal
y
But at tme t, future values of are predictable because all changes during small intervals have expectation equal to y. nis means
,.
ln this section, wc consider some examples of continuous-t'nc
-r
dyg
xt +
.
6 Examples of Martingales
(43)
f,
u
that is,
'
:
'
0,
=
as
Leaving aside formal aspects of delining such a process Xt, here we ask simple question: is Xt a martingale? a ziaccumulation'' of infinitesimal increments over ne process Xt is the
:'
.
can be expressed
i:.
j'. l'
of AA'', should be proporlional
vvtx.vr) to anmer
=
tzzolz?
to
.
(41)
For e'xample, is it
(u)
han it seems. It wul be at tlze core of tlae next
have not yet defiaed integrals of raudom
incromcntal cuanges.
.q
C H A PT E R
132
*
and
Martingales
6
But, this last result gives a clue on how to generate a martingale l X t J Consider the new process;
: '
.
Martingale Representati
p
.
of Martingales 6 yxamples
:
last equality follows because increments in St are uncorrelated with and past St. As a result, the cross product terms drop. But this cavrent meaus that
-
'rjje
:.''
.
z/
Xt
=
xt.
-
( ,
EX'sv
=
=
2.
+ F)1
ttf
-
xt-vv-
E'(.A'2+
which mcans
E
(
It is easy to show tbat Zt is a martingale: J7rl&+zl
133
zY/)! - gt
+ F),
slh,
gaz/j
=
(s7)
whichproves that increments in Zt are predictable. Zt cannot be a martingale' atransform'' the But, using the same approach as in Examplc 1, we can change and obtain martingale. the In fact, following with a a mean z/ equalityjs easy to prtwe:
(4 i (5
' .J
.;
FlEZ',+z.1 Xt + EfXt+v - -Yfl - y-t + F). (5) But the expectation on the right-hand side is equal to #,F, as shown :' Eq. (47).nis means
tz2(w
E / jzrow
=
-
+ /)j
zt -
=
'*
f /EZ/+r1
Xt -
=
(5 That is, z, is a martingale. Hence, we were able to transform Xt into a martingale by subtra function a deterministic function, Also, note that this determ inistic settings result general time, holds in as well. more increasingover nis
;
q
';, 'E r.
6.3 Exumplc 3: An Fxxponcnfwl Process '
:
tut
N(0, c'21),
'w
.#. ,9.
Now consider a process St with uncorrelated increments during s
intcrvals
'imcan,''lo
'
6.2 Fxumple 2) A Sqtuzl-el Pgocess
(
.'
ne third example is more complicated and will only be partially dealt w1t11hcre. A gain assilrne that X t is as detined in Example 1 and consider the trans-
:(
E
where the initial point is givcn by
formation
'
.
.stl
(5
0.
=
j.
.:
..
Detine a nem random variable: z
(5
J2. t
=
/
(5g)
from Z/ wc obtain a martingale. again illustratcs the sarne principle. If somehow a stochascxamplc njs it tic process is not a martingale, thcn by subtracting a proper transformed into be one. can nis brings us to the point made earlier. ln linancial markets one emnnot expcct the observed market value of a risky security to equal its expccted Value discounted by the risk-free rate. nere has to be a risk premium. Hence, any risky asset price, when discounted by the risk-free rate, will not be a martingale. But the previous discussion suggests that such securities prices carl perhaps be transformed irito one. sucha transformation would be vezy conven ien t for pricing nancial assets. B su ytracting
(
Ia't
tr2/
aat.
Wj4ere
a--sj ,
-
.
,
gq)
a is any real number. Suppose thc mean of Xt is zero. Does this tralsformation result in a martingale? 1 We answer is notice yes. We shall prove it in later chapters.l However stil'l xmethug otjd. js itself a martingale. Why is it that has one to ne xg subtract the hmction of timc glt),
..
According to this, Zt is a nonnegative random variable cqualing l .r ls Zt a martingale? . square of 5,. The answer is no because the squares of the increments of Zt are pr asmall'' consider the expectation of the inc k intewal dictable.Using a in , ment z,: 2 J, &2) ' Srlu/+, E 1((,% )j2 t (s t rml
s
.(t
h
.7
,
-
X1
=
=
- s
-
SIEXI-I-A
-
xVI1
.z
-
#(/) lrj..jj
'.
,
jjouee
-
:
: '
.
'
:
'.r,
.
'
(
at js t,ly suuractiug ,
,j
:
..
..
we jeara about
a2
-f,
=
2
from it a function of time, say, gt). Io's txmma.
(60)
:
.
) '
.
C H A P T ER
134
*
Maninga les
6
an d
Martngale Representatio
j
in order to make sure that St is a mmingale? Were not the increments X impossible to forecast aomay? t ne answers to these questions have to do with the way one takes deri tives in stochastic environments. This is treated in later chapters.
y
t'
l
.
basically, that
N*t
Nt
=
:
,
)
z.1 An
r
...
'.,
,
:l'
.
.
L
denoted by Mt,
Of
the same
(6.
'
;
Ml
=
EP
IFw
,
LA/'t''6b 1lt
=
(6),
Mt.
is a martingale.
' 1) Lz(
I
-
y(s).
(69)
'
-
.
=
.
' vN
expec ' This resu lt comes from the recursive property of conditimal rando chapters. For several any later wil) whicb tmes in see we tions, variable Z, we can write: EP .1/1 EP E.Z I .1,1, (6j s > 0, E .fd+x1
=
,
,
It turns out t hat the sequence of forecasts, for 0 < x'. EP
(
I61
some probability #,
isy,
E
G 1T.
IT-L
Wforecasts,''
Next consider SUCCZSSWC ma de at diferent tnes,
with respect to
.
of the logic used in the previous
jz' r a e (70) nis is a sum to be received at time F and may be random if c is stochastic, ueresz is assumed to be known. l'ina'ly, consider the ratio cr/sz, wluch is a relauve price. In this ratio, we have a random variable that will be revcaled at flxed time F. As a we get more information on thc underlying asset, St, successive conditional eoedations of this ratio can be calculated until the Gv/B,r is known exactly at time F. Let the successive conditional expectations of this ratiox Calculated using different information sets, be denoted by M t gT EP Mt (JI) j It Bv Where 1? denotes, as usual, the infonuation set available at time /, and P is an apmopoate mobabgity. Accordiug to the previous result, thesc successivc conditional expectations shouz form martingale: a Bv
i
646+1
Applkafien
Next, consider the investment of $1 that grows at the constant, continuously compounded rate rs until timc F;
'
1t
martingale.
gz
There is a Simp le mmingale that one can gcnerate that is used in pricing complicated interest rate derivatives, We work with discrete intervals, distribution #. yz bability considera random variable l'w with pro Suppose we keep getting n be revealed to us at some future date F. denoted by z, concernirs v. as umepasses, ,, t + 1, w ? such tuat. w 7
(66), (68)*
payoffs at rmiteexpiration dates r. Many Most derivatives have do not make any interim pamuts until cxpiration either. Suppose this is the case and let the expiration payoff be depcndent on some underlying asset price % and denoted by
. frcquen
inormation 1
=
nere are manywithhnancial applications one common case. we deal random
. ,.L
..
=
side of
section.
'
.
.
(6,)
,
(65)on the right-hand EP lrr Iz,J Mt. I
gyyy
ntus,Mt js a
gsz'gyyj gtysj j y'jj
J
that are unm x,* a lso has increments clearly,themartingale. willbe a martingale. variance finite, and it is lts right-contkiuous It is a d ictable. . square integra ue.
7 The Slmplest Martingale
=
whjch is tr jviajjy true. But Mt-vsis itself a forecast, Using EP gF# It-vsj J,j
(6..
l,
-
best forecast of a future forecast is what we gMisxl we have =p?p
ik the Poisson counting process Nt discusscd in t
We consider again chapter. Clearly, Nt will increasc over time, since it is a counting pro will grow as time passes. Hence. Nt cannot b? and the number of jumpsupward trend. has clear l t a martingale, Yet the compensated Poinon Fzrt?ce.f denoted by N:,
135
whicj, says, Applying the Z this to forecast now. r E E u t+s j gl j
/'
1:
6.4 F-rramp!c 4: Rig ltt Centinueu.s Muuingales
Martingale vhesimplest
( .
..: ,
C
Mt
::t
'Bm.
'' ?
? k
J :.
(67)deleted
=
,
p E g.$J/+, I fij
in proofs, all other equaton
,
numbers
s
>
(),
remain
(72) unclaanged.
r'
S'1
i'
C H A P T ER
.
t
Martingales and Martinga le Rep xesentati
6
S'
! :'
7.2 A Rerrzfrk
'2:J
.i
.,
Gy
J!
'
,
, (,
value of the bond. Then, the M3 is the conditional expectatitm tf the discounted payoff maturi undcr the prtabability #. It is also a martingale with respect to
the par
,
. according to the discussion in the prcvious section. g whethcr azbitragetake Mt the The intersecting question is as we can Price of thc discount bond at time /? ln otbel- words, letting the F-maturi default-free discount bond price be denoted by Bt, T), and assuming th #(f, F) is arbitrage-free, ca,n say tbat ; '
Bt, F)
8.1
book wc will see that, if the expectaon In the secmd half of t.1:1,s under probability #, and if this probability is thc real Af'/ calculated a will gcneral, Mt equal the fair pricc B(t, Tj. robability, then not, in # used calculating in Mt is selccted judiciouslyas if probability But, the arbitrage-free =equ ivalent'' probability # then
l
FJ3 =
100 1 It p
.
UT
previous examples showed that it is possible to transform a wide varie appropria ,of con t inuous-timeprocesses into martingales by sttbtracting
ne
meang ln t itis
sectim, We formalize thesc decomposition.
D 00 b-Meyer
gpecial
cases and discuss the so-ca-f
.
,
<
/#.-j
<
tk
=
:F)
(77)
.
st '
with probability p
= -1
:
with probability
(1-
p).
(78)
It is assumed that these changes are independent of each othcr, Further, if p 1/2 then the expected value of LS ti will equal zero. Otherwise the mean of price changes is norlzero. Given these conditions, we first sbow how to construct the underlying
'
8 Martingale Representations
,
,
-
:
(7.
that is. the Mt will correct ly pr ice the zero-coupon bond. The mechanics of how # could be selccted will be dscussed in later cha ters.But, already the idea that martingales are critical tools in dynarnic should also be clear that we can defino pricing should become clear. lt (Wi era l M using different probabilities, and they will all be martingalcs t martingal Yet. only one of these particular proba bgities). respectto theirarbitrarpfree Bt, T). of cqual the price will
.<
(
ddliquid,''
l
Mt
/:
at times
the price of a financial asset St. If the intervals between the timcs f/-1 and ti are vely small, and if the market is the price of the asset is likely to exhibit at most one uptick or one downtick during a typical fj t-: We formalize this by saying that at each instant ti, there are only two possibilities for St, to change!
,
,
=
obsen'es
() <
.
Bt, F)
137
a n svamp!e
a trader suppose
(7i
Mt
=
je Represenrations
will be introduced. The example is imporFjot, a ftmdamental example three least) rcasons, tant for (at is practical. By working with a partition of a continuous reason rac nrst jnterval, we illestrate a practical method uscd to price securities in time %ancial markcts. sccontl,it is easicr to understand the complexities of the lto integral if with such a framework. onebegins (j finally, the example provides a concrete discussion of a probability M assign probabilities to various trajcctories associated how one can space a xd prices. wjth asset
;
default-fr '
100,
=
8 Vartijagu
..
,(
.h .
Suppose rt is stochastic and G'r is the value at time F of pure discount bond. lf T is the maturity date, then
'
=
,
Probability space. We obseae tst
at k distinct time points.lz We begjn with the notion probability. T'llc jp, (1 pl refers to the probability of a change in Stj and is only a (marginal) probability distribution, 'wliatis of interest is the Probability of a sequence of price changes. ln other words, we would like to discuss probabilities associated with various ''trajectories,MlB Doing this Tequires constructing a probability space. Given that a typical objcct of intcrest is a sample path, or trajectory, of Ptice chan es g we first need to construct a set made of all possible paths. Of
.
-
' '
.
:
E
3'
'
n
,
),'
'
!z
hble tlw important assumption tlnat k is fmite. or eumple, the trader may be intezested im the length of the current uptrend treotj ZSSG
;
:
)
':
; . ( .
u
r: : .:
'z' x.
prkces.
or down-
r' j'
: '
.. '
138
CHA PTER
Martingales and Vartingale Representati
6
.
'
t.xyj
uhv
-1
=
,
,
.
.
).
+1
=
,
8
,
.L
:
This space is called a samplc space. Its elemcnts are made of sequences +1's and -1's. For example, a typical sample path can be
typical St. is made of the sum of Lut probabilities such Note that sincc a used to obtain the probability distribution of the St, as well. be as (:j) can would simply add the probabilities of different trajectories this we js dojng .H Stg the jeatj same to that be more precise, the highest possible value for st, is stv+ k. This result if all incremental changes will value h, i 1, k are made of of probability this is outcomc Tlle +1's. >s,
.
;
'
( ::. E
'ro
k is srtite,given an initial point st, we can easily determine tbe addkng incvemental changes, This by the asset price by all trajectoyes, i.e., the sample x.puc possible of construct t set can Next we dennc a probability associated with these trajectories. the price changcs are independent (andwhen k is tinite), doing this is The probability of a certain sequence is fotmd by simply multiplying probabilitiesof each price change. 5'' (hat begins with +1 at time) For example, the particular sequence it, . and alternatcs until time tk,
since followed jectov he we
surtitvaleRepresentations
'
''
=
Pstk
1
.(k-$-
will have the probability
tl
Lkb'tz
+1,
=
=
1
-
,
.
.
-
,
k is even) (assuming P el #/2(1 - pjkz =
tutk
=
-
1),
(
)
St +
=
.
,
(83)
#.
=
the lowest possiblc value of Stk is St, - k. similarly, is given by
g
k- *
.
'
':,
=
.
j p 1*
=
k
y 1)
.g
;
(j .pjk
::zz
ne
probability of this
(g4j
s
.
+ k In thcse extremc cascs, thcrc is onlv one traiectorv that eives 5'. or st st' k. - the price would bc somewhere between these two extremcs. In gcneral, Of the k incremental changes obsewed, m would be made of +1's and k m made of -1's, with m k. ne stI wjll assume tlle value s =
.
, (@
Tl'ie probability of a trajectory that continuously declines durv the periods, them continuously increases until time /,, wgl also be tile s Sinc.e k is hnite, there are a hnitc number of possible trajectories) the sample space, and we can assign a probability to every one of th
J'
v
E
:
'
'k
uto
-
k/'z
St
'.
trajectories.
=
Sto + m -
;
(k -
(85)
mj.
Note that there are several possible trajectories that eventually result in the Same value for Adding the probabilities associated with a11 these
It is worth repeating what enables us to do this. ne hniteness of 'i plays a roie here, sincc with a Iinite number of possible trajectories assignment of Probabilities can be made one by one. Pricing deriva produds in Iinancial markets often makes tbe assumption that k is !' and exploits this property of generating probabilities. Arlother assumption that simplilics this task is the independence of cessive price changes. This way, thc probability of the whole trajedory be obtained by simply multiplying the probabilities associated with c
.z
&
incrcmental
c'
*'
=
combinations,
,
wc obtain
plst,
,$j,
=
+ 2m - k)
=
ml cjk-mb 11
1
-
pp-m
(86)
,
where
change. this point, we have dcalt with the sequencc of changes in the Up to pricc. Derivative securities are, in general. written on the price itself. cxample, in the case of an option written on the S&P500, our interest wit h the level of the index, not the change. . from subsequ 0ne can easily ttain the level of thc asset pri changes, given the opening price St2 : '
Ck(k-m)
=
k! m jjk
-
m)!
.
.
Tls probability is given by thc binomiul Jfnbuljtpn. dtribution colwerges to normal distribution.ls
.
.
=
k
5-/ +
st i
=
1
vbj-
),
(
.....>
x,
this
:
::
k
As k
14
':'
' .
j. ..'
;
'..
..
'.
.
Addition or probabilities is permitted if the underlying cvents are mulually exclusive. In p artic-ular case different trajectories satisfy his condition by lefinition. exam' Ii-aks is aa Ple of weak convergcnce.
vua
-..,0
t
7
!:
. '''
:
140
C H A PT ER
*
Vartingales and Martingale Representati
6
. ingaltt Representations 8 Mart
r
!
.
shown earlier, we can write w as ylk 1 - 2p)(k + 1) +. z(, (94) aje. jyj Hence, decomposed mart we g a submartingale into where Ztk a o components, ne srst term on the right-hand side is an increasing deterministic variablc, The second term is a martingale that hs a value of st, + (1 2 ) at time /u. Th.e expression in (94) is a simplc case of Doob-Meyer decompositioml;
$ 8.1.l Is Stk a Mttrlrlgtsd? martingale with in.f respect thc delined in Eq. to Is the Lutk (82)a J price changes LSL k ? mation set consisting of the increments in Consider the expcctations under tbe probabilities given in (86) E
NO
-(
,
,
,-tk-jl
.
.
=
,
.
+ (-1)(1 ((+1)17 -
+
-fk-l
#)1
(
,
side is the expectation of hstk, here the second term on the information 1/ at time ltk-, Clearly, if p increment given the un known and have this term is zero, we Iight-hand
W
/'Es
E
tk
I.s ?c'
hst .
,
.
.
.
,
a.sk-ll sz -
1
.;)
-
,
=
,
nte General Case decomposition of an epward-trending submartingale into a deterne ne istic trcnd and a martingale component was aonefor a process obsewed at a jinite uumber of points during a continuous interval. Can a similar dewhen we wok with continuously observed O mposjtion be accomplished processes? ne Doob-Mcycr theorem provides the answer to this question. We state tlw thcorem without proof. Let VrJbe the family of information sets discussed above. THEOREM: If Xt, 0 s l s x is a right-continuous Numartingale with respect to the family j.1fj.,and if E z't-jq < x for all t. then Xt admits the decomposition
r
x7/
u
r
(
1,
which means that ) will be a martingale with respect to the info tion set generated by past price changes and with respect to this parti
t5l
probability distribution. will cease to be a martingale with respect to 1/2, the lf # k) However, the ccntered process Ztks defincd by
t&
.S.r,) + (1 2,)1 -;i!:',r ,: 1) -
k
J7 gksl+ (1 -
+
::::::
.
2#))
:
jh' '
' k,
(
f=l
; ;'.
it Zt,
willagain be a martingale 8.2 Dot-Meyer
=
z,,.16
. .3
(
1/2.
>
p
:
( !
:tE
which means, EPLSL,
Stb Ixsla, .
since lp Inlt
1Z,k-
>
1 according to
can be checked I
1.
lhat
ktk..ll -
(91).This
-
-
,
>
implies that
the expectat jo j of
.(z/ J, k
(
Sv-, AlJ
I'.
.'.
:
j'
cw
,
rtingale-
'::
S
will
on past .(Zrk J7
''
at expjration date F
4..
jy
'. '..'
.
(95) .p,
=
qstkJ is a su jy
conditional
..
-d2,
8.2.2 The Use of Doob Decomposition We fact tbat we can take a proccss tbat is not a martingale and convert t into oue may be quite useful in pricing finucial assets. In this sedion we consider a simple example. m Re assume again that time / c (0,F1 is continuous. The value of a call option c written on the underlying asset St will be given by the function
:
Thcn, as shown earlier,
Mt +
M: is a right-continuous martingale with respect to probability and vtj is an increasing process measurable with respect to z,. This theorem shows that even if continuously obsewed asset prices contairj occasional jumpsand trend upwards at the same time, then we can convert them into martingales by subtracting a process observed as of time 1. lf the original continuous-time does not display any jumps,but process i: continuous, tlAc.n tlne resulting martingale will also be continuous.
vb
so that we expect a genera 1 upward trend in observed trajectories:
=
Where
,?
Consider the case where tbe probability of an uptick at any time t somew hat greater than the probability of a downtick for a particular as >
XJ
:
Dectmvpositimz
1
'*''f'
(
Stk + ( 1 - 2#)(k + 1),
wit1arespect to
,
-
'
k5';:,sk Ufp
,
=
tdpast''
EP
141
E
.
'.'.
(..! ..
.(j. t).
kqq'.rK, (jj -
(96)
.
Tbi: term is often used for martingalcg in continuous p artjtjoo of a continuous-tme intewal.
(liKrete
r.
max
time. Here we are working with a
'
: :!
:
.
C H A PT ER
142
*
6
11
=
lslmaxlkr
K, 01 1T,1,
-
'
(
9
h..
.,4/
;:
Acall
.
.
C
' .
t ?
(
E
where thc expectation is taken with respect to the distribution ftmction governs the price movements. Given this forecast, one may be tempted to Pask if the fair market v Ct will cqual a properly discotmted valuc of E (max (.V- K, 01 i121. risk-free intcrest rate r. the cxample, For (constant) suppose we ese ' write to A', 01Ifr1, discountS'Imaxgs'r
=
-
give the fair market value Ct of the call option? is a martingale with depends on whether or not e-rt The an-er is, have the It, pair P. lf it we spect to
wouldthis equation
Fplc-rrcwltq
or,after
e-rct,
=
/
results thus far to define a new martingale Mt,. be any random variable adapted to lt Let z, bc any mar-tingale wxtlt rcspect to It and to some probability measure P. the processdelined by
Let Ht j
EP
j-r(T-8
G'rj
cwj
Cf
=
(E; .'
'fhen
&
Mtk
a
(1
(1 I.
-rts r
will bc a submartingale. But, according to Doob-Meyer decomposition, wc can decompose c -r/
k
j
/
.f
(1,. .
e .-rts t
aa
t +
zt
is a,n increasing lt measura ble random variablc, where tingale with respect to tbe infonnation 1t. .,zl,
(1
,
and Zt is a
y. y) u
Fyf
1
j
i-,
f=1
jgy .gy ,
f-1
yj
.
t
ulnside
,
..
.
j
s
gz z j -
/j
.
Eto
((
i
j
jj
..
j
=
gatpj .$J,., =
'i
lawe
uq..,
..
r.
t
s
.
;
:
(1.:7)
.
I@'E)ti..
..6 :
1
remirjd ttje reader tlaa tus means, gs'en the information in wgj 4x, kaowa exac-tly.
qe member ttjat
s js v-. y.j)
.s
J(,
za
j.j.
jjgy
(),
5). thus has tbe martugale property. ,
:
jjtjtg
ju zj are unpredictable as of time lj-2..1..9 A1so,,,)Hcj is pted Tltis means we can move the E 2f-, I-j operator
-ada
yjjjs jm jus P
.j.
4: ;
sa
)....
.
..
k
.Htj
(.
''
'.
' .
x o j&j 4
sut jjwyemosts
I
;
.
to obtain
rjahen,
,
k
:;
tl
(105)
'rl,e
dfconstantf'
Fj
'5
That is,
)
,
i-'
(1
sf.
Zq
-adapted
b
>
-
j
.
jyj
(zt
with resped to 1t. The idea behind tlus representation is not difticult to describe, z, is martingale and lms unpredictable increments, fact that Ht is incremenf-tl in means Ih given h.L are h-j Zq will be uncorrelate with Ht as well. Using thesc observations, we caa calculate
;
gc-rlT-dlyz,
Hli.L
wul also be a martingale
'g
Then e -& C/ will be a mmingale. Bet can we expect e-r'st to be a martingale under the true probability As discussed in Chapter 2, under the assumption that irwestlzhrs risk-averse, for a typical risky scctlri'ty we have '1 EP
A/lrl+
=
f=1
'
j
,
;
,
(
:r)
,18
,
E
by e-rt,
mult p ly ing both sides of the equation
i
<
Flrst Stochastic Integral
wecan use the
-.
oj I /,!.
'rhe
9
-
e -rtz'-llxrjmaxgqs. K,
Tjw First Ssochastic Integral
can be obtained exyicitly, we can usc tlae decomnoIf the f'unction along with (101)to obtain the fair markct value of a sjtion in (104) 1. time option at However, this method of asset pricing is rarely pursucd in practicc. It is and signilicantly easier to convert asset prices into martinmore convcnient by subtracting their dzift, but instead by changing tlie undcrlying #ales, not distribution r. Probability
'
:
e-'lcy.lfjl
) ' :
Martingales and Martingale Represenmti''
According to thfs, if the underlying asset pricc is abovc the strike p K the option will be worth as much as this spread. If the underlying value, price is bclow K, the option has zero < /, T, the carlier time t At an exact value of Cw is unknown. But calculate a forccast of it using the irtformation h available at time can
.'
..'.
z
,
that
rhe vaquc of
7
c H A PT
144
ER
ts Martingales and Maningale Representati
*
'
.
It turns out that Mt deqned this way is tlle hrst example of a st question is whether we can obta.in a similar result w ticinteval. sup/gff f!.-: 1goes to zcro. Using some analor, can wc obtain an exp sion such as l
Mt
Mv +
=
Hvdzu
0
.
:
..
,
.t
0
=
to
<
,
.
,
ti
<
.
.
.
,
fs
f .. 1
We obtain
gsj syj-,j + g5',,stk.., - () +
-
.
j- )
a
,(j
. : . :
s
jfl
+
pt(j
ysjg
jtzrjjsy
+
y
j=j
j
,
sjj
-
+
p jyyoj yyjj -
(110)
.-j- /;,j.s,j,
mayjsj
wlwrc the right-hand side is the wealth of the decision maker J/er time ti trading. A aoselook at the expression (110)indicates that the left-hand side has exactly tbe same setup as the stochastic integral discussed in the previous Kdion. Indeed, the av and ptj are J'zjo, and they are multiplicd securities prices. by increments in Hcncc, stochastic integrals are natural models Ior formulating intertemporal budget constraints of investors.
'
:
hq
-adapted,
'
:
: :'.
T
'
10 Martingale Methods and Pticing
f Shares of riskless and risky and ptf-j bc the number Let at fright before held by the investor t im e 1. trading begins. Cle will variablcs be 1 at p and ptoare the nmrand these random 2i and the denote jnjtial holdings. Let #jf prlces of the riskless and :: h E at time ti. sccurities we now consider trading stateges that are seLtzfinancing. y suppose where time ti westments are Iinanced solcly from tlw s trategies are ceedsof time fd-l holdings. That isa they satisfy
Doob-Meyer
decomposition is a Martingale Representation Theorem. at the outset seem fairly innocuous. Given any they say that we can decompose it into two components. G, Ofle is a trend given the information at time f the other is a martingale with respcct to the same information set and the probability P. Mus statement is cquivalent, under some technical conditions, to the
'
curities
=
=
.
f
(109)
,
jrj
integrals stochastic
=
.
s sj s:,
.
9.1 Applicuferz to Finuncc: Trnm'ng fltzin,s
we
Pricing
-%,.,
'
have interesting applications in linancial theory. in this scction. f these applications is discussed consider a decision maker who invests in both a riskless and a security at trading times f!
,
T
(1.
,
Methocts and saxingale
We can now substitute recuzsively for thc lefl-hand sidc using Eq. and using the dennitions for tf-.! j.-a,
(
where dZu rcpresents an ivnitesimal stochastic increment witb zero m giventhe information at time !9. The question that we will investigate in the ne xt few chapters is whe such an integral can be delined meaningfully. For example, can Kheme appromation Riematm-stieltjes bc used to delinc tlne sth integral in (108)?
o
i :.,
'
'rhe
jc
nese typcs Jemartingale
;
-adapted.o
.
of results
dzknown''
,
'
representation
'
tzf.-LBt;+ #li-:st
=
a, i s,, +
pbsti,
(1
.
=
,
.
.
.
'
.
11
,
westor
,
( j J j)
.)
,
'EE J
E
J
;
at t ime t the
jdMs
ere the o is known given the information set 1s, the g(.) is a nonanfunction of C and Ms is martingale give' the inlbrmation a sets n probaulit; fzx ) and the p zl show' In this secon, that this theorem is an abstract venion of some we very important market practices and that it suggests a general methodology for martiugate methods n Nnancial modeling.
.
knows his holdings of riskless and risky scctirities.
gjf t
udpative jjjj
'
,
'C
y.
Dgds +
,
Wj),
:
.
:.
,
z
+
=
i 1 2, n, According to this strategy, the investor can sc us holdings at tim With alll for an amount equal to the lefl-hand side of t hc equation, and lmits d securhits. ln of risklcss an risky these proceeds purchase ati pt, Sense his investrllcnt today is completely linanced by his investmen t Z . CViOXS Pcriod. Pf
wherc
C'w
5
alAs we will see Iater, thc nonanticipative nature v are uucorxjaterj.
:i
aud
' ..(
.
(. .:
'
:
'' .
7r:.
.
>
of the function
#(-) implies hat g(c7,)
7
;:' :.'
.$
.
'
.' )
(2 H A P T E R
146
*
N'Iarcingalesand Martingale Representati
6
..
E'
t' !
First, some motivation for what is described below, Suppose we would likc to price a derivative seculity whosee pri denoted by C;. At expiration, its payoff is Cz. We have seefl in Chaptev , that a properly normalized G can be combined with a martingale me% . .!%to yield the pricing equation:
jhtcing Methodology
l1 y
:
.
discretc equivalent of the martingale by the following equationl
rjajw
Cr
n
.
cT
(11.
.
cr +
=
ut
wherc
.L-jJy
;
t f;
$'
(1
()
=
.
/
,
t
=
<
.
.
,
<
/,j
z:
=
(117) the
p
g B.
d
,w
x
=
0
(1
,
jj
.
.
.
sedg:
j A
'Ihe Erst step in such an endeavor
.
'
can be obtained for a derivative security's price Ct. and explain the notion' we look at the implications of this representation 4 a self-snancing portfolio.
c ti
''.
'
r:
C1
.
=
the function G(.) is known and the Sv is the underlying assct at time F.
price (unknown)
s ti +. p
u$
?;
(jjyj
,
dsweights''
,
taj
.
x
11.2 Time
' , .
)
E
.j
'
:
We now conside changes in trivia tjy.
q:
'
.
'
,
.,
.
:
(1
G(,V),
lf
.
(
:.
a
Sure
.t'
,
a
,
the a t,. p Ij are the of the replicating portfolio that cnthat its value matches the C Note that wc kmow the terms on thc right-halad sue, given tlle informat'ion at tne fj. Hence, the pt, ) are kmnlicati%le. We can DOW apply the marthgale representation theorem tls j ng this hedgej j o yjx xsjratjsg poujojjo. Were
j.
-
,
,
ion
> 0 feprcscnt a mall, linite We proceed in disclvte time by letting ZtelYals teWal and WC Sllbdivide as in the Period (l,F1 into n such tf a deriva pricc revious Section. current represent and St the The Cf P sec urity and the Price of the underlying asset, rcspectively. The Ct s unknown of the problem be low. Thc F is the expiration date. At expira the derivative will havc a market value equal to its payoff,
ahedge''
.
.
A Priclng Methodology
is to constnzct a synthetic
for the securify Ct. We do this by using the standard approach utilized in Chapter 2. Let Bt be the risk-free borrowing and lending at the short-rate r, assumed to be constarit Let the Sti be the price of the undcrlying security observed at tille ti. Thus, the pair fBt Sti is known at time ti. wl now, suppose we select the at. pti as in the prcvious section, to fonn a t'eplicating portfolio!
'
W here
Mb
-
'x
v..-.
of the normalized Ct, i.e., of the ratio Ct/Bt. ' This suggests a way of obtaining the pricing Eq. (112),Given a deriva security Ct, if we can write a martingale representation for it, we can try to lind a normalization that can satisfy tbe conditions in (113)and (1 under the risk-neutral mcasure # We can use this procedure as a gen ! way of pricing derivative securities. ' . In the next section we do exactly that. First we show how a ma
11
Mt,-'
=
be of any use in detennining How could this representation arbitcage-free Pfi Of the derivative sec-urity Ct'l
..
,
T
(116)
f,
f=j
that
and n is stlcb
wherc tbe b is the trend
repegentat
#tcij lNj )--q
+.
'
W
l
t
ot
j=)
Xv'rf
.
Et/
FI
means
;'
It turns out that Shis equation can be obtained from (111) Note 'that Eq. (112),it is as if we arc applying the conditional expectation omra EP of Eq. (111)after normalizing the Ct by Bt, and l (.1to bot.h sidcs letting ;J3
(111)is then
'
> -.1 Et Bv #, =
in
gwen
(p
'
C
representation
G ; during the period (:, 7-j. We can
write
..
i
of .
n
cv
: 'F
' :
rr .
,. 't
.
=
c
/
+
)'')hct J=o
i
,
(j19)
y'
:.
; :.
:
. :
148
IaT E R
cHh
becauscLCt,
=
n
c7,+
=1
Pro duct
1-1lle'''
Bq 1+ gt,,.
f=1
'
E
'rlms
n
.%1
f=1
+ j=(
.?
cz
.;.
(1
and third terms on thc right-hand
n
(trjlj)
q
i==l
(S)
1'1
(At:'&)
k& Emfkh-/fl f=tJ =
Y
f,xu.tl
+
.:
('%)
t'/f
.
f=1
(-4
kr.s,.1 gt,,+.2$..
a Atz,i
=
fe2s+, -
./,j
ati
=u
way of obtaining
/.?I.+,
-
Bt
,;
= 5 1,.- 1.
-
Sti
.
', 9
'$ .
..:
.
equations beltw is b )? simple algebra. Given
j,;#q1 -
and subtract a,(s,(+,
a dd
a:.Bz;.t
- a.i
.
t
%
.
-
cr,,)
=
=
(cr;... -
) (.::a/f
B%+ +
srj., +
ati
(B,, -,
-
S6) :
.
..:
'
j ,
,j
'
.
.j
ic
' .
..
L 1. .
l
yjye jxacketed terms i.n (126)will not, in eneral, vanis.h under such an operation But at this point there are t'Wo tools available to us. <. rirst, we can divide tlze (G. B(, 5'j) in (126)by another arbitrage-free price, aatj wrjte tlae martingale representation not for the actual prices. but instead for normalized prices. sucha normalization, if done judiciously.may .
.'E
:
(.#Jj)
ati
t-$'qlj
.
w,,c E gc j
:
.
...
a t ; i' 1 1,,+, a%B(,
.
,
side, factor ou t sOiI:T
jlgtjj .
>
E
,
on the rip-hand
q%;,
p'ven the information set it involves the price changes hz,, as,, that occur aper ,j. and may contain new information not containcd in h However, although own, thcse pricc changes are, in general, predictable. Thus we cannot expect he second term to play the role of dMt in the martingalc representation leorem. The second bracketed term will, in general, have a nonzero dri and will fail to be martingale. a Accordingly, at this point we cannot expect to apply an expectation operator E /p (.j where P is real-life probability, to Eq. (126)and hope to end up with something like
..
1he
jj
(Aa,f)
tlnknown
&,because hetlcc
i ;,
-
and k'a/, '
The second bracketed term will be
i.
#p.! #&
=
gatj
hnttj + at,
(111).
j
'
.
and obtain:
,
: t'
' ..
(#,.1
-
gtari)
)+j+
'
.
-
(lksjj ) .
k%;
' fs
used the notation, .1
(12j)
cvq. i-
)
'rhe
-
'
r
LBtj
'rhc
,
n
ati /-t)
Now consider the terms on the right-hand side of this expression. Ct is tbe un known oj tjw problem, We are. in fact, looking for a method to dctermine an arbitragc-free value for this term that satisses the pricing Eq. (112). two othcr terms in the brackets need to be discussed i:l tltlttil. Consider the first bracketed term. Given thc information set at time Ij-yja every clement of this bracket will be known, The Bt are prices obkpri is the rebalancing of thc rcplicating scwed in thc markets, and the iatt, PO rtfolio as described by the financial analyst. Hence, the first bracketed term has some similarities to the Dt term t-le martingale representation
(1
and
Can
i=
rc:ct,
k.'
We
Ct + +.
-
.4;
r,
(tr,) .Bfo:+ f==tl
=
j==(I
lhat
=
(..
)
of (121)22
n
r;
n
N
.
'
+ (aaj)sb-vj
n
-
:r
du.v + u-pv.
tzcq
j Bt,.j +
Regrouping,
.
,
:';
l
Applying tbis to the second
note
f-tl
r;
..
(1
.
G+
...
.
#(u,.t))
z'Another
=
.
lchange''
=
wherewe
bc rewlitten as;
(121)CM
Cz
(1
Z/3
149
t .
the operation of taking first differences. in a product, &.v, can be calctzlated u
where the A represents Now, rccall that the L<
i/r Bq -1-pr-jil n
n
= Ct +
th e
A
11 h Pricing Methodology
q
port folio:
using tbe replicating
.
=
Martingales an d Martingate Representati
6
Gf Or,
Ct+, -
cr
.
'
7
.'
..
y' '.
1.50
c HA I
'r
ER
-
Martingales
6
and
Martingale Representati
'
q:
I
.
jj y pricing Methodology
!
'.
by the drift of ensurethat any drift in the Ct process is quite convenient given that indeed be variabic. nis may normalizing the f'uture payoff, Cv, anmay. discount want to ' may second, when we say that the second bracketed tcrm is in general p ' dictable, and lience, not a martingale, we say tltis with respect to real-worldprobability. We can invoke thc Girsanov theorem and swit othcr words, we could work with risk-neu probabilitydistributions, In probabilities,z3 we now show how these steps can be applied to Eq, (126). t i'compensated''
exp
'
Risk-Neutrul Pgobzbity
(-) '
=
c --(
#'
Bt
Notice immediately that tbe We w have
ill
t
/,
.
--(
=
Bt
B --1
=
Bt
'
is a constant and does not grow over
!
sut
z
=
t!-,+
n
gttzfjl
with (jae new restriction
''
imal
'
gyj
s,/
nus, applng
?.! ''
EIT
the operator
Etp
,'.,
Iwq -
=
g.lto r
2:
('r--+
e't
n
ajy
f
-
:
c
s
+.,,,
sf
st
we substitutea r
'
asecause
(136)
..
.. : :
(13,)
p
BtEt
Cv Bv
( 138)
,
(138)?We do
i ,
L(At''.) y
'.-.
+ (Ac/.)
,j
,.-.1 -
(1,q)
(). ,
f-c
*at is by making sure that tlze replicating portfolio isseltpnancing. In fact, the last equality will be obtained if we had )
.
v
+ ().
the azutrage-free value of the unknown Ct. So, how do we eliminate this last bracketed term in Eq. tu s j)y cjy j yjje oos ng j.j g:,) s,o oat
lst#, #,r#l, (j3 , = -.!; t, from chapterz, that or ssr/s,.z-lRemembe, market normalization,
discllssion :. t3Girsal1ov tlleortm Will b discused ill dethil in Chaptel 12 and 13. '' provides a mo tiva ion tbere is no Ito correction wrm lwrez s- is determiuistic and st enters linearly, .
jyyjj
&,.- (aa,,)&,,.-j gtaa,l
=
l
'l''hc
+
Clearly, if we can eliminate the bracketed term, we will get the desired
s-
here bitrage condition, and witli mosey dertite no-ar
(134)gives:
yoyujt
..
W
(1a5)
+
izzzf)
.
-
Eq.
,
(
during an infini
.z'
-
(134)
probability 15
p.
-,.-
j jkurl )
=
=
,
E=1
E(a,,) (aa,..)&..-j
Jrccc: ()
.y. yfp
6IB rBtdt, (1 t because the yield to instantaneous investment, #/, is the risk-free rate ( We now use this in'. , dBt dSt S ds St (1 : dt =
J'-lgal. (-$.)j
that under the risk-neutral
:
,
ider
+
'
ftr a1l ti. 0 (1 t. = by Bt has clearly eliminated the trend in this v aykajj.#' -,
(133)
*
jj
(a,,) #/;,
+
,,+j
f=l
.LJ
The normalization tlwre is morc. next the expected changc in normalized Cons sntewalu. we can write in continuous time,
0,
=
ut
,
(1
1.
=
(r#,(/f -
where thc / is the risk-neutral probability, obtained from state-prices as also has zero mean under #. discusscd in Chapter 2, Hence normalizcd tbe discrete timc equivalent of this logic to eliminate now use can we wqiting bracketed in unwanted W e by terms start the (126),
'
Ktnorma
(132)
trdt
-
Nt =
E
In order to irzlplement the steps discussed above, we first conveni every asset by an appropr iately choscn price, ln this casc, a value of and defme corrcsponding Bt normalization is to divide by tbe
ds l,j,
E:
-
.rat)
'
11.3 Nonntzliwt';en
.
ected return from St wil1be the risk-free rettzrn r:
s,>j-j
:
(tnd
151
. l
..;
))
a,;+,a,f., +
.
.LLt . '..
pt,.wsbw, a,;s,,+, -
+
#,f.s,..
,
(14n)
7
E
152
C H A PT ER
.
6
:
.
Martingales and Martingale Representati
..'
.
Let Sf be the price of an assct obsen'ed by a trader at time f, During jfjjijaitcsimal periods, the tradcr rcceives new unpredictable information on These are denoted by l
is, the time ti4 1 value of the portfolio choscn at time tt for all f. suficient to readjust the weights of the portfolio, Note that this exactly writtcn lbr the nonnormalized prices. This can be done beca is equation :. normalization the we used, it will cancel out from both sides. whatever 'rhat
153
yj gcfrences
E .,..
'
'
k..z
*'
cjst
.
tyst
a(o
,
.?
11 4
A
szzmmury
whcre h is volatility and JH( is an increment of Brownian motion. Note tjot volatility has a time subscript, and consequentiy changes over time. Also no te that dst has no predictablc drift component. longer period, such unpredictable inforrnation will accumulate. overaintenral F, the asset price becomcs Aftt:r ajl
'
E
we can now summarize the calculations from the point of view of pricing. First thc tools. ne calculations in the previous section dcpend basi martingale reprcsentation tll on three important tools. The first is the it into decomposc kn that, given This a a proccss, we can says rem. trend and a mart ingale. This rcsult, although teclmical in appearanma'' in fact quite intuitive. Given any time series, one can in principle sep . it into a trend and deviations around this trend. Markct participants , work with real world data and who estimate such trend components tincly are, in fact, using a crude form of martingalc representation theo The second tool that we used was the normalization. Martingale resentationtheorem is applied to the normalized price, instead of the served price. 'This conveniently eliminates some unwanted terms in the tingale represcntation theorem. The third tool was the measure change. By ca 1cu la ting cxpectations J ing the risk-neutral probability, wc made sure that the rcmaining unwan terms in the martingale reprcscntation vanished. ln fact, ut ilizatitm of Of risk-neutral measure had thc effect of changing thc opected IrcaJ St process, and the normalization made sure that this new trend was e inated by the growth in Bt. As a result t'f a1l this, the normalked Ct en up having no trend at al1 and became a martingale. This Sves tlze PH .1 . Eq. (126),if onc uses self-hnancing replicating portfolios. t, '
d+.z'
St-vv
#Bi.
o
(108).lf evely incremental ncws is unthen thc of incremental news should also be unprcdictable sum Pre (aS ( j tjme ,). But this mcans that St should be a martingale, and wc must '1h
js equation has the same form as (jjctable,
jave
.
/ iT
s
fr
t
u
dw u
=
().
This is an important propcrty of stochastic integrals. But it is also restrica tion imposed on linancial market participants by the way information Ilows in mazkets. Mattingale methods arc ccntral in discussing such equalities. They aAe also essential for practitioners.
,
.
'
St + ;
.
.:
v
=
,
.
.'
13
?
qq
A reader willing to learn more about maningale arithmetic should conthe introductory book by Williams (1991),The book is very readable aad Provides details the mechanics of al1 major martingale results using on Simple modcls. Revuz and Yor ( l 994) is an exccllcnt advanced text on mar
;.
', 12 Conclusions ( This chapte dealt with martingalc tools. Mmingales were introduced ' processes with no recognizable time trends. Wc discussed severa 1 exam that will bc uscful in later chapters. ) This chapter also introduced ways of obtaining martingales from Pl ' cesses that have positivc (or negative) time trcnds. We close this chapter with a discussitm that illustrates why theore E concep ts introduced hcre are relevant to a practitioner,
Slllt
:
.'
:
@ (.
'
ij. 1!' :
.,( ,
( ..
..
References
.
).. ;:
?
..
l54
(2 H A P T E R
*
Martingale Representatio
and
Martingales
6
'
14 Exercises
3. Lct F; be a Wiener process and martingales? Stochastic processes
:
1, Let F be a random variable with
.
,
<
x.
('b) Xt
'(
=
2))) + t
=
lFl
E IF
=
'J
If,1
%
=
A
Tlz - 2
...u,'
.
r
Mot
=
+
'
j
'
Where
X
Bi
* )7 f=l
Bi ,
(b) Let
=
.
.
.
,
'';.
=
gdt +
(y.gp).
(b) Mv(X7.4 (c) MvlXvt
'.
FIX4
26..:%
If/1,
r&1.
'i
=
=
=
I'V
Izliz cp'z
.
5 Given the representation: '
'
L:
T MI'lxtj
:
'J '!
P) = Bi +
uylxvj
'
Where
( k
kn -'lP; =
.
Is K a mmingale? d) Can ( you convert Jr into a martingale by an appropriate transfor- $ : mation? 2 ides (e) Can you convert P; into a martingale by changing the probabil assodated wit.h a coin toss? '
;
'
'? j J
. j.
j
Mozkj
#(f, X/IJI#;,
+
Can you determine the g(.) if the Mvxkt European call option at expiration? nat is, f Mvxvj is given by:
:
N
=
o
:
4, a martlgale?
i uu; 1
g(.) in the above reprcsentation for the case where M(.) is
(a) MwtA-zl
'
Tail
and
=
flt. ne
given by',
yt
Head
F(A-4 I /21, Zi
Is zj, i 1, (c) Now defme!
the oeterminc
E 'r.
of the toss of a fair coin:
+l -1
dxt
'Cip'
;
(a) Calculate the .E1.A'4I
PrOceSS
.'
4 and consider X4. .f11,
of information sets Xt is known to follow thc SDE'.
the equa jkty jaous gjven the sequence
underlying
lt
n
=
where each Bi is obtained as a result =
'
'
#(f, Ar/)dW$,
o
'l.
Consider the random variable!
=
()
Mvxt)
.
'.
We let n
S'I/P;JJ
4. You are given the representation..
EL
(b) Does this mean tlmt every conditional expectation is a mart ingal4, .It #iventhe increasing sequcnce of information sets (Jp 2,
denote the time. Arc the following
)1.
is a mart ingale.
6+1
t
o
XI
(c)
f
2
'
(a) Show that thc Mt defined by Mt
xt
(a)
'
f(F1
155
lj Exercises
. : '. .
0
<
K
<
=
mx
is the payoff of a plain vanilla
gz -
K, 01
,
x is the strike price. Where is the dflsculty?
'.
. .' jjr.
==
.
y
'
,..
-
-
.
1f erentlatlon toc astlc nvlron #
.
*
@'
@
.
gj
y
'
j
.
y.
j Moti:
yatj
soximated reascmably well t)ya deterministic model. But, in the case of app dcrivative assets, the randomness of the underlying instnlment is espr icing eptial. Aftcr all, it is the desire to eliminate or take risk that ieads to the s of derivative assets. In deterministic environments, where everyexistence bc fully predicted, there will be no risk. Consequently, there will thjrjg can need for loancial derivative products. sut if raadomness is essential, t,e no would one deline differentiation in a stochastic environment? how one simply attaa random error terms to ordinarydifferential equacan derivatives? Or are there new diffiand use thcm in pricing snancial tions stochastic d@erential dctining equationv (SDE) as well? culties in differentiation in stochastic environments using the chapter treats nis stochastic differential equations as the underlying model. We tirst construct the sDE from scratch, and then show the difficulties of importing the difl ferentiation formulas directly from deterministic calculus. Morc precisely, we first show under what conditions the behavior of continuous-time processa St, can be approxlmated using the dynamic,s a described by the stochastic dlyerential equation
.
zE j i
113.
'
ents
.
' '
b'
' ) ',
.i
; tj ; :
.,
.
=
ax
ax
-
,;.
'
ds t
.;;
Differentiation in deterministic environments was rcviewed in Chapter lnformau u ne derivative of a function flx) with respect to x gave us . about the ratc at which f (.) would respond to a small change l x, deno bydx. TMs response was calculated as
d/.
tI:
E
1 lntroduction
.
(0n
ast,
t)
at +. yst,
tj /p;,
(z)
ljrt is an innovation term representing unpredictable events that during tlle inhnitesimal intewal #f. ne att, tj and the bst, tj are occur the dnp and the dffusion coeflicients, respectively. They are f/-adapted. we study the properties of the innovation term #H(, which drives second, tlle system and is the source of the underlying randomness. We show that is a vcn, irregular process and that its derivative does not cxist in thc Knse of deterministic calculus. Hence, increments such as d.Sl or #W( have to be justihed by some other means. Constructing the SDE from scratch has a side beneht, nis is one way we On get familiar with methods of continuous-time stochastic calculus. lt may Prvide a bridge between discrete-time and continuous-time calculations, and several misconceptions may be eliminated this way. W jwre
:
y
(
'
=
-
$ where fx is tbe derivative of f (x) with respcct to x. We need similar concepts in stochastic cnvironments as well. For how wo ple, givcn the variations in tbe price of an underlying asset the price of, say, a call option writlen on St read? ln dcterministic en ments one would use '' standard ru les of differentiation to investigate s questions. But in pricing Enancial asscts we deal with stochaatic variabl arid the notion of risk plays a central ro le. Can similar formulas be when the underlying var iables are continuous-time stochastic processes? ne notion of differentiation is closely lirlked to models of ordinary (ODE), where the effect of a change in a variablo ferent ial s anot her set of variables can be modeled explicitly. In fact, (vector,utu determi ential equations are formal ways of modeling the dynamics o f this. : processesa an d the existence of the derivative is necessaly for doing of Can diferentia l e quations be used in modcling the dynamicg Erst difficulty in doing this is becausc of the rando prices as well? red in a metal rod may ness of asset pliceg. The way heat i: tran sfer '
.%,
'
''
.
:
'lzflrfozz.
.,:jv
-
!k
:
'rhe
..
.
)E
a u o tjvation
'
t
'lhis
section gives a heuristic comparison of differentiation in dctcrministic and s toc1)ast j c env j ronmen ts lt f be the price of a security, and let F(,%, tj denote the price of a derivative instrument written on St. A stockbroker will be interested in knowiug ds tl:e instantss next ;, incremental change in the security price. On the other hand needs dF the incremental change in the derivatives desk a
s:,
F
y. )!:
.
.
,
:
S.
.
!( ; ,.
11
j: ;
j! rI
.?
; '
.
58
C H A PT ER
.
Differentiation in Stochastic Environmen
7
.
1 59
2 Mtxivation
T J
::
expansion will yield
price of the derivative instrument wlitten on St. How can one calculate dF l departing from some estimate of dst . change ) What is of interest here is not how the underlying instrumznt but, instead, how the fmancial dcrivative responda to change in the price the underlying asset. ln other words, a zuchain rulc needs to be utilized. the rules of standard calculus are applicable, a market participant can u : the formula JF dFt dst, (3 DS J or, in the pmial dcrivative notatien, '
i
fxj
=
''
,,
.':
=
(4E
But are the rules of deterministic calculus really applicable? Can this cha i. rule be uscd in stochastic environments as well? Below we show that the rules of differentiation arc indeed different stochasticensironments. We proceed with the discussiop by utilizing a ftm of x. As discussed in chapter3, standard differentiation is the limiting opef;
.x012
(6)
-
x()j: + Rlx, x(j),
-
(7')
,
.:
Fs #.%.
1 2
.-xxtxljltx
where A(.f, xu) represents all the remaining tenns of the Taylor series expansion. Note that this remainder is made of three types of terms'. partial derjvatives of fx) of order higher than 3, fadorials of order higher than :$ and powers of @ xn) higher than 3. Now s'witch to a Taylor series approximation and consider the terms on the rght-hand side other ,.1)a.11Rx, ak). The fxj can be rewr itten as fxz + Ax), if we let
,
=
IL 31 .-yxxxtxolgx
+
''
dFt
xlj1 f (&)+ Xtxtllgar - +
'
'
x
'
x
=
Tllcn the Taylor series approlmation
,
will have te
1
.f(x)
tion
,
ationdenned as
wherethe
f(x + hj lim h --,0
fxj
=
'.
',
.:
J'
:
x'.
@
+ k)
-
fx)
.
,
discussionof this important issue.
by k, we obmin a ratio. nis ratie tells us !he derivative is mJe of change. a zF'or he sake of notational simplicity, we omit tbe time subscript on x.
lBy aividing J(.x+ changcs per Hence,
) - fxj
interested rcader
'-rl:e expausions.
is reforred
back to
chapter3
for a review
: :
of
'raylor
sez'
'-fxxtxs z +
+
2 representation.
1 '-fxxxLx) a 3!
(9)
.
,
,
z > t). A-E'-rl
,
uowmuch
[email protected]
.
T(x0)2 fxx)
,
.
3
-
.r
.x
Suppose f (.x)is a function of a random proccss x. z N()w suppose want to expand fx) around a known value of x, say xz. A Taylor seri
';r)
term f x x, Consider the second term lafxxtA.Ylz. If the variable x were deterministic, One could have said that the (Ax)2 is small. This could have been term normegligible, yet small enough that its Xsttied by keeping the size of Square (.r)2 is negligible. In fact, if x was small, the squae of it would be even smaller and at Jopze point would become negligiblc. However, in the present case x is a random variable so changes in x will also be random. Suppose tbese changcs ave zero mean. nen a random variable is random, becauseit has a positive variance:
'
represcnts the change in thc function as x chang time, then the derivative is the rate at whic if represents Hence, x by interval.l ln this case, time is inhnitesimal changing during is /x) an ' tandard'' calculus. variable and s deterministic one can use << But what if the in fx) is a random variable moving along a contin. ! axis? Can one deEne the derivative in a smilar fashion and ous time C nzles? standard The ansaver to this question is, in gcneral, no. We begin with a heuz's -'
fx
H ere,
+
fcrm4
asmall'' On the rigbt-hand side of this x represents a change in the random variable x. Note that although this change is considered to be smtll, we do not want it to be so small that it becomes negligible. After all, our purpose is to evaluate the efed of a change in x on the fxj, d this cannot be donc by considering negligible changes in x. Hence, in anpotential approximation of the right-hand side, we would like to keep the a
(,
X,
limit satislies
X<
fxb
(8)
xa.
-
. E' '
Rtn
,
.; '
''
c.
ZL
q
'!
''.g j
Ih the following, for notational
./x(a.a).
.
....
(10)
Rtlt read literally, this equality the average,'' the size of means that, (xxjzjs nonzero. In other words, as soon as x becomes a random variable, treating (ax)2 as if it were zero wiil be equivalent to equating its variance
.i
.E.
'
.
simplicity,
we ornit the arguments
tyf
/'zxt.rLjl,
fxqxz),
:
''
160
C!il A P T ER
Differentiation
7
.
in
Stochastic Environmen:
to zero. nis amtxnts to approximating the random variable x by a non. random quantity and will defcat our purpose. After all, we are trying tof tiod the effect of a random change in on '? Hence, as lcmg as x is random, the right-hand side of the Taylor series approximation must keep the second-order term. On the other hand, note that while kceping the first- and second-order.' terms in S-v on the right-hand sidc is rcquired, one can still make a reasoaable argument to drop the term that contains the third- and higher-ordey powers of z'-r. This would not cause any inconsistencjr if hicher-order mo- 1 ? ments are negligible. 5 Taylor-style approximation result, candidatc writbe for a As a l can one ten as
Once x becomes deterministic, we can assumc that for small ;r, and use
.
'
.f(x).
f (.t()+
.x
ffxa'J
2
.
(11) L.
211d
idaveragc''
7,
+
.Yl
7 fxLx +
1 -afxxfllirlzl
f (.rt)+
.f(ab)
,a?l -
J
1 fxhx+ -/xx(x*1, .
wherc x* is the mean square limit of (A.r)2.
2
;
z
(13).. q
j :
0
..'.
marreaders may remembez the dscussion nvolwng vvn.atms o txmtouous-time showetl that for continuous square ntegrable martimgltlesy Ei: in There, (-.hapter 6. wc tingales mcaningful ran- '2 thc irst variation was ntinite and the quadratic variation converged to a ''' vanished. Hcnce. if tlle x is a continuoU varations a1I whte higher-order variuble, the dom E equal lx, mme set martingale, n in higher-order to zero the can terms squarcinlegrable
ysome
Gltemember
X. to '
jx.
.x
'.
... '
.
(Aa:)2
x-..t
i.x
,
(16)
=
f)dW?)-
&(S(f), ttdt + bst''
(17)
-
/t)
<
/1
<
---
<
tk
<
---
<
tn
-
r
(18)
that in probabilistic convergencc wc arc ntcrestcd in finding a random variable tslargc', variables X converges. For a the a sequcnce oz a' mily of random approl often' variable mation random used since X* then be Xn, tbe 'trnitin for S can as an 1:.... ''tting variable would be eusier to landle than Xn itself.
'
approximateserise.
2
jkm
7:*
'rhe
''
ES .
A
Iconstructed''
!
'
.1y
.y.
ln ordcr to understand the way differentiation can proceed in stochastic environments, the SDE will bc from scratch. Thc construction will procced from discrete time to continuous time. wewill consider a time interval t e (0,r1. x axis, (0,Fj, is partitioned into n intervals of c Ons ider y'igure 1. equal leugth h. ln terms of the notation used in previous chapters, wc Cfmsider ntewals given by the partitions
(12)) '
()r
ytak+ ax) fxoj u y kx
d-$(d)
:-
(Af))
-
x.
.
''
fx
(15)
%
The concept of differentiation dcals with incremental changes in infinitesimalintervals, In applications to tinancial markets, changes in asset prices over ineremental tlme period8 are of interest, In additon, these changes are Z'WVW'CII to be random. nus, in stochastic calculus, the concept of drivative jaas to use some typc of probabilistic convergencc,t' The natural framework to utilizc for discussing differentiation is the stochastic differential equation (SDEI:
.
.
txoj u' p
3 X Framework for Discussing Differentiation
,
is random, we can write
,
dehne a derivative. This is discussed next.
,'
.'
.x
x' -
'
-
jjm ax--.
.
,
If
(14)
fxhx.
.r
:'i
where the (l.x)2s replaced with its expedation, nis is equivalent to revalue as a method of ap Placing the term zfxxixjl wit.h its proximation.In the second part of this book, we introduce tools that take exactlythis direction. . A second possibility is to usc: instead of Flt/.rj, some appropriate limitof the random varablc (x)2 as the time inten'al under consideration ( 4, It turn! goes to zero. Such approximations were discussed in Chapterresult would in the under conditions, these procedurcs that two somc out In fact, if represents the timc pcriod during which th , exprcssion. same changc is obsetved. and if h is ( sma 11 undcr some conditions o'lh may.r:: bc close enough to (.v)2 in tlie mean squarc sense. nus, we have /51* possible approximating equations. depending on whetherx is random or not. .x
7
w.r But with stochastic Ax, it is not clcar whether we can ignore the third term, --+ 0 in (9), 1et
L
t fxLx + -/=S((,'x)2j,
.f(x0) -
+
a x
,
-rl
-
is negligible
One result of all this is tbe way diferentiation and stochastic envircmments. For example, in the case of Eq. (14),we can tly to divide both sides by obtain the approximation x and
-
f (xp+
f(xl
.:r)
(x)2
is handled in deterministic
:.
j.
161
Framewcrk for Discussing Dikrkntiation J A
(
'
.
R . ;
wlich
:
.1
CHAPT ER
Differentiation
.
in
Stoclstic
L'
.
Environm
J
, ':
:)
)
SJ
-I(.1 zepresents the expectation conditional on inthe symbol end of intewal k available 1. ne kl,f is the part in the at formation that is totally unpredictable given- the information available at &-.:1 k trx right-hand side repthe k llth interval. The Erst of term on the cnd the ( the duzing the interval. Stj kth price asset in actual change f esents that a market participant would have predicted second term is thc change 1k-L.7 We call unpredictable components of new information set givcn the tjt)yl informa.tk0n Nutc the following properties of the irmovation terms.
jpre,
:
.
,' ' j
&n
E
-
'
rrhe
..
; ' .;'
'k ,
utp'l is unknown at the cnd of the interval (k 1). It is obsen'ed at end of interval k. In thc terminology of measure theory, /li is said to the bc mewurable with respcct to Iu. That is, given the set Ik , one can tell the 1zl' exact value of k&lli given the information set of Values of arc unpredictable, .
'
Srs
?
. '
.'.
./1
'
sre
,,
w
: ,.1
Ja
time k
One major difference in this chapter is that we have, for all k, fk-j
-
Eg.L
=
-
h
Whtre
Sk
Ek -
l
'l'he
' 7
and 'rhe Iatter represents inten/al h.
Slkhj
-
..
( .( .
)
t
during a
J
i)
.
(,% .V-11
Ek-j
E&-
.Sk-11.
-
u..;ju jj,j .y.
gal;fh+.
.
.
.
+
,
.
,
jgtjj
.y. uju j
1/-1
j + Ek..j grqj
=
latter is true because Ft.-1(,l'P't.1 equals zero and the 1 are known gjven Jk-j.
(29)
;,q-j, i
k'rx,
=
1,
.
.
.
,
'' j::
''news''
4
''live''
',:
fashfon;
=
r::a
at js t he j mportance o ran dom valiables such as l.fk? Consider a Nnandal market participant. For this decisitm maker, the imNrtant information contained in asset prices is indeed I#k. nese unP'edictable occ'ur continuously and can be obscrved on all major networks such as Reuters or Bloomberg. Hence, movements
a
.
A';'4
(2J)
W'II
'lE
Now pick a particular inten'al k. As Iong as the corresponding exp tions exist, we can c/wcy,s define a random variable H$ in the follo -
!
j.
.
-
k
:(
1)).
- Slk the change in the security price St Lvk
=
jjgk
=
(
(26)
+ ll4.'v
assume that the initial point W' is zero. can we show that 111 is a martingale:
E
vkh)
.
Wtz
We deline the following quantities observed during these hnite inte =
,
j-.l
r '.'
(
.
.
)--)ap).,
=
'
F
=
(25)
.
k
$7
we have the relation /1
for al1
,
11,r,g+
k wz
(.
kh.
=
0
=
'
;
tk nus,
Hi j
AW/;C
' j;
whichmeans that
g
represents changes in a martingale process and is called a martingale d,yewnce. ne accumulatcd error process 1#i will bc given by
,
''
'
(
h,
=
1:
:t
*
t#
-
%L
.
F 1G U R E
.
.
i
p'rl
o
s.,A
uiykyll&jj
.
/
s3
163
Fmmework for Discussing Differentiation
?.
?
I
(
'U tbe information unjnfbrmative about the future movemtmts in set is completay slt), then us jxedictjou wilj l,e zero. undarthese cooditions, (,5'v I will itself be the unpre-
'
:y'
'
.5,.-,
' .
dicuejecomponont.
.
.. .
s: :
E. ,:..
t.
:
. t
-
C
.'j:'
.'
:( (
164
C H A PT ER
in asset prices will be dominated by AHi. This implies that to discuss diff entiation in stochastic environments, one needs to study the properties Alli. In particular, we intend to show that under some fairly cccptable //162 cannot bc conside sumptions, l,frq2 an d its intinitesimal equivalent ln Taylor-style approximations. as (fnegligible''
.1
:
7 Differentiation in Stochastic Environm
*
'
4 Tjw
,
:.,.)
tgg;
..'.
and thc variance of cumulative errors, Je'awill be positive. nat is, morc aad more frequent observations of securities prices will not eliminate all ('risk.'' Clearly, most fmancial market participants will accept such an the aisumption. Uncertainty of asset prices never vanishes even when one obes the markets during fmer and hner time intervals.
.''. '
of lncremental Errors
Rsize''
'
i ne innovation term kl4'zi represents an unpredictable change. (AHi)2 is square. In deterministic environments. the concept of diferentiation de , with terms such as Ah1$, and squared changes are considered as neglip-b Indeed, in determin%tic calculus, terms such as (Al#;;)2do not show up d ing the differentiation process.s On the other hand, in stochastic calcut'' the variation in the second-o one in general has to take into aount of thcsc terms. terms. This section deals with a formal approxlrnation There are two ways of doing this. One is the method uscd in c,o ;E ' on stochastic processes. The second is the onc discussed in Merton (190 we use Merton's approach becausc it permits a better understanding the economics bchind the assumptions that will be made along the Merton's approach is to study the characteristioq of the information ;ow . hnancial markets and to try tt model this information flow in some pr 1 way. , need notation, We first to define some variance of kIl'r: be denoted by l'$ : Let the (unconditional)
serv
ASSUMIVION
2: ;z <
wherc
.:
,,12
'
-
Je'
'
Pk
'oEAlvo1. ;
n
p.
=
t :,
Eo
)-Talfl
k
=
1
:
)-qu,
k
=
1
AsssNjmuox
i'
(3ly
with
:
!,
is inaepemaent of n.
here .,4,
W
8ney
1
arc ccnfined
>
0,
(3
F. ' .
K
L
to highcr-order
derivatives.
=
l,
.
.
.
,
a,
jnjependent
... '.. .! (
:.
aj.
(35)
of the asset price during thc most volatile
w zlg
j
(j <
..4,
<
j
j
(g6)
of a.
Centmted
.
V>
k
According to this ssumption, uncertainty of financial markcts is not conin some special periods. Whenever markets are open, therc cxists at Ieast some volatility. nis assumption rules out lotterylike uncertainty in nancial markets. Now we are ready to discuss a ver.y important property of ( Ui)2. e following propositio. is at the cemer of stochastic calcuus.
' ;
.'1
(g4)
x,
wn a x
'y;
1:
u
L'
. .'.
wherc thc property that fH'': are uncorrelated across k is used and t expcctationsof cross product tenns are set equal to zero. Wc now introduce some assumptions. following Merton (1990). ASSUMFHON
a:
maxgli. k
nat isa p'ruwxis the variance subiritewal. We now have
,;
#T =
=
mdX
' j..
The variance of cumulative errors is delined asl
.<
is indcpcndcnt of n.
,:
=
az
This assumpt jou jmposes an upper bound on the variance of cumulative errors and makes the volatiliiy bounded from above. As the time axis is chopped into smaller and smaller intenrals, more frequent trading is allowed. Such trad jjs does laot bring unbounded instability to the system. A large majority of market participants will agrec with tMs assumption as well. ajter ajj, aljowing for more frequent trading and having acccss to on-line sereens does not lead to inlinite volatility. For the third assumption, define
:
.
165
otcnalsvg
'
''.
4 The
lncremental Errors
Thjs assumption imposes a lower bound on the volatility of sccurity prices. when the period (0,F1 is dividcd into tincr and finer subjt says that
E:
'
(a.f
tsize''
(E
'. (' .
9.u
...
:.:
.
.. ..
ememtx,r
that the subintewals bave the samc lcngth
.
7
:' '
.j .! '
j
?
r.
l66
c H A P T ER
.
.'
Differcntiation in Stochastic Enviro
7
Under assumptions 1, 2, and 3, the variance of Aet.
preportional to
,
=
.E
(
.
pivide
y. .j :t:
''
(45)by n: (46)
(j
,
Implica:ion
orw
j
.. :
;
PRoeosrnoN:
F( A''Fk 12 a'lh k
'1.
.'
,
Then,
is a linite constant tbat does not depend on h. It may dep here n on the information at time k 1.
W
pz
PROOF: Use assumption
@
-'1
3 P'pwx
>
.
=1
-42
,, (P;;)
>
( . .
(
''A3lz'rrw'
2
h'max
>
F
.
a'lla'1s h. F
u
>
(49) (5p)
.
r
clearlythc variance
term lzk has upper and lower bounds that are proportional to h, regardless of what n is, This means that we should be able to find a constant n depending on k, such that Jz'vis proportional and ignoring tlle (Smaller) to higher-order terms in h, writc'. j
(4
(;
.
h
xalgva!l
>
-v -!k >
.;
1,1
;
FglHzkj
=
2 rkh.
=
(51)
t..
i. Then,
'
;
1 > n X h -7. > -
r
4:
v'fs
-43
Jzrnlx
>
(
p'k
I'-k .
.
n
Fi
W
z4
nOW
Obt
,
k=1
,
.
1.
,
k=z 1
lz'k >
(4
-4
)
.'.
.'
prk.
r
'
.
--1u
.:'
.
: :
-
,
#-j
U
..
'lhis
=
a)p-now has variance k
-
,
''
sk
yg .yg.) /z
:
tnle Wc11 >
Where g
p
;
..
nl.z p1 zN
Titis proposition has several implications. An immcdiate one is the following. First remember that if the corresponding expectations exista one can always write
:
(
j
5 One lmplication
:
(4
We rfhis gives an upper bound on lzi that depends only on also depends only on h. We lcnow that that bokmd lower a
.
.
zla .,4
f
azlz
iS
h
1
zls
3 '-' u 1
Rllerefores
.1
ar4
n
p-p>
)'. ,
Now divide both sides by n,,4:2 j
A
>
means that
'fus
)
k= 1
-
a pux
.,
(3.
(48)
.
-''''
.4
Jzjr>
:
>
z'lauslux
.
Sum both sides over al1 intenrals: n Fi > l-dnzr'dx' k Msumption 2 says that the left-hand side of this is bounded from a
Note that n
.:'
'
Pk
=
K. >
..
3:
(47)
Use Msumption 3:
E,
of the protf.
>
MIJX
.
.
smaller.
Sinc,c this is a central result, wc prtwide a
'.
'
According to this proposition, assct prices become Iess v olatile as Sketch
X1 lt T
'$
-
.
::a
Ek.3
guk y-j j +
.C0
After dividing
yg
.j
.
-
jju g
.yv
.y
j +.
(u.ijq
(5g)
,
both sides by
2
cygy ju
jy
.
this equation, the parameter rk is explicitly made into coemcicnt of the H?ktcrm, a is a trvial transformation, because tlx term rzklppk w-ill now have a vaziance equal to h.
q
(' y'
:'
, .
'
f2 H A P T E R
*
Differentiation
7
in
Stochastic Environm
..3'
E
(
=
tlse
attirg t lae
6
.'
.
Results Together
169
'
.E'(Al,''f,$ g,1 h. this to
:
.
,
But, according to the proposition,
Suppose we
'
C
'
.
j
justifythe approximation; apz2 k x
justructivc
qujtc unpregjctablc
,
: ...
2
'.
ddnews
because it shows that the fundamental characteristic of in infinitesimal intervals, namely, that ,,
k k jz slo-auz
.
:
('
.
ajh
=
'
insurmountable difhculties in dehning a stochastic equivalent
Iead to may of tbe time derivative,
.
(In Chapter 9 we show that this approximation is valid in the sense of m , square convergence.) ln Chapter 3, w hen we dehncd the standard notion of derivative, we h go to zero. Suppose wc do the same here and pretend we can take of the random variable:
6 putting th e It esu jts Together
''
Gllimit''
W%-1)s+/: Bk-ll lim h ->() Then this could be interpreted as a time derivative of M. The appro tifm in (55)indicates that this derivative may not be well dehned:
Up to this point we have accomplished two things. First, we saw that one stochastic process St and write its variation during somc fmite can take any interval h as
-
.
(
.
t '
'
1
hJ'J,(.-1)&+/, - '1,#r(k-:)/l1 - h --o
Um
.+
az..p;-;
'lsmall
fhl'.
, ? .'
,
h;
l
-
VrtHil
E
t
.
h
t :
t
s
E
Ek..j
.,
Where
:.
hm
,
J. .
. h ()
.14
().:1
0 6
FlGURE
.
0 0 .
1
..
....
' .. r :(
'
2
. :
zk-jj.
(5p)
-
ps-jj
=
.,4
(&.j, ).
z1tft-l,
..
...:
hb
=
-4(f,.-l,
t)) + alk-klh
'
:
t1
Agstlmirlg hat the corresptmding
.
:
.
.:
.
(60)
-d(.)
,
L.
.
:
(k%
!1
..;
'.
.
(kS-
represents some hmction. ucwedthis way, it is clear that 'f is a smooth flmdion of it will have a Taylor series expansitm around (j
a4(')
=l&h
f(b)
2
-1
j
i
E) .
f (h) explcxles
(58)
.
Ths term is a conditional expectation or a forecast of a changc in asset prices. 'The magnitudc of this change depends on the latest infonnation set andon the length of t'he time intewal one is considering. Hence, E'v-:g.% k-.1)can be written as
.'
f 0
=
In order to obtain a stochastic difference cquation delined ovcr hnite intewals we need a third and final step. We necd to approximate thc first term on the right-hand side of (57),
,
4
5yj
''
,
:.
8
sg..jj +. aawk.
-
whcre the term JP''k is unpredictable given the information at the beginning tjw time ilatervttl,ll the unpredictable innovation second we showed that if is term has a variance that is proportional to the lcngth of the time inten'al,
Clearly, as h gets smaller f () goes to inhnity. A well-dehned llmit d :;' anotexist. of course, the argument prcsented here is heuristic. The iirniting o ation was applied to random variables rather than deterministic f undio and it is not clear htw one can formalize this. But the argument is s
,
j,u
Oj
hll =
-
K.
Figure 2 shows this graphically. We plot the function /()
s# sk - 1
':'
cxpectations
cxist.
-I-.A(z:-,,
).
(61)
7
?.
.
:
. 7..
C H A PT ER
170
Differentiation
7
.
in
Stochastic Environ
.
'
:E
t
h) with rcspect to h evalua is the Iirst derivative of Hcre, J(&-j) the remainder of the Taylor series expansi ht is Rlk-j h 0, ne at and predicted chang in asset p thc will not if h 0, time Now, pass words, other In will bc zero. X(Q-1,0) 0. (
.'
.
7 Conclusions
J
..4(f1-:
,
=
'
:
,
piffercntiation in standard calculus cannot be extended in a straightforStfchastic derivatives, becausc in infinitesimal intetvals the ward fashim to random processes does not equal zero. Further, when thc tlow variance of Of neW irdbrma tjorj oyeys some fairly mild assurnptions. continuous-time become very erratic and time derivatives may not exist. random proccsses As thc latter becomes smaller, thc raju small intewals, AH$ dominates tio of llfrp to is likely to get Iarger in absolute value. A well-dehned limit czmnot be found. On the other hand, the difliculty tf delining the differentials notwithstanding,wc needed few assumptitans to construct a SDE. In this sensc, a that can stoc jostic differential equation is a fairly gcneral representation be written down for a large class of stochastic processes. lt is basically constructed by decomposing the change in a stochastic process into both a predictable pa,l't and an unpredictable part, and thcn maklng some assumptions about thc smoothncss of the predictable part.
*
F
=
'
; .
=
' Also, the convention in the litcrature dealing with ordinary stb having that equations is differential any deterministic terms powers o 8reater than onc are small enough to be ignored,l3 Thus, as in standard calculus, we can let '
)
A(-l,
and obtain the first-order Taylor series approximation: Ek-Lf-k
-
17
kk-j
?(/k.-1
,
.
'.
(
0,
.
.
khlh.
Utilizing thesc results together, we can rewrite (57)as a stoc j.ja suc ence equation: j4 X 445//r-) khlh + tzylllz Sk jh 1. - rr'(k-1 u$)-1),
,
(j.
171
9 gxercises
(E
.
g :
;,
(
0 and obtain the inlinitesimal versio'' In later chaptcrs, we let : (57), which is t'he stochastic diffcrcntial equation (SDEI: -->
8 References
'.
dstj
=
alt.
This stochastic diffcrential equation ion (J.; component. di
t ) dt + h dkrt
The proof thata under the three assumptions, unprcdictable errors will have Thc chapter in Merton a variance proportional to h is from Merton (1990), (1990)on lhe mathematics of continuous-time finance could at this point jye use s! to tjae xader.
(
).
is said to have a J8# alt, tj aa
,
,: !(.
6.1 Stochzssffc Differentzls
.,
. At several points in this chapter we had to discuss limits of ran ch incrcments. The need to obtain formal dc*nitions for incremental such as d.t, #1Ft is twident. E.k How can these tcrms be made more explicit? It turns out that to do this we nccd to desne the fundamental conce the 1to integral. Only with the lto integral can we formalize the notiol ' vtlchastic ff//regzltfl: gtlch as dSI JW?),and hence glve a so 1id interpreta of the tools of stochastic differtmtial equations. This, howevera has to .2 until Chapter 9.
q Exexjses 1. We consider thc random process S(, whch plays fundamental role a in suck-scholes analysis'.
.'t.
,
st
.
wure p,t js a
*
wjenerprocess
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(lzlj
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with Ht Jfr,)
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factor, and
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wlcll says that the increments in have zero mean and a variance equal I,IZI l to s Thus at t the variance is equal to the time that elapsed since 1#' observed we also know that thesc wienerincrements are independen't over time Accordin to this s/ can be regarded as a random variable with lg-normal P distribution. SCN we would Iike to work with the possible trajectofojjowed tjs t)y process.
.
-
lzcliverl fjr-j, wr arz dealing with norjrandom quanlites, and the dcrivativesin the ;' seriesexpansion call be taken in a standard fashion. ,:. l3sirlce ' is a deterministic funclion, this is consistent witll the standard calculus, m '' igneres all second-order terms in differeniatioll. for X and hK Tls shpws tlle the l ill tht! ntAtatit:n we are reintroducing
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7
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172
C H A P T ER
*
Differentiarion
7
in
Stochastic Environm
fr and f Let Js 1. Subdivide the intenral subintervals and select 4 numbers randomly from: .01,
.15
=
=
=
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(a)
Constnzd the JF; and St over
bers
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). the (0, 11using
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these random a
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Plot the H( and St. (You will obtain piecewise linear trajecto b that will approximate the true trajectolies.) (b) Repeat the sme exercise with a subdivision of (0,1j into 8 ia i vals. (c) What is the distribution of l.. '.
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log
for
LGsmall''
(d) Let
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=
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log- - logs'/ . -, as time passes? 0, what happens to the trajectories of the
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randomvariable?
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At every instant of an ordinar.v trading day, there are threc states of thc worldz prices may go up lAy one tick, decrease by one tick, or show no ehange. In fact, the pricc of a liquid instrument rarely changes by more than a minimum tick Hence pricing hnancial assets in continuous time may proceed quite realistically with just tluee states of thc world, as long as one igl orcs a rare events. Unfortunately, most markets for financial aisets and derivative products may from time to time exhibit behavior. nese periods are exactly when we have the greatest need for
.
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logs,
ar ets
5
2,5
.000001.
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l.n
iE'' .%-.z5
represent? ln what units is it measured? How does this ran variable change as timc passes? (e) Now let k = How does the random variable,
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< A What does the term log log
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accurate prjcirjg. W'hat makes an event Is turbulence in Iinancial or markets the same as eventsn') In this chapter we intend to clarify the probabilistic structure of rare events and contrast them with the behavior Of Wiener processes. In particular, we discuss thc types of events that a Mtner process is capable of characterizing. This discqlssion naturally leads to tjje charaderization ()f rarc events, we show that events'' have something to do with the discontinuity Of Obsen'ed rice p processes. This is not the same as turbulence. increased variance or volatility can be accounted for by continuous-time stoclmstic sses. What dstinguishes rare events is the way t:eir size and tlieir probabilty urrence changes (or does not change) with the observation interval
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174
U
c H A PT ER
.
The Wiener Proceas and Rare Even '
8
j Twt Generic Models
ln particvular, as the interval of observation, hs gets smaller, the size of nort ma1 events also gcts smaller, is, after all, what makes them nis ln one month, several large price changes may be obselved. ln a wee 'kw fewcr are encountered. Obscning a number of large price jumps during a) period of a few minutes is even iess likely. Often, tbe events that cur dtuul ing an minute are not worth mucb attention, This is the mait characteristic of events- They become tmimportaut 0. as h On the other hand, because they are ordinaly even in a very small timt interval h, their probability of occurrence is not zero. During small timef intervals, there is always a nonzero probability that anonnoticeableo some news will anive. ' A rare c'ptd/?l is different. By dchnition, it is supposed to occur nfrequently. ln continuous time, this means that 0, its probability of as h occurence goes to zero, Yet, its size may not shrink. A market crash such j as thc one in 1987 is xtrare.'' On a given day, during a very short periodyl', there is negligiblc probabtl' ity that one will obscae such a crash. But wllen it occurs, its size may not be vcry different whether onc looks at an intewalk .: of 10 minutcs or an intcval of a full trading day. previous chapter established one important result. Under nc some vev mld assumptions, the surprisc comptment of asset prices, tnlfl, had
i
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idordinan'''
GKnormal''
--?.
7
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=
2
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movement. on tjle otlaer hand, if sporadic jumpsare a s'ystematic part of asset price changes, then value-at-risk calculations become more complicated. Attaching a pro ba bilie to the amount is likely to lose in extreme circumone stances requires modeling the ; ( rare event process as we jj
(1) :
during a small intcrval. : ln heur istic terms this means that unpredictable changes in the ase' ' price will have tlle cxpected sizc o-tgtr-'h1 . But emember how a deviation'' is obtainctl: one multipliesg possible sizes with the corrcsponding probabilities. lt is the product ot l+# terms, thc probability multiplied by tlte of the event. A variancdl proport ioll tl to 11can be obtained either by probabilities lhat depend on Iik whilc the size is independent, or by probabilities tlaat are independent of h, while the size is dependent. 2 nc lirst case corresponds to rare evcnts, amd the second to normd' : . cvents. ,
1*1 Relamce
oj tl
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.
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.
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.
expccted size'' refcrs ozuy to the atsolutc l--r'lw by delinition, unprcdctablc, one knows notlting are, combination of la thc a aor two.
value of tiae chmve. Becausc surprises'. about thc wlel of tlwse changes.
J'
'
.
.
.
Two Generic Models
There are two basic building block,s in modeling continuous time asset prices. One is the Wiener process, or Brownian motion. This is a contittuous stochastic process and can be used if markets are dominated by events while occur only infrequently, according to the probabilies in the tail areas of normal distribution. The sccond is the Poisson a Poess which can be used for modeling systematic jumps caused by rare events, Tlw poisson grocess is discontinuous. By combinirtg these two buildihg blocks appropriately, one can generate lt model that is suitable for a particular application. Before discussng rare and normal events, tltis section reviews these two bliildingblocks.
This chapter is focused on the distinction between rare and normal :. events. The reader may be casily convinced that, from a technical poiti:
.
'z
,
.;
.
such a distinction is important--especially if the existence of rare events mplics discontinuous paths for asset prices. But are tltere practical applkations of such discontinuities? Would pricing financial assets proceed djff6rerltlyif rare events exist? Thc answers to these qucstions are in gcneral tf' srmative.One has to tlse different formulas if asset prices exhibit jamp discontinuities, This will iptleed affect the pricing of fmanciai assets. As an eumple, consider recent issues in risk management. One issuc is capital requirements. How much capital should a fnancial institution put losses asidc to covcr due to adverse movements in the markct? The answer depends on how much value is a t risk There are several ways of calculating such value-at-risk measures, but they all try to measure changes in a portfolio's value when some underlying asset price mtwes in some emreme fashion. During such an cxercisc it is ven' important to know if tbcre exist raz'e events that cause prices to jump discontinuously. If such jumps not are llk'ely, value-at-risk calculations can proceed using the nonual distribution. Price changes can be moddcd as outcomes of normally distributed random Processcs, and, under appropriate conditions, the value-at-risk will also bc normally distributed. It would then be straightforward to attach a probability to tbe Hmount one can lose under some extrcme price df
-.
varancc
175
'
.
.
'
('
l 76
c H A PT ER
.
The Wiene: Process
8
and
Rare Events
j Twc Generic Models
2.1 TH Wicnex Pmocess
Tbe formal defmition of Wiener processes approached DEFINITION:
l
derlvinc wie-ne '' one that
-'one
'time
-al
tjon sets
''.
u$ t) and =
g(.)2j ct E gtT.px t trajectories of p; are continuous over ,
,,
2.
.
.
(2)
A Wiener process W(, relative to a family of informais fz,J, a stochastic process such that:
W'ri' a squarc integrable mmingale with
,
i
is
ZS
ln continuous time, (fnorm al'' even ts can be modeled using thc Wiener : process, or Brownian motion. A Wiener process is appropriate if thc unrandom variable, sav wr,can onlv claance continuouslv. with a ? in gener observes inte al process, during a s ismall changes in v,, and this is consistcnt with the events being ordinary. s There are severa I Ways One can discuss a Wiener process. t approach was introduced earlicr. consider a random variable arg, takes one of the hvo possible values J-hor -J-h at instances -mall
as martingales
'nw
.
/.
This deh'nition indicatcs the following properties of a Wiener proccss.' JF)has uncorrelatcd increments because it is a martingale. and because evefy martingale has unpredictable incrcments. p; has zero mean because it starts at zero, and the mean of cvery increment equals zero. W(has variance t. Finally, the process is continuous, that is, in inhnitesimal intervals, the movements of Wz;are infnitcsimal.
;
.
where
for all
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Then the sum
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Note that in this delinitionj ntlthing is said about incremcnts being normally distributed. When the martingale approach is used, thc normality
(
willconverge wenkly to a wienerprocess as n goes to infinity, Hcuristica process will be a good approximating model fl thismeans that the wienerside,3 right-hand the thesum on In this desnition, a wienerprocess is obtained as the limit, in som. sense, of a sum of independent, identically distributed rando probabilistic th y The variables, important point to note is that possible outcomes forchan i ucfunctions of the length of subinten-als. xs h--x 0, increments inT4',become smaller. withtlzis approach, we see that the wicnerprtxe ''
follows from the ssumptions stated in the delinition,6 The Wiener proccss is the natural model for an asset price that has increments but nevertheless moves over time continuously. unpredictable Before we discuss this pont, howevcr, we need to clarzfy' a possible confusion.
,
2.
l .? wiener owc,sxor Brownian Asfjsn? Thc reader may have noticed the use of the term Brownian motion to describe processes such as p;. Do the terms Brownian motion and Wiener Process refer to the snme conccpt, or are there any differenccs? The definition of Wiener process given earlier used the fact that 1-2 was a Square integrable martingale. But nothing was said about the diattibution Of H(. We now give thc delinition of Brownian motion, ,
.
'have
a Gaussian (normal)distribution. can also approach the Wiener process as a ctmtinuous square in martingale. In fact, suppose H'')is a process that is continuous, h grable . variancea d has increments that are unpredictable given the f 4 an linite of information sets (J'/J.5nen, according to a famous theorcm byF; these properties are sucient to guarantee that the incremcnts in ' normallydistributed with mean zero and variance c'2 dt. W
glOne
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DEFIMTION: A random process B t. t s nian motion if:
..
:j
3As n goes to inf-lnity, the expression on he, right-iland side will be a sum of a .E large number of random var iables that are indcpendent of one arlother and that are all infinitesimal size. Under some conditions, the distributin c)f the stlm will btl app.roxima aormal. This is typical of ctntral limit theorems, or, in continuous time, of wcak convere s'Fhis aLst means that the inczements are uncorrelatcd
=
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over time.
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1. The process begins at zero, Bt 0. 2, .B jjas stationalyy independcnt increments.
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10 rj
.( .
js the f'amous Lvy tlxorem.
is a
(standard)Brow-
178
C HA PT ER
.
j Two Generic Models
Tle Wiener Process and Rare Events J
8
k
,'
3 'F
case of Brownian motion is that, this time, the size of Poisson outcomes does not dcpend on dt. lnstead, the probabilities associated with the outcomes re functions of dt As the observatitm period goes toward zcro, the incrementsof Brownian motion become smalle/ while the movements in yj rema in oj tlte same size. The reader would recognize Nt as the Poisson counting process. Assuming that the rate of occurrence of these events during #/ is , the process
B l is continuous in f.
4. The increments Bt zcro and variance It
distribution with mean
Bs have a normal
-
.:1
-
:
.
.$1).
(Bt - Bs) N(0, If (6j.E, . nis de:nition is, in many ways, similar to that of the Wiener process. There is, howcvcr, a crucial diference. F) was assumed to be a martingale, whileno such statcment is made about Bt. lnstead, it is posited that Bt haq' Ig E a normal distribution, nese appcar to be very important differences. ln fact, the reader may $ think that W';is much more general than the Brownian motion, since no ' .j assllmptionis made about its distribution. well-known Lvy theorem states This hrst impression is not correct. The that there are no differences between the two prosses. f. 'w
.t
,.
de:ned as
u,
xN, - f wjll be a discontinuous square integrable martingale, It is interesting to notc that
'
.
'
THEOREM: Any Wicncr process motiorl
PFOCCSS.
E EMrl
=
(jj) !)
0
(9)
and
I/FIrelative to a family It is a Browniah tf J:
sgsy( j2
,
'.
This theorem is very explicit, We can use thc terms Wiener process and. Brownian motion interchangeably. Hcnce, no distinction will be made 1..
(j())
/.
nus, although the trcctozies of Mt are discontinuoes, the Ilrst and second moments of Mf and W( have the same characterization. In particular. small time intervals of length with both have incremcnts , ovcr processes variancc proportional to we emphasize the following points. ne trajectories followed by the two processcs are very different. One is continuous, the other is of the pure Aump''type. Secmd, the probability that Mt will show a Jump during a very small inten'al goes to zero. Heuristically, this means that the tralectorics of Mt are Iess irregular than the tralectories of lT'j,because the Poisson counting proamost Ccss is of the timc.'' Although Mt displays discrete jumps, ctmstant .
) i..
.l0
;
2.2 The Poisson Proccss
C,
Now consider a quite different type of randlm environment. Suppose :, Nt represents the total number of extremc shocks that occur in a fmanno. I j ' eve nts occur in an unpredictable marketuntil time f Suppose these malor
.
,
,
.
' t Either they E;
fashion. 'rhe incrcments in Nt can have only one of two possible values. willequal zero, meaning that no new mwor . event has occurre (j oj. they will r '/
,
,
equal one, implying that some malor event has occurred. Given that male't events are rare, increments in N t that have size 1 should also occ'ur rarcNt J We use t he s)rmbol dN t to represent incremental changes in Nt d p (tf Cxmsider the following charactre an injinitesimal time period of length . ization of the incremental changes in Nt :7 ''
'At a
propmional to q%. zu 'M is calied a compensated poisson is rcfcrrcd o as the cvmpensatory process. The term. It for the positive trend in N 9 and converts it into a process
'
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Speed
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hveen these two concepts in thc remaining chapters.
R
179
s'trcndless''
-.k,
.
':(
dN i'
1
with probability
0
wit.h probability 1
dt
=
(n'.
.
-
'C'Aheuristic w'ay of calculating
.'
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-
q'hich
.t
:J
Note t hat here we have increments in Nt that can assume two possibl VZUCS dtlf i 'ng an inlinitesimal interval dt. The critical difference from thdr'
ln
cl
.'
is heuristic nis 'Objec-ts''
'.
12 z dt. =
(jc)
because we do not know whethcr we carl treat inczements such as dMt as similar to standard random variables. To make the dscussion precise, one must j)egitz witla a snitesubdivision of the umeiutcrval, and then present some type of limiting argumcnt.
.'
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.
(11)
gives Efdsf
.
should be considered swnh)lz ?A! this point, the use of dN, and dt insteaz of 1.x! and later cuapers, it is lloped tlaat tlw meaning or lae notatio,n gx., and d, will become e aros:
the variancc of dMt is as folltms: 12z dt + O'EI - dtl, .E'Ed,'-z,12
E.
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('
L.
. :
2H A P T E R
180
*
8
The Wiener Process and Rare Eventsi
Two Generic Models
z'
.(t.
it will not have unbounded variation. H$, on the other hand, displays in. finitesimal cbanges, but these changes are uncountably many. As a restllti: the variation becomes unbounded. Hence, it may be more difficult to defin integrals sth as T
f (H'',)Jl1$
)
to gcnerate a trajedory for the Poisson counting process Nt, t (E (0,11. trajectol'y is displayed in Figure 1. Wc note the following charactcristics of the Poisson Pathsl 'rls
;
. . . .
J
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.
than intcgrals with respect to Mt'.
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t(p
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definition may '
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=
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A
Evaxnplcs
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re 1 displays a Poisson process generatcd by 13.4 was Excd. ne was selected. Next, h =
Fi
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=
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.2
0.3
.4
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FIGU R E
0.B 1
.1
0.8
0.9
1 h
'.
v'
.
r.
.!; '. :t!
'
.. ''
.' '..
.
. The path shows occasional jumps,due to the Poisson component. . Between jumps,the process is not constant; it Euctuates randomly. This is due to the Wiener component.
i'.
a computer. Firsq computer was ask
The trajectory has a positive slope. (Hence, Nt is not a martingale.) Changes occur in equal jumps of size 1. Thc trajectory is constant between these jumps. In this particular examplc, there are 14 jumps,which is very close to the mean.
Figure 2 displays a mixture of te Poisson and Wiener processes, First, a trajectory was drawn from thc Poisson process. Next, the computer was asked to generate a trajectory from a standard Wiener process with variance The tw'o trajectories were added to each other. h We see tlze following characteristics of this sample path:
:? .':
lndecd, it is true that, in general, the Riemann-tieltjes applicd to this latter intcgral.
181
)
(
X
:
Cl APT ER
182
.
8
The Wiener Process and Rare Evenu
* nc noise introduced bv the Wiencr process is . due to the Poisson process. This may Jumps Wiencr process with higher variancc. nen, it to distinguish between Poisson Jumps and noise *
-'
.
.
j 5DE in Discrete lntenrals, Again
,
much smaller than the 2. change if we select a.' could bz very difficult caused by the Wiener'
singlc ncws itcm in determining the pricc or interest rate or currcncy derivatives is significantly smalier, although present. In the following sections, we characterize normal and rare events, and Iearn ways of modeling price series that are likely to exhibit occasional jumps.
,
''
.
(.
!
Componcnt.
183
';
2.# Bttck to Rzwe Ftents
3
Compared to events that occur in a routine fashion, a rare event is by sizc, This classihcation seems obviu, deinition something that has a is not very etsy lo justify.Consider the Wienet ous, but at a cltaser look, equation that is driven by a Wiener pr. differcntial process. A stochastic small intervals of length h, unexpecte'.-'''''''''''. assuming in that cess amounts to variance of o-lh, where the z- may depend on tll: with cliangcs price a occur Further, well. thc distribution of these unexpectedf available information as normal, chafiges is price l A normal distribution has tails that extend to inlinity. With small b nortzero h, there is a positive probability that a very large, unexpected pxi/.. changc will occur, Hence, with a nonzero h, the Wiencr process seems to tW diffcrentiz perfectly capable of introducing big 5, events in thc stochastic or b equations. Why would we then need another discussion of
A deeper analysis of normal vs rare events is best done by considering a stochastic differential equation in finite intervals.ll Consider again the SDE that ws introduced for discretc intenrals of uqutl size h in Chapter 7:
ttlarge''
: .
vk
'
ask-L,
trts'k-l,
k4h +
lHi,
k
=
1, 2,
,
.
.
,
n,
(13)
Gtsurpzise''
*
,
''
dnormal''
r
-
=
-
:'.?
CVC n(S?
Sk-j
kjh is the drift component which determines how, on where the the average, the increment Sk Sk-L is expected to behave during the next interval. zHi is the innovation tcrm, determining tbe component of asset prices, lt was shown that, under some assumptions, the variance of the innovation term is proportional to h, the length of the intcrval, The k)2 s the factor of proportionality, term tJ'tu%In order to study and arare'' events in more detail we makc a further simplifying assumption.lz
:.;.,
7>
-
t7(&-1,
.1..
idrare
SDE in Discrete lntervals, Again
F
ne problcm with charadcrizing rare evcnts using a Wicner process j'..1 . the tails of the normal distribution c h the following. As goes to zero, less and less weight. At thc limit. h = 0, these tails havc completely v . ; ished. ln fact, the whole distribution has concentrated on zero. nis is :' be expected because thc Wiener proccss is continuous with probability on As h --> 0, the size of price changes represented by the Wiener pro It has to become smallcr and smaller. ln this sense, the Wiener process is suitable for represcnting situations where, in an cxtremely short intew priccs can move in some extremc fashion. What we need is a disturbancc term that is capable of generating events in extremcly small intewals. ln other words, we need a process depr may exhibit Jumps. Such a process will have outcomes that do not ' shrink. will : not of outcomes size the and the small. h gcts as on h, sample pa Thus events correspond to occasional jumps in thc
ASSUMNION 4: AWi can assume only a hnitenumber of possible values. ne possible outcomes of lli and their corrcsponding probabilities
'
.
are 15
zL':
.
.
tv, Hi
,
=
with probability
1:72 with probability
,1 pz
:
'.
uu
,
'tlkernemberfrom Chapter 1 tlta i.n order to obyain the SDE in discrete used scweral approximations. ybr small but noninlinitcsimal h, such equations approximate sensc only.
:
:.
1.
L. of the process. Several markets in derivatives exhibit jumps in prices. This is more O the Case in trmmldities, where a single news item is mtre likely to C crO important information for the underlying commodity. Reports tn . the same comm S. for example, are Iikely to cause jumps in futurcg on of. ity. In the case of financial derivativeg, this is less likely. The weight ,;
.
f'
.:
L
with probability
'lHere also we follow Merton
''qnere are
intomals, we lzold n an
(1990).
two rcasons that we iulroducc tllis assumption. First the Jistindion betwecn 'Ttl'e and nonnal cvents will be much easier to introducc if the Second, are hvnite. mssibilitics - lIlYiual asset prciug in flnancial markcts often proccods 81t,11 eithcr binomial or trnornial trees. the case of binomizal trces, thc market participant arxsumes thal, at any instant, there are Ottly wo possible moves fot the price, Wih tzinomial trecs, possible moves are raiscd to three. Xence in ractical situations, the total pumber of possible states is selected as linite anyway. P
,
@
(
,
.-
'
.
;' )
184
C!l i A PT E R
*
. .
4 C'haracterizing Rare
'Fhe Wiener Process and Rare Evenu.
8
and
Normal Events
18.5
!
Although it is not clear which event will occur, the set of possiblc evenl is known by al1 agents. A typical wi reprcsents a possible outcomc of the; innovationtcrm trjrp'k.. while pi denotes the associated probability. Tlw .( parameter m is the total number of possiblc outcomes. lt is an integenl4 tlutcomes- $ There arc two types t')f wi. The hrst thrcc represent and the N>.q;. For example, z&I may represent an uptick, u?a may be a downtick, certainly change'' in asset prices. ln real time, tbese are may represent routine developments in linancial markets. 'I''he rcmaining possibilities, z%, w5, are reserved for various types of special evcnts that may occur rarely. For cxample, if the underlpng semlrity t is a Jerivativc written on grain futures, 1174 may be the effect of a major drought,thc 1.% may be the effect of an unusually positive crop forecask # and so on. Clearly, if such possibilities rcfer to extreme price changes, and % if they ar rarc, then they must lead to price changes greater tha,n one tick- f) Othemise, price changes are caused by normal events Iz;l, ztla, uy. ' t h e pro babilistii This sctup will be used in the next scction to determlne
Using the important proposition of the previous chapter, this means m
dtnormal''
nno
.
(17)
'rhe
Rweights''
,
!
..
nlh,
=
woerethe parameter m is the number of possiblc statcs. The left-hand side oj Eq. (17)is simply the weighted average of squarcd dcviations from the arc probabilities associated mea n which in this case is zero. outcomes.ls with possible side of (17) is a sum of m Iinitc, nomncgative numxow,the left-hand of such numbers is proportional to h, and if cach element yers. If the sum zero), then each term in the sum should also be proportional js posive (or should equal to zero. ln other words, each #/w2j will be given by or
,
.
piw;
j=1
:
.
p i wli
=
cih,
(18)
',
structure
Of
wjwre () < ci js some factor of proportionaliyl 2 are linear functions of h. s quatjon (18)says that all terms such as rjlslpr Tjwnj one can visualize the pi and the wi as two functiotls of h, whosc
'j
rare events.
j . . .
'
)
: .
product is proportional to h.
'
4 Characterizing Rare
r. :':'. .
and Normal Events
.,i'
assumptions
''.
/Jf
.
Fgty'k H'r k 12
o-lh k
piht
(19)
u?ytl,
(20)
and
.
wi
.t
(15)J ?
,
=
'
:
=
is,
J
,. '. ( '.
1-3 of the previous chaptcr, an important result wa: . $ proven. lt was shown that the variance of tzkH'k, Under
rrhat
such that
.
:!
.'
=
pihtuhhtz
was proportional to the observation interval h where o'k was a known Pa'd rameter givn the information set 1k-L. Thjs rcsult can be exploited furthcr if we use assumption 4. In fact, , tjjjj c . very explicit characterization of rarc and ntrmal events can be glven Howevetk) unpleasant. the reader hnd the notat ion a bit may ' way, although tMs is a small price to pay if a uscful characterization of rare and norm a1'k j events is eventually obtained, ;), valr number of fnite assumption lAP'v According to 4, can assume only a explid f ues, ln terms of tz;f and the corrcsponding probabilities, #j, wc can 'j' write the variance as 'r (16 varlfqlfrkl piwi2
cih.
=
(21)
)L
We follow Merton ( l 990) and assume specilic exponential tjyese junctjons pjhj and u?j(J;)..
.
?.tjl
.
lj)
j
hr'
(2g)
pihj
=
pihq
(;g)
,
Where
r f and qi are nonnegative constants. zi and jt are constants that may depend on i or k but are independcnt of the size of the observation interval. ,
,
,?
f 1
J
=
'
t
-
..(I
l4Both u?i and /7. caa very wcll be made tt'ydcmnd t)n he inforrnation sGt 1k. H()w cumber*mgz this would add a k subscript to thesc variables and make the notation more of and independent k. make this, wc n, p, avoid
=
and
...
=
)
forms for
.
' .
.,.:
'blrlgencral,
subscript.
:. :
( :
: .
;
lswe
show the potentia! dcpendence of 4: tbne passes, L)yadding the subscript to
':
.
,,,..,
p on the irtformation hat becomes available
vk.
c,. wiil depend on k as well. To kecp notation
simple, we eliminate
the k
E
186
C H A PT ER
The Wiener Prkxess and Rare Events
8
*
1.4
z-.'''''
1.2 G
'',
Rare and chaxacterizing
Btlt this implies that
.
>
qi + 2rj and
:
-..-
,
.,''
.,''
x.
cj
. '
.,...---7 h
.
''
0.4
0.6
1 .2
1
0.8
1.4
,
j
2/
j
(2:)
.
(29)
-
:
h
=
Thus- the parameters q;, ri must satisfy the reseictions 1 () r/ s s 2
i
..-v<
. '. '
.t
F 1G U R E
and
.
() s qt
,
.L
E'
Figure 3 displays some choices for hr'. Three examples are shown: the when case when r; 1 (notallowed in this particular discussion),thatthe case when particular, for small 1/3. In and the case we see rj h hr > h. Aording $o Eqs. (22)and (23),both the size and the probability of the As h gets larger, then the : event may depend on the intcwal length, change and its probability will k magnitude obsewed price of the (absolute) get larger, except when rj or qi are zero. y To charaderize rare and normal events we usc the parameters rf and qi. Both of these parameters are nonnegative, ri governs how fast the size of the event gocs to zero as the obsezvation intewal gets smaller, qi governs r .t how fast the probability goes to zero as the observation iriten'al decreases. It is, of course, possible that ri or qi vanish, although they cannot do so at ; t he same time.17 We now show explicitly how restrictions on thc parameters ri, qi e-rm ' distinguishbetween rarc and normal events, ', made such variance is of of ll'rpin tcrms as ne (18)
wefind that
=
=
ri
,
:
(30)
1,
:s
there are, in fact, only two cases of interest-namely,
.5,
=
(27)
1
=
..
-.--
.2
182
Normal Events
.,---M
9.6
0
.,..----h''2
f(*) h 1/3 .M
4
4
rn
0.8
0.2
.
1/2,
=
qi
(3j )
0,
=
and
'k
ri
',
.
vjahc
:.
of
(),
=
qi
tirst case leads to events that we call
t
.'
1
=
(32)
.
ne
events, We discuss these in turn.
second is thc case
.
:
.
:.
4.1 Nonnul Etnts
ne
:?
condition for
evcnts is
LGnormal''
j
'
-
2
.
.
2 Pi'u?i
But we know that each piwi2 is p roportional
piub2
.g
.
cih
=
i
to h as well:
(33)
0. =
.?.?
zp' y-jgzri lgqi
>
1/2. To interpret this, consider what happens when wc select rj First, we know that the q; rnuuf equal zero. jtr As a result, the fu uctjons tbat govern the size and thc probabilify of the outcome u become, respectively,
(24)
.2
=
A:rj
b
( :!.
=
hh
1f z
.tg
=
i
y-y
(gj)
<
;
.
(O) )
,
Pi
;
=
(36)
lh.
According to this, the sizes of events having ri will get smaller as the intewal jength l gets smaller. On the other hand, their probability does not .5
=
,i
Hence,
-
i 2/i /;(n+.2r;)
=
(26).
cih.
,)
E.
;
I7Rememlyer that the product of nlIand pi must be proporional to allowed. equalzcro, thesG producls will not depend on and th s is not ,
.
lf both ri an d #i .. k (. .1
l8Remember that
'.
: .
j...
2.ri+ q
i..
i.
aud tluat qi cannot be negatjve. . '
.a : .
:
=
1
(34)
':.'..'
.' :' 11:*
188
CHAPT ER
'.(
i.
4 Characterizing Rare
The Wiener Process and Rare Events
8
.
'(
but have a constant probability They are aG tlccklrence get Now suppose all possiblc outcomes for llrl are of this type and have the sample patbs of the resulting M( process will have a ri nUm ber of interesting properties.
depcnd
O11
.
These
Outcomcs
Obsenation
Of
.5.
are
(
'
d'ordinaly''
Smaller,
k.
'rhcn
=
If there are no rarc events, then all wi will have r; wi
=
.5,
=
)
VifW
h-h 11141 --.p
(37) k
=
p; +. h
lim
$. .J
p;
-
u)
....j
1im
=
p;-+0
--'o
h
,
(42)
s ':
or ' after substituting for wi,
LL,
(38) $
0.
(41)
h
.
h h
lim i
=
-.0
. This will be true for evety evcnt uj. ln the limit, the trajectodes E of H( will be such that one could plot the data without Iifting one s hand. Infinitesimal size. Each incremental value will have 0 for events, the probabilides F On the other hand, since q oftliese wi will not tend to zero as Jl 0. ln fad, the probability of tese will be independent of h..
.
j
'e
.1,
jjm
h-..u
1
(43) (44)
x.
..y
p;
''normal''
=
This means that as thc intcn'al h gets smaller, tlze J#; starts to change at an infinite rate. Asset prices will behavc continuously but crratically, (Here wc assumed, without any loss of generality, that b was positive.) nis concludes the discussion of trajectories that are generated by events of normal sze. we now consider paths generated by rare events.
''?
-+
,)
events
(39) t. )
pi A. =
j'
It is in this scnse tlaat normal evcnts can generate continuous
umepaus.
.,
?
#. l.2 Smoothness of Sample Pashs ne sample paths of an innovation tcrm that has outcomes with ri 1/2 are continuous, But they are not smooth. deteTFint remember what smoothncss means within the context of a ministic function. Heuristically, if it does n0t a function will be change abruptly. In tather words, suppose we select a point x() where tlm the if for small function/(x) is evaluated. J(x) will be smooth at
y' k: )'
=
. :
4.2 Rure Evcnts Assume that for some event kzlf, the parameter rj equals zero, nen the qi equals 1, and the probability of this particular outcome will by dtmnition bc given by corresponding
'
Itsmooth''
.. '
$
.
.x1)
,.
,
pi
.
E ;
.(I
fxv
=
for some i. Tnklng Iimits,
'
and their size
'lhis
=
....%'
-
ji E
will shrirlk as h gets smaller. At the same time, as h goes to zero. the values of tt;j approach each other. mcans that the proccss Pi will. in the Iimit, be cfmtinuous, The steps taken by tlzv will approach zero:
limwi ..-.0
W';+/, 1F;
'
;
4.1.l Continuous Paths
ratio
l89
proportional to /z1/2. In other words, as time passes, thc new events that affect prices will cause changes of thc order Xh. At any time t, the unexpcctcd rate of change of prices can be written as
'small''
intelNals
Normal Events
and
+ h4
-
(x())
g
events wi that have a rz 0, qi 1 are events, sincc, according to this equation, their probability vanishes as k (). O11the other hand the size of the events will be given by =
jfJ
is, the function is smoo th stays finite as h get smaller and smaller, has a derivative at that pint. fundions ls the snme delinition Of smoothness valid for nondeterministic suchas p; as well? In the particular case discussed here, there are a hnite number m possiblcvalues that Hrk can asgume. The sizesof these events are
,
;
it
''rare''
=
-..>.
:L rrhat
(45)
.
'l'he
t
yxm
V-'-'
lh g
=
':
7.
uf
::
).
=
ti
'
.
that is they wgl not depend on thc Iengtll of thc inten'al h. We make the following observations conccrning rare events.
'
,
,
.5 .'
E L,b
. .g.
:'!: ' .. ,
L
.:
.
(46)
;'E'
c H A PTE
190
R
.
8
Tbe Wiener Process and Rare Events
5 A Model for R.are Events
Paths 4.2.1 sample of an innovation term that contains rare paths snmple In discontinuous. fact, tbe sizes of those wi with q 1 do
events will be = not depend on As h goes to zero, kp-p will from time to time assume values that do not . get any smaller. The sizc of unexpected price changes will be independent a jump. of whensuch rare outcomes occur, H'; will have of probability these 1, the other hand, if the jumps will depend qi on obsering a jump VII on h, and as the latter gets smaller, the probability of also go down. Hence, although the trajectory contains jumps, these jumps
that is predictable given thc information at that time. and another unprcdictable. In small intewals of lcngth h, we write
:'
Sk -
:
As
qi
<
1
.
,
.
(49)
n.
,
.
.
..
t
dt + J(St,
/)
#W$.
(50)
errors :W'). In the case of rare events, thc deEning facttrs are that the size of thc event is not inlinitesimal cven when h is, while its probability does become negligible with h --> 0. Accordingly, the new innovation term should be able to represent (random) jumps that occur rarely in asset prices. Further, the model should be iexible enough to capture any potential variation of the pro bability of occurrence of such jumps. One can be more specific. First, split the error term in two. lt is clear from the previous discussion that changes in asset prices will be a mixturc of normal events that occur in a continuous fashion, and of jumps that occur sporadically. We denote the first component by AHS.ne second component is denoted by the symbol &Nk. To make this more precise, assumc that the event is a jump in asset prices of sizc 1. At any instant
.
Lq'
(48)
1)
att,
=
in order to take and their varionly differfact, js proportional h, the time interval, ln the also to an ce ence from the case of a Wiener process occurs in the continuity of samPIe paths. Hence, the same SDE represcntation can be used with a simple modification. What is needed is a new model for the random, unpredictable
.
(47)
.
.
JJC
i
<
1, 2,
really mean. d jfferentials such as d. t or nere js po need to adopt a diferent represcntation into accoun t rare events. These also occur uncxpectedly,
l
and 0
=
In later chapters we stu y the soEs more precisely and show what the
'
'
1
k
,
(j
i
4+J 2 Further Comments What can be said of the remaining values for r/ and qi? ln other word cons ider the rangcs -
klH$
gets smaller, we obtain the continuous-time version valid for inintervals! dSt
C
common. Clearly, if the random variable AI'Vcontains jumps,its sample paths VII 0ne would need a modcl other than thc Wiener process not be Gmtinuous. of such random shocks. to capturc the behaor
<
frt.%-l,
a.%k-k, k)h +
''
are not
rf
=
that is
,
=
0<
&-1
jsitesimal
.
,
191
,
' What types of sampk paths would thc Hz)possess if the possible outcomes ! : havc rj and qi within these ranges? It turns out that for aIl rj, qi within these rangcs, the samplc paths will ! be continuous but nonsmooth, just as in the casc of a Wiener process. is satissed, thc size of wi WII nis is easy to see. As long as 0 < rj < be a function of h. As -->. 0, wi will go to zero, In terms of size, they are .: not rare evcnts. E Note that for such outcomes the corrcsptmding probabiiities a 1so go to S zero. Thus, these outcomcs are not obsen'ed frequent ly. But given tat E their size will get smaller, they are not qualified as rare events. l:, .
.
.5
,
k
-
1, one has
,
5 A Model for Rare Events
Nk - N&-1
Where
we jrt
with probability Ah
1 =
with probability 1 - /z A does not dcpcnd on the information set available at
hxk
j'
(51)
,
()
=
Nk
-
time k - 1.
(52)
Ng.k.
-such
What type of models can one use to rcpresent
asset prices if there are rare
tNk
(
events?
Consider what is needed. Our approac 1,tries to represent asset prices by E'?' observed changes into two components: one an equation t ha t decomposes ' .
.y :
represent
jumps of
size 1 that occur with a constant
.19
rate
m-rlw rate of xcurrcnce of tlw jump during an intewal h can be calculated the corresponding probability kh by
E
.
r.
.
K
'' ..
by dividing
:'
192
H A PT ER
The Wiener Process and Rare Events
8
.
193
Moments That Matter
..
'g
It is clear that Nk can be modeled using a Poisson counting Poisson process has the following properties:
process. A
6 Moments That Matter
, ,
1. During a small interval h, at most one event can occur with probability 21J vcry close to 1. 2. The information up to timc f does not hclp to predict thc occurren (or thc nonoccurrencc) of the event in the next instant 3 cvcnts occur at a constant rate
and rare'' cvents ia important for one Tlw dfstinction betwccn other reason. Fractical work with obsen'ed data procecds either directly t)r indirectly of the undcrlying processes. ln Chapter 5, lv using appropriate representing various expectations of the the term as defined we uaderlying process. For example, the simplc expected value F(-Y/1 is the rst moment. The variance fntrmar
:
'.
;
:
.
JKmoments''
,
K'moment''
,
.
'l'hc
L
.
:
ln fad, the Poisson process is the only process that satishes all these con- : ditions simultaneously. It seems to be a good candidate for modeling jump discontinuities. We may, however. need two modihcations. First, the rate of occurrence of jumpsin a certain asset price may change and cannot i over time. ne Poisson process has a constant rate of ourrence L . accommodate such behavior, Some adlustment is needed. Sccond, th incrcments in N/ have nortzero mean. The SDE approach dcals with innovation tcrms with zcro mean only, Anothcr modillcation is needcd to eliminate thc mean ol dNt. Consider the modised variable
.sg-v,112
varlxr)
:
.;
is the second obtained by
'
t
'
b
N, -
=
f
).
(53) ('' ;
'
The increments h will have zero mean and will be unpredidable. Furtber, if we multiply the Jt by a (time-dependent) constant, say, r.rzt%-j kj, the sizc of thc jumps will bc timc-dpcndcnt, Hence, e(&-1 kjuk is an apPropriate candidate to rcprescnt unexpected jumps in assct prices. nis means that if the market for a tinancial instrument is affected by sporadic rare events, the stochastic differential equations can be written as
';,
,
,
,
Sk - Sk
1
=
aLuk1. +
As h gets small,
k)
trzt-v-:
+ ojuk ,
ktij-lk
la ,
klH''k k
=
1 2. ,
s
:
(54) -
.
.
.
,
.
n.
'$..
E 1rn.'H'k Vargfn
*
1kl
=
ast
,
f)
dt +
tn
St
,
/)
#H?;+
(55)
tj #./;.
and This stochastic differential equation will be ablc to b an dle (drare'' events simuitaneously. Finally, note that the jump componcnt dh and the Wiencr component dv t have to be statistically independkmt at every instant /. As h gets smaller, the size of events has to get smaller, whilc the size of rare evonts remains the same, Under these conditions the tw't types of events cannot be to each other, Thcir instantaneous corrclation must be zero.
.
6h-lj
IzP-z. + h-lk
.
1
=
(,I
+
?Jp.j
2+ ) gpl'tt'j =
.
.
.
-
+ pm zt7ull
=
.
.
(58)
0
2,
(59)
+ pn, 137,,,1,
independence of Hzrkand uk is implicitly used. Now consider the magnitude of these moments when all events are of the anonual'' typc having a size proportional to hll That is consider the case when tll qi (). 'Fhe hrst moment is a weighted sum of m such values. Unlcss it is zero, it will be proportional to /11/2:
'
,!
'
:!
'
11
=
.; '
' .
? .' .
SlGl
j'.
drelated''
0, this probability will become .1.
(57)
11',
wherethe
:.
idnormal''
-->
+
't
Qnormal''
zilAs
Ekx,
are
-
this becomes Jzvb't,
-
(centered)moments
where k > 2. As mentioned earlicr, moments give information about the procegs under consideration. For example, varilmce is a measure of how volatile the pices are. The third moment is a mcasure of the skewness of the distribution of pricc changes. The fourth moment is a measurc of heavy tails. In this section, we show that when dealing with cbanges over inlirtitesimal intervals, in the case of normal events only the first rwwmomcnts matter. Higher-order momcnts are of marginal significance. Howcvcr, for rare cvents, ali momcnts need to be taken into consideration, Considcr again thc case where the unpredictable suzprisc components are made of m possible events denoted by wi. The first two moments of such an unpredictable error tcrm will be given byzi
'
Jt
Higher-ordcr
Tr.v',
,'
.
-
(centercd)moment.
Et
(56)
FE.A-,
=
pj
kll'rk j
=
1/2(yp
1 t'?j
+
.
.
.
+ pmlmj.
(60)
.t'
2'In the remaining
.J
.E .j' :E,
.
>
:
(.
.
part of this section,
fnlur,
t, i
=
1, 2 wll be abbreviated
zhi%..
.:
194
C H A P T ER
*
Ctmclusions
Te Wiener Process antl Rare Events
8
As we divide titis by we obtain the average rate of unexpectcd changes in prices. Clearly, for small the X-his lalyer than and the expression ,
representing the lirst moment and the second representing Parameters, tne will variance, be sufflcient to capture al1thc rclevant information in price thc The Wiener process is then a natural choice if there are small for data no rare cvcnts. lf there are, the situation is diferent, Supposc a1I events are rare. By defmition, rarc events assume values wi that do not dcpcnd on h. For the second rnomcnt, wc obtain
,
,
E l'H$l
6
(61)
h
.
!:
gets larger as h gets smaller. We conclude that when the hrst mornent is not and cannot be ignored even in small intervals h. equal to zero, it is The same is true for the second moment. The variance of an unpredictable change in prices contains terms such as w2. i When the wi are of normal typea their size is proportional to hlll. Hence, the variance will bc proportional to :
, ''
Vtutn
)
h
=
p/ti i
=
zf
:;
i
(62) zg'
.
1
nz
u p-i J-.2
=
(67)
,
j=1
where the wi do not depend on h. As we dividc the right-hand side of the last equation by h, it will become independcnt of h. Hence, variance cannot be considered negligible. Here, there is no difference frtam Wiener
.' ';
2
tzc zuk
'
11
k)pt
2
'
E
As we divide this by , we obtain the average rate of variance. Clearly, the h will cancel out and the rate of variance remains constant as h gets smaller. nis means that the variance does not become negligible as --> 0. ln the case of anormal'' events, the variance provides signilicant information k i about the underlying randomness even during an irifmitesimal inten'al Now consider what happens with higher-order moments,
PCOCCSSCS,
Howevcr,
the higher-order moments will bc given by
.'j
Efth-lkj''
.
lzl''j
(#)
=
+.
.
.
.
+
rysui''.j ,
(63) F
with n
> 2. Here, when the events under consideration are (f the normal type, raising the nh to a power of n will result in terms such as
v)'l But when n
>
tnkL/ztn.
=
zi. .
'r
.(.I
(64) .;..
2, for small h we have
'
'
j
(65)
A-IJLAM'S.J (s-2)y2 = h '
h
Titis rate will depcnd on h positively. .As
tv''
by h, we obtain t he
( .E
..'.,
.
f=l
(66)
'
g
;
gets smaller. hn-1)/2 will co'n-
to Zero. :2 higher-ordcr moments of unpredictable Price Consequently, for small will useful information if the underlying events aro changes not carry any model that depends only on fA probabilistic of the type. a1l
?
J
/E
,
zzWherl n is grcater than 2, tlze exponent
of
will be positivc.
(68)
.
'.
Verge
''normal''
)'-'!zqpi
tznormal''
'
Consequently, as we divide higher-order moments corresponding rate:
h
Tis is thc casc because with rare events, te probabilities are proportional to /7, and thc latter can be factored out. With n > 2, higher-order moments /I. As we divide higher-order moments of Llk by h, thcy are aL() of order smaller will not get any as h ---> 0. Unlikc Wiener processes, higher-order of moments h cannot be ignorcd over inhnitesimal time intcrvals. nis that if prices are affccted by rare events. higher-order moments may means prtwide uscful information to market participants. This discussion illustrates when it is appropriate to limit the innovation tcrms of SDF,: to Wiener processes. If one has cnougb conviction that the events at the roots of the volatility in linancial markcts are of the typc, thcn a distribution function that depends only on the Iirst two moments will be a reasonable approximation. The assumption of normality of #P; will be acccptablc in the sensc of making little difference for the cnd results, because in small intcrvals the data will depend on the hrst two momcnts anpvay. However, if rarc cvents are a systematic part of the data, the use of a Wiener process may not be appropriate.
i
E (C'L Hi j/
=
3=1
',
7 Conclusions
.
'
',' '
..
.
111the next two chapters, we formalize the notion of stochastic differential equations. nis chapter and the previous one laid out the gmundwork
;' (.
!
c H A PT
196
ER
.
8
The
wienerProcess ana
8 Rare and Nonnal Events
Rare Eveno
in
Practice
197
'.I
for sDEs. We showed that the dynamics of an captured by a stochastic differential equation, dSt
=
a-t,
tj dt +
f) #'F; + !6.1-1((.5'1,
asset price can always be htt
,
/)
t' ,,
8
.
.
,
<
.
tn
r
=
(70)
'n
(g1)
=
This gives t-l'lediscrete time points (fjl. We next model tlle values of St at thesc spccific time points, Ij. For sake of notational simplicity, we denotc thcse by S:
.
,.
i '
j
qj
l
=
g
1,
'
=
(j j. '
'
,
.
.
j
LL
the
t'ygj
n.
..
,
.
'
:
In small inten'a ls thc random variable W) is described fully by the flrstand second-or der moments. Higher-order moments do not provide any addit ional information. Hence, assuming nkrmality and letting W( be the Wicner process prov idcs a good approximatitm for such events. Rare events cannot be captured by the norm::tl distribution. If they are likely to affect the financial market under consideration, tbe unexpccted The Po isson pr componen ts should be complemented by the dh process. CCSG WO11 ld represent the Properties of Such a term reasonably well. ejlst, 1) arz Given that the maC ket Participant Can Pick the Parameterg O-qn G21 s 1, tj at will, the combinatitn of the Wierr and Poisson processes marKets. represcnt a 11typcs of disturbances that may affect nnanclal
<
na
,'.,
rarely,
/:
with
;
idlarge''
-
J. '
,
Wsmall''
Occtl'r
rfj () < =
.
and right-hand side is the expected change in whcre the firqt term on the unpredictable given surprise component, brackets is the in the secon d term the information at time /. The sttchastic differentialswerc not tskdelined and increments, forma11 y, so the discussion procceded using .I#i. ne unpred ictablecomponents of SDES are madc of two parts. #H': capregularly. dh capteres ttzresevents of insigniflcant size that happco
that
''simple.''
)
(69)
#Jf1,
'isystematic''
As usual, wc divide the time interval of of equal length k such that:
and lergth T into n subinten-als
,',.
.%,
even ts
to be
'
j
binomial modcl implies that once it reaches a certain state or node, at cvely discrete point f, thc immediate movement in Si will be limited to only tw'o up and down states, which depend on two parameters denoted by (h.z.Lq u. and way these two parameters arc chosen depends on the types of movemen ts ; is believed to exhibit. We will discuss two cases. of aj and di will bc made to depend on the In the lirst case. the W hcrcas the probabilities associated with them will be independent of ln the second case, the revene will be tnle. ui and #j will bc independent ne of while (hc probabilities of up and down states will depend on it. Clcarly, thc ftrst will corrcspond to the case of events and will eventttally be captured by variablcs driven by the Wiener process. ne second will correspond to rare events ayjtj wgj jead to a Poisson type behavior. rne
.;
.
,.2
1
'rhe
'
y
.
.
g
:
;#'
.jza
,
i,-
.
i ,,
,
: !:. :;
u'
' t'
llare and Normal Events in Practice
,
(dnormal''
s L
,,
and rare evenl ' ln this section, we treat how t he distindion between normal mics. ln parcudyna' modeling price of asset k will exhibit itse lf in practical made be it more :,. curiosity, thcoretical or can lar, is tls dist inctiononly a discussion? above-mentioned concreteby cxplicitly ta king into account the within This is best the class of question is seen yes. ne answer to the last bino mial discuss We with Chapter 2. two models in dealt binomiaiplicing t E. j,, events, representing random norma a driven by a being term dels, one mo
,,
':
y 2 .
xo
Ewcnjs
Suppose the Si has an instantaneous pcrcentage trend represented by the parameter t, and an instantaneous preccntagc volatili of o'. For both cases considered below, we assume that Si evolvcs according to the following:
''
.
'''
tbe other tbat incorporates rare events. First wc nee d to review thc standard binomial model for a hnancial assd although a process sucn ( Price. Wc work with an underlying stock price S/, considcred. be could also a9 instantaneous spot-rate r/ . 't
sf.j.l
..
c
PTOWe are intcrested in discret jzjryg the behavior of a contzutlls-time his discretizatilm time interval (0, F1, T < :x). We also want t cess st, over
be
Lip down '
.
.
q
w ith probability pi
dikh with probability 1 s'up''
'
'f
i
xj yi
-
pi.
(.u)
and Ju'e labeled as llut in practice, both of the movements may or onc of hem may stay the same. nis choicc of the, terms should be regardcd 011jy as a symbolfc way of naming the two sates. .A1so, the parameters u: and d, may also demnd on the obscnrcd at that node or even at earlier nodes. Here we adopt the simpler Pase tlf state-irldcpcndent up and down movements.
' i .:
8.1 Thc Bfnomzl Model
Jo'Ltese,states
.
; h
'..
Sfdownp''
j
:'
l98
C H A PT ER
*
The Wiener Process and Rare Events
8
ts Rare and Normal Events in Practice
,
?.
is inlluenced only by For the case wherc coeflicients Ll; and di can be chosen as:24 .%
ui
di
'v%A C
=
1 pi j =
gl +
(74)
,
12 s
.
,
.
..
nrarc''
,
,
=
lim
--
r
A-.0 Z L
1(
,/--al
!f.
1+
TF
-
J
-Z
di
.,
Clearly, this way of parameterizing a binomial model is consigtent with the notion that the events that drive the S; ovcr various nodes of the tree are iinormal.'' These events occur frequently, cven in small intervals, bet the.ir size is small.
8.4 Tlve Bchtzl?or of Accumxlzted Clzangel
r '
The discussion abovc dealt with possible ways of modeling
.g
,
:
There is another interesting question that we can ask: Leaving aside the one-step changes, how do the accumulated movements in Si behave as Passes? ln other words, instead of looking at the probability of one-step changes in as increases, we might be interested in looking at the behavior of
(f the binomial sctup, except : N()w wc keep the same characterization changc thc way &j, d j, an d pi are modelcd. In particular, we change the depcndcnce on the time interval A. nus, in place of Eqs. (74)-476) wc assume that the parameters () f tlze model are now given by'. (78) . bli for al1 i, .
Gtime''
'
: rp
ui
;:
di
=
Ld
-%i+n,
:
,
ecA
(79)
for al1 i,
?drrhis is not the only choice that will characterizt
'snormal
A-f
,
.
.i
:j'
evepts.
'
'
'
EE
for some integer. ehangesin after uh
.;. :.
.
!
the proba-
bilityand thc size of a discretized two-state process Si s a fuction of the discretizationinterval A. We were mainly interested in what happened to one-stepmovements in Si as is made smaller and smaller,
2
3 Ruge Ntnts
=
(yo
.:
? 8
e a,
with close to zero. Clealy, this way of modeling the binomial paramcters is more in line with the rare event characterization discussed earlier in this chapter.
'
(7-p
=
(83)
't
.
=
,
:!
y!.
(80)
Gup''
%
.
for all i,
,
Ktdown''
'lhis
.
=
is chosen as!
,
-
;
First, some comments. The parametcrs ui, di, and pj are chosen so that i' is thc case because on the right-hand thcyare the same at every node i. 'j i. According side of Eqs. (74)-(76) there is no dependtmce on Si, i 1, ' of f hion is homogeneous in Si is discretizcd that to this, the dynamics a as #F, or pi across time Clearly. this need not be so, and more complex ui, modeled is kept in (74)- ) as can be selected as long as the dependcnce on .'1!.: (76), nus, in this particular casc we can even remove the i subscript from . .! uj, di.' what and happens note Second, to more imptmant for our purposes, parameters u;, di and pi as goes to zero. From the dehnitions of these parameters we see that as .->. 0 thc ui, di go toward zcro. Hcnce, with a parameterization such as in Eq. (73),the k movements in Sf become negligible over infmitesimal intewals. Yet, the LL. i probability of these moments go to 1/2, a constant; =
movement
'size''
..
(76)
Wup''
where 0 < A and 0 < a are two parameters to be calibrated according to and probability of jumps that one is expecting in Si. The # 1 the is also a positive constant. It represents the behavior of Si when there is a jump. di is the case of no jump. Corlsider the implications of this type of binomial behavior. As tbe time interval, is made smaller and smaller. the probability pi of the aup'' state will approach zero, whereas thc probability of the state will approach one. This means that Si becomes less likely to exhibit changes, and .->. smaller consider As smaller time intervals. 0, the Si will , as we follow a stable path during a tinite interval. Yet, even with vcry small there is a small probability that a event will occur becausc according to (80): eas.s i ) 1 A., Pro b (<j+j (81) which, depending on is perhaps very closc to one. nis is the case because in small intenrals'.
C
:'
for al1 i.
,
',
(75)
,
(sx/-j /f.
?j
i
for alI i, e and the probability pi can be chosen as: =
and the probability, pi, of an ;
for a11f
r
-cvZ
events, the growth
ddnormal''
(:4)
1, which, in a sense, rcpresents thc accumulated n successive periods of length / has passed.
rI
>
:'*.* .-'..' .-.' .111* .11* I....lll....-d .l.d -.'
200
C H A PTE R
*
The Wiener Process and Rare Events
8
:
ts Rare and Normal Events in Practice
20l
.)
('. First, some comments about why we nced to investigate the behavior of such a random variable. Clearly, the modeling of as a two-state process may be a reasonable approximation for the immediate future especially if the is small, but may still ieave the market practitioner in the daz'k if the trading or investment L horizon is in a more diatant that occurs after n steps of length For example, the interest of the market profcssional may be in the value of ! %., t < 7) at expiration, rather than the immediatc Sf, and tbe modeling of I immediate one-step probabilities may not say much about this. Hence, a market professional may bc intcrested in the probabilistic be- lt havior of the expiratitm point value Sv as wcll as in its immediate behavior. And the probabilistic behavior of thc accumulated changes may be quite different than the #;. that govcrns the immediate changes in i. l n1s 19 tlle because in n Peiodsr thc Si may assume many Values different from .Li case just u 2.Si or dui. . Thus, we considcr the probabilistic behavior of the ratio: .
With a linear equation we can calculate the mean and variance of the Si random variable log J. eas jjy:
.
a
t
k
ui
.
''
E log
,
.flzpglrc
.
Ftzr log
F'(7r Herc, rcmcmber that
.z'
jr
'::' u
::
)
=
.
&
= Z log d + -
Z) log (/
zs ltg
d.
(%) t .
E
:
(87) h
As discussed in thc previous paragraph, this last equation is rlow a linear t function of thc random variable Z. 1og converts a produc! into a O'I'he ut, 4 are multiplicative parameter's. Taking whichis easier to analyze in asympotic theon'. Central limit theorems ure formulatei general, in terms of sums.
m
'
:
nptl
- p).
(92)
F
=
-
.
g.r
(94)
(rl'l
(95)
,%..
-
value is easy to calculate. If wc have rl independcnt trials, each with a tlacn he total number of apected movements will be np. The Vrf Z) is slghtly variance of z ftyr a singlc trial is: morc cornplicatcd. 'tup''
'ne
pl 1
.
.
..
j
:
'
(91)
'tup,''
F n trials the rlpt1. p). -
'E'
in
log #
b-l-heexmcted 7rtbabilityp of
.;:
:
+
-f
'(
= I
2
=
:'
'
lhc
=
p).26
One can also get the approxmate (asymptotic) probability distribution of 1ogStwh/si. First, note that the log Si-vn/ui is in fact logarithmic changc in the underlying process: S-vn 1og log Si-n 1og (96)
,
(n -
jog
u pj
xf-q.r sj
-
This is equivalent to a process that takes steps of expected size JJ.,' over rtl, F1, and whose volatility is equal to (Z'UX at each step. Hence, the mean and variance of the rate of change of Si modeled this way will bc proportional to ,'. suchstochastic processes are called geometric processes.
'.
Z log u +
(89)
g grjy;j n
2
:.
=
l
Iog
=
N
$
'Si-n
lerlzj.
logj(,lJ?l + n log d pu1 n#(1 - p4 A glog
=
log S;
-jj
(88)
2
t.f
=
+ n log d
(93) k Replacing this and the values of u, d, p in both (91)and (92)we can get the asymptotic equivalents of the mean and the variance. In other words. with u, d, p, given in Eqs. (74)-(76),the fkst order approximation gives:
' r.
which depends on the way the main paramctcrs of the binomial-tree are modeled. The discussion will procccd in tcrms of an integer-valued random ). variable Z, which represents the number of '' up movements obsen'ed beE twccn points and + n. According to this, if beginning at point i, Si expericnces only movements, then Z r!. If only half of the movements n12 !' and so on. (.: are up, then Z We investigate the probabilistic behavior of the logarithm of Si-vnlsi, j L instead of ratio itself. because tMs will Iinearize the random cffects of uj, yrjri in terms of Z.M Before we proceed further, we eliminatc the i subscript from ui, #i. rf ( given that at least in this scction, thcy are assumed to be constant. We can now write; Gtup''
Si-v''
n
..
''
sj
S-vn - Si
E
'r::
(85)
glog gj glog
u pj Flzl
=
But we lnow that the f'gZ) is simply np and the rtzrgzl is npz Rcplacing these!
''
S.s-l-zl b.lq:
Si-vn
?:
.
.
varianc.e
-
.p)2 +
(1 -
p)40 - #)2
of the a indcpendent
=
p41
lmovenlents
-
.p).
(90)
will be n times p(1 - p),
r
E' ;
.
'j'
(--1.IA p'r E R
2c2
-
Tle
8
Rare Events
Process and
wiener
It can bc shown that if we adopt the parametrization in (74)and that corresponds to normal cvents, then the distribution of (log 0 by is approximated, as fr2), N(>(nA), glogSi-vn log-/j ks/.j-u-logl
10 Exercises
;
:
(75)
(c)
.
.; '
-+
-
-
2o3 ,
(1 -) -
n
1
-.s
Ixt the random variable X,t have a binomial distribution:
(97)
n
y
That is, (logs-h. logs'/l is approximately normally distributed. ' that lf, on the other hand, the parameterization in (78)and (79) correlogM) spondsto rare events is adopted, tben the distributionof (log given by; j approximately be 0 will as a .
-
. a
Whefe
'
vi-vn
-
Cach
Bi is independent and is distributed according
-.
'
11
,
glog
1ogi/j
-
vi-vn
'w
Bi
(98) :
Poisson.
These are *0 examp Ies of Central Limit Theorem, where the sum of a large number of random variables starts having a recognizable distribution. this divergence between thc applications of central hmit what
yt1 s i' izzz
E
!
with probability p
t)
wth probability 1
=
to
p.
-
we can look at Xn as thc cumulated sum of a serics of events that occur over time. The events arc the individual Bi. Note that there are two paramcters of interest here. Namely, the p and the n. 'T'he Erst governs the Probability of each B i, whereas the second govcrns the number of events. The question isa what happcns to the distribution of X,t as the number of events go to infinity? nere arc two interesting cases, and the questions below relate to these.
C ,
t ?
caeses tbeorems? Of independent .! It turns fmt thata in order fOr a Plrperly scaled sum random variables tt converge to a normal distribution, each element of negligibk. The condition for asymptotic the sum must be asymptotically l negligibilityis exactly the one that distinguishes normal cvents from rare events. Thus, with thc choi of para meters for ui di pi for rarc events, ;; the events are likely to be asymptotically nonncgligjble, and, convergence i, will be toward a Poisson distribution.
Gtevent''
,
.
'
,
(a) Suppose
,
9
'
)
--+.
x,
-.+
=
=
(19+:,no rare events 'rhediscussionthatcharacterizing innovation terms have a snitenumber of possible values ; tion assump the discussion signincantly. x reader interested in the formal 3. simplined justiying t laestatements made in tus chapter can consider tl,e arguments approach martingale dkscuss to Bremaud (1979). sremaudadopts a of poim processes- which can be labelcd as generalizations of . dynamics ,, Poisson processes, is covered iri Merton
now, n
=
.
References
wlle
0 such that A np remains p constant. nat is, the probability of getting a Bi 1 goes to asequencar, of getting a zero as n increases. But, the expected one remains the same. nis clearly imposes a certain spccd of ConvrBence on the probability. k) Write the implied formula What is the probability PrlXn as a function of #, n, and k. k) as a functon of the three (b) substitute np to write prxn 1. tcrms shown in ouestion (c) t-ct n -+ x and obtain tlae poisson distzibution..
.
.,
-
-
,
,.
prxn
,E
(d )
!,
:
10 1, Show that as n 1 ( a) 1 -u
...+
tx
(1 ) (1 (b)(y -,,) -
-
.
A
-
n
-->
.
'k j
k
-
.
c-
sxercises
:
p
1
)
-->
1
,
? ,,
'
.
uemem yej. tjaat jjurjjs
-
k)
=
c-
k!
, .
tjljs jjmjting process, the p --> 0 at a certain speed. How do you interpret this Iimiting probability? Where do rare events ,t in?
r'
1
1 Introtluction Wuw.
G-ww
.w.
''':*v.
x .
w
.
:
,m4X.
expansions. After taking into consideration any rcstrictions imposed by the theory under consideration. one gets the differential equation. At the end of thc agenda, the hmdamental theorem of calculua is proved to show that there is a close correspondence between the notions of integral and derivative. In fact, intcgral denotes a sum of incrcments, while dcrivative denotes a rate of change. lt seems naturat to expect that if one adds changes dXt in a variablc Xt, with initial value A% 0, one would obtain the latest value of the variable:
E *
*
te ratlo
tOc
ln
:
*
!
astlc
,E
e ts
V1rO
=
1.
'
@
.
l
dXa Xt. (2) () This suggests that for every differential equaticm, we can dcvise a corresponding integral equation. In stochastic calculus, application of the same agenda is not possible. unpredictable arriveg continuously, and if equations representing lf the dynamics of the phenomena undcr consideration are a function of such noise, a meaningful notion of derivative cannot be deMed. Yet, under some conditions, an integral can be obtained successfully. This pelnnits replacing ordinary differential cquations by stochastic differential equations =
('
Tlte Ito lntcgral
(' 1
.'
.'t :;
..(.
LKnewf'
; ' .'! i
:: .
;.
'I. .
1 lntroductlon
? '#!..
and integration operae'. One source of practical intcrest in diferentation nelil tions is thc need to obtai' n differential equations. Differential' equations used to describe the dynamics of phpical' phcnomena. A simple linear dif-'t ' ferential equation will bc of the form dXt AXt + Byt (1)
dXt
.L+h- xt
::.i
''
,4
,k'
'''
::
204
r
j
fsatu
=
(4)
l
s;
dSt
ast,
1)
dt + rst, t4 JWt, t Afler we take integrals on both sides, this equation =
l
I0,x). implies that
/
g'tuu, zls'u, u) du + dSu u) Jlzl'u. (6) t3 p Wbere the last term on tbe right-hand side is an integral with respect to increments in the Wiener process 1. =
=
)
(3)
doing
l
'
(0,x),
Now, consider the SDE which rcpresents dynnrnic behavior of some asset price
.
.
f G
St.
E
1If B 0, thG cquation is said to btl homogenous. Wlhen yt s indepcndent of /, the spte,m ; becolnes autonomous. Otizemise, it is nonautonomous. Xt. future path mind ftar 2For examplc, the engincer may have in somc desired nen :,. issle is to f nd thc proper (y,) which will cnsure thtt X( follows this path.
t/@(,
Can be used to give meaning to dXf. In fact, at various earlier points, we made use of differeatials such as dSt or #11$but never really discussed thcm in any precise fashion, The definition of thc Ito integral will permit
'
-'
n
r+
where dzjldt is the derivative of Xt with resnect to f and where y, is all.. and B arc parameters. 11 exogenous variable. Ordinary differental equations are necessarv tools fol' orac tical mod- .? . eling. For example, an enpnccr may think that there is some variable h' of Xt, determines future changes in Xk values that, togethcr with the past approximated by the differential equation, wlch r'-qft This relatitmship is ' c applications. be utilized in various equatbm differential obtain used the ordinary The following agenda is to First, a notion of derivativc is defmed. It is shown that for most ftlndioilj! of interest denoted by Xt, this derivative exists. Once existence is es F'CI'i lished. thc agenda proceeds with approximating dxt/dt using Taylor ''''
at dt +
.
,
''''
=
wherc futume movements are expressed in terms of differentials dxt, dt, and 4/14,; instead of derivatives such as dxtldt These differentials are desned using a new concept of integral. Fbr example, as h gets smaller, the increments
,
dt =
205
1
+YVVFW'VXYW
:.
: ''
( r.
*'
.:
206
C HA PT ER
.
9
Integration
in
Stochastic Environments
Introduction
.'
.'
The intcmretation of the integrals on the right-hand side of (6) is not immediate,As discussed in Chapters 5 through 7, increments in p) are too erratic during small intervals h. ne rate of change of thc H'; was, on the smaller.3 If average,equal to h -1/2 and this became larger as h became inlinite? would their be not erratic, incremcnts sum these are too nis chaptcr intends to show bow this scemingly dilcult problem can be solved.
rcasons- First, the Etfst-vk 5)Jwas set equal to a jrt-order Taylor scries approximation with respect to : -
;
,
Etst-k (
,
second,the
'
asu,
Ito lnterul . 1 The
and
:'
EL
dst
i'
'it
g.t,s,
;y)
l ''l-
esu, ;
,
tJL,
.
'
,.
:
ysu, ;
uj lrzj', ,
(1g)
-
Taking the intcgrals in a straightftarward way, we would obtain the difference approximation: /)J? + ojz,
r)(p)+, -
.p;j.
EEfIiv
E .
:
. ..(
(1(9'' t
aLut,t4h
+
G(k,,
fl.f#;.
.
(11)
This is the SDE rcpresentation fn finite intervals that we often used i# previouschapters. The representation is an approximation for at least twe
:L(7
',yA; mde ocbangc we mean tue staozard dewaton o .;., - ;,; divded oc avera,e standard deviatons In Chapter 6 it was shown that under fairly/IL/'general assumptions, thc ). Unpfedictzble shtcls Werc Proportiona1 to
ssy
ne
.
>
...
''
'
.
f
: t. :.
.(
zl/rp;.
(14)
u .Practiee :1 the 1to integral is used less frequently than stochastic differemial equations. Practitioners almost nevcr use the Ito integral directly to calculate derivative asset pris. z.ts will bc dismzssed later, arbitrage-free prces are calculated either by using partial differential equation methods or by using marungale transformations. In neither of these cascs is there a jxkz . caleujate any Ito integrals drecuy. lt may thus bc difficult at this point to see thc practical relevance of tbis concept from the point of view of, say, a trader. It may appcar that denning the Ito ntegral is essentially theoretical excrcise, with no practical a implications.A practltioner may be willing to accept that the 1to intcgral existsaad prefer to proceed directly into using sDEs. reader is cautioned against this. Understanding thc delinition of .tbe Ito integral is important (at least) for two reasons. Fjrst. as mentioned eartier a stochastic diferential equation can t,e desned only in terms of the lto ititegral. To understand the real meaning behind the SDES, one has
..
?
6
u) J')p;zz v(-$,,
1.2 The Pxucf-ictzl Rclewtmce OJthe It() Infegrul
.
:$
side is deEned in the lto sense and
.measurable
'
&
;+/,
,
.
Rcwriting.
tjgj
That is, the diffusion terms of thc SDES are in fact 1to integrals approximatcd during inlinitesimal time intervals. For these approximations to make scnse, an integral with respect to J#; jy jhrst jd j)e Jejined formally, Second, we must impose conditions on the s ou way ast t) and ojS(, t4 move over time. In particular, wc cannot allow ty cse y.t parameters to be too errac.
(8) g(
j :, intemal. linite time h is where some t//-crczicl7 apprimation that wo From here, one can obtain the hnite . k smali, and h is lndced, if 8. several Chaptcrs times in 7 used auu, uj and , 1.f) (rt-u, may not change vcry much during u E (/, t +. J,cspecially if they 1 are smooth ftmctions of Su and &. Then, we could rewritc tbis equation a? t+k /+n JFx du + abt, /) (9) l r) au l 1'+ /1 k E :
s-vs st < az,
Jjjt,
2+
7.1
0'6Su, u)
alSo, u) #a +.
-
(
,',
/'-.1-/1
=
crqvhg t)
'
f.ho
:.'
.t
ayy tj dt +
tjw second jntegral on the rfght-hand that as h --> 0,
:
(p)
tju
precise way, then onc could integrate both sides of
St-vh-
=
..
0
is defined in gome soE in (5)..
(
j
I
(St, tjh.
wjl jn fact mean that in the intqral equation, t+h a/j ay, u) du + du 'z =
,
Obtaining a formal defrtititn of tlle Ito tcgral will make the notion of a stochastic diffcrential equatitm more precise. Once the integral
=
u), tz.t,5k, ula u + ) were approximated by their 1. Both of these approximations require some smoothness value at u conditions on asu, u) and u). AIl these imply that when we write su,
;
SDES
)1 e g/,f
=
'
1
-
.
E'
c H h P z ER
208
-
lntegration
9
in
Stochastic Environmens
of the Ito integral. Otherwisc, errors can be made in applying SD& to practical problems. This brings us to the second reason why the lto integral is rclevant. Given that SDES are defmed for inhnitesimal intewals, their use in linite intervals require some approximationa. In fact, the approximation in (14)may may Thcn a new approximatitm will have to be not be valid if is n0t delined using the lto integral. This point ig important from the point of view of pricing fmancial deriva(IOCS Calclzlations using tinite inteNals. tivess gince in Pradicc O11e aiWal's dayo is Ciearly nOt an inhrlitcsimal inten'al, and the utiFor examplep lization of SDES for Such periods may require approximations. The preciK approximations will be obtained by taking into consideration form of these of lto integral. the delinition To summarize, the ability to go from a stochastic difference equatioa defined over the linite intezvals,
to havc some
a-spakk
k
u,-
+ h
k,
1, 2,
=
.
.
.
,
(2&
ast,
#f
1)
ojsu
4.
,
JW';,
t
e
'
E ,
y, 7
2.1 Th e Riemunn-:feltjes
.
L' .
and vice versa, is the ability to interpret (/H?;by dehning Jj in a meaningful manner. 'rhis can only be done by constructing integral.
In this partic-ular casc where the derivative sticltjesintertl can be written in two ways:
, u) #1Fk'97 a stochastik :
z .
.? ! '
'
..1 f
(;
/ (x;)dxt
t
rnw 1to integral is one way of desning sums of uncotmtable and unpret dictable random increments over time, Such an in tep-al cannot be obtain by utilizing the method used in the Riemann-stieltjes integral. lt is use ! ' to see why this is so. rando As seen earlier, increments in a Wiener process, lpl rcpresent variables that are unpredictable, even in the immediate future. The value the Wiener process at time f written as W(.is then a sum tf an uncfmnt.a ) number of independent increments: r 0
a
.
0
(1 4
.
/
'
f) (Remember that at time zero, the Wiener process has a value f zn d writc Hence, Hz;) 0.) This is t he simplest stochastic integral one can
i
.'.e
l
).
(20)
gxt) dyfxtj.
(gj)
Rere, we have an interal of a ftmdion glxt) taken with respect to F(.). Kmilar notation occurs When we deal with expectations of random variables. l7or example, F(.) may rcpresent the distribution function of a random variable xt. and we may want to calcuLate the expected value of some
=
.:
dFh
F
.
tj)!/
=
jj
'l
,
,
16
=
exists, the Riemann-
ittegal, ln the notation on the right-hand side, the integral is taken with remect to F(.). lncrements in F(,) are used to obtain the integral. Wc can complicate tlle latter notation further, For example. calculating we may be interestcd the integral
.
l
v
.
/(.)
'rhen,
'j '
,
(19)
The integral on the left-hand sidc is taken with respect to xt, where t varies from () to value each the multipleu of at is by the x, y(.) w, ieuitesimal increment dxt. nese (uncountably many) values are used to Obtain the integral, nis notation is in general presea'cd for the Ricmann
:;
'j
s
) jjxt). =
dx l
,i
2 T e Ito lntegral
Integ'rul
Ljyjyt
'
l+h o'v
(18j
.
we have a nonrandom fundion F(x/) where xt is a deterministic suppose variable of time #(.) is continuous and differentiable, with the derivative
(16)
(0,
X),
u) Jp'x
7
..: :
',
f)
by integrating the inno-
ne jntegrals n (17)and (18)are summations of very erratic random variables, since two shocks that are e > 0 apart from each other, JW/Iand (yjjj nl.f are still uncorrelated. The question that ariscs is whethcr the sum of such erratic tcrms can be meaningfully defmed. After all, the sum of so m aa:y erratic elcments can vely well be unbounded. (uncountable) again the way standard calculus desnes the integral. consider
)
+ rrst
,
()
L'
stochastic differcntial equations, =
integral is obtaincd
.
(15)7
n,
Integral
?
tone
-
ICo
A more relevant stochastic vation term in the SDE:
understanding
Rsmall.''
to
2 The
.
:
1.
<
:
C H A PT E R
*
2 The lto lntegr'al
lntegration in Stoclastie Environments
9
L.
.
glxt) for favd r:4
.) is the sum of alI such rectangles. If consccutive ti, i 0, n arc distant each from other that is, if have partition of we not very a ((), 'J-) -this approximation may work reasonably well. ln other words, if the ftmction g(.) is intcgrable, then the limit
;
=
.
.
.
,
.#z;c
F.(-r,)1
',
(22) !
!.(.;r,)dljxt).
=
-x
Heuristically, in this integral, x; is varied from minus to plus infinity, nnd the corrcsponding valucs of g(.) are averaged using the increments in #F(.). #F(.) in tbis case reprcsents the probability associated with thosc values. in (21) and (22). Note the important difference bctween the intcals value of x/ for a t' The In the lirst case. it is the t that movcs from 0 to F, . variable. weil random unspccified. could be a It particular f is left very ( would make the integral itself a random variable. The integral in (22)is quite different. ne t is constant, and it is xt thnt random variable. goes from minus to plus inlinity. The integral is not a variables in tle picture the :L F or the case when there are no random Riemann-stieltjes integral was dcsned as a limit o some infiltc sum. 'Fhe j' integral would exist as long as this Iimit was wcll dehned. To highlight (111- :E: methodology on E fercnces with lto integral, we rtwiew Riemanntieltjes 't again. Suppose wc would iike to calculate ';
n-1
lim supj.I -?'i I
:
..=v0
.
+I
f 0
F(.xj)J
r
(/#(x/l
gxt)
=
-
=
?
o
(26)
will exist and will be called the Riemann-stieltjcs integral. ne readcr should read this equality as a defiriition. ne integral is defned as the limit of the sums on the right-hand side.s The sums L are called Riemann
rrhis
.
gx,,.-, )(F(a7rj+,)
:'
.'
6
Su1lS+
':
'
'.
,
2.2 Stoclusfic lafcgrutm und Riemcnn Sams
'
'
Hence, the value of the Riemann-stieltjes integral can be approximated using rectanglcs with a base and varying heights. Can wc adopt similar rcasoning in the case of stochastic integration? We can ask this question more precisely by considering the SDE written over hnite intewals of equal length h11 (dsmall''
'
E. '
:.;
.
T
0
)
Ixt )/F(-r/
.
'
.L'
'
Sk
i 6.. 'The formal calculation usiog the Riemann-stieltjes metbodolor based on the familiar construction where the inten'al (0,T1 is partitioned r into n smalkr inten'als using the times
Sk-l
-
-,-11
f.
'''
N-
P;,
= =:0
!
)LF(x,,.1) g(.;t-ri+,
F(.rt..)1,
-
:(
ne
right-hand
:'.
7>
k
jv
..(,
)
side of this equation is a sum of clements such as
-
+,
.
)
-
F(x,..)1,
EIJ(sk-:
'll
-f-
,
k=l
J&
=
rl sl.-yilpxa :)7.j(tg(5't-1,
,
,
n.
(27):
t,'(-$,,-1, J'''I k=1
klEafK,l,
(28)
:1
kll
+
:
J'''ltrtus-j
k=1
(25
whereas
.
2
and (F(x,,., ) F(xj)1. The srstterm re which is the product of The second term rcsembles the point ted at g(.) l xt a rescnts eva ua g(.rto,)gF(a:2.,) F(xlf)1 can be visu elemcnt dFlxij. Eacb crcments g(x2j.I ). as a rectangle with base (F(a';,.( ) - Fxti )1and height
.
n-1
: (1
.
.
Sk on the left-hand side of
.'..
I (..r, ')IF(.x,,.
1, 2,
=
a limit
(24
f
kllJFk,
,
Can wc use a methodology similar to the Riemann-stieltjes approach and defzne an integral with respcct to the random variable St as (sometype otl
k
.
(rsk-j
,-1
l-lf-sk =1 -
$!
Thcn the finite Riemann sum Ji is deined:
k4h +
.l,
n-1
'
.
au
Suppose we sum the incremcnts
';.
(O)
=
usual, it is assumed that F
,
kll/.&B$1
,
(29)
nh'?
=
'
.g(x;f-,)
-
'
s'That is, if this limit convergcs. are Inauy different wap rcctangles can approxirnate tllc arca under a tmn'e. one t'an piek the zectangle lhr same way, but change the height of the rectangle to tlase of tlle +.x'i - eithet 1l ). ., (.vti ) or to #( a ?By considering intenrals of equa: Iength, the partition or ) Tl can be matle fmer with & x. Ot/lerwise, the conditio.n sup.. 1I/ - t.-,I 0 has to be used.
,
.
+,
.
G'riaerc
-
4Whcn the function #( second or third moment.
.) is the
squarc
or the
(n.11x,
.
:
.k '
of xt this intcgral will simply be ..? ,
:.
(.: ;.
(
iz'i
.
' 't... :
->
-...
.
.L .:
:
L
l
!
..
2: ';
.
C H A P T ER
212
*
lntgration
9
in
Stochasric Environments
2 The 1to Integral
'
@ 'J
21J
J:
'
Erst term on the right-hand side of (29)does not contain any ram dom ttrms once information in time k becomes availabk. More impon increments in time h. By deftnitantly,thc integral is taken with respect to r zt ' is a smooth function and has finite variation. nis means that tion, bme the same procedure used for the Riemann-stieltjes case can be applied to de fine an integral such as8 nc
55.
0
'
The use of mean square conveqence thc sum
11 ''
j
n
1: '
p
i)
a(Sa, u) #1,F.
'''
aS
1: 1
ujdu
lim
=
n ...s x
(31) '
lftxs'k-l1)1. ,
k= 1
; .E.
tj
.:.
jjm
,
'.'
term
Ep'k
-
prk-1
.G (32):
1
:)1pk
H,',-1J
-
:::::1
(33t
:.
to a random variable.
is an integral with resped We can ask several questions:
;
Sk-Sk.j
.
,
* Which notion of limit should be used? nc question is relevant beca the sum in (33)is random and, in the limit, should converge to a r dom variable. The deterministic notion of limit utilized by Rjem Stieltjes methodology cannot be used here. the s . Under what conditions would such a limit converge (i.e.,do in (33)really have a meaningful limit)? What are the propcrties of the limiting random variable? .
: E (.. '
.
.j'
#li-l
,
k=1
1.
(
side ca'a bc written
-'it,-
-..
T(z(-(,-I),
'= 1
kllltl
,
kljp'k
t) be nonandcipative,
aru
.(j.
g-yj
ltt Ztegral within the context of
- pk
..jj,
k
r:,.
1 , g,
of the
.
.
.
,
n,
(:!y)
f
in the sense that they are inde-
) be
Rnon-explosive'':
T
E
gjuj
aen
,
0
.
the Ito integral
t)2 dt
<
g9)
x.
W
f)
U'st, ()
(
L)
Jl''l'),
(40)
s the mean square limit,
'
.
?-
in more detailed lbrm as
n
-
k - 1)/11,
. as .
.
., ..y,
1 . :K . .'
llrK: pk-lj
i.
-.
. ..
'
.
l .
.
y
..o x
m--.---
jy ti--t
crst,
-.+
-
k=1
..'. s
y
tr(&-1,
J1
,,
,
....
This integral would be a random variablc.
lirn
kjh...qtwty%-y
'.
tz'tk%-:kllH' -
Of thC
pendent of the future; and 2. the random variables tEr(x$,,
.k
tl
u)
we
1 the o'kh
,
*
0
,
The Ifo Integ'rtzl
Jtkk-j,
=
,
:
.
tt
wherc (H,': J-.j j js a standard Wiener process vith zero mcan and variancc let
.
s
We limit our attention to a particular intcgral determined by the e ssib ter'ms in the SDES. R turns out that, under some conditions, it is po rand of the the stochastic limit in integral rzzcczzsquare as to define a
8Th: sum on the right-band
k=1
ys.
.
DEFINITION: Consider the finite inten'al approximation stochastic differential equation
.
,
jy) ju .pj,.j
,
We can now provide a definitim stochastc differential equations.
y
zrlsk-L,
(pt,s-y
2.3 Denifm:
' t
.t.
u
s
.
is a random variable, and the sum #
p-..x
2
T
''
-
,
(36)
has a variance that goes to zero as n increases toward inhnity. Formally;
Howevcr, the second term on the right-hand side of (28)contains ran- : 1 tlmf dom variables even aftcr Ik-j is revca 1e d In fact as of time k .
(35)
and the random variable called the 1to integral,
z
:1
H1-1 1
-
1=l
'
'
k)(T#-:
a'sk-L,
Lt
implics that the difference betwecn
t) gp),
()
..o (; jy -
!(,,w,i-
p-xiti---d
i-t-
,-
--.?-,i-t-,-,-I-,
wit,
w
-
,-,,
(41)
;'g'
C il A P T E R
2 14
.
lntegration in Stochastic Environments
9
'nae
)
r'
Ito Integral
'
According to this definition, as the number of intervals goes to inlin and the length of each inten'al becomes inliniteslma - l tbe finite sum w.ill approach the random variable reprcscnted by the Ito intcgral. Clearly, the definition makes sense only if such a limiting random variable exists. ne assumption that c(&-:, ) is nonanticipating turns out to be a fundamental ) condition for thc existence of such a limit.io To summarizc, we see three major diffcrences between dctcrministic and stochastic integrations, First, the notion of limit used in stochastic integra- ), tion is different. Second, the 1to integal is dehned for nonanticipative fkmc- ; tions only. And third, while integrals in standard calculus arc dehned usmg' the actual i'paths followed by functions, stochastic integrals are desned ; within stoclmstic equivalence. lt is essentially these differenccs that make = some rules of stochastic calculus differcnt from standard calculus. .j) ne following cxample illustrates thc utilization of mean square conven i; gence in delirting tbe Ito integral. In a second example, wc show whv the .1 ,, Ito integral cannot be defined
215
f(x,) ftxtl-x.
''
,
Equal Triangles
: l
:
.
'1 'v
I
l
-
: $
X
:
'ktz
tl
-
FIGURE
L(E '
(t
where, as usual, T deline the sums
1, rrhe Ito integral is a limit. lt is the mean square limit of a certain ; appropriate exist, the Ito integral sunw for in order to some sum. nus, ' 'hnite
.
,,
must
W.
l :
i
2.4 An Expositoky Exumplc
:
r:
:'.
-
,Q.
i
''
dTpathwise.
: 2
converge.
=
nh and for any f,
/j+!
//
-
/1.12Second, one would
=
u-i
.r,
Given propcr conditions, one can show that lto sums converge and that !, Ito integral exists, Yet it is, in gencral, not possible t, the conesponding aplicitb calculate the mean square limit. nis can be done only in some sI)ecial cases. ln this section, we consider an example where the mean sqlmm 'J(7 limit can be evaluated explicitly.ll i Suppose one has to evaluate the integral ) z' Xt dx ?, (42) 0 g ' 0. Where it is known that x() ' If x, was a determlnist ic var iable onc ctuld calclzlate this integral' usinr the Enite sums delined in (24).To do this, one wou ld lirst partition th@. in tewal 10 F1 into n smaller subintewals all of sizc h using
L
'1
=
.x:,
-
f=0
LC
xjq., (.x(.,
(44)
j
and let n go to inlinity. The result is well known. bntegral of (42)with xc 0 wll'l be given by
'.
=
T
:.
x t g.xt
=
1 -x2
g z
ne
Riemann-stieltjes
(45)
.
'Ihis situation can easily be seen in Figure 1, where we consider an azbitrary function of time x Z and use a single rectangle to obtain the area under the
?
'
L'kn'e. jg
=
lf x ! is a Wiener process, the same approach the L'a must be moded to
,
*11,
cannot
be used. First of
'
n-l
,
to =
0
< /': <
-
.
.
<
K
=
Pk
(4:.9.'
z;
,.
f-o
,
$2
Zteg!'v 10One technical point is Whcther the lim iting random variable, that is, the Ito choiO shown 7'1. the hat depends bc It can t. on thc choice of how tlne partitiorls the (tl, parltion dOeS IlOt infueFlct tbc Value Of th() ltO tegral. ,
-
-
'
11
This is in ctmtras
to a prt:t f wilere it iS
!lIROWn
that the lllit
.'-
'Texiyts.''
.
': ,.
,
.'
:
-
(46)
lzEqual-sized subntemals is a convellierlcc. ne same result can be shown with unequal ti as well. 13A 3jyj# j; rectangjo works because the function being integratcd, flxt ), is just the Vs-degree line f(x t ) xt.
(.4.,
.
xb.(;r,.+2 .,:,f1-
=
=
1
T! @
.: '
.
.
(2 H A PT Jf(R
216
Integration
9
.
in
! i' E
Stochastic Environments
2
The lto Integral
217
i. 1
'
I
.
which gives!
In other words, the first xt has to be cvaluated at time ti instcad of at xt tt.t- ), because othemise these terms will fail to be mmandcipating. wll1 be unknown as of time ti, and will be correlated with the increment .. one could use 5. (x,. x,,1. In thc case of the Riemann-sticltjes+ integral. + type o f sum and still get the same answer in the end. In the case of either intevation, resu ltswill change dcpending on whether one used stoclmstic As will be seen later, it is a fundamental condition of the Ito. ' x,. or xb. that thc integrands be nonanticipating. integral 1.$is second, now a random variable and simple limits cannot be taken. In taking thc limit of L, one has to use a probabilistic approach. As menj tioned earlier, thc Ito integral uses the mcan square limit. !E Thus, we havc to determine a limiting random variable Pr such that '.
'l'he
1 2
'
jzrl
f+j
.
=
''-1
R-1
y: c
.
rl'' l
g,
zj .
.
.-I
yyja
j uuu ()
i az:()
(g4;
.
f-l
,
i u=-()
-
the xt)
.
+ ,
the first and sccond summations in (54)are the samc except for tirst and last clcments, Canceling similar terms, and noting that
XOW
,
Veor
() by dennition,
=
,4
'
p-n
.
2
(p-u- Je'l
lim E
D-->
pa-i
(4D. '
r;-1
limE
x/,x/..
N'-Km)
.,l
,' -
jjm
t
l''
-
=
s-sx
..j
(48)y
0,
::ct
=
x,.
*1+1
x. .-
v. . *.
Below we calmzlate tMs limit explicitly. ''f
'Fhis
:
cquare
Ward
be a good candidate for Z. Takng expectations
Will
:
x2
;,.,
ab From
(44),and
lctting a Je;
=
-
1
2
=
1 2
-g(J
=
i=
and b
x
n- 1
+
ll.rtf+
b)c =
xl,+1 .
xri.., )z
-
..-:: :
gives
.'
c,
A.Y/j., :)-
(5zg.
But
xt. 1
-%'Ii+,
,
'
(5
,. .
:
:2
.2 .
t *'.
't 1
(t +. j
jj
.jy
(,jjy;
..
=
...
E j=()
?. 'h .:
'4sy
.:.. ':
.
constructon
fj)
=
z, we can
T evaluate
(59) the expectation
yv
,,-., ...
;, E
A
MA
f=0
,
k
a:,f+, =
E
(j+j -
this as a candidate for
/1 ' -- 1
tty
zs
+
x ow using
:' 4 -
f=0
to
j=o
-
xtiz,
=
Fl ..j
gj
a z - b z 1,
-
(+, j
2=0
f=tl
which simplgies
,,-1
E'j.,$xz
=
>.
.
OC
,:-1
u-1
E
(5
in a straighttbr-
Way,
:.
a + b) 2 = a z + yl + ga pp,
(57)
.
jx()
Limit Explicit Calculation of Mean random variable #' stcp by step t4': intcnd calculate the limiting to we clarify the meaning of the Ito intcgral as a mean square limit of a random'' ' sum. 'T'he Erst step is to manipulate the terms insidc P's. We begin by noting that for any a and b we havc ;J
2 4.1
2 A.rrk..!
j .
.
(56)
(j.
j
D-
E
:
.'
=
.
(49)yi!'
.
*z
a.x2 t+z ..z
f+l
.-b.L?
.x/ih1
i'=()
2
''squares''
: i.
whcrc for simplicity we let
s
1
Ij.j tjafs exxresgon, there arc two on te left-hand side, onc s due to the random variable itsclf, and the other to the type of limit we are using. Hence, the limit will involve fourth powers of x, First, we calculate the expectation:
:.
J=-fl
(55)
.
kxa
:
2
'Vl
In other words, we now have to find the Z in
.q
Or, equivalently,
a.v2 w,
-
-r'
Note that xv is indepcndcnt of n, and consequently the mean square limit of I'?k will be detcrmined by the mean square limit of thc term
.
0,
=
,,''1
1 xz 2
.-
4 x t .j. 1
t.
=
a. .j.
c z'.
,,.:
j pj
i=
j'-xg
(6n)
,,..j
jkwjz
jjax;a
+ g.z .gz y..f-,j j
z aayy i.v y
jrxt)
,
r
!
j
.,
i!
.
i
,
;'
E
'j
c HA rT ER
218
Integration
9
.
Stochastic Environments
in
-x'z/
:!
.
f+
j
l
The Ito Integcal
-
.
webtain
l)2,
3(fi+: -
-
I.. 1
Pk
.
t
Ef-('
.
l-1
2
r;-1
J.*2
E
li+l - F
f=()
=
f =p
n-1
+
i
+F Now we use the fact that (.+: ti size. We bave the following!
same
2
(f+l rl - 1
- 27-
f=0
- 3(f +l
fj)2
-
=
/./)
-
Tjle tenn on the right-hand
(63)'..
n-1 a-1
2
J=0
/j)(/y+:
j
)
-
-
'
we see
'
...'
'.? '
''f
r ()
(64)
'
,
nn - 1)/12,
J
''
(65) .
variable 2llxz F
.1
=
-
-F2
=
.-nnhD
(66)
.
-
zl-1 .';,
jjm E ''*X
2
.(
1
T
-
i+ !
1)/12 3nhl + nn -
=
hl
-
which means that
(67) r .
f=0
kt '
2
Fl-1
E y=()
This implies that a,s n
.-.+
Ax2 F (-1 -
tx),
=
(O)
2T.
2i,
-
I
F
=
1im lhT h -.+ n
=
0.
'
.
F
(73)
,
.' '
- F
=
().
(74)
w
a (tsxt)
(75)
,
()
which can be interpreted as the sum of squared increments in xt. lf this integral exists in the Ito sense. then by dcfinition,
'
2
kx, i, ()
lnhl
thc size of the intewals will go to zero, and
n-l
E lim l-.+0
=
-
2
hx2I
j=tl
t)
.'.
1 c xz 2
lt is interesting to convert this into integral notation, Assume that xt is a wiencrprocess and consider the integral
4
,
n- 1
.
r'
,
-
r).
.E.':
Put a 11these together E
2- 27- j=0 (fj.j-j /j)
=
2.4.2 An Jakywrllyu/ Remark I F1 thc previous section it was shown that
l
'
rI-1
xtdxt
In the case of the Riemann integral, there was no additional term F, This is one example where the Ito integral can be calculated explicitly using mean square limits. We 'Iind that the Ito integral is the limiting random
E':
,
F
that the ,to integral rcsults in a diffcrent expression from that in ne Ito integral is given by
standard calculus.
'
an (j
(72)
xt dxt.
()
;
(/j+,-
side is the Ito integral,
. Ei
lf).
(7j)
.
T
=.()
f
=
. 'E
(l+1-
(70)
,
1 -L x2z - T 2
F(p;Ij2
/.
?nhl
2 A-r/.a. =u0
:-+CM
Et
1
N
ff)(+1
h. for all i, since al1 intervals are the
=
Ijm
'
;,
j< /
xv
-
.
;.
z-1
=
rl-1
2
''
g, 3(/j+j - t;j
2
2
.
..xr.+1
.''
o
-
is F
jq.j
wen-1jind the mean square limit of p; by using thc mean square Irnit of c just ojxaiaed.
. .
.x2
/.()
.t
=
.
(62)
-1
Ed.
Thus, the mean square limit of Going back to )'.,
!.'
and
.E7E.;r4 1
219
(61) '
(l+.t- n)(/+1 bt,
-
F
.
'
We consider the components of the right-han d s ide of (60)individually. Reltlizing that Wiener process increments are independent, Fl.-rz,
.
.
( :
,E
..
.
.
'
(. .
(69)
N-
.
E ljm -.+
.
..j ..
:
r :
. j
n
:
. ';
-
: '
x
1
fx()
1.x2,f.I
r -
0
2
Ldxtjl
=
4).
(76)
'
?t J .:
C i4 A P T E R
220
lntegration
9
.
But wc know that
z'
in
Stochastic Environments
=
(7X
T1
;
we obtain a result used to working
|
r
2,21
()
aa
Et
h
(78)
#/,
=
..E' .) J
1-
(dxf )2
()
Ito Integral
is a sum of unpredictable disturbances that affect asset prices during an jstewal of length Now, if each increment is unpredictable given the jaformation set at time t, the sum of these increments should also be unpredictable, This makes the integral shown in (82)a martingale #reace'.
'
(74),(76),and (77)together,
Gunusual''
of the
.
dt
0
Putting the equalities that may seem a bit calculus:
3 Properties
'j'
Jlr'.
tra
=
(83)
0.
l
Then, tjw jntegral
L,
). ';
where the cquality hoids in the mean square sense. It is in this sense that if H') represents a Wiener process, for inlinitesimal #f. one can write: j, (l4() 2 #/. (79) :, lt In fact, in all practical calculations dealing with stochastic calculus, it a d%n by dt. The preceding 1: common prac tice to replace the terms involving . discussion traces the logic behind this procedure. The equality should be t in terpreted in the sense of mean square convcrgence. ' j;f ;. '', 3 Properties of the lto lntegral ; Consider the stochastic diffcrential equation
t
'
ty'. twz u
0
:
(:4)
=
becomcs a martingale:
:f
''
t
Es
L
lntegrating
aut
/)
,
dt +
a.X, g0,F1, we
ltelval
this equation over an
g.
T
T
dSt ()
t) d'pl
alst
=
,
f)
dt +
cst
(
1)
,
(80)
,
tj #'Uz),
tion set
,E
obtain
0
.%
<
<
t.
(85)
h.
J. l.1 Case ) Assume that the volatility parameter c'st, of the level of asset price St, and of time t
E
gide is defned in the lto sense. wbere the second integral on the right-hand integral? this properties of W'hat can we say about the
tz'udBi,
0
we consider two cases of interest,
'$
(81)
=
Hence, the existence of unpredictable innovation terms in equations describing the dynamics of asset prices coincides well with the martingale Property of the Ito integral. The condition that ensures this martingale propcl'ty is the one that requires o't bc nonanticipativc gjvcn thc informa-
.:
=
tW?'.
0
.
#.:
w Gu
tj is a constant
independent
,
(y.(,%
'
,
.;
/)
=
(86)
c.
.1
Then tlze Ito integral will be identical to the Ricmann integral and will be #iven by
',.
3.1 Tle lto lntepr u! js u Mxjrtzagzg.
-
PE i
useful lt turns out that the Ito intep-al is a mmingale. nis property is for and theory in modeling the innovation terms of asset prces in snancial practical calculations of asset prices. Models that describc the dynamic behavior t:f asset prices contain ilm of vation terms that reprcsent unpredictable new's. As a result, an integra 1 ) the forml:i
(: :'
r-f.
.
t :5we arc
q'Implifying
tllc, notation
7
I+A
'
by lctting e(Su
,
Ll)
s
(82) ; .;
=
au.
'/ ?'
. j
(87)
Consider a forecast of the integral
.
z,.#p;z
J'EWs-j.a- W?(1.
=
t
)
/+.A
t7'dlt
0
x #p;
t
t g.
0
#p, u
.
J
k '
(yy)
tu,;
() =
p. :
g.
=
(y.(
J#)
-
jf,/ )
j
(gp)
:
'( :
L
(2 H A P T E R
222
9
.
lncegration in Stochastic Environments
=
-
1#)1(JT;
SEG(l#;4
H$) +
-
(zpul
Hi)I(l - W$)1
-
(9p) .;; :.
H$). (91) = rtWziproperyl6 the martingalc We see again that the lto integral bas Thus, whcn tr is constant, thc Riemann and lto integrals will coincide and both will be martingales.
,
r)
=
(97)
of the rectangle, W and then multiplying these by the Riemann sums would then involve terms such as Ha + H$ (H6+A Rbase''
7.
e.
i
-
2
.
t ;
dSt
;.. .
=
dk't
fz'l!l
H$.
(98)
61.
of
.
be of the form
Bill
The innovation tezms in this equation
*.4,0
then the lto integral will be different from the Riemann integral, and using Riemann sums to approximate the lto integral may lead to self-contradiction. This is illustrate.d by thc following example.
-
Clearly, the expectation of such terms is not zero, since the argument .(.) and the base of the rectangle contains terms that are correlated. We consider the simple case where the SDE is given by
'
i:
(92)
(rtt,
'
W6
l#;+a+ 2
as
:.
3.1.2 Case 2 On thc other hand, if g' dcpends on St. which in turn depends on W(,the lto integral diverges from the Riemann integral and remains a martingale, whercas the Riemann intcgral ceases to be onc. F()r example, if the price of the underl/ng asset has a geometric dfstribution with the diffusion term Gtz,
Ito lntegral
Note that the term t7'5(F;) depends on J#; indirectly, through Now consider what happens when we try to approximate the second side using Riemann sums. integral on tbe light-hand One approximation used by Riemann sums uses the values of the Wicner amidpoints'' of subintenrals. nis amounts to calculatprocess obsen'ed at ing lirst thc terms
'
H$)1
-
the
ut.l
where k > 0. This is the case because increments in the Wiener process have zero mean and are uncorrelated', Fl(z(B$+a
Pxoperties of
:,
l+A
tyl'l'ulH''u
.
.
t
('
.
To approximate such an integral with a Riemann sum, a rcctangle with basc B$+a W6an d hcight frgR-''lflj may be used218 z
;
-
Fxumple J. 1.3 Suppose asset prices follow the SDE xdrl
dSt
=
avh
,
f
) dt +
3
a'bbj , 1) #Hz),
0S
(93)
f,
=
J,
j .
::'
:@
(94)
sg('':+ac-''':)(w;+a
!
E
an d
.'!
dsu
lha
B$
#Su du +
=
t
J'
=
t)- l
tA-u
0.
l
'
evsu dll/u
p;2)1w;1
-sgJtw$ta- (99) (100)
and + 0. This means that tlze approximating sum has a conditional exPectation that is not cqual to zero, It ispredictable. Clearly, this contradicts
:'.
.
t-blk
H$).
1 = -,2
,
That is, both parameters are proportional to the last observcd asset price St. and integrate this SDE'. Consider again a small interval of length
lGlteznernhcr
,4$)11,;1 -
EE
(95)
=
( /p)ma-
operator F/(.1 to the right-hand
.
t) ast. zz.(-&,,
p;
g
side,
'(
IJ'S:
w;+ a +
fw
But, applying thc conditional expectation
:
where the drift and diffusion parameters are givcn as
asuh,f)
f+ac.pvadll'u
t'
:
.
(96) .'
R7llere
''I%T
.
i
:, ; :
!C . -.
;'
we abuse the notation
in writing
simplicity, we tsse one rGctangle. canbu used.
.s'(1&;)
=
5't. But it simplic,s
thc exmsition.
In fac, much fmer parlitions of the intewal
(t, z+A1
.? !'
224
c H A P T ER
.
lntegracion
9
in
stochasticEnvironments
'
3 Przperties
an innovation
i
.
the
tcrl'n.
claim that the integral on the left-hand side represents
q
q drf.:t
# ()-
()
(101)
<
In order to presen'e the nonanticipating property of rst, tion of the Ito integral must use rectanglcs such as
g ''
F.
(102)
r)(p;.a r) will, by delinition, be uncorrelated
with t'he incre-
Replacing the 5',+,
=
(103) j ?$.
There is aal additional comment that relates to the samc point. If tho functionsbcing integrated are not nonanticipating, thcn there will be no guaranteethat the partial sums used to construct tbe Ito integral will convergein mean square to a meaningful random variable, Hence, there is an even more fundamental problem than losing the martingale property; the integral may not exist. ne next section discusscs this point brielly. 3.2 Ptahttee
)
.
E 'J
,
5'/ f ''
5'/f
'--h
.
-x-'k
with probability 1 p -
v'-i -s,fi
a
.
vr
.)., .
,
.
.
z.l o y((.))(
K, depends
.
(107)
,
.
.y-)
.j.
on a particular
y(o/-ltx/-l
,j
+
.
.
(j(sj
.
trajectory of St. lf l''s converges, it
.f(.)
ts,, ) =
In other words, thc sign of St,-,
h
.,
-
(j.(s)
hj.
assumes thc value of plus or minus one, depending on
.(.)
-
s jguty
=
j
.
means that a1l ekmcnts in P'u are positive, so
nis
a-1
p-
vf
=
n y-
=
(11())
.
Fuuc ()
using
w
=
nh,
r;
I.z
N%
=
r
(11J )
.
'!.
t
,
(119
'.
s-A
(105)
Clearly, as 0, Ie'swill go to infinity. If such paths have a positive probability of occuncnce, then therlslwfsc sum cannot converge in any probabilistic sense. example is imponant for two reastms, Firsta we see the meaning of a pathwise integral. In calculating thc integral pathwise, we did not usc the probabilities associated with Lut ne in te gral was calculated using the actual realization of the prtcess. n'''e Ito ntegral, oyj the other hand, is calculated using mean square convergence, and the integral is determined wiu,in stochastic equivalence. .->
'lhis
=
,
valuc of
'/'l.
-<-,
can be called a pathwise inteval. lt turns out that there is no guarantee that such pathwise integrals cons'crgejrk stochastic environments. We consider a simple example. in L be given by Let the f'unctions
)
,
,
nh. usual r A pical path of this process will be a sequence of + XS and -N followingeach other. For examplc, a typical realization may look like
l ,/.i
(106)
-,.1-
,
with probability p
=
)E-%,-I
in Je;jwith these observed values, we gct
.
vh
.
(.f(ovA)(
=
t
ln stochastic calculus, one occasionally encountcrs the statement that stochastic intcgrals cannot be defined pathwise. What does this mean .? St,, i 1, 2, Considcr thc binomial process n, measured over discreteintcrvals of Iength during a period g0,Tq! .
-,f
;
!
=
ne
-
,
lntep'uls
-
l''n
) y,
-,
lvi, -V'',
,
tr(&, 1)
J-1f-.,
Suppose P;; is calculated using a particular path for St. For example, consider the path where plus and minus VXaltcrnate:
.
',
tz'(f).
n- 1
=
dSI
f=()
Thc preccding discusgion shows that the Riemann integral is not consistcnt with assumptions made in asset pricing modcls, except in the very spccialcase when
..where,as
P;;
',
f), approxima-
p;),
f.r(<2,
terms ajst,
using a Nnite sum such as:
., ,&,
flslj
0
.
'/
5 1+,'
E/
ments
1'
.!
expectationsl
the herefll'll
Suppue a Enancial analyst has to approximate an integral of the form
'
If such correlations are not zero, evaluating the Ito integrai using Riemann sums will impiy innovation disturbance terms with nonzero
W
225
lto Integral
of the
'
.:
@
u
,
.
L
J
'( 'j:
C H A PT ER
Integration
9
*
in
Stochastic Environment.s
lntegrals
'
:
second, we see the importance of using nonanticipative functions as f(.). the future,'' it anticipated the si> of ln fact, because f.) was able to all made in the summation sign positive and elements the y,;. nat st to an exp loding Je;,as n increased. leid'b'
E.
t
.
E
,
) .
and I
E
;
.
,', :'
..''
mean square to
(j
fvbi fll-ri...,
-
random
variable
J
that we call the 1to
r'
:f . .
)
4.2 Covrenlon Propeuie.
,
: It should not be forgottcn that the lto integral is a random variable. (More precisely, it is a random process.) nerefore,it will have various ; ' moments, ' of the integral of a moment gives lirst the 'l'he martingale property nonanticipating /'(.) with respect to a Wiener process i' ..l 7> . 0, (114) E /'(p), flJH'; () , '
fvh,
f) +
#(r, -
)j dst
s
:'
'-xloougs
not
suarastoed.
w
.
.;.
:
.)-
=
i uuc ()
dt discussed earlier,
=
()
yt,y;,t) trs;
r
.j.
gqyt,
j;
#.;.
()
tjj'y;
JumpProcesses
f(Mt, )(A/rf+, - Mt,j,
(118)
pathwise, 'Fhis P;, will convergep and the variation of the proccss Mt will be tinite With probability one. Under thesc conditions, we say that P'u converges
:
it may exist, determiniss sucil a Iimit expliciusis
4/+/2
x-aN-1
P$
.'
=
.
9
What complicated the dcfinition of a stochastic integral was the extreme irregularity bot of contineous-time mmingales and of the Wicner process. This made a pathwise delinition of the integral impossible. Do we havc the same problem if wc have a stochastic integral with reSpect to some jump process? Could one use the Riemann-stieltjes integral when dealing with, say, Poisson processes? the answer to this question is aflirmative under some surprisingly, conditions, Supptue a process Mt is a martingale that exhibits hnite jumps only and llas no Wiener component, Trajectories of such an Mt will exhibit occasional jumps, but othenvise will be very smooth. 'Thcn, one could deline a J(,,
2
E asome''
u)2 du
.jtm,
E
5 lntegrals with Respect to
(113) 7.*..
S/f1
t =
'
k
,
convergein integral.lg
v
. '
n-1
i zzz()
a
-
Ito integral also has some propeztics similar to thosc of the Riemann-stieltjes intcgral, In particular, the intcgral of the sum of two (random)functions of St in (6) is equal to the sum of t'heir integrals:
.;
(112)
wherc (,%Jis given by (6),exist? It turns out that if thc function f (.) is contjnuous, and if it is nonanticipating, this integrai exists. In other words, the finite sums
'
rrjae
.
.
fsu,
jtp-v, u) JJ,p'.
azuiflmz
#.a
:
l
u) dsu,
-
a Note the recurling use of the equivalence
E. ,'
One can ask the question: when does the lto integral of a general random function fluts l),
ulj Ju
(j
.'
4.1 Existence
0
aE'g/'tr#-, utgl p,
=
E
The Ito integral has some other properties.
,
dW?ld #(W'u, t.) #Bi
&)
t
'
4 Other Properties of the 1to lntegral
'
/(Bi
a
';.
are given by the vari-
second moments
'rhe
where 1#)is a Wiener process. ance and covariances
.
''see
227
JumpProcesses
Respect to
with
pathwue. . .
'
!
228
C H A PT ER
Integration
9
.
in
''.
Stochastic Environments
8 Exercises
:i '1.
6 Conc lus ions
in the following-.
,
!.
nis
chapter dealt with the dehnition of the Ito integral, From the poit of view of a practitioner, there are tw'oimportant points to keep in mind. First, the error terms in stochastic differential equations are delined in the sense of the Ito integral. Numerical calculations must obey the conditions set by this delinition. Second, the stochastic differential equations routinely used in asset pricing are also dehned in the sense of the lto integral. Above all, we saw that the lto integral is the mean square limit of some random sums. These random sums are carefully put togethcr so that the resulting integral is a martingale. We also discussed several examples and showed tlaat the rulcs of integration are in general very different in stochastic environments, whcn com. pared with the deterministic case. This was the result of using mean square
(a) Write the Riemann (b) Write the (c) Calculate (d) Calculate
E. j F '
E
/0 , /'j ,
FI
i
:'
./:::u1
) ;)
i
.
.
.
,
tn-
:)
). :'
2 !
'
? :.'.
'1
.L .
... (
=
fM -
6. Can we say that this is an application
y
tn
dujv + a(Jr)
=
J'JSJS. ()
4. In the above equation there are hvo integrals. Which integral is dclined in the sense of lto only? 5. Can we say that this is a change of variables?
.:
.
1,
.
l
s#H(
''
-
-
j=1
l
''
.z.
,
j= l
rl
rj-llpq g(/./ L/.,(u$-. p;,-.lj +
3. Now use this information to show that:
L'
(0.fI:
gf/p;, -
-
'
:'i.
consider the
:1
f/-lwzt--j
#(ur)
. '2k) ;'.
8 Exercises
,
f'u--.t, tn
How is this different from the standard formuia for the differentiation of products:
.:
nere are several excellent sources on the derivation of the Ito integral. Karatzas and Shreve (1991)and Rcvuz and Yor (1994) were already mentioned. Two additional sources that the readcr may find a bit easier to read The former source could be a very are Oksendal (1992)and Protter (1990). good manual for quantitatively oriented practitioners and for begming graduate students. lt is well written and easy tt understand. Technicalities are avoided as much as possible.
tbt fi
,
wc can always write:
.( .
7 References
0
.
:,
!
: 9
Usc thc subdivision of
-
y''
examples of evaluating thc lto intcgral,
W62:M
.
and
'.
.
r
intep-al in discrete time using an lto sum. the expectation of the three Riemann sums, the expectation of the Jto sum,
2. Show that given
;
Fortunately, in evaluating Ito integrals, the direct route of obtaining the mean square limit will rarely be used. Instead. Ito integrals can be evaluated in a morc straightfozward fashion using a result called Ito's Lemma. 'Ihis will be discussed in the next chapter, where we will also discuss further
(0,FJ and
of the above integral as threc different
approximation sums.
h,
Convegence.
1. Let J1( be a Wiener process defined over integral:
229
(
.
;
.
of integration by parts?
E'
E
2 Types of Derivatives
.):
'j
In (2),dFt is uscd as a shorthand notation for dF(St, J). This should not be confused with Ft, the partial of F(.) with respect to t The third is the chain rl/tr:
'E
.
gyqst tj dt
. :.
lto's Lemma
231
.
tusy
,
=
F, + #/
(3)
FI.
A fnancial market participant may be interested in thcse derivatives for various reasons. The partial derivative has no direct real-life countemart, but gives tipliers'' that can be used in evaluating responses of asset prices to obsenred changes in risk factors. For example, L measures the response of Fut tj to a small change in St only. As such, Fs is a hmothetical concept, since the only way a continuous random variable St can change is if somc time passes, Hence, in realitjr, f has to change as well. Partial derivatives abstract from such questions- Because they are simple multipliers, there is no difference between the way stochagtic and deterministic environments define partial derivatives. A classical example of the use of partial derivatives occurs in delta hedging. Suppose a market participant knows the functional form of FS le f). Then, this mathematical formula can be differentiated only with respec to St, in order to End the partial derivative Fs. nis Fx is a measurc of how much the derivative asset price will change per unit change in St. In this sense, one does not have any of the dficulties encountered in dehning a time derivative for Wiener processes. What is under investigation is not how Fst, t) moves over time, but how F(, ) responds to a hypothetical change in st,with time tixed. notion. It is assumcd that both The total derivative is a more time r and the underlying security price St change, and tben the total rcdifferential sponsc of Fgt, t ) is calculated. The result is the (stochastic) dFt. This is clearly a vezy useful quantity to the market participant. It repTesents the obsen'ed change in the price of the derivative asset during an itjten'al d(. The chain rule is quite similar to the total derivative. ln classical calculus, the chain rule expresses thc rate of change of a variable as a chain effect of some initial variation. In stochastic calculus, we know that operations such as dlldt, dstjat cannot be detined for continuous-time square j ntegrablemartingalesa or Brownian motion. But a stochastic equivalent of the claain rule can be formulated in terms of absolute changes such as dl, udSt, Jf, and the Ito integral can be used to justify these terms. nus, in stochastic calculus, the term rule'' will refer to the way stochastic diftrentiala relate to one another. In other words, a stochastic version t)f total differentiation is devcloped.
:
.
dfmul-
' : l '? t ('. 5
1 lntroduction
:.
As discussed earlier, in stochastic environments a formal notion of derivative does not exist. Shocks to asset prices are assumed to be unpredictable, Wtoo erratic,n Thc resulting asset pris and in continuous timc they become smooth. Stochastic differentials need they not be continuous, but are may to be used in place of derivatives. lto's rule provides an analytical formula that simplihes handling stochastic differentials and leads to explicit computations. lt is the main topic of
'
i'
'..),
, '. 'r
this chapter. We begin by discussing various types of derivatives.
2 Types
of
Derivatives ut
,
pF(.%, nS
/)
Ft
,
=
c?F(,%,/) pf
.
The second is the total derivative dealing with difcrentials:
dFt
=
r '. '
i
,
=
idsmall''
Rrealistics'
and Suppose we have a function FlSt !) dcpending on fwtp variables t, whcrc St itself varies with time /, Further, assume that Sf is a random process. ln standard calculus, where aIl variables are deterministica there are tllree sorts of derivatives that one can talk about. ne lirst are the partial derivativesof Fst l), denoted by
F,
:.
k :' ): ?p '
F, dSt + Ft dt.
' ' i', :
' G.
(1) '
ichain
E
(2) : :k
.
'
)
: 5':
230 .
)
:.
f)
H A PT ER
.
Ito's Lemma
10
E
2.1 Exumplc
Ftrj, tj Let us calculate
c
-r,(
r-/) jtm
1 .
.
ut
.',
(4)
.
..
the partial derivatives Fr, Ft: Fr
al'lu
=
JF
=
=
trt
-(F
/)Ie-G(
-
obtains new information about JF; and obsen'es a new as time passes, one jjwrement, d.%t.nis will also make F(.) change. The sum of these two effects is represented by the stochastic differential dFvh, t ) and is given by the stochastic equivalent of thc chain rule. be obsewed in continuous time. We again Let the random process the time intewal F1 artition I0, into n equal pieces, each with length h. P and use the finite difference approximation. However, we write this as an equaiity
''
We discuss a simple example before gtaing into lto's formuly. The examle P will help clarify the mechanics of taking various derivatives. Let Frt, tj be the price of a T-bill that matures at time F, and let r; be a hxed, continuously compounding risk-free rate. nen
t
w-t) jjsj
(5j
tYk
.
:
.fz l
IF Jf
'
=
rl
(e-r,(T-J)1(mj
/)
-4r
=
-
/)Ee--r'(T''01()(.)j
dr l + r,(c'
r,(T..-l)1()()j
at.
E'
f (a) ftat) =
':'
whereR denotes the
(p
'(
nis example suggests that when rt is random, we may be able to define the counterpart of total derivative, using the lto integyral, which gives a meaning to stochastic differentials such as drt. This intuititm is correct, and .. ; the result is Ito's formula. However, with stochastic rt not only does the E).. of drt change,: but the formula will also be different. interpretation '
,
.
3 lto's Lemma
'
k
'
The stochastic version of the chain nlle is known as Ito's Lemma. Let Si E be a continuous time process which depends on thc Wiener process E . Suppose we are given a function of St, denoted by Flut, f). and suppose w? J.E tyf passe:i change when time #/ amount in F(.) wouldlike to calculate the Clearly, passing time would inquence thc F(&, /) in two different wayK First, there is a direct inquence thrtugh the / variable in FS t /). SeconG '
.
'
,
,
. . ' Recall that such qllantitie,s are dcfined i'fl terms of mcan square ctmveqence and w Etlti. ' .. S
i.
.
..
+
f'(.b)@ -
.Yo)
+
1
f
,.?
,
(8)
n,
(Ao)(;r
-
x(,) .z + R,
(9)
remainder.
lncidentally, some readers may wonder if h-erneirregularty of st. z''(.) can be a smoothetis function Tsmxthncss''
'
(L
,
We apply this formula to F(,;, I). At the outset, F(.) has to be a smooth functionof st.l But there are two additional complications. First, the Taylor scries formula in (9) is valid for a f (.x)which i.s a function of a single variablex, while Fvt, f) depcnds on 4w variables, St arld t. Second, the formulain (9) is valid for deterministic variablcs, while St is a random process.Before usg Taylor series, these complications must bc addressed. ne extension of a univariate Taylor series formula to two variables is Onc adtls the partials with respect to the second variable, straightfonvard. With two variables, cross partials should be included as well. The applicability of the Taylor series formula to a random envimonment ig a deepcr issue. Firsta it should be remembered that some of the terms in Taylor scries are partial derivativcs. With respect to these, one does not haveany difliculty with differentiation in stochastic environrnents. Second, We have differentials such as dst. Here, wc do need an adjustment. which is in tenus of thc interpretation of the equality and not in the Taylor series txpansionitself. ne formula for Taylor series expansion will remain the
:.
..'
.
.p,
,
E
,
dljr 1:
.
.
.:
.)
1, 2,
=
,
(6) g.
Note that these partials will be the same regardless of whether rg is deterministic or random. By taking these partial derivatives, we are simply calculating the rate of change of F( with respect to small hypothetical changes in rt or in 1. O n the other hand the total derivative relates to the actual occurrenc,e of random events. In standard calculus, with nonrandom r/, thc total derivative of this particular F(.) will be given by
k
akh + JklWzrp,
=
using thc mean square equivalence between the left- and right-hand side as --z 0, nis notation will be presen'ed throughottt this chaptcr, Also note k4 k) to ak and for trtxk-j that wc shortened the notation for tzt,v-j, to n We calculate Ito's formula in this setting, using the Taylor series. Recall the Taylor series expansion of a smooth (i.e.,inhnitely differentiable) ftmction /'(x) around some arbitranr point
r
''
..I
=
2J3
3 lto's Lemma
L
.
)..
Sxhastc
u.. ' .
Statemerlt :
.E .,j
does not contradict the exor sf and still be a ver.y irrepllar process. lrrcgtzlariy hure is in the sensc of' how F(.) claangcs over time- It is not a abou! laow relates to F(.). s
!
(-2H A
PT ER
.
Ito's Lemma
10
3 fto's Lemma
.
.'
snmc, but the meaning of the equality sign would change. ne equality would have to be interpretcd in the context of mean square convergence. where the 1, 2, We apply the Taylor series fonnula to Fuk k), k Sk is assumed to obey sk ak h + (10)
Wllat does this equation mcan? On the left-hand side, F'(k) indicates the total change in F(Sk #) due to changing k and Sk Hence. if F%, k4 is the price of a derivative security, on the left-hand side we have the change is the derivative asset's price during a sho'rt intewal. This change is exPlained by the terms on thc right-hand side. The jrst-order effects arc the effects of time, represented by FI g/zl,and thc effects of change in the underlying asset's price, Fstakh + tqHij. In thc latter we again see that changes in security prices have predictable and unprcdictable components. Second-order effects are those changes that are reprcsented, ftar the timc being, by squared tcrms and by cross products. Higher-order terms are grouped in the remaindcr R. In order to obtain a chain rule in stochastic environments, the tcrms on the right-hand side will be classed as negligible and nonnegligible, It will then be shown that in time inten'als, negligible terms can bc dropped from the right-hand side and a chain rttlc formula obtained. ln addition, as h --> t)aa limiting argument can be used and a precise formula obtained in the mean square sensc, formula is known as lto's Lemma. 'I'hc Iirst step of this derivation is to separate thc terms on the right-hand side. This requires an cxplicit criterion for dcciding which terms are negligible, Aerward, one can consider the size of thc terms on the right-hand sidc of (11)individually and decide which ones are to be dropped.
'
=
,
.
,
.
.,
'
AJP;I
n
=
.
.
:'
First, tix k. Given the information set 1k-L, Sk-L is a known numbcr. Next, apply Taylor ,s formula to expand Fuk k) around Sk-j and k 1:
.
: '
-
,
Fsk
,
k)
Fsk-j
=
,
k - 1) + Fagkk -
!L
+ Fathsk -f-i'F,,E/z12
X-11+ Ft (1 +
;
'g
1
.'
.jF,,(kV-
kk-ll
c $ t; k,,
.
- 5'-1)1 + R,
,
(11) $
where the partials Fx, F,,, Ft, Ftt Fst are a1l evaluated at Sk-j k - 1. R represents the remaining terms of the Taylor series expansion. Here we are keeping the F/, Fst, Ftt notation for convenience, although these pnrtials are with respect to k. Transpose Fuk-j k - 1) and relabel the increments in (11) as follows: ,
,
'j?
E:
rrjs
)
',
,
Fsk, kj -
k - 1)
FSk-L,
sv Notice that Eq.
vsk-j
-
.
=
FsLsk + Ft gl +
hsk
Wc can substitute the right-hand expansion of (11): F(k)
=
Fsfakh + .j
+i
qa
'
F.,IA&1
'
+ R.
ak h + o
.'
(10),and
+ .H'kl + R.
f
=
fl,ol
'
.
+
.
d
Wbere
'
R is the remainder.
=
This is equivalent
j
q
< .'%
;
(total)differendation formula (19).
.,
q. t
't;
i j.,(&)(q$)
1 y/ut,%ltA5'l
.
.2
(18)
a + R,
total dcrivatives is just
fs dS.
to assuming that whilc in the Taylor (18), is small and nonncgligible, the terms involving ae smaller and carl be ignorcd as A. -.+ 0. Consequently, termfsds is preserved, whilc all othcr terms are dropped.
c .q
+.
But the formula for
JJ
;
..
(1
.,
:
in the Taylor soriYf' tuWil
=
;'.E
.'
,k
.f(&)
-
that ...) .(
1
F.vfllllA
f (.)
(10 t
.
side of this for
t' ,
p-k
+ F?(1 + glslakh + tzkHz'l;I
F,,IA21+
:
(15J
j
=
.. t
'
stochstic Culculxs
This section discusses the convention used in determining which variables can be classilied as anegligibk'' in stochastic calculus, In standard calculus, the Taylor scrics expansion of some function fv arouud gives
j1
'
. .
1 a jyFx.vr.%l
that the dynamics of St are governed by Eq.
we have
3.1 TW Nofimz of fGze'' in
.
;
knoW
):
r;
'j
'
+ jF,?I/?1+ We
.;
(14)
.
1
ZUt
'
. '
>
kh - (k 1) Now substitute these into (11): =
!;
.I
(11)already uses the increment for the time variable:
iFjkj
..
(13)L
isk.
=
'E7 r!
(12)
F(k)
=
,
KGsmall''
'P
.
135
(19) serics expansion (A,5')2, (x)3, .
.
.
in the limit, the is the
Thc rcsult
E
!
(2 H A P T F,R
*
10
lto's Lemlua
j Ito's Lemma ?
To see why such a convention makes scnse, note that as tY gets smalier, get small faster.This is shown in Figure 1, terms such as (.u$)2 (15,)3 where the functions .
,
.
.
#1(Ak)
=
hst is rando Jm its variance will be positivc. But variance is the of tlpical (z%). Hcnce, on avcrage, assunting that negligible will is a )2 be equivalent to assuming that its variance is approximately zero-that St s, approximatcly, not random, This is a contradiction, and it defeats thc purpose of usjng soEs in markets for derivative products. After all, the objcctive is to pzice rhk, and risk is gcnerated by unexpected news. Hencc in contrast to deterministic environments, terms such as Luk jl carmot bc irored in stochastic diffcrentiation. Oiven that the terms of size h are of lirst order, and that these are by convention not small, the following rule will bc used to distinguish ncgligible terms from nonnegligible ones. IEsize''
,
t,%
47
.
'
(20)
LS
'
.:
and
t
gzut
2
(A5'1
=
'
(21)
,
ks-
'
'
approaches zero much faster and smaller.
are graphed. Note that thc function gzt/j gets smaller than the hznction :1(5') as
2,37
:,
Thus, in standard calculus, a1l terms involving powers of du highe t.11%L 1 are assumed to bc negligible and are dropped from total dcrivatives. ne ; questionis whether we can do the same in stochastic calculus. The answer to this important question is no. In stochastic settings, te : timevariablc t is still deterministic. So, with respect to the time variable, te . ) samecritcrion of smallness as in deterministic calculus can be applied. termsinvolving powers of dt higller than one may be considered negligl-ble. .' on the other hand, the same rationale cannot be used for a stochastid .J differentialsuch as dSt2 chapter9 already sbowed that, in the mean squaTe -. sense, we have #%2 dt. (22)
coxvsxrlox: Gjven a function #(.T#;;, ) dcpendent on the increwjenerprocess ;#), and on the time increment, consider
'
ments o j tjw t jw ratjo
,
:'
rtBi, h
.
h4
(23)
.
Ij tus ratio vanishes (in thc m,s. sense) as 0, then we consider , hj as negljgjbje in small intervals. Otherwise, gtNlFk, h4 is nong negjigoje. .->
'
tajjj
.
,
:' '.
=
Tus convention amounts to comparing various terms with h. In particular, if the mean square limit of the function j'(H'k, h4 is proportional to r with r > 1, it will go toward faster than h thc zcro (i.e., square of a small number is smaller than the number itselt). the other hand, if r < 1, then on the mean square limit of , ) will be proportional to a larger power Of h than h itself.S The following discussion uses this convention in deciding which terms of stochastic Taylor series expansion can be considered small, a
j'
Hence, terms involving ds; are likely to have sizes of order dt, which wal' as nonnegligible. lf terms wolving dt are preserved in Taylotk considered j.Iq.,:E the same must apply to squares of stocbastic diferent approximations. point. If is a random incremen t!;E we further emphasize this important variance of W this increment. Sin w ill be then E the (15,12 ithmean zero,
.j,(.f<
..%
.'ff '
i
git.as)
.
'r
1
.4
'
Z.
12 .
( ':
1
Tcnzzs
N()w consider Eq.
(11)again:
;
;. 0 8
,
J
.
o.6
3.J Fivst-ovjev
'c'.:
p( Asl-as
y (k ) = y'sga g Jj + j. + ..F y
.t
a
0.4
x ',
pa2(AS) 0.2
(j,2
0.4.
0,6 FIGURE
=
J
(%S) a
0.8
+
L:
1
1.Q
'
..
i'
',
L
1
.
,tj
fy
juk h
py j + +.
g. k
y jigj
auzk j2
F // j/;j2 + F st g/jjju: +
aud that these alsty determine whether the ralio Stlcll is llae case wlacn we deal witll cross-produu
(c4) gj
k
rjjyj +
R.
(&) becomcs negligible as h gets smallcr. terms of Tayor serics expansions.
n
!
1
:'
c H A P T ER
238 ' -'
''''
10
*
.
:
are ciearly first-order incrcments Here, the terms that contain h or 1/1,,1 + negligible. As not or Flh are divided by h, and h FvLakl are these smaller, do not vanish. For examp l e, tlze smallcr and made tcrms is
Faakh
lim pl--sf)
:
vartn
t
Fsx
.
)
l :
cv
rogx ku''J !
using consequently.
ojk
(g1)
.
1 1 ak h F x. + 2 2
'
>
j
dt.
(g2)
(),
(gg)
j ttuhpkla + ztzjytv/iauzk 2 ,-Fssoka
jg
h
z
(34)
.
Again, this approximation should be interpreted in thc m,s, sense. 1, in small intervals, the dl ference between the two sides of equality (34)hms 0. a variancc that will tend to zero as Before one can write the Ito formula. the remaining terms of the Taylor serics expansion in (24)must also be discussed. 'rhat
..
?
-..>
side of (24)by h, s ,:
E
( ..
3t
Ftth :y (>) ...jE.E 2 sincc in the numerator we liave an This term remains proportional to that dcpends on h1, a power of h higher than one, and tbe i is not random. Hcnce, this term is negligible: '
.
F h 11:, -->0
=
(24)involving cross products are also negligible in small under the assumption that the unpredictable components do not contain any The argument rests on the continuity of the sample patlas fbr st. Consider the following cross-product term in (24)and divide it by h
intenuls,
''jumps.''
(29):
().
..
,
',:
Next, consider the second-order term that dcpends on l,klz, 1 z -2 Fwgkupl expanding thc squarc, an d dividing by h, Substituting for
crossProdxcjs
The terms in
.:
tt
4 Terma Intlolttiag
,
,
incremcnt increment
.,9
Fxil/zjgtzk +
fzymj
-
.
'l''le
gaknhWk ttzq,ljr/rlz + h
yz
.
v
Rsmall-''
Gsmall,''
Qh
(x)
The numerator contains In tbis equation. the fil'st term is and not is the term ran dom. ne third te p owcr of h greater than onc, scction). The Seco next prodklct It involves a cross (see is also This is term. on the otlwr hand, contains the random va riable tAH'-k)2.
; ,
.
t
.
r
E.
..;
'
g
y'z/ ju: +
aauzpj,
(:35)
right-hand
side of (35)depends on l#k. As --> (), pk. goes to zero. ln particular, Hz-p becomes negligible, becausc as --> 0 its variance goes to zero. That is, does not change at thc limit h 0. anotjxr way of sang that the Wiener process has continuous sample Pq1jys, AS long as the processes under consideration are continuous and do not (upjay any jumps, terms involving cross products of AI'IIand h would be lcsizible, accoding to the convention adopted earlier.
usk,
az /;2 1 k + -aF x;
=
. .
'. .
Wz;I
'lnhis
=
r
1
i .:
the criterion of negligibility, we write for small h'.
-
,
terms on thc right-hand
=
;)
c!
Now divide the second-order and consider the ratio
=
/q is a term that calmt:t be considered negligible, since by dehni(jon, we are dealing with stochastic Sk, and the nonzero variance of xS implies:
!:,
Terrrts sectmd-or'der
'f
2
rj-hus,
..,
k gets larger (in a pro babilistic sense) as h becomcs smaller, since the ten'n ; /2 taf l'li is the order h 1 r A1I srst-ordcr terms in (24)are thus nonnegligible. )
3.3
H,k)
dkkz
7:
g6)
''
given thc
jt was also shown that in the mean square sense discussed earlier,
y
=
l@
lj
.
(25) r
Fh t s; /).-.s(J lg vanish as h gets smaller. are c learly independent of h, and do n0t On the other hand, we already know that the ratio
'-
-
t
and lim
''-
: :
;
..q
Fsak
=
239
square of a random variablc with mean zero that is unpredictable nast. Its variance was shown to be
E
'
ratios
3 fto s Lemma
'
lto's Lemma
,$,
that
j
.
'
,
(-aH A P T E R
240
.
1
Ito's Lemm
l0
'
5 Uses of lro's Lemma
7
3.5 Terrmxin tlkc Rcmufnder
5 llses of lto's
.
?
All the terms in the remainder R contain powers of h and of H'kgreat than 2. According to the convention adopted earlier, tf the unprcdictabl events''-powers ot type-i.e., therc are no shocks are of r,f'i greater than two will be negligible. ln fact, it was ghown in Clta/ J . ter 8 that continuous martingales and Wiener proccsses have higher-order, E 0. moments that are negligible as
'hc first use of lto's Lemma was just mentioned. ne formula provides a fool for obtaining stochastic differentials for functions of random proccsses. For example, we may want to know what happens to the price of an option if the underlying asset's price changcs, Letting Fvh, t) be the option the underlying sset's price, we can write Price, and
'drare
(Tnormal''
:
st
-->
dl-ut
J'r'
4 The lto Formula
'
.i
.
l p2F tr,2 dt. dt + z?r 2 slf ' ss / or, after substituting for dSf using the rclevant SDE, @F 1 p J.F z JF JF (rt dt + + dFt h #/P) at + pkq; 2 ch%c Jf 6st =
where the equality holds
l
,/ '
(%j' ..
:.
J :
.
t' '
,
(37 .
the mean square sense.
Fss z dt.
o
(39)
rnis
')
HF
1
j.
,
r.
=
ds ' +
Fs dSt + Ft dt +
=
,
,
PF
t)
If one has an exact formula f0r Fst f ), one can then take the partial derivatives explicitly and replace them in the foregoing formula to get the f). Later itt this section, stochastic deerential, dbut we give some cxamof of this lto's 1cs Lemma. use P The second use of Ito's Lernma is quite different. Itcfs Lemma is uscful in evaluating lto integrals. may be unexpected, because lto's formula was introduccd as a tool to deal with stochastic differentials. Under nonnal circumstances, one would not expect such a formula to be of much use in evaluating lto integrals. Yet stochastic calculus is different. lt is not like ordinary calculus, where integral and derivative are separately dehned and then related by the fundamental theorem of calculus. As we pointed out cariier, the differential notation of stochastic calculus is a shorthand for stochastic integrals. Thus, it is not surprising that Ito's Lemma is useful for evaluating stochastic intcgrals. We give some simple exnmples of thesc uses of lto's Lemma. More substaatial examplcs will be seen in later chapters when derivativc asset pricing is discusscd.
';
()
-->
ITO'S LEMMA: Let Fst f) be a twice-differentiable function of t and 5 of the random process St: .i dSt at dt + o JM, 477.4 Then we havei' with well-behaved drift and diffusion parametez's, at, =
,
,
As h We can now summarize the discussion involving the terms in (24). and we drop a11negligible terms, we obtain the following result:
dF
LenznAa
: ',
ln situations that cail for the 1to formula, one will in general be giv an SDE that drives thc process S): f ) dt + o'lut f ) #Wz; (38: dSt a Vehicle YIIUS, tl1C ltO ftrmlzla tat takes thc SDE for x can be Seen J.S a corresponds t). In fact, Eq. (37)is (. that Fbsl the dctermines SDE to l). '; F(&, for stochastic differential equation with fman dcaling useful in have tool to lto's formula is ckarly a very derivatives. The lattcr are contracts written on underlying assets. Using tb ( lto formula. we can delermine the SDE for tinancial derivatives once are given the SDE for the undcrlying asset, For a market participut of t wants to price a derivative assct but is willing to take the behavior Itt's formula is a necessary tool. u nderlying asset's price as exogenous, 'ty''
=
,
,
5.1 lfo's Fonnua as a Clutn Rule
.
'
A discussion of some simple cxamples may bc useful in getting familiar
withthe terms introduced by Ito's formula.
and
,
'
5.1.1 Ekumple l Consider a function of the standard Wiener process JP) given by
/-(1, f)
:..
'
,
.
H'2.
Remember that F; has a drift paramcter Applying the Ito formula to this function,
:
irregular. Sqll 4with this we mean that ttle drift and diffusion parametezs are not too c(.J, write t) as at notational smpiicity, we condion. For satis woud this integrability Glx Q.S G( r j t ( .
=
')( .
#Ff
=
1
g (2
dt
+ 21
(40)
0 and a diffusion parameter 1. (/p)
;'
:
:
C H A PT E R
242
.
10
.
5 Uses
'
Ito's Lemma
'
or dt + 2H/; tl4(.
=
:
(42)
,
;
Note that Itt's formula results, in this particular case, in an SDE that has
ah,
1
1
.
FLJ,#;,
s
and
r)
E: E
(44)
2W')-
=
.
dcpends on tlw inormation
the dria is constant and the difssion
uence, setz
and apply the Ito formula to F(p;, d,',
-
-
F(l#I
,
i'
f)
3 + t + chpi
=
.
F(p;,
(45)
1
0 + p; dw,i + - dt.
-
,
1 ;f, dt + ehls#H') + il e #f. '
..
(46) )
1 z j l4
' :
=
-
2
cz
eB6
dt +
dl'l').
(47)
i
t
,
a SDE for F(St , f ) with T/-dependent drift and dif-
y
()
('
al', f)
1+
=
1
2
(53) side, and using the
'
t48)
.k
.)
t
and
t
1 t 2
H?;d 1#k+
=
()
-
.
(54)
XJW$
j =
-
2
1 1C/z - h 2 -
(55)
in Chapter 9 using mcan square convergence. It is important to summarize how Itozs formula was exploited to evaluate lto integrals,
!., ,,$
ds,
wuch is the same result that was obtained
.,
'-e
o
Rearranging terms, we obtain the desired result
y'
j
1+
+. 2
:
Grouping, KIFt
t
1
u, Jp;
or, after taking the second intep'al on the right-hand delinition of F(E, f):
.
=
t =
(5z)
r#).Writing the corrcsptmding
:
! dFt
f)
;,
We obtain
thig hasion terms:
/):
nis is an SDE with drift 1/2 and diffusion integral equation,
E' .,Li'
5.l.2 Fwvzrrlr/e2 Next, wc apply lto's formula to the function
(j.j)
,
,,
/*
CaSC, We Obtail
.
z
2 rp)
=
'
rltb r)
ln
243
oejue
.,
(43)
t) = 1
:
Ito's Lemma
In Chapter 9, this integral was evaluated dircctly by taking the mean square limit of some approximating sums. nat cvaluation used straightforward but lengthy calculations, We now exploit lto's Lemma in evaluating the same integral in a few stcps.
.,
dFt
of
''
Jj .
1
/,
t)
=
eHs .
(49)
.j
.
t
dH;, 0
rrltis
(:.
suppose one needs to cvaluate the following lto integral, which was discussed in Chapter 9: ra< (JuJ
1. we guessed a form for thc ftmdion 12(14:),f).
2 Itors j-emma was used to obtain the SDE for F(H'), I). 3. wc applicd the integral operator to both sides of tMs new SDE, and Obtained equation contained integrals that an integral equation,s were simpler to evaluatc than the original integral, 4. Rearranging the integral eqtzation gave us the desired result. ,
5.2 Ito 's F tymnu ltz a.s un lntcgrfztion T'xl
' : :
. ..
g bjn
;.
amounts .
.t .
'J
1 .
.
..
factj SDE notation is Rimply a shorthand to writing thc sos i.u f'ull detail.
jbr
integra.l equations. Hence, this stc,p
7
f'
.
I .
'
E;
. :j
cHh
13T E R
is indirect but strakhtforward. ne ne technique exact form of the functicm F(I#;, f). t he of using Ito's guessing tec hnique t-emmain evaluating nis ploitcd in the nex't chapter.
.
10
:'.
J;'
.
(56)
F(W'7, f)
,
1
f dls
B/)dt +
4 j (59)
.V W;kds + JWI.. f) o Rearranging, we obtain the desired intcpal:
Jlul/l'l
=
0
t t)
=
JI'FI
.
(*);
=
-
0
't
:'
;,
...x
,.
t
r
Lz )
Again, the use of lto 's Lemrrla yields the desired integral in an indiredt but straightforward series of operations.
.
?
:
'j
L$ . ..7 'S
6
lntegral Form
of
'
lto's Lemma
q
f*' : As repeatedly mentioncd, stochastic differentials are simply shorland j Ito integra ls over small time intervals. One can thus write tbe lto form ! in integral form. ; E.'( Integrat ing both sides of (37'),we obtain
7 j Mujjsuvute
.
. .
o
.
Fst,
tj
=
Ft5a, 0) +
t 0
Fu +
F ss c2u
du +
1)
we
F
Fs dsu,
yx
ku
,..
.: .
::r
; . '
.
.
j64l
gxcsg
now extend the Ito formula to a multivariate framework and give an example. For simplicity, we pick thc bivariate case and hope that the reader can readily extcnd thc formula to higher-order systems.
r:
t
,
,
''' 't'
Ol/-
dg
Our discussion has cstablished Ito s srmuu in a univariate cage, and using that unanticipated news can be charactered Wiellcf PTOCCSS iIICFCYCIItK We can visualize two circumstanccs wherc this model may not apply, Undcr somc conditions, the Rlnction F(.) may depcnd on more than a sing le stochastic variable St. nen a multivariate version of thc lto formula needs to be used. The extension is straightfolward, but it is bcst to discuss t brieqy. Tjae second generalization is more complex. One may argue that tinancial markets are affected by rare events, and that it is inappropriate to Consider error terms made of Wiener processes only. One may want to add jllmP Processes to the SDES that drive asset prices. The corresponding Ito formula would clearly change. This is the second generalizatitan that we discuss in this section,
L
Herc the first term on the right-hand side is obtained from l //6 0. J/ll'rtl
7
under the assumption
'''
0
y uru
(2.
nancial markcts.
!
:j
H$ ds.
-
()
j -
ut
''
Jt
f
s #F;
j.
,
Fu +
Ito's fonnula is seen as a way of obtairting the SDE for a function F(.;, /), given the SDE for the underlying process St. Such a tofl is very useful when F(St, r) is the price of a snancial derivative and is the underlying asset. But the lto formula introduced thus far may end up not bcing suGciently gemeral undcr some plausible circumstances that a practitioner may face in
:(:
''' i. '
/
l
l
0) - F(&,
order to
7 lto's Formula in More Complex Settings
(58) '
integral equation,
Using the dchnition of dFt in the corresponding
Fluh, tj
(62)in
,.j
E. . :. . .'F
+ 0.
(63)
.
.,
,),
App ying Ito's Lemma to F(.),
F(&, 0).
is equality provides atl expression wllere integrals with respcct to wiener processesor o ther continuous-firne stochastic processes are cxprcssed as a srjctionof intcgrals with respect to time. lt should be kept in mind that in (j2) antj (64),Fs and F,, depend on u as we jj
:..j#: ..;':
(5X
fW';.
=
=
t)
''
'
,
-
j'
Fs dSu
';i
F( 16 f):
F(.%, f)
t
.. .
.:#1'P),
-
Wc can use t jje version of the Ito formula shown in obtain another characterization. Rearranging (62),
@ ''f E ' .,
We use 1to5s Lemma. First we de line a function
=
dFu ()
:';
.
t'
245
l
:
.. .
where J#; is again a Wiener process,
More CompLexSettings
integrals will be ex- ' .S
.E
/
in
where use has been made of the cquality
'.'
only difliculty is in
5.2.1 Another Faample Suppose we need to evaluate
dFt
ito's Formula
lto's Lemma 2.
.
'
r
246
(a-H A P T E R
.
10
Ito's Lemma
Suppose is a 2 x 1 vector of stochastic processes obeying the following equation:6 differential stochastic vh
dS, (f)
f.'qltfl
?(!) =
d-bbt)
tu +
r?z(f)
t,-21(/)
J1(r)
=
dt
+
,'22(r)
JM(f)
J'r4itfl
-
gg'jltf)tf#J(f)
+
tnc(l)
#W,$,(/)1
tjute jntewal
,
(65)
dszt) ojjL't ),
(tW21(/)
) d/ +
dlfh (f) +
tr22(l) #B$( f)1,
Hence, a limiting argument can be constructed so that in the mean square
..j ,
(/<(f)#;,p;(j)
:.
=
(67) t
#5'z(I)2:
)2 (Vj(/)+
d-Llt
'j
dytll
x%(f)
=
=
Fr dt + Fst dSj + Fs,
2 1 + gF,,,j dS, + FxaxadS1c + llh
; ')
lu
':
(69)
''
,
tp
t '
l
(j; co + jgj j tjjty;j,k (/) .j.
.u)
(,5)
g.at/ld.jj(tjlj
aud
% id
dRt
=
/24/)
dl +
EtW21
'
(f) -1-tr22(/) (l)6IW'S
#Wl(f)1.
'rjaus
the short and the long rate have correlated tew' n a j of leugth h. this correlation is given by
;
.j
.
:
:.;
',,-7 :
t. is a slight claange irl the noation dealins with tlw time urable following equation wc write tlw stochastc diffcrentals witlwut slaming heir do o.n t.
s'rlere
Corrllrf -rerms
:9
dence pen
.... ' .$ lk ; ?
(g4l (69)
'rhese
(i:
rrhis
lhe
t
,'
JSl 521,
at.
fyu(/)gu(/)j
..k.
7.1.1 vzln Eaample fr/rzzFinancial Derivatives Options written on bonds are among the most popular interest rate dcrivatives. In valuing thcse derivatives, the yield curve plays a fundamental role. One class of models of interest rate options assumes that the yicld Ctlnre depends on rw,pstate variables, rt representing a shot ratc and Rt representing a long rate. The price of the interest ratc dcrivative will then be denotcd by F(G, Rt J), t g0,Fj, intcrest rates are assumed to follow the following SDES!
'
where the squared differentials Et,%)2, ((5z12and the cross-prodtld term dh (/.% need to be equated to their mean square limits. We already know that dt1 and cross products such as dtdv' (/) al'ld #/4/14$(1) are equal to zero in LIIC mean square Sense. Point WaS disnoW is ntwelty The only d univariate Ito's obtaining Lemma, in the cusse
'In
(7g)
These expressions can be substituted into the bivariate lto formula in dsLtjl, dsztll, and dsqtjdszt). to eliminate
..;,
z
dt.
.(gjy(/)ajj(/)
(68) ...
for a1l f. function of Supposc wc now have a continuous, twice-differentiable f). How can we write tbe 51(l) and Szt) that we denote by Fvhtj, &(/), stochastic differential dFt ? ne answer is provided by the multivariate form of Ito ,s jxmma,; #F/
g.azgtrlj .j((yj2j(/)
dqsv!tjcjsgLtj
r
.
0,
(72)
''y
b
=
dt
The cross-product term is given by
s
k%(f),
c'a,l(f)
and
(rjzzt/lj
=
')
0,
for
and
.
.$'1
=
(71) dSLtlz
';..:
=
trlztrl
().
=
This gives the following mean square approximations
:
1, 2, j 1, 2, are tbe drift and diffusion parameters where possibly depending on Sitj, and wherc +1 (f), 11,$(/) are two independent # ? Wiener proccsscs. f hastic 52(/) represent stoc ' In this bivariate lamework, two pro- ( (/), ccsses that are inouenced by the same Wiener components. Because the 'E ihjt) parnmeters may differ across equations, error terms affecting the two equations may not be identical. Yet, because the have common will correlated, they in general be ftlr the special t, except components, errtw when L case t7j(l),
(7p)
0.
=
scnse:
;
t72(f
expect
.sgap-k(/)aJ.pk(/)j
@ .
and =
x we
:?
(66)
247
Here we have thc the existence of cross products such as JM(f)#Fi(/),8 product of the increments of t-w47indepcndcnt Wicner processes. Over a
!
of thc following forms:
This means that we have two equations dsjlt)
t72(f)
7 ltcl's Formula in More Complex Settings
Fk
;
'
,
'
such as dstwszt)
,
hRt4 *411
=
((zI(/)dJ.a,1
errors,
(f) -I- t,.:cllltnzt/ll.
depend on #pk(/)dw(,).
(76) (lver a linite
(77)
c
248
H A PT ER
10
*
lto's Lemma
ne market participant can select the parameters hjt ) so that the equations capture the correlation and volatility properties of the obsen'ed short and long rates. ln valuing these interest rate options, one may want to kmowhow the option price reacts to small changes in the yield curve, that is, to drt and dRt ln other words, one needs the stochastic differential dF!. Here the multivariate form of the lto formula must be usedlg ,
dFt
=
FR
Ft dt + Fr drt + FRRCO'ZI,
+
+
t7'lcl
1 dRt + jgFrr( c a.) +
2 f.r.tz
)
(78)
+ tz'lct'zkzll #f.
+ zFrattnlul
The stochastic diffcrential dFt would measure how the price of an interest rate dcrivative will change during a small intcmal #/, and given a small variation in the yield cunre, the lattcr being causcd by drt and dR:. 7.1.2 W'lf/1/z An invcstor buys N;.(l) units of the fth asset at a price #;.(/). nere stochastic are n asscts, and both the Nj(f) and #j(f ) are continuous-time shocks. random processcs, potentially a function of the same Thc total value of the investrncnt is given by the wealth Ftf ) at time f:
Ito's Formula in More Complex Settings
suppose we obselve a process St, which is believed to follow the SDE dst
i uuc 1
Suppose we would like to calculate thc increments in wealth passes. We ust Ito's Lemmm M
n
Jy-(f)
=
Nitjdrtlt) J-l /=1
s time
+
J''ldsitlptq
+
)7 lNit4dlqt). f=1
J=1
(80)
It is clea,r that if one used the formulas in standard calculus, the last term of the equation would ntt be present.
7.2
we writc
as
q.
:
0,
(81)
0.
=
(82)
=
.
,
.,
with sizes (/j, 1, kJ.The jumpsoccur at a rate :,/ that may depend Once a jump occurs, thc jump type is selected on the Iatest observcd randomly and indepcndcntly. The probability that a jurnp of size ai will occur is given by pi.l Thus. during a finite but small inten'al the increment h will be given (approximately) by =
.
.
.
,
st
.
,
ut
ax, -
-
g,7,()''-2 uj/ajjj,
i uuc 1
(83)
where Nt is a process that rcprcsents the sum of a11 jumps up to time t. More prcciselya iNt will have a valuc of ai if there was a jump during the k h, and if thc value of the jump was givcn by a;. The term (Ej=: aipij is th the expected size of a jump, whereas rcprcscnts, loosely speaking, the Probability that a jumpwill occur. nese are subtracted from .N; to make lt unpredictable. Under these conditions, the drift coefficient at can bc seen as representing the sum of twta separate drifts, one belonging to thc Wicner continuous component, the other to the pure jumps in S, =
a, + 1/
k
111 aipi
(84)
,
i uuc
Where
of thc continuous movements in St. ar is a drift coecient lt is worth discussing one aspcct of the jump process again. Thc pro-eess has rw'osourccs of rarzdomness. The occurrence of a jump is a random 10In lhc case of the somdard
gAgain, for notational simplidty,
t
We need to makc this assumption, since this tel'm is part of the unpredictable innovation terms. This assumption is not restrictive, as any predictable part of the jumps may be included in the drift component at. We assume the following structure for thc jumps. Between jumps, h 1 , 2, remains constant. At jump times Tj, j it varies by some discrete and random amoum. We assume that there arc k possible types of jumps,
fzl
tny.(r)
#H') + JJ/,
F161
Ito': Formulu and Jumps
nus far, thc underlying process St was always assumed to be a ftmction of random shocks representable by Wiener processes. Tls assumption may be too restrictjve. There may be a jump component to random errors as well. In titis scction, we provide thig extension of the Ito formula.
n
where tW/; is a standard Wiener process. new term dJt represents possible unanticipated jumps.This jump component has zero mcan during a tirtite intcrval h:
(79)
Ntjpitt.
=
at dt +
=
'lnhe
N
y'(fl
249
redundant.
Poisson process,
all
jumps
have size 1.
nis
sep is thus
'(
(
250
C H A P T ER
.
10
Ito's Lerrkma
Exercises
,
251
.; .)
event. But once the jump occurs, the size of the Jump is also random. Moreover, the structure just gven assumes that thesc two sources of randomness . are independent of each other. Under these conditions, the 1to formula is given by ,
t) =
dFst,
dJF
where
Ft-vht
k
(FSt
+J/,
f j-FS
3=1
,
f
))#j+
1
,7-
Fssp
o
=
for Chapter 9 also apply here. lto's Lemma The sources recommended and the 1to integral are two topics that are always treated togcther, is the book by One additional source thc reader may appreciate Kushner ( 1995), which provides several eumples of lto's Lcmma with jump processes,
;
L'
.
dt A-Fsd't +.JJw,,
'
'-'
:
(85)
Ej
is Svenby
(F(&, t) - FLSt-, f)1
..j7
Fbb't + ai,
A,d
-
Exerclses
10
2
3
d-lk
9 References
.
:'
/)
-
F(St. tjlpi
j uuc 1
#l.
(86)
1, Differentiate the following functions with respect to the Wiener process 14z),and if applicable, with respcct to /. (a) f(W';) 72 1(H':) 1#r/
7:
Finally, Sf- is dehned as
?
= =
st-
=
1im%,
s
<
r.
(87) .'
(b) f(H() (C)
nat is, it is the value of u at an inlinitesimal time before 1. How would one calculate thc dly in practice? One wouid first evaluate the expected change duc to possible random jumps,which ig the second side of Eq. (86).To do this, one uses both the rate term on the riht-hand occurring during dt and the expected sizc of jump in F(.) of possible jumps If during that particular timc a jump is obsenred, caused by jumps in St. right-hand the side is also included. Otherwise, tho hrst then the term on
:
'
(d) g
: '
i
.?'
(a) x (b) X (c) z
;
E .
:
:7
tool in stochastic
.(
calculus. There
Ito's Lemma is the central difcrentiation are a few basic things to remember. First, the formula helps to determine
.g'
(d) X:
;
?
'i.
=
(71)4
=
(H
+
=
t2 +
eH
e'Z'b =
12)2
B?rz
=
gtz
l
:'
:
=
,
.
j
')
lexyected
'.'
.
'2
frG-'W)l+'7'W u
0
-
(a) Calmllate dut (b) What is the rate of change'' of St (c) If the exponentlal term in the delinition of St did not contain the 1+2/ term, what would be the dst What would then be the 2 expected change in
:
E
1
S/
;
.)
W$-
3. Let W( be a Wiener process. Considcr the geometric process St again'.
'''
stochastic differentials for tinancialderivatives given movements in tlze tmderlying asset, Second, the formula is completely dcpendent on the deflnig tion of the lto integral. This means that equalities should bc interete within stochastic equivalence. ! Finally, from a practical point of view, the reader should remember ' bnt standard fonnulas used in deterministic calculus give signiiicantly diffefent results than the Ito formula. In particular. if one uses standard formulas, this wuld amount to assuming that a11 processes under observat ifm have ) zero inhnitesimal volatility. This is not a pleasant assumption when one tryingto pric,e risk using fmancial derivatives.
e('D
=
=
''
'
=
f f
2. Supposc the H(k,i 1, 2 are two Wiener processes. Use Ito's Lemma in obtaining appropriate stochastic dilerentiat equations for the following transformations.
:
8 Conclusions
d(Wl2'
) F'rl' ) JJJKds ,
,
:
term will equal zero.
f (W'; f (W';
=
'
':
..
. '
' , :.
.
'
:
E ?Y-m-'eW
-
-
o
t-'
m.+-
-
X'-'XJ..-
'
'
=-
=
-.
.
-
z
variable perfedly, and #H?; spE wc would get
: .'
(.
:
na
e *
erlvatlve
dSt
;
r1CeS
;':
. Dtfferentlal Equutl Xtochzz-qtlc
.
'', E
dS
=
ast, t4 dt + u'lSl
,
;
)
f ##J( ,
f q
? ; :,'
(1)
(0vx).
7E '
was justilied as a symbolic way of writing z+
dsu
=
/
auu,
a) du +
o'vb'u , t
lf)
ll4i,
(a)
r :,
t when is infmitesual. We repeat some aspects of this derivation. First of all, no concept from ').. @ theory was uscd to obtain (1),The buic tools : Xallcial markets or snancial L split used were the lto integral and the ability to some increment in a and unprcdictable components. predictable price into random This brings us to another point. Given that the decomposition in F..q. : (lJ is done using the information set available at time t, then to the extent ' playtrs may bave access to different sets of information, the SDB different in (1) may also be different. For exampie, consider the following extremeJl1 .( irlformation'' and learns Case. Supposc a market parcipant has changes in advance. Under tbese the random events that inlluencc price (unrcalistic) conditions, the diffusikm term i.rl (1) would be zero. Sinco tllr 9he calz Predict tIIW ; Participant knows how dh ig going to change, he or :
:
1.1
O-LS:t) j
(/4,6
(4)
.
,
clmdfjona(m
'
dk
'.
..selves
Winside
:
=
af rmtj
n
E) $ . '
(St'
) dJ
-1- el'%, J)
dW?l'
J 6
!0,tx'),
t5)
were ailowed to depcnd on St and f. Hence. these parameters are themrandom variables. Thc point is that given the information at time f, they are obsewed by the rnarket participant. Conditional on available information. tey constant. This is the consequcnce of the imPortant assumption that these parameters are fj-adapted. At several points ftbecome''
,
)
j-
We drift and difhlsion parametcrs of the SDE
%'
'
,
.1
.h
252
a(Sf tj dt
d'new
'
'J.
t+h
t+h
(,3.)
) dt-
awst,
.
E'
'T'hc concept of a stochastic differential equation (SDE) was introduced in Chapter 7. In Chapter 9 wc used the 1to integral to formalize this concept. nc notation
=
,
'
1 lntroduction
were to write this participant's
In these two cquations the drift aod t:e diffusion terms cannot be tlie SDES are different, which makes same, The error terms that drive the shows /) that the exact form of a' (.$ difercnt from aLst r). This example the sos, and heoce the dermition of the error term Jp), always depends on the family of information scts jh, t e (0,TJ). If we had access to a different family of information sets, we would make different prediction errors, and the probabilistic behavior of the en'or terms would cbange. Given a different family of information sets f;*, we may have to dcnote the smallcr variance errorsby #Jf7 instead of ##J';. It may be that tfHz;*bas a than #Hz). In stochastic calculus, tbis property of B') is formally summarized by saying that the Wiener process 1.F)is adapted to the family of information scts If. The SDES are utilized in pricing derivative assets bccause they give us a formal model of bow an underlying asset's price changes ovez timc. But it is also true that the formal dclivation of SDES is compatible with the way dcallvs behave in hnancial markets. lri fact, on a given trading daya a trader continuously tries to forecast the price of an assct and record thc events'' as time passes. These evcnts always contain some parts until one obseaes the dut. After that, they become unpredictable that are known and become part of the new information set the trader possesses. This cbapter considers some propcrties of stochastic differential equations.
.
:' .
.
=
1. If wc
whcreas for al1 other market participlmts,
t,
#
*
0 for all
=
tst
.,t
1CS O f
253
) Introduction
J
'''*-%
i
I
'
.
.
2
r
'
254
c
1. A P T E R
The Dnilt-rtitu of Derivative Prices
l1
.
suggesting that these
we made assumptions
0
u)2 du
<
x
l
=
) #/ +
sa
,
f
) J 14/),
.
E
aah
: : : : :
: : :
3h
4h
:4
(0,x),
2h
-
.L
>'loe/'''E,
.
Timv
t
y
l
::
;.
E
g
:;
SDES rrhat
;
, !
SDES
solutionof
is, it A stochastic differential equation is by detinition an equation. unknown. notion unknown is the stochastic St. contains an process nis ne of a solution to an SDE is thus more complicated than it may seem at the eutset. What we are searching for is not a number or vector of numbers, lt is a random process whose trajectories and the probabilities associated With those trajectories need to be deternned exactly,
E.,
'.:
: '
e
error
h
)(..
l
r
':
s2
aw,
h
''3
.
..
equation
a'vh
: : :
$
.
2 A Geometric Description of Paths lmplied by f
E--XX
I
.,too
,
#
!
:
J
These conditions havc similar meanings. They requirc that the drift and mucla,, over tme, diffusion parameters do not vary Note that the intcgrals in these conditions are taken with respect to time. di ln this sense, they can be desned in the usual context. According to thlq, the conditions imply that the drift and difhlsion parameters are functions of bounded variation with probability one. In the remainder of this book, we assume that these conditions are a1ways satisflcd and never repeat them.
consider the stochast ic differential
,
1.
and
aut
s
j
't N ,
: : :
'-...w
g3
error AWa
-
X1K
,...'''''
0
=
Sr
k
=
tz.45-
255
(6) L ;.
wherc the drift and diffusion parameters dcpend tm the level of obsewed
'
' tm 1. asset price St and (possibly) would such an SDE imply for St'? What type of geometric behavior An example is shown in Figure 1. We consider small but discrete intcr- 1 vals of length h. We see that ovcr time, the behavior of S: can be decomP0ged into hvo types of mtwemcnts, First, there is an ewpected path durh the intewal. These are indicated by upward- or downward-sloping arroe. nen, at each I kh, therc is a secon d movemcn t orthogonal to the Pre- (' changes.l dicted These arc represented by vetical arrows. Sometimes tIIG Over are negativea' other times they are positive. ne actual movement of Si i indicated and is of these two components t ime is determined by thc sum t by the heavy lirie, Of Ej This geometric dcrivation emphagizes oncc again that the trajecto ries $ likely to be very erratic when h becomes ininitesimal. . x t are
3.1 %ut
$
Docs a Soltitimz Metm?
Fjrst, consider
the
intewals:
snite difference
in small. dscrete
approximation
,
:
:
Sk
-
&-l
=
ask-j
,
k4h +
twtkk-j
,
klAI'V,
k
=
1, 2,
.
.
.
,
n.
(7)
=
,.
T13eSolution to this equation is a random process St. We are interested in hriding a sequence of random variables indexed by k, such that the increments Luk satis (7). Moreover, wc would like to know tlze moments and tjje djstribution function of a process Sk that satisses Eq. (7),At the outSet, it is not clear that, given a particular J(.) and tr(.), we could fmd a sequence of random numbers whose trajectories will satisfy the equality in (7)for a1I k.
,
,
'.'
here implies
i<'orthogonal',
'xupcorrelatcd.z'
.
..l
t.
.
E '
:L
E
' :
.
L
vh,
f
SDES
*4
''
r
of
:!
parameters should be wcll behaved. It is customary to specify thcse '' regularity conditions cac h tim an SDE is proposed as a model. t) parameters are assumed to satisfy thc condine algt, f) and tions t If) 1 P jt7tk%, Idu < tx)
dst
.3 Sglution
'E
duringthe previous chapters,
;
'
. E
j
j
'
256
C;H A P T ER
*
'
The Dynamics of Derivative Prices
11
i''
More importantly, our purposc is to look for this solution when h, the inten'allength, goes to zcro, If a ctmtinuous time process satisfies the
argue that nding an St and a JI';,such that the pair qt, )j satisfies this cquation, is also a type of solution to the stochastic differential equation. In tlis type of solution, we are given the drift parameter ast, /) and f). the diffusion parn mcter gst, we then find tlie proccsses #, and 1i', such satiscd. in Eq. contrast to strong solutions where one is that (8) is nis considers solve it another given of the problem. but for the J#;, does not confusing potentially points hcrc. First of all. Clearly, there arc some tf11.) and l'F/ bctween both if are Wiencr proccsses what is t-he diferencc #/? Are these not the samc objcct? W ktjj zero mean and variance If looked at in tcrms of the formof distribution functions, this is a valid question. The density functions of tr.fl and Jl'f''j are given by the same formula, ln this sensc, there is no difference between thc two random errors. Thc difference will be in the sequence of iriformation sets that define #T#; 2 the same, the tw'o ranand d 11zt. Although the underlying densities may bc real-life phenomena if could indeed represent diffcrcnt dom processcs very informatitm sets. mcasurable with different respect they are to re-emphasizes made it precise because This has to be an impormore order clarilied for the nccds be in earlier that made point tant point to a reader to understand the structtzre of continuous-time stochastic modcls. Consider the following SDE, wherc the diffusion term contains thc cxogeuously given tp;:
-$,
equation
t
I
J&
for aIl
f
o'vu,
0 tl 0 solution 0, then we say that St is the of
>
d'St
a(St, t) dt
=
';
t
u) du +
auu,
=
+
J)
U'CSI,
uj #JFa,
(8)
j' '
tfM,
(9)
7
SDES
Bccause the solutions of are random proccsses, the nature of these solutions could be quite different when compared with ordinary differential equations. In fad, in stochastic calculus, there can bc fpyw types of solutions. 3
.2
Types of
i 6
; 7
soll4tolu
!
'
The Iirst type of solution
to an SDE is similar to thc case of ordinary differential cquations, Given the drift and diffusion paramctcrs and the random innovation term JT#;, we determine a random proccss paths of which satisfy the SDE;
.
:;
(
ut,
dSt
ast,
=
t) dt + rfst,
t) /))),
/ t:
j0, x).
on time 1, and on the past clearly,such a solutio'n 5', will depcndvariablc values of the random 1, as the underlying temporaneous illustrates: equation
,/
=
vb
+
t
o
rlsu,
u)
'
;
?'.
and con-
integral
/
u) du +
asu,
'
(10)
J
.f(f,
=
.
l
) j,
is a Wiener process whose distribution is determined simultanewhere 'r'lz; SDES, thc givens ounly . with t. According to this, for the wcak solution of f!(.) and tr(.), on% thc drift and diflsion parnmeters, of the problerns are respectively. The idea of a weak solution can be explained as follows. Oiven that solving SDE: involves linding random variables that satisfy Eq, (8),one can ''
dW'.
%,
.
z-, could generate some othcr information some process (, Th.e process Ht. necessarily be f/-adaptcd. But Il''r will will The corresponding not set still be a martingale with rcspect to histories Ht.*
,;
(12)
f)
'
&
'
*,),
df + G(S/,
.
',
,
f)
Gut,
=
Hcuristically. the error process J;'I'Isymbolizes inhnitcsimal events that af,, cc fect prices in a completcly unpredictable fashion. ne history generated tay such jnjisjtesjmal cvtmts is the set of illformation that we have at tirrle t This we dcnotcd by ;t.3 Solution Tbe Strtng then calculates an St tbat satislies Eq. (13)with #H( given. That is. in order to obtain the strong solution St, we need to know the family h. This means that the strong solution S will be f/-adapted. The weak soiution on the othcr hand, is not calculated using the PFOCeSS that generates the information set lt. Instead, it is fotmd along with
:;
a for all f > 0. The solution determines thc exact form of tMs dependence. When H( on the right-hand side of (8) is givcn cxoacnouslv and ,S. is then xo/ufftfzy of tfe sDE.' This is we obtain the so-called strong dctcrmined, similarto solutions of ordinary differential equations. ne second solution concept is specinc to stochastic differential cquat ions,It is called the weak solution. ln the H?dtki solution. one detcrmirles thc process &
,jl
d%
'.
(11) ..
fp-u,
257
3 Solution of SDL
::E
!.
E
L 2./u
'
;
.
.'
OVCS
E'
:
(
we see in On
Can
J
,.L
k
chapter14
,
tize two
wicner procesws
As mentioned earlier, mathematiciarls
4necatzse '
d't.
of tls martingalt prowrty, still be defmed the same way.
may imply different prebability mea-
call such inlormation ,
sets c-lleld
G-algebra.
or tbe Ito integrals that are in the background of
SDES
j' r t
C H A P T ER
*
11
.
3 Solution
'Fhe lxnamics of Derivarive Prices :1
Hence, the weak solution will satisfy
dk
=
ak.
r) J,rr,
(14)
where the drift and the diffusion components are the same as in r'I''J where is adapted to some family of information sets Ht.
3.3 Gch
equation
,
EE
(8),and
dxt dt -
:
.
Notc that the strong and the weak solutions have the same drift and - will have similar statistical propen diffusioncomponents. Hence, St and S: ties. Given some means and variances, we will not be able to distingktish betweenthe tw'o solutions. Yet the two solutions may also be diferent.s 'T'he use of a strtmg solution implies knowledge of the error process F'1. If this is the case, the snancial analyst may work with strong solutions. Often when the price of a derivative is calculated using a solution to an SDE, one does not know the exact process H'r/.One may use only the volatility and (sometimes) the drift component. Hence, in pricing derivative products under such conditions, one works with weak solutions.
can
.$
tion
E
:
'
xt
,
.!
'.
:
?
.( ').
' '
o/ Stl'ong Solteions
E
'
=
&+
asu,
u) du +
.:
rqt-u,
u4 Jl'u.
t
6
p,
x),
(15)
ao
..: '
'
'ng
E(I
.
''
'
,2
;
:.,
rjwo
sof course, the rcader may wonder how he candidate soluion was obtained to begn w'ith. This topic belongs to texts orl diffcrcnlial equations. Here, we just deal with models routinely
.'
1 tion i
llS0
2 We'ak
Solutin.
Bu!
lhC.
,,
,
:'
SO B
=
.
Say
..
Anjr Strtng
first condit j oyj js satisfed.
Hencc, the candidate solution satis 'Iies the initial condition as wcll. We thus that Xt solves tbe ODE in (16),This method verilied the solution using the concept of derivative. 6 If there is no differentiation theory of continuous stochastic processes, a similar approach cannot be utilized in verlfyl' stlutions of SDES. ln fact, if that onc uses the same differentiation methodology, assuming (mistakenly) j tjons it holds in stochastic cnvironments, and trics to averlfy soEs to so u by takin g derivatives one would get the wrong answer, As seen earlier, the rules of differentiation that hold for deterministic ftmctions are not valid for gunctions of random variables.
71*
is valid for a1l t. ln other words, the evolution of St, starting from an initi al !.' point &, is determined by t-he two integrals on the right-hand side. The s0- h !. lution process st must be such that when these integrals are added together, y( . they should yield the increment St - &. This would verij. the solution. ; This approach veries the solution using the corresponding inte equation rather thm using the SDE directly. Why is this so? Note tbat accordingto the discussion up to this point, we do not have a thcog .( candiof differentiation in stochastic environments. Hence, if we have a E'. date for a solution of an SDE, we cannot take dcrivatives and see if tlm alternative routes exst. corresponding derivatives satisfy the SDE, 5
ne
,
(..-()c) x (j.
':
0
ugy atyadj
=
=
l ''l
)
which is indeed a times the function itself. Letting f 0, we get
.
jr
0
j
':r
'j
at
-t.xg
E,
Lfunknown's
st
d dt
y. ,y
Thc stochastic diffcrcntial cquation is, as mcntioned earlier, an equation. This means that it contains an unknown that has to be solved for. ln the under consideration is a stochastic process. case of SDES, the By solving an SDE, we mean detennining a process St such that the interal equation
ut
x()c
=
js a solution of (16).Then, the solution must satisfy 2$$5 conditions. First, jf we takc the derivative of Xt with respect to t, this derivative must equal = 0, the function a t jmes the function itself, Second, when cvaluated at t which is assumed to bc equal initial point, value -Ya, the jloultj b give a to known. weproceed to verify the solution to Eq. (16).Taking the straightlbrward derivative of Xt,
!
.
.
jj
(16)
axt,
=
where a is a constant and A-()is given. Therc is no random innovation term; this is not a stochastic diffcrential equation, A candidate for thc solution be veriticd directly. For cxample, supposc it is suspected that the func-
:
Solution Is to Be Pveferved?
3.4 A Dtctuuion
259
SDL
The process of verifyiog solutions to SDES can best bc undel-stood if we start with a deterministic example. Considcr the simple ordinary differential
;'
t) dt + c.r(.,
of
reverse
is n0t true.
.'.' :i
Dged in
.'.'
' ' (-
. ..1: .
f. :
.? .; .
(
j
L.
. :
'
.'.
'
snance.
';
.
.1
260
C HA PT ER
11
*
The
(
Wnamicsof Derivative Prices
f
dt
!
(20)
ax l !'
dSt
2.
x
0 ea'
;'
=
a dt +
tr
JH(,
f
:
0
)
tr,
k0,
f,
r ds ?
E
t)
.:11
(24)
mdt+ GJHI.
=
(25)
$'
&
'
'
mduy
=
0
u
0
vJ<.
k
(26)
.'
Since the first integral on thc right-hand side does not contain any random terms, it can be calculated in the standard way!
?
Jki..
l
,,
y't. gdu (27) 0 nc second integral does contain a random term, but the coefficient of #H''a is a time-wariant constant, Hcnce, this intep-al can also be taken in the usual way
.:..qi
=
2
: .
.
.'
.
,','
q . jj ;..
(23)
..
l
%.
where by defmition 8$
=
!
.;
s
.L
:
=
y.t + tr/l,
,
(29)
=
sf)
g((c-)f,.2)/+crp$) '
(,3c)
Note that this solution candidatc is indeed a function of the parameters a and o., of time t, and of the random variable W(, Clearly, we are dcaling with a strong solution, gince vS$depends on JP; and is J/-adapted, How do we verify that this function is indeed a solution?
.
.' '
'.' :J
'
',
''.,
7
(28)
Consider the candidate
p
:g
(k
,
Any solutions of the sDE must satisfy this integral equation, In this particular case, we can show this simply by using lto's Lemma.
',
.' ;'
..3'.
H( - rJ/l)j
0, Thus, we have - dtu 0 su
.
(kY
7Te drift and diffusiorl pazameters are constant and do not depend on the information available at Time 2. Slt is important to keep in mnd that the SDE discussed alxwc is a special casc. In generalv t h e, strong solution S( sown iri (23)will depend on the integrals of asu, u), Gtsu, a). afld depcndence will be on tlle wlzole trajectonr of 11.). ffH';. Hence, the
t7'l
; 1
:,,
>
tr #JF,k
=
(1
'
.
,
(0,x),
First, we calmate the implied integral equation:
?',
Hence, the solution will be a stochastic process becausc it dcpends on the random process +:.8 Using deterrninistic differentiation formulas to check whether this f (') satislies the SDE in (22)means taking the derivatives of St and JF; with respect to f But these derivatives with respect to t are not well defined. Hcnce, the solution cannot be vered by using the same methodolor as in thc dcterministic case. Instcad one should consider a candidate solution and then ushlg lto's Candidate Satislies thC SDE Or the Corresllondiv YCDFIIZ :' ty to SCC if this integralcquation. ln the example below, wc consider this point in detail. :l'
f (E
,
./
(22)
H( ).
tlz/;
.
h'll
=
dSt
V,
.
given.; When a strong solution of this SDE is obtained, it will be with functitm /(.) that depends on the time /, on the parameters (d?,fr, S'*J. somc und on thc H( :
-% fa,
dt +. o6L
1
.i
Vcrifying the solution involves diferentiating this function Xt with respect to the right-hand-side variable f, and then checklng to see if the ODE s satistied, Now consider thc spccial case of the SDE given by ds r
gvh
=
.r.
(21)
.
to SDE:
which was used by Black-scholes (1973) in pricing call options. Here, St represcnts the price of a security that does not pay any diWdends. Dividing both sides by St, we get
!'
.j =
solzttitzn.s
!.'j
. with ZVpven, both sides of the equation contain terms in the unknown functitm Xt. That is why the ODE is an equation. ne solution of the equation is then a specihc functionthat depends on the remaining parameten and known variables in the ODE. ne parameters are (f?,A%), and thc only known variable is the time t. Hencea the solution expresses the unknown function Xt as a function of the known quantities'.
.A't
261
Again consider the spccial SDE,
'
=
SDES
3.5 Vegi/icctielz of
-'
Some zrther comments on this point might be useful. Note that in an ordinary differential equation,
dx '
Solucion of
' .
262
C H A PT ER
l1
.
The L-hrrlamicsof Derivative Prices
Consider calculating the stochastic differential dSt using Ito's Lemma'.
ds t
=
1 z g&etta-lsz2l/mtrl#;lj a a.
dt +
fw
t/pl
+
1
i
ty a
a
the very last term on the right-hand side corresponds second-order term in lto's Lemma. Canceling similar terms and replacing by St, we obtain
where
dSt
,%gtz
dt +
=
tr
(31)
,
to the
#JF;1,
a-, -
s,jtu
-
of
SDES
263
This would make the current price equal to the expected price at time F discounted at a ratc r. This martingale property is of interest because it can be exploited to calculate the current price S3. Wc now calculate Et (,z,l.The rst step is to rcalizc the following: gT
=
)fOlrj
g(r-
g;j0
gtywT'p-r
j
(g8)
>
so that expectations of Sp depend on expectations of the term
(32)
the original SDE with a equal to y,. It is interesting to note that the whichisfrldt la are eliminated by the application of Ito's Lemma. lf the rules terms of deterministic differentiation were used, thcse terms would not disappear in Eq, (32),and the function in (30)would not verify the SDE.
In fact, if we had used ordinary insteadgive
3 Solution
calculus, total differentiation would
e
fwllz'w
(gg)
,
where, for future reference, the expression is a nonlinear function of l#'w. Hence, the Sp is a nonlinear function of Hz-ras well. This means that in taking the expectation thc 'jg,J front operator we cannot of the random term W'rr. We can approach the expectation Etfeeikb 1 in two diffcrcnt ways. One method would be to use the density function for the Wiener proccss JFr the cxpectation directly by integrating arld
tzrl,
L<move''
Itake''
jlvzl dt
+.
crduzij,
(33)
and this would not be the same as tbe original SDE if a equals p.. Hence, if we had used ordinary calculus, we would have mistakenly concluded that the function in (30)is not a solution of the SDE in (24).
Et
gcf'B'r
)
c-H'r
= -Y
ytrpk.I ptlj
d py,
(40)
where the term in brackets inside the intcgral is the (conditional)density function of H''z. The (conditional) mean is 1, and thc variance is F - 1. Calculating this integral is not diflicult. But we prefer using a second calculus.'' nis method will illustratc method, which is specilk to Ito's lemma once again, and will introduce an important integral equation that sees frequent use in stochastic calculus, According to Eq. (38),Sv is given by the function ftstochastic
3.6 An lmporftmt Exumplc is gome Suppose other words, we have et
dSt
asset price with a random
=
rSt dt + xsv J7,
t
rate of appreciation,
this
SDEI
s/
s()
=
(?1(r
(34)
j0, x)-
QE
The previous section discusscd a candidate for the
In
(strong)solution
ltr2)t+crH$)
of
(35)
'
Now, suppose is thc price at some futtzre time F > f As of time 1, this Sp is unknown, But it can be predicted, and the best prediction will be given by the conditional expectation: uv
.
.E',E,z1 FE-zk/rl. =
(36)
ln asset pricing theory, one is interested in whether the following equality will hold: s/
.g-rlr.
/).s /
j. T j
.
q5p
Sg.
=
e (r(k$a
(r.2
)r
1(t
tsrrty
1 -
(4j.)
The idea behind the second mcthod is to transform tMs nonlinear expression in 1,FI into a lirtear one, and then take the expectaticms directly without having to use the density function of thc Wiencr process. The method is indirect, but fairly simple, First, denotc the nortlinear random term in Eq. (35)by Zt:9 Zt
=
c'Ws
(42)
.
Second, apply Ito's Lemma:
dz /
=
tzztrli',
dw J +
9The next fcw dcrivatons use the t subscript geleraiity. It simplilles cxmsition.
1 -c c e
.1
2
nstead of F.
dt. nis
(43) loes not cause any loss of
W
q
;'r'
fe
nird,
c H A PT ER consider the corresponding
zd
=
t
z
+
Finally, take expectations
interal
t)
ds.
'
(44)
.
t
E
eGH$
Jp'
1,
0
'
0 since the incremcnts in a Wiencr process are independent from the obscrved past. Consequently, A'gz';j cf'Wl
t!
.J(;
1+
=
.j.tr
g,sjz ,
j g,
ookcj.l jmyjes
so
r . 6
. .7.
'
s
/
L'
-t7' 2xs d n 2 to the ordinary differential equationlo
is equivalent
1+
=
.
=
Xt
with
FgZI)I =
=
1. Going back to
E / 25' F) Using the result
,
=
=
I''Q'
e
' j:
E .::
g-wl =
t
j gus-
j56)
'r
.
gz'wpi
=
(5J)
p;,
vjv
T
j
=
s.
te
(r-h ) crz )(T - l )
.
(5g)
(51)
4 Major Models of Ics.B:el''OCT-/)j
(52) '
f, of
(48). .?
:
.
.
SDES
There are some specitk soss that are found to be quite useful in practice, ln this sedion, wc discuss these cases and show what types of asset prices they eould represent and how they could be kksefui.
'
to
-,)s
'(
(ks-a/r-lWltrl)
10By Saking the derivatives, with rtspect
.
Thc use of standazd calculus implies that today's stock price is not equal to the expected tuture value discounted at a rate r. Wc losc the martingale equality.
c l
(55)
.
just derived for fgzl, E
Ej
':
Qgzz),
I),$()e(r-)tr2xr)1.; izz').
agusz'j.
and the expectcd value of thc stock price would be written as
(50)
,
e-rl
=
dz,
L
(49)
E EZt 1
t
.
.
dxt 1 7 = dt 2 1. ne solution of this ordinary differential equa-r-xtt
with irtitial condition xll tion is known to bc
:. '(r
(48)
e-rTE
=
It is worthwhile to repeat the way Ito's Lemma is used in these calculations. By using it we were abie to obtain an integral Eq. (47)lear in Zt. This way we could move the Fg.1 operator in front of Zt and use the fact that increments in Wiener process have zero expcctations. nis eliminates thc integral with respect to the random variable. ne second integral was with rcspcct to timc, and could be handled using standard calculus. At this point, if instead of Ito's Lemma we had used the rules of standard calculus, Eq. (43)would become
:
xt
(54)
.
)
(47)
('/er(z'-/) 1,
=
That is at time f = 0, the asset price equals thc cxpcctcd future price discounted at a rate r. For any tlme t we have, correspondingly,
. .rl
.
f 1
j5yj
,
tjaat
... '' ;
',
llxrl
f
. ..'
with the substitution Z which is true by the definition of Z. Note some interesting characteristics of Eq. (47),First of all, this equa- ' tion does not contain any integrals desned with respect to a random vari- j able. Secondly, the equation is linear in E (Z/j. Hence, it can be solved . in a standard fashion. For cxnrnple, we can treat S(Z'/1 as a deterministic .$' ''', variable, call it xt, and then recognize that L . =
0
:L
(46)
T
gtp-ytwglyoys;;r
ajj,
we obtain
.(
(45)
=
$
st
.
0. Also,
=
whcre the 1V)term on the right-hand side appears due to conditioning on information at time f Recognizing that
k
on both sidesa and note the following: =
SDES
.
2
()
4 Major Models of
'E
--trzgsr'l
J14$ +
.7(2aj
since, by deinition, Hi
equation; 11
d''H
tw
t.
Tlze Wnamicsof Derivative Prices
ll
*
.E.E.
)
c 1-1A PT
ER
:'T
l 1 The Dynamics of Derivative Prices
.
4 Major Models
k
of
SDES
:
4.1
E'
SBES
Lineur Conatant Cocclnt
The simplest case of stochastic differcntial equations is where the (11-1h2 and diffllsion coeficients are independent of the information received over time: =
Jt #/ +
/H(,
(7'
t e
(0,x),
S
(59)
';
ne
expected variation
.
, ,;
:J ;
Sk
E'
(60) .
in hSt will be Varlks'/l
.
lh.
1
(61)
J
C
Alt example of thc paths that can be described by this SDE is shown in t Figurc 2. Computer simulations were used to obtain this path. First, somo desired values for p. and (r were selected: .;
!)
Then, a small but
=
G
=
: .
:3
.
(63)
h
.001.
;
(
.001
)+
.03(/Hi
),
k
=
1 , 2,
.
.
,
,
1000.
PT
yr
j.ces -
'
(64).q
4.2
'
=
.01
u
; .'.' ..((
sniteinterval size was decided upon:
zS-.j+
fttrend''
(62) )
.01
g
=
with the initial poiot s', given, onc substitutes randomly drawn normal random numbers for Wi and obtains the Sk suessively. As can be seen from this tigure, the behavior of Sf seems to lluctuate around a straight litte with slope y,. 'T'he size of tz' determines the extent of the tluctuations around this line, Note that these Cuctuations do not ecome larger as time passes. ajs suggests when such a stochastic differential equation is appropriate in practice. ln particular, tls SDE will be a good approximation if the is linear bchavior of the asset prices is stable over time, if the and if the Nariations'' do not get any larger. Finally, it will be a good approximation if there do not appear to be systematic jumps in assct
:' '
=
(65)
'
,
ut
=
100
=
ne
Finally, a random number generator was used to obtain 1000 independent, normally distributed random variables, with mean zero and variance 001, The fact that JP; in (59)is a martingale permits the use of indepcndcnt (normally distributed) randtam variables. A djscrete approximation of Eq. (59)was used to obtain the & plotted jja sgure 2. The obsewations were determined from the iterations
'.
.:
where Wzlis a standard Wiener process wit variance 1. In this SDE, the coeocients t and tr do not have timc subscripts t. This means that they are constant as time passes. Hence, they do not depend on during a small inteaal of lcngth the information sets It. The mean of is given by
F,EA,%Ipz.
is assume d to be an approximation to the inlinitesimal inten'al dt.
initialpoint was selected as
:'
':
dSt
'INS
r
: .'
.(
dst
.5
=
p-$-, dt + r-
Jl4.),
t e
((),x).
(66)
This model implies that in terms of the formal notation,
: E).
/)
=
o-qst, t)
=
ay,
:
':
: '.
ss,
and
:' k
r
yst
.
((a)
llence, the drift and the diffusion coefficients depend on the informathat becomes available at time f However, this dcpendence is rather
'
tion
FIGU RE lx
..
.k
SDES
Thc standard SDE used to model underlying asset prices is not the linear model, but is the geometric process. It is the model constant coecient exploited by Black and Scholes:
. 1; k'
.'(.
cometric
.1
'
.
'.),
!
..
.E .!
268
c
H A P T ER
.
4 Major Models of
'
!
26p
=
St
'
(.
.
15& dt +
JJ#;
.30k$
4.
dst
,
. i
'
just discussed is the
=
gst dt + cw St tH(.
t
square root process,
((),x).
(72)
UE
ifvariance's
:.
ut.
'J
KGtoo
,.
r :
'
/
!
h
:
gs t
::) :E.
',
-
j5,j, t gt
o
.g()
st tuj,
j,sj
as s'o in ygure 4 are more subdued than the ones in pj gure 3, yet the sample paths have trends.
clearly,tlae quctuations
: ,'
:
(<similar''
'
,
.E.
st
.
whcretbe drm and diffusion coeflicients are as in thc case of Figttre 3, but wherc the diffusion is now proportional to St instead of being proportional to St. We select thc initial point = 100.
,k
(71)
y.
'
Here the St is made to follow an exponential trend, while the standard deviation is made a function of the square root of St, rather than of St iself. nis makes the of the error term proportional to Hence, if the asset price volatility does not increase much'' when St increases, this modcl may be more appropriate. will, of course, be the nis case if St > 1. As an example wc provide, in Figure 4, the sample path obtained from tje same (/H/: terms used to generate Figure 3, We consider the equation
E'.
'rhese
c2&2
SL
i' s
(. ly'
.. '
St
.
.r '
;. ).
'
100 Time
.
(
.-
. .,
l()c Time
'
1
K
FI
c;v R E
'
!.
''
=
=
' 'i
'
sqxuge Root Iarocess
$
100. As can be seen from this graph, the St is with the initial point made of two components. First, there is an eponential trend that grows at 15%. Second, there are random lluctuations around this trend. variations increase over time because of higher prices. What is the empirical relevance of this model when compared with constant coeocient SDEP lt turns out that the constant coefficient SDE described an asset price that Cucttzated around a linear trend. while this model gives prices that Euctuate randomly around an exponential trcnd, For most asset prices, the exponentialtrend is somewhat more realistic. But this says nothing about the assumption concerning the diffusion c,0efticient.Is a diffusion coefficient proportional to St more realistic as well? To answer this, we note that the ''va riance'' of an incremental change in times tk and fk-j could be approxlmated by between . st
VartkV- ,T:-1)
.,9
A model close to the one
'.
(70)
,
In
,
E
ut
=
-z.
'
,
This means that although the dri.ft and the diffusion part of thc increment in asset pr ice changes, the drift and diffusion ofpercentage change in still has time invariant parameters, Figere 3 shows one realization of the St obtained from a Iinite difference Vpfoximation Of dSg
'
.
''
(69)
/a,
Hence, the variance increases in a way proportional to the square of some practical cases, this may add too mucli variation to s ,
,
,
1
'
;
$.
strailtfozward. ne drift and the standard deviation change proportitmally with &. In fad, dividing bot h sides by st we obtain ds-1 dt + = :H.:.
ka
ji
SDES
j
j
k
'
The Dynamics of Derivative Prices
11
'
.
FlGU RB
3 : (
5
4
q
'
;
j
. ' .
,
((2H A P T E R
270
*
Stochastic Volatilit'y
The Dynamics of Derivative Prices
11
271
S1
Finally, another characteristic of this process is the meaning of the parametez a'. Note tbat with this speciication of the diffusion component, volatility of St. Markets, on the o- cannot be intcrpreted as perccntage other hand, quote by convention the percentage volatility of underlying
t
aSSCtS.
Process
4.4 Metm Retrting
10
An SDE that has been found useful in modcling asset prices is the mean modelr'' rfdvtrrzw A(p, St) dt + O'S:#M. dst (74) (Js St), valuc'' p,s thc term i.rlparenthcses, As St falls below some will St wiil become positive. nis makes dS; more likely to bc positivc, eventtzally move toward and revert to the value y.. A retated SDE is the one where the drift is of the mcan reverting type, but the diffusion is dependent on the square root of =
idmean
-
vh
1$/
A(/t - Sl) dt +
=
g'
ut
(75)
Jl,f?;,
Thcre is a significant difference behvcen the mean reverting SDE and the two previous modelsThe mean reverting process has a trtmd, but the deviations around this trend arc not completeiy random. ne process St can take an excursion eventually reverts to tbat trend, but the away from the long-rtm tzend. lt time, take excursion may somc ne average length of thesc excursions is > 0. As this parameter bccomes smaller, tlze controlled by the parameter excursions take longen Thus, assct prices may exhibit some predictable periodicities. nis usually makcs the model inconsistent with market efEciency. Am example of a sample path of a mean reverting process is shown in Figure 5. We selccted ,
.05,
p,
rfail:lf)
-
A=
=
.5,
(76)
.8.
tr
=
This implies a long-run mean of 5% and a volatility of 80% during a time inten'al of lcngth 1, The implies an adjustrnent of 501. l According We then selected the length of linite subintervals as h 1000 Stts. will observe length 1, intenral of wc to this, during a timc obtained, and and variance were with numbers mean zero Random thC sample path was generated by using the increments 1000, (77) 1, 2, k zp-:)(.(X)1) + .8Wi, sk
FlGU RE
5
The trajectory is shown in Figure 5. Because the diffusion term does not depend on St, in this particular case, the process may become negative.
4-5 flrntein-tlhlenbeck
Pgocess
Another useful SDE is the Ornstein-uhlenbeck dSt
=
-mSt dt +
GJM,
process,
(78)
where p, > 0. Herc the drift depends on St negatively through the parameter /j, and tlze diffusion term is of the constant para' meter type. Obviously, this ls a special case of reverting SDR'' This model can be used to represcnt asset prices that Euctuate around zero. ne :uctuations can be in the form of excursions, which eventually revert to the long-run mean of zero. 'T'he parameter Jt controls how long excursions away from this mean will take. larger t'he p., the faster the St will go back toward the mean. imean
'l'he
,00
=
.
5 Stochastic Volatlllty
,001
.5(.05
=
=
-
wherc the initial point was ut Ilerhis
is oftell used
10 model
=
100.
interest ralo dynamics.
.
.
.
,
Al1 previous exnmples of SDES consistcd of modeling the drift and diffuparameters of SDF-S in some convenient fashion. ne simpiest case lhowedconstant drift and diffusion. complicated most case was the ne mean reverting prxess. A much more general SDE can be obtained by making the dri and the diffusion parnmeters random. In the case of financial derivatives, this may Sion
272
'zH A P T E R
.
The Dynamics of Derivative Prices
1l
Exercises
have some interesting applications, because it irnplies that the volatility may be considered not only time-vaying, but also random given the St. For example, consider the SDE for an asset price St, dSt
p. dt +
=
th
8 Exetclses Consider tle following SDE: d(J#;3)
(79)
JW(j,
=
ttyb -
=
3gF)Jf +
/
(80)
rt ) dt + art J1'l''i,,
where the Wiencr processes 1.f'k/4r'i! may very well bc dependent. Note what Eq, (80)says about the volatility. The volatility of the asset has a lonprun mean of tyb. But at any time 1, the actual vtalatility may deviate from this long-run mean, the adjustment parameter being ne increments #HGare unpredidable shocks to volatility that are independent The tr > 0 is a parameter. of the shocks to asset prices The market participant has to calculate predictions for asset prices and for volatilty. Using such laycrs of SDES, one can ohtain more complicated models for representing real life, Iinancial phenomena. On the othcr hand, stochastic volatiiity adds additional diffusion components and possibly new risks to be hedgedmay lead to modcls that are not ,
,
I#;2JJI .
(a) Write the above SDE in the integral form. (b) What is the value of the integral
wherc the drift parameter is constant while the diffusion parameter is assumed to change over time. More specilically, tr1 is assumed to change according to another SDE, do't
273
0
l#'. 1dw.$
.
2. Consider the geometric SDE: dSt
gstdt
=
whcre St is assumed to reprcsent index is
+ fr.S'/tfH/;,
an equity index. The current value of the
ut.
idcomplete.''
rrhis
.t
6 Conclusions
z-dayintervals
denoted by &.
(a) Use coin tossing to generate random errors that will approxmate the term Jl4z),with
SDES. We distinguished rrhis chapter introduced the notion of solutions for sliila.r solution is solutions. The to the case of strong bctween two types of wcl/ novcl. stlution is equations. ordinary differential ne We did not discuss the uzw/c solution in detail here. An important example will be discussed in later chaptcrs. nis chapter also discussed major types of stochastic differential equations used to model asset prices.
ln this chapter, we followed the treatment of Oksendal (1992),which has sevcral other exa'mplcs of SDES. An applicaticms-minded readcr will also benest from having access to the literature on the numerical solution o SDES. ne book by Kloeden, Platen, and Schurz ( l 994) is both very acccssible and comprehensive. lt may very well be said that the best way to undentand SDES is to work with their numerical solutions.
940.
It is known that the annual percentage volatility is 0, 15. The risk-free interest rate is constant at 5%. Also, as is the case in practice. the effect of dividends is eliminated in calculating this index. Yottr interest is conhncd to an 8-day period. You do not see any hnrm in didding this horizon into
four otnsecutive
7 References
=
H F
=
+ 1,
=
-1.
(b) How can
you make sure that the limiting mean and variance of the random process generatcd by coin tossing matches that of #H( as 0:/ (c) Generate three approximate random paths for ovcr tMs 8-day period. ..-.
z
vh
3. Considcr the linear pdce:
SDE that represents
dSt
.01
=
the dynnmics of a security
Stdt + .05,$,#J4$
with 1 given. Suppose a European call option with expiration F 1 and strike K 1,5 is written on tllis security. Assume tlmt tlle risk-free interest rat is 3%. xlj
=
=
=
274
The
CHAPTER.H
Wnamicsof
Derivative Prices
(a) Using your computer, generate fivc normally distributed random variableswith mean zero and variance ,.
(b) Obtain one simulated trajectory for the q Choose r L (c) Determine the value of the call at expiratlon. (d) Now repeat the snme experiment with Iivc tmiformly distributed random numbers with appropriate mean and variances. 1000 times would the (c) lf we conducted thc same expernent calculated price differ signihcantly in tw'o cascs? Why? (9 Can we combine the two Monte Carlo samples and calculate the option price using 2000 paths? ,2.
=
,
Pricing Derivative Products Plrtfal Differenticl Eqlzatnu
4. Consider the SDE: dSt Suppose #H') is appr/ximatcd
+ .1#F).
.05#f
=
by the following pocess + A with probability .5
ls
1 lntroduction
=
- l
with probabilitjr
.5.
1. Calculate the values of St begin(a) C-onsider intenrals of size k 1, ning from t 0 to I 3. Note that you need S and repeat the same calculations, (b) Let f (c) Plot these hvo realizations. (d) How would thesc graphs look if 1, and obtain a new (e) Now multiply the variance of St by 3, let #/ realization for =
=
=
=
.5
=
.01?
.
=
=
ut.
(To generate any needed random variables you can togs a coim)
nus far wc have learned about major tools for modeling thc dynamic behavior of a random process in continuous time, and how onc can (and cannot) take derivatives and intcgrals under tllese circumstances. nesc tools were not discusscd for their own sake, Rather, they were discusscd because of their uscfulness in pricing various dcrivative instrw ments in linancial markets. Far from being mere theoretical developments, these tools are practical mcthods that can be used by market professionals. ln fact, because of somc special characteristics of derivative products, abstract theoretical modcls in this area are much mol'c n menable to practical applications than in other areas of finance, Modern inancc has developed two major methods of pricing derivativc Products. ne first of these leads to thc utilization of partial differcntial equations, wbich are the subject of this chapter. The second requircs transffrming underlying processes into martingales. This necessitatcs utilization Of equivalent martingale measures, which is the topic of Chapter 14. ln Principle, botb methods sholzltl gve the same answer. However, depending 0n the problem at hand, one method may be morc convenient or cheaper to use than the other. ne mathematical tools behind these two pricing methods are, bowcver, very different. Firkt, we will brielly discuss the logic behind the method of pricing seCulities that leads to the use of PDES. These results will be utilizcd in Chapter 13.
27$
J(, .!.
$
C H A PT ER
276
.
12
.J :
Pricing Derivativq Products
.j
,t
2 Forming Risk-Free Portfollos
'
instnzments are contracts wzitten on other securities, and these have flrtite maturitics. At the time of maturity denoted by T, the contrads derivative contract should depend solely on thc value of the of the price Fr Sp, the timc T, and nothing elsc: security underlying
:
tive of FS f /) with respect to f,1 Our maln interest is in the price of the derivative product, and how this price changes. nus, we begin by positing a model that determines thc dynamics of the underlying sset St, and from there we try to determine how Fl-f f ) behaves. Accordingly, we assume that the stochastic diffcrcntial dSt obeys the SDE ,
'
r
jy :E' ,g
L
,
. !!
This implies that at expiration, we know the exact form of thc function Fsv, F). We assume that the same relatiooship is true for timcs other than T> and that the price of the derivative product can be writtcn as
,)
dS,
:.
!'k
i.,'
(2)
r)-
l
,
(1)
Fq. = Fl5'w, F).
,
important remark about notation. dFt should again be read as the change total in the derivative price Fut, tj during an intenral dt. 'This should not be confused with Ft, which we reserve for the partial derivaIAII
'j
Derivative
FS'
Forming Risk-Free Pordblios
f
'
auh, t ) dt + ajutb tj #H(.
=
(:2
dFt
s''
. j.
dFt
,
,t,
,
(3) T
t) + tst,
.Ju
;;
'
=
(4)
d1 dFt + 0z tk'r.
,'
i
;.
subscript ; general, , % will val'y over time and hence will carly a time Sucb dependence. ln this equation, both d% .t YS WC11. At this Point wc ignore thr . due unpredictable to component that have increments an an d # are '
eL
ln
k
t
innovation term #5F; in dst.
:
i weightg Ortly wben the portfolio Strictly speaking, this stvhastic dferenial is correct Will qlliff #OiIH be Will bc fMIXICF right. lhcR 'J. O11 tflrrkhi tile T1()1 dcpcfld OD Othenvisc! rclevant when we discuss the Black-scholes framework below. ! ..f db
1
-1
Yl1i:
l
'j.
Ft dt +
=
1
2
(6)
Fsso't dt + Fx dst.
(5),and
obtain the SDE for tbe derivative
1 Fsat + jFa., o'ta + Ft dt + F, o d l1(
.
,
where p1, 01 are th e quan tities of the derivative instrument and the undorlying security purchased. They represent potfolio weights. ' The value of this portfolio changes as time t passcs because of chang/s 1 , t?), % as constant, we can write this change as in F(Stb r) and St. Takmg .
d#,
(5)
Note that we simplised the notation by witing at for the drift and et for the difhlsionparameter. If we knew the form of the fundion Flut, l), we could calculatethe corresponding partial derivatives, F,, F,,, Ft, and then obtnin explicitlythis SDE that governs the dynamics of the inancial derivative. The functional form of F(St /). however, is not known. We can use the followingsteps to determine it. We fimt see that tbe SDE in (7)describing the dynamics of dFt is driven by the same wicnerincrement JHSthat drives te St. One should, in prindple, be able to use one of these SDES to eliminate the randomness in the Oter. ln forming risk-free portfolios, tMs is in fact what is done. We now show how this is accomplished. First note that it is the market Pmicipant who selects the portfolio weights pl, %. Second, the latter can always be set such that the dPt is independent of the innovation term (sH( and hence is complete% predictable. ne reason is as follows. Given that dFt and dSt have the same unpredidable component, nd given that pj, % can be set as desired, one can always climinate the dl'pkcomponent from Eq. (4).To do this. consider again
.>
et-st,
=
We substitute for dSt using Eq. asset price:
'i
-
I0,x).
Using this, we clm apply lto's l-emma to hnd dl?
.7
Thc increments in this price witl be denoted by dl. At thc outset, a market .k participant will not lnow the functional form of Fst, r) at times other than ;:, cxpiration. nis function needs to be found. .i.. 'Fhis suggests that jf we have a 1aw of motion for thc St process-i.e-, if .;L. determined-thcn we can E we have an equation describing the way dut is would Lb and #5'/ dFt that But this obtain dlt. means lto's Lemma tf use /: namely be incrcments that have the same sourcc of underlying uncertainty, example, the irmovation part in dSt In oher words, at least in the present j, innovation term. E we have fwtp increments, dFf and dst, that depend on one continuoY ( Such dependence makes it possible to form risk-freeportfolios in fs timc Let Pt dollars be invcsted in a combination of F(St /) and St:
l
t e
dP3
.
=
0j dFt + dz dSt
(8)
r'
'j (
'
S
c HA
altdsubstitute
for dFl using dPt
=
P T ER
12
*
7
Pricing Derivacive Products
(6):2
dt terms are common to all factors, they can be obtain a partial differential cquation:
t
:
(9)
2 *1 ez the
0
r(F(-% f) - Fauh) ,
(wereplace
...,
(10)
1
,
and
-
#(f) in
rF + rFsct + F/ + - Fssg'/2
2
(11) !
-F,.
=
to
'
0,
-a
sm
g.a ,
(1:)
.
We remite 0
:
St
Eq. 0
,
2:
t
(16)as s
7)
rnte
:1
'
',
'
=
j.
y,t +
where the de rivative asset price Fs t, tj is denoted simply by the Ietter F for notational convenience. We have an additional piece of information. der jvative product will have an exp sationdate F. and the relationship between the price of the ffl derlying asset and that of the derivative asset will in general, be known exac tly at expiration, nat is, we know at expiration that the price of the derivative product is given by
:': 'aese partic-ular values for potfolio weights will Iead to cancellation of tbe e tcrms involving dst in (9) and reduces it to ,: 1 'E F t dt + -F rt2 dt. dP' (12) 2 Clearly, given the informat ion set 1t, in this expression therc is no randeterministic increment foT dom term. The dP is a completcly#rcc/fzsle, aIl times f. This means that the port folio Pt is risk-free/ PI, its appreciation must equal the earnings of since tliere is no r isk in f isk-free westmtnt durixtg an interval dt in order to avoid arbitrage. As3' exped od t) sumingthat tbe (constant)risk-free intercst rate is given by r, the ( =
=
(15)by its components.l
,
ez
'eliminatcd''
'
:
, way we wish. Suppose we ln this equation we are free to set ignore for a minute that F depends on St and select =
279
ssncethe
J
1 z dt + 01 dst. t?l Ft dt + F, dSt + ,-Fsset .
? FO rming Risk-Free Portfolios
,
).
..
':)
'
:274
F(.V, F)
,
,
ip th e case where St pays no
#t
and must equal
rp ;
dt
-
Gs
(13) ;
Rdividends,''
(14) # .
i
(r
in the case where St pays dividen ds of per unit time. In the latter cmY, the capital gains in (14)plus t he dividends carned will equal the risk-fl'o
rate.4
,
a
s dt
L
t !
utilizingthe case with no dividends, Eqs, r#ydt
=
Ff dt +
(12)and (13)yield
..)(
?.uj
.
t
.
%-'
2
) l
E 11 y if t)1, e2 do not depond on S,. nliaear'' F will or n tizne. value at vary of over pout: Note this mportant ezset -t e o no opuons, the zl will be a funcuon ot ot. C Producs such as options, or stnlcttlres conta ining yet it wll gi!* that tlae rksk-fyee portfoo metbo d is not satisactory mathematicay, PDE. the f*Ce ie: 4xoc the role of Jl. Some innitesima l tlme must pass in order to earn interest orearnin#i the interest lnteres! ra tes r, dividends.If no t 9mepasses, regardless of the lcvel of E W i1lbe zero. The same is tr'ue for dividcnd earninp. zlkecall that this WII be correct mathematica
3
means
.)
'
'vvk;o
.
.'xw-,
,
''corrcct''
u
'
r)
=
maxlsy
-
K, ()j.
(1p)
-
u/rrlftrgfe,
'
r:
T
d
'
1 '-Fsso z dt.
(18)
According to this equation, if at cxpiration the stock price is below the strike price w K will be negative and the call option will not be exercised, lt will be worthless. Othenvise, the option will have a price equal to the differential betwccn the stock and the strike price. Equation (17)is known as apartial #l//reliftzl equation (PDE). Equation (18)is an associated boundary condition. The reason this method works an d eliminates the innovation term from Eq. (4) is that #(.) represents a price of a derivative instrument, and bence has the same inherent unpredictable commnent #1z#; St. as nus, by combining these hvo assets, it becomes possible to eliminate their common llnpredictable movements. A-q a result, Pt becomes a risk-free investment, since ts hlture path will be known with certginty. This cotlstructon of a risk-free portfolio is heuristic. From a mathematical point of view, it is not satisfadory. In flrmal approach, a one should form self-unancing portfolios using completeness of markets with respect to a class of trading and using the implied ynthctic e quivalents of the assets under consideration. Jarrow (1996)is an excellent source on tjnese concepts. Next section d jsmzsses t jus p ojnt jn more detail.
'
,'
F),
j s a known junction of %. and F. For example, in the case of a call option, G(.), tbe expiration price of the call with a strike pzice K is
capital gains must equal
r
Gsv,
whereG(.)
:.
dt
=
'
>'
f'
''
r'
C H A P T ER
280
*
Pricing Ixrivative Products
12
3 Accuracy
':
3 Accuracy of tlze Method
Method
281
the third derivative of F is there becausc we are applying Ito's Lcmma to the F already diferentiated with rcspect to St. After replacing the differential dst, and arrangingt
wbere
'
1: ;
E
The previous section illustrated the mcthod of risk-frce portfolios iri obtaining the PDE'S corresponding to the arbitrage-free price Fst tj of a derivative asset writtcn on St. Rccall that the idea ws to form a risk-free portfolio by com bining the underlying asset and, say, a call optitm written on it:
of the
'',
,
Jzv (st, t) '
5.
,
obdlj-t,tj
1
Fsssszf
jjjjj'
;i7
'
Pt
'j
(21)
+ (hdst.
i
ysspstzvjt
dt + y
jI
g'xjlyyj.
.T;2T2
E:
,
=
+
Thus, the fonnal diffcrential of
where 0L 0) are the portfolio wcights. Then we took the diferential during an ininiteslmal time period dt by letting: dP,
tsrt/u)
+
F t + Fnmh +
=
(20)
OLFIVS:!) + %St,
=
j
pgtdt + yssgstat
't
.:
Pt
=
when
.;:
.: :1
% is equal dPt
Mathcmatically spe-aking, this equation treated the :1, % as if they are l ctmstants, because thcy were not differcntiated. Up to this point, there is .E' really nothing wrong with the risk-free port folio method, But consider what ' ; Izappen, when we sclect the portfolio weights. seleded thc portfolio weights as: ) .,
+ Fsdst)
.
-
,
03F(St, t) +
hut,
(25)
to -Fs will be given by:
tdt
-
=
S
Fsdst
-
1 Fst +' Fssla.st + jF.zsslrlsl
dt +. F ss tykztlj/i
t
.
.
we
clearly,this portfolio
aavc;
.(.( ''
0j
=
%
1,
=
(22)
-Fs.
S,
dpt
tGunpredictable''
in the sense that it eliminates the makes the portfolio risk-frec, but unformnately it and component random violates assumption that 0, pz are constant, In fact, the % is now the also general, F, is a function of St and f. Thus, in because, dependenton st Erst replacing the :), 0z with their selected values, and then taking tE differentialshould give a very different result, Writlg the dependence of F on St explicitly:
rlnhisselection
,
will not be self-hnancing
Ttworks''
.j
..j
Pt
(F,df + Fgdstj - Fvdst
:
F st + F ss r. t +
q
..t
Using this equation eliminates a,e ssll Ieft wfo,
(24)..
utdl. -
.
and,.,% .at we now have a thira term simce the ,, is epen uenton s, is timc dependem aad stochasuc. In geiwral, tisis term wul not van- .1. which is a . ish. In fact, we can usc Ito's Lemma and calculate the dFs, functionof and r. nis is equivalent to taking the stochastic differenn .s1 of the derivative's DELTA: )' yj '. Fssdst 2sldt Fstdt + Fssso+ dFsut. t ) t
d
xote hence,
,
vt
dn
hdst.
,
.2
lbat + p-ds-)
1
j
r)d/)
-
0,
=
F s6s
g.2,42 /
+
(x2y' ss
st
most of the unwanted
- F-.ds-
''
-
,,
(,,,.,(.-
jtzapi + fg
whiclxwill not hold in general.
,
.
-
sly t
:
)
(M
rjstdo
=
(). terms in
2.
. .
.r;w)
.(),
(26).But we
+ ,;-.,.s,-d.;.
Thus, in order to make the portfolio Pt self-linarlcing we need
'.j
=
-
write
.j
..
=
/)
wuch will, in general, not be the case. ln order to scc this, note that differentiatjng the Blackvcholes PDE in (17)with respect to St again, we can
:
'rhen, differentiating yields:
dpt
e
&FxxIGY11(+
(23) ',.
.
,
..
;'
..
Fstk r) - Fsut, t l-f
dyst
in general, since we do not
On the right-hand side there are extra terms, and these exlra tenns will not equal zero unlcss we have:
.'
,
=
=
(26)
(2a)
:
.)
C H A PT E R
282
)(
PricingDerivative Products
12
.
'
4 Partial Differential Equations
j.
3.1 An Intevpvtation
J
from some underlying assince derivative securities are always setts), the formation of such arbitrage-free portfolios is in general quite straightfonvard. On the other hand, the boundary conditions as well as the implied PDES may get more complicated depending on the derivative product one is working with. But, overall, the method will ccnter on the solution to a PDE. nis concept should be discussed in detail, We discuss partial differential equations in several steps. d<derived''
Although, formally spcaking, the rsk-frec portfolio method is not satisfactory and, in general, makes one work with portfolios that require 111't' fusions of cash or leave some capital gains, thc method still givcs us the result? this correct PDE. How can we interpret The answer is in the additional term, S1tFxstc'dM+ (p, rjjdt. Wis term has nonzero expectation under the true probability P. But once wc switch to a risk-frec measure # and deEne a new Wiencr process H(* undcr this probability, we can write: '
.
'
'
-
'
VZY
(GdWl ''F (X
=
rjdt
-
5'
).
(-$-/
-
:
: '.
l
:.
t ?
4 Partial Differential Equations
4.2 Whut Is the Beunxo
(17)in a general form, using
k
=
0,
0
::
St, 0
:;
l
(29)
t .S T;
.
''
.'
with the boundary condition
.1
F(qV,T)
GCST,
=
(30)
F)>
k
G(.) bcing a known function/ nc method of forming such risk-frce portfolios in order to obtnn PDESarbitrage- free pr ices foz derivative instruments will always lead to 5In the literature, thc PDE notation is diffcrent than what is adoped the I'DE, in (29)would be writtcn as exnmple, aFCX,
f) +
azlx,
l)-Y + iI'FtCX, t) + /hFv<(A-, 1)
=
0,
in this section. Fof
0 S X, 0 S
I
5 71
)
l
!. .:'F
'''
(31)
with the boundary conditlon
..
,
F(xY, F) In this section,
done.
Condifiou?
Partial differential equations are obtained by combining various partial derivatives of a hlnction and then setting the combination equal to zero. The boundaly conditions are an integral part of such equations. ln physics, boundary ctmditions are initial or terminal ytates of some physical phenomenon that evolves over time according to the PDE. ln hnance, boundary conditions play a similar rolc. They rcpresent some Contractual clauses of various dcrivative prodpcts. Depending on the prodtt and the problem at hand, boundary conditions may change. The most Obvious boundary values are initial or terminal values of derivative contracts. Often, linance theory tells us some plausible conditions that prices Of derivative contracts must satisfy at maturity. For example, futures prices and cash prices cannot be lvel'yldifferent at the delivery date. In the case Of options, option prices must satisfy an equation such as (19).ln case of a discount bond, the asset price equals 100 at maturity. If there are no boundary conditions. then Iinding price functions F(St 1) that satis a given PDE will, in general, not be possible. Furtherv tbe fact that derivative products are known fttnctions of the undcrlying asset at (qpiration will always yield a boundary condition to a market participant.
t
=
atF + ajFsst + azlt + aaFss
fquutionM?
tm
=
:j,
r
,
PDE
In what sense is the PDE in (29) an equation? With respect to what is this equation to be solved'? Unlike the usual cases in algebra where equations are solved with respect to some variable or vector x. the unknown in Eq. (29) is in the fcrm of a hmction.It is not known what type of function FSt, t ) represcnts. What is lnown is tbat if one takes various partial derivatives of F lSt, !) and combines them by multiplying by coeflicients ai, the result will cqual zerta. Alsoa at time f F, this function must equal the (known) G(&., T)-i.e., it must satisfy the boundary condition. Hence, in solving PDES, one tries to fmd a fundion whose pmial derivatives satisfy Eqs. (29)and (30),
t
r)A)1 2 0,
nus, in small intervals. the extra cost (gain)associatcd with the portfolio P3 has zero expectation. It is as if, on the average, it is sclf-financing. But, it aaverage'' is taken with respect to the synthetic Iiskis interesting that tbis neutral mesure and not with rcspect to real-life probability. See Musiela and Rutkowski (1997)for more details.
We rewrite the partial differential equation F, the shorthand notation Flut f)
fhe
'Iunknown''
We will have: E z: cFs,(f'AW'1+ (Jt
4.1 Why ls
=
G(X, F).
wc keep using S; irlsead of switching
to a gencric variable
'
X, as is ustlally
'.E ..:
g
.k
.i
.
'j.
c H A PT
284
ER
.
Pricing Derivative Product.s
12
To see the role of boundary conditions and to consider some simple PDES, we look at some examples.
5 Classifcation of
PDES
E
-'1
5
:
: :..
of
ass iflcation
285
PDES
We can immediately guess a solution!
: :
Fuh, t)
.. ; :
where a,
given by
'
One can classify PDES in several dihkrent ways. First of all, PDES can be linearor nonlinear. This refers to the coeficients applied to paztial derivativcs rl the equation. If an equation is a linear combination of F and its
-
p are any
d/z
' a.nd
JF
.
.
'
.''
,%,
-a.
,
idsolve's
=
; . '
Flst, t)
.
,):
0
(32)
T1
Et
''
@'
-ni-
-,--.i.-zi---
--
--,t
h--ai---
-f
1(j, -1a
s
J,
s
1p.
(56)
=
-2,l
+ 2/
-
4.
-10
:; f
'g
10,
-10
S St
:s
10,
(37)
The examples of Fvh, tj given in (36)and (37)are very different-looking functtons. vet they botlt solve the PDE in (32).Tltis is because Eq- (32) (joes not contain sufhcient information to allow tlle function Fst, t ) to be determined precisely, nere are uncountably many functfons Ftu/ r) whose jrst paruals with respect to St and t are equal. j.f jja addtion ' to (33)we are givcn some boundary conditions s xow Well, then we can determine the F(St, t) precisely. For example, suppose we know that at expiration time l = 5 (theboundary for t) wc have ,(,s,
s. .
s
,
k
p-ti-l
s ts
jjxs.
:
-f.,,--
-j()
We a Vain see tbat FS /' tj is a plane. But in this case it increases with respect to z, and decreaseg with respect to St, the contours ate again straight
,
--ski--t-
..#.4,
.
i
ts-t t,--
.,32
-
c
Flkt, 1)
i
Aording to this PDE, the negative of the pmial of F(.) with resped to . and ! l ig equal to its partial with respect to St. lf f were to represent tim, would St were to represcnt the price of the underlying security, then (32) interval . mean that the negative of the price change duriag a small time rwithSt tixed, equals tbe price change due to a small movement in the price of the underlying asset when f is Exed. .' In a financial market, there is no compelling reason why such a relaonis derivatives. But partial suppose exist be-een thc (32) sMp should tw'o nevertheless written down and a solution Fst, tj is sought. What would this function Ftt, tj Iook like? ----
5s:
Hence, this ftmction satislies Nole that in this case F. = 3 and Ft = the PDE in (32).This solution is a plane that increases with resped to St, but decreases with respect to t. y'igure shows another example where
,r
:s f
=
-3.
',
.s
=
=
'rheso
consiaer
(35)
.
Their sum will equal zero, and this is exactly what the PDE in (32)implies. suggested by the function (33) is a plane in a The Solution three-dimcnsional space, If no boundary conditions are givena this is al1 We kIIOW'. We would not be ablv to determinc cxactly which plane F(.%, !) would represent, sincc we would not be ablc to pinpoint the valueS of a, # given the information in (33).AII we can say is the following; 0, So 0 the interccpt will equal p. For a sxedSt, the Fut, t) has at l Contours that are straight lines with slope For fixed /, the contouz's are Straight lines with slope a. thc PDE in Figures 1 and 2 show two examples of Fvh, 1) that of the plot the plane: Rgure 1 is (32).
'.
oe pos sr a szction,.(.sF l A-Fs 0, 0 s
=
?''/k
'i
5.1 Example lr Ineavs Figat-trtlcr PDE
(34)
ot
''
',:
(33)
p,
-a
=
,
af +
-
With such a function, the partials will be
constants.
,:'
. partial derivatives, it is called a linear PDE.6 order differentiawith of thc has second do of classification to type ne tion, If aIl partial derivatives in the equation are lirst-order, then the PDE willalso be Erst-order. If there are cross-partials, or second partials, then thc PDE becomcs second-order. For nonlinear nancial derivatives such as options.or instruments containing options, thc resulting PDE will always be second-order. E difernus far, these classcations are sirnilar to the case of ordinary PDES. is specc to ne t ential equations. The third type of classcation PDES
Iatter can also be classiticd as elliptic, parabolic. or kyperbolic. ne PDES. are similar to parabolic we encounter in fmance We hrst consider examples of linear Iirst- and second-order PDES. in linance. Yet they may help establish examplesllave no dircct relevance PDES undezstanding of what intuitive are, and why boundary conditions an important. arc
ast
=
). .
5)
-
,,
-
zss.
(a,)
!
h
CHA PT ER
286
Pricing Derivative Products
12
.
5 Classiicarion of PDF.-$
I7( lt't )
then there will be no meaningful solution because Eqs, (41)and (42) overdetermine the constants a and p. Thus, when F(St /) is a plane, we need a single boundary condition to cxactly pinpoint the fundion that solves te PDE. This is easy to see geometrically, since a boundary condition corresponds for I (orSt) and then obtaining the interto frst selecting the section of the plane with a surface orthogonal to the tue axis and passing from that 1. ln Figure 2, the boundary condition at t 5,
:
K
,
:
20
O
fendpoint''
..
.2
' L
. ... ' . ' -
.
.
.
.
.
.
=
.
-10
-
?
10
:
-5
.5
0
F(55, 5)
'
10
..
Fl G U R E
l
5.1.1 Remark Thc solutions to the class of
.?
we can
now determine
the unknowns a and p in Eq. (33)., a p
2,
(39)
=
4.
(40)
,
This is the plane shown in Figure 2. On the other hand, if we had a second boundary condition, 100 rl Sl
Ft + F,
..'.2
'i.'
Fvt,
.
.i
.).
..
It was easy to guess the solution of the first-order PDE discussed in Example 1. Now consider a second-order PDE J2F J2F (45) = d/2 as tz
3: .
.40 .
-
''
.
..
.jg
.
.. .
.. . .
.
:
.
>5
,
'
*
S=l
..
.
,
'
or, more succinctly,
(
:'
lipc:
.j
j
.
'
-5
0
o 5
s 10
.1
) S
'
.h
.S
,
';
10
.3.
i
..
F IGURf
2
(46) - .3Fss + Ftt 0. First note that w are again dealing with a linear PDE, sinct the partials in question are combincd by using constant coeftkients. Again, ignore tlze boundary conditions for the moment. We can try to guess a solution to (46).It is clear that the function F(.) has to be such that the second partials of FlSt /) with respect to and t are proportional with a facor of proportionality equal to relationship betwcen Fs nis and F!/ should be true at any St and t. W'hat could this function be? =
.
-2,s+10t-4=0
t=5.S= l 0
-5
...
. .. .
-!
' ..
.g
'gj
''
..
(44)
.
5.2 Exumple 2: Lineug, .Nectmd-ordex PDE .:
.
q,St-qt
'
:
-;!() :9()
e
=
.
F(Sr,t)
-10
1)
This ftmction will also satisfy the equality (43). lt is the boktndary condition that will dctermine the unique solution.
say, at
(41)
1 ()
(43)
0.
arc not rcstrictcd to planes, ln fact, consider the function
h
'
.31,
=
J
=
20
PDES
.
=
F 100, f ) = 5 +
(42)
,
=
5
10
6 - 2%
=
is shown explicitly. Note that the other candidatc for FSf, t) shown in Figure 1 will not pass from this line at t 5. Hence, it cannot be a solution, Also, when Fst, tj is a plane, the tenninal conditions with respect to t or St will be straight lines.
.
0 5
287
.
288
C HA PT ER
Consider the formula 1 Fst, 1) jat,% =
-
Pricing Derivative Products
12
*
6 A Reminder: Bivariare, Second-Degree Equations 0
.3
k$'())
z+
289
/()) z +
ytztf-
pvb - 5'clt!
-
/n),
where &affl are unknown constants and wherc thc parameters a and again unknown. Now, if we take the second partials (f Fvh f ):
(47)
p are
-500 .10
.15X -2000
,
p2F t?/2 =
-
(48)
.3a,
10
-5
J 2F
1a. (49) JT2 = Hence the second partials Flf, F,. of the Ft&, /) in (48)and (49)will satisfy Eq. (45).nus, thc Fst, t ) given in (47)is a solution of the partial differential equation (45). Note that for flxcd Fst, !), F(St,
/)
Fo
=
=
100 + /2.
=
=
=
50 +
.902.
(52)
equation is another parabola. But the relevant plane is F, St. We give an example of such an Ffut f) in Figure 3. The figure displays the three-dimensional plot of the function
nis
,
Es t Es 10, -10 S St :!! 10. -10(5-, -4)2 - 3(I -2)2, (53) conditions, of boundary In surface The has contours as ellipses. terms we 10 as the tcrminal value for l and get a bounday condition can pick f that has the form of a parabola: 4)2 192. F.%o, 10) (54) -
Fst,
/)
5 10
-10(5b
7see the next section.
-
10
FIG U RE 3
The boundary condition for F(0, Thcse
&
=
=
ut
f)
=
0 will be another parabola! -
l60
-
3(/
2 )2
-
(55)
.
-20,
two boundary conditions are satisfied for 4, /a 2.
a
=
p
=
0,
=
6 A Reminder: Bivariate, Second-Degree Equations lt turis out that frequently encountered graphs such as circles, ellipses, parabolas, or hyperbolas can a1l be represented by a second-degree equation. In this section we brieqy review this aspect of analytical geometry, since it relates to the terminology concerning PDES, For the timc being, let ;r, .y denote two deterministic variables. We can desne an equation of the second degree as zfx2 + Bxy
-10
=
0 5
=
=
-1 o
(51)
rrhis is a function tbat traces a parabola in the E t plane. Yet sucb a botmdary condition is not sumcient to determine ::111 the parametel's a, p, &, k. One would need a second boundary condition, say, at / 0: F%, 0)
:2=0 -5
,
f)
-1()(X-42-1
0
(50)
,
the contours of this function are ellipses.; Again, the solution of (45) is not unique, since the Flut f ) with any a, p, &, tg could be a solution, as long as it is of the form (47).To obtain a unique solution we need boundary conditions, One boundary condition could bc at St 10: F(10,
Parabola:
Here
ad,
+
B, C, D, E, F reprcscnt
Cyl + Dx + Ey
+
various constants.
F
=
0.
(56)
The equation is of the of y is a squazc. or By choosing different valucs for B, C, D, A'. F, the locus of the equation can be in the form of an ellipsc, a parabola, a hyperbola, or a circle. lt is worth digcussing these brielly.
seconddegree, because the highest power of a4,
.x
C HA PT ER
290
Pricing Derivative Producrs
12
.
whichwill be
6.1 Circlc
at A:fl,
Consider the case where and
C
=
B
(57)
0.
=
equation reduces to Ax2 + Ayl + bx + Ey + F
second-degree
=
(58)
0.
After completing the square, this can always be written as (.Y &)2 + R, .yt))2
(.y-
-
=
(59)
whichmost readers will recognize as the equation of a circle with radius R and center at (xo,a). To see why, expand (59): x2+
.x2(j
.y2
-
2.x()x 2yf)y+ -
+ ya2=
as the equation of an ellipse, where the center is
recognized
l Given values for B, C, D, E, F, we can always determine the values the parameters xc, yt), tz, p, R. since by equating the coefhcients of of of (66)with those of (5$, we will have six equations in expanded form the unknowns. six ,
1 -R
6.2.1 Faample ne method of completing the Jsllre is useful for differentiating among ellipses,circles, parabolas, and hmerbolas. We illustrate this with a simple example.8 second-degree equation
Consider the
9x2+ 16.:2 - 54x - 64y + 3455
(67)
0.
=
Note that
(61)
..4
=
.
2m) D, - R
(62)
E,
(63)
=
- R
'y,
(60)
R.
ln tMs equation, we can always let
=
B2
9(x2 -
2 + )j)2 'tl (64) = F. R and the y that satisfy the second-degree Hence, with C, B (), the cirde will in the x, y plane. equation always trace a when R special 0, the circle reduces to a point. Alother In the case the circle has obtained when C 0. degenerate case can be second-degree. equation straight the is line, but degeneratcd into a n0( ./4
.r
=
=
4,4C
-
(68)
-576,
=
with an ellipsc. We directly show this by
so we must be dealing the squarcsr':
and
=
291
ad,
-4 ne
A Remindert Bivariate, Second->gree Equations
6.r + ?)
+
16(y2
-
4y + ?)
(69)
3455,
=
KGcompleting
By filling in for thc question marks, we can make the tw'o terms in parenthescs becomc squares, We replace the Erst question mark with 9 for t-he requires adding 81 to the right-hand side. ne secfirst parcnthesis. ond question mark needs to be rcplaced by 4. This requircs adding 64 to thc right-hand side. We obtain 'rhis
rrhen
..4
=
=
9(x
-
3)2 + 16(y ;)2 -
=
3600
(70)
1,
(71)
6.2 Fllflue The second case of interest is whcn B1 4Ac -
@ - 3)2 (y 2)2 400 + 215 -
<
(65)
0.
nis is similar to the case of a circle, except B is not zero, and the coecientsof and y2 are different. We cazz again rewrite the second-degree equationin a different form, R, ab)(A' (66) + p(y y49)2+ y(.r a(x
=
Tltis is the fonuula of an ellipse with center at
.r
=
3, y
=
2.
-3:2
.b)
.r0)2
-
-
-
-
Srrhe
metric
metllod SDES.
of
Kscompleting
the square''
is used frequently kn calculations
involving geo-
292
C H APT ER
12
*
Pricing Derivative Products
8 Conclusions
6.3 Plzabohz
293
I7(Elt
,t)
The second-degree
(56)reduces
equation in B2
4.447
-
0.
=
(72)
The easiest way to see this is to note that B 0 and either 0 or C 0 satishes the required condition, But, when this happens, the second-degree equation reduces to ,4
=
,4..2 + Dx + Ey + F
=
0
to a parabola whcn we have
=
-5 -100
=
- 15 o
.
...
.
.
.
. . ..
..
.
J.
'
..
. . . . .. . . v . .. . . . . .
. .
%'$ .
.
' . .
. ..
'
2
.
'
M
. ' ' J: ' . . . .. . ... .w... ''
.
.
' '
:Q'.. .
x
w
.w'
'.
' ' '' ''
.
.
.
.
.
'
.
'.
.
-10
(73)
0,
0 n f!
-
.5.
.gc
which is the general equation for a parabola.
0
()
5
20
10
6.4 Hyperbolc ne general second-degrec the condition
equation in
(56)represents
> (74) - 4AC 0 satis*cd, will is This case havc limited use for us, so we will skip the dctails.
7 Types of
PDES
Example 2 suggests that the contours of Fst t ) would in general be nonlinear equations. In case of Example 2, they werc ellipses. ln fact, partial differential equations of the form . ,
J0
are called ellipdc
+ aLl PDES
+ asls
+ azl
=
0
(75)
if we have a25
The PDE in
+ asFtt + asft
-
41:4u4 < 0.
(75)is called parabolic al5
-
2
45
4(7,414 0, =
4/ g a 4
is satisfed.
/4
/)
=
-
>
0.
<
0
4)2
3(I
-
defined as
-
2).
(80)
Note tbat the contous of this fundion are parabolas. This Fuh, t j will have boundary conditions as parabolas with respect to /, and as straight iincs with respect to Sf. Such an FSt, t) is one of thc solutions of the PDE 1
The coeocients
/52 =
5
Fsx + jFj
0.
=
(81)
of the PDE are such that
() and as
=
4/ 3 a 4
=
0
,
(82)
0. Hence, this PDE is parabolic. 8
(7:)
=
4J3
Fst,
-10(u%
/)
(77)
Clearly, Fvh, t ) graphed in Figurc 3 is a sllution to a PDE that satisfies the condition of an elliptic PDE, since a4 0 and both la and an are of the same sign, As a result, the condition a52 -
Figure 4 gives the graph of the function Ftxi,
since a4
Finally, tbe PDE is called hyperbolic if -
7.1 Exumplc: Purubolic PDE
(76)
if
40
F 1G U R 13 4
a hmerbola if
B2
.s
(79)
Concluslons
ln this chapter, we introduced the notion of a partial differential equation (PDE). These are functionalequations, whose solutions are functions of the underlying vari'ables. We briefly discussed various forms of PDES and introduced the related tenninology. This chapter also showcd that the relationship between Nnancial derivatives and the underlying assets can be exploited to obtain PDES that derivative asset prices must satisfy.
r'
Lb. LL
.
i
294
C H A PT ER
.
12
Pricing Deriyative Products
;
)0 Exercises
:.
9 References
k
Most of our readers are interested in PDES because, at on point, they will be applying them in practical derivative asset pricing. nus, rather than on the tlzeory of PDES, sources dealing with tbe numerical solution books PDES will be more eseful. In most cases, these sources contain a brief of of the underlying theory as well. we recommend two books on summary PDES. smith(1985)is easy to read, nomas (1995)is a more comprehenS and ive recent treatment.
:
D
considerthe
PDE:
fx + wkth
,
; ,
'
=
x e y, lj and F q gO,1j. (a) What is the unknown in this equation? (b) Eelain this equation using plain English. (c) How many functions fx. y) can you lind that (d)
.
&
Exerclses
fxx + According
to this, in Laplace's respect to the variables in the
fyy-1' Xz
. .x,
z, y. The
; q
0
r5
equationv the sum of second partials with t'unction must equal zero.
(a) fx' (b) fx, )') (c) fx, )')
4z2.y
=
=
xl
=
A'
(d) ytx, z, y)
./
-
x2y
E
:J
:
function satisfies Laplace's equation? Is it to an equation in general?
alf
X;r
+
TD
+
A,
z, y)
x
s
1z
antl 0 rs t rs 1.
e
=
3x2 + 3y2 - 6z2 +
.x +
y
-
9z - 3
the single boundary condition sufscicnt for calculating a oumerical approximation to ytx, !)? (b) Impose additional boundary conditions of your choicc on T(0, t) and
J(12,/).
(c) Choose grid sizes of Ax ical approximation to have imposed. ./'(x,
.
'
)
!29,vr2J+w.(3x+2y+4z)i
=
r
/zz),
According to the heat equation, Erst patial witb respect to t is proportional to the sum of second partials with respect to the variables in the hmction. Do the following fundions satisfy the heat equation? A')
::
z, y, f, that satisfy tlle
where a is a constant.
fx, (b)fx,
6, O1.
-
(a) Is
(.y+ z)
2. A function fx, z, y, f) of four variables, following PDE is called the heat equation:
(a)
(),
=
mmxl.x
=
()s
y
.x
whyis it that more than one '' ood', to havc many solutions
'z'r
fx, 1)
E.
-
=
.2
with the boundary condition
;
-
1
+
zx
.L
3 3xy
=
1.
=
4. Cfmsider the PDE:
,
./
-
such an
find a soluticm to the PDE? Is the solution unique?
t
Do thc following cquations satisfy Laplace's equation? z, )')
caa you
Er
1. You are given a function fx, z, y) of three variables, following PDE is called Laplace ' s equat ion: =
((),y'l
,' :
will satis
equation? Now suppose you know the boundary condition:
' 10
0,
.lfy
.
,
=
3 and
.25
it
1) under
and calculate a numcrthe boundary conditions you
=
E
2 The Black-scholes PDE
E. . .
297
and, more importantly,
'
cst la
C
*G<
o
o
.
V'AUD
,
E7
An Applicutitm
p)
cst,
=
ts
((),x).
We occasionally write to denote (rst. Undcr these conditions the funn of and PDE Black damental scholesand the associated boundary ccmdition are given by
aloo
<
,
0
E;'
yj gaj
j
''
:s
maxjys
=
(3)
,,.
tjj.
.y
(4j
Equations (3) and (4) were first used in finance by Black and Scholes fundnmental PDE of Black (1973).Hence we call these equations Scholes,rsl and Black and Scholes solve this PDE and obtain the form of the flmction yqst tj cxplicitly: L
. .L
1
lntroductlon
,
ln this chapter, we provide some examples of partial diferential equation methods using derivative asset plicing. One purpose of this is to have a geometric look at thc fundion that solves the PDE obtained by Black and Scholes (1973).ne geomcty of the Black-scholes formula helps with the understanding of PDES. In particular, we show geometrically the implications of having a single random factor in pricing call options. Next, we complicate the original Black-scholes framework l7yintroducing a second factor. This leads to some major difficulties, which we VII discuss brielly. Finally, we compare closed-form solutions for PDES with numerical approaches. We conclude with an example of a numerical asset price
Fvh
,,
E
dk
:
dz
. 5'
x(#.) j
)
l
We
aut, tj
SDE =
Ia'st
-
Ke .-rr-txa
(r +. ) t a. r
(s2)(w
jnj,s l jK) +
a)
,
r)
-
-
=
=
(/1
r-
tr
-
(8)
t.
1, :.:are two integrals of the standard normal density: (f
N(#j)
= ..x
.
1
.jxa
e
2,;7.
dy
jq)
.
To show that this function satislies the Blackucholes PDE and the correcondition, we have to take the first and second partials of (6)with resped to St, and plug thesc in (3)with the F(St, t) and its partal with respect to t. result shoeld equal zero. As / approaches F, the ne functionshould equal (4).
: '' l s
spondingboundary
,
1
onlyone of thz seccnd partals, namely the one with rqspect to is present in this PDE. Also uote tsat there is no term. nder lhese conditions, we can casily calculate the value of tlae expression somclaapter,2,
''
,
.
Special
7
r
:
In Chapter 12 wc btained the PDE that thc pricc of a derivative written on ZSSCt Sj mtlst Satisfy tlrldc'l' 9O1MC Clmditifms. The underl/ng risk-free interest rate was assume d securitydid not pay a dividend, and the onsider the
=
,L
tl1C kmdcflying
r,
)
-
i
at
stNd,
=
'
2 The Black-scholes PDE
NOW, Supplst
/)
where
.!
calculation.
tobe constant
,
Where
E
aj
''
(1)
a: zeto. the '
296 : :
- 4fz,u,,
nis means that leaving aside the preyencc aack-scholesPDE is of the parabolic form.
: :. y'
ur,
Jonstant
.
j5) of St
aEd
u/,
which are always postlve,
C'
:
298
(2 H A P T E R
lxmk ut tl 2.1 A Qeomctric
13
*
Bluck-choles
299
The Black-scholes PDE
3 PDL
Fonnto
surface'' defined by the axes labeled St In Figure 1, we have a where the latter represents and F /, to expiration.'' These two axes represents an underlying asset price of fonn a plane. For example, point By going up vertically toward 130, and a to expiration'' equal to the surface we reach point B, which is in fact the value of the Black-choles formula evaluated at
in
Asset Pricing Whorizontal
Wtime
-
We saw in Chaptcr 12 that functions F(5'f , f) satisfyg various PDES could be represented in three-dimensional space. We can do the same for the Black-scholcs PDE. The solution of this PDE was given by (6). We would like to pick numerical values for the parameterg K, r. tr, F azld
-e1
'
''time
1....
E ,
represent this formula in the three-dimensional space F x S x t. We pick
Er
,80.
ad1
'
B
=
F(130,
.2).
(11)
'j .065,
r
K
=
=
100,
.80,
a'
(10)
=
'
j
and substitute
these in formula (6).nese numbers imply a 6.5% risk-free borrowing cost, and an 80% volatility during the interval t e (0,11. 'Ihis markets. But it mnkes type of volatility is Mgh for most mature snancial the graphics easic.r to read. The life of the call option is normalized to 1, 1 implying one year, and the initial timc is set at 10 0. Finally, with F strike the prie is set at 100.2 To plot the Black-scholes formula with these particular parameters, we must select a range for thc two variables St and !. We let range from 50 to 140, and let t range from 0 to 1, The resulting surface is shown in Figure 1. =
.
.%
;
,
.4
.',
;
=
',
.L
ut
.
1
C
b
We display two typcs of contours on thc surface. First, we flx St at a particular level and vary f. nis gives lincs such as aa', which show how the is flxed at 100. call price will change as t goes from 0 to 1 when 'The second contou is shown s bb' and represents Fbh f ) when we and move St from 60 to 140. It is interesting to see that as f goes flx t at toward 1, this contour gocs toward the limit shown as cc'. ne latter is the usual graph with a IdII.Itat K, which shows the option payoff at expiration. We would likc to emphasize a potentially confusing point using Figure 1. The Black-scholes formula gives a surface, once we tix K, r, and F. nis surfacc will not move as random events occur and realized values of 4/11/; bccome known, Realization of the Wiener increments would only cause random movements on the surface. One such example is the trajectory denoted by fla. Cz. in Figure 8. Because the increments of the Wiener process arc unpredictable, the movement of the stock price along the t dircction steps.'' Over infirtitesimal intcmals these steps are will proceed also infinitesimal, yet still unpredictable, The trajcdory C(),Cr is intcrcsting from another angle as well. As time passes, will tracc thc trajcctory shown on the St x t plane. Going vertically to the surface, we obtain the trajectory Ca, Cv. Note that there is a deterministic correspondence between the two trajectories. Given the trajectory of St on the horizontal plane, there is only one trajectory for Fvh, t) to fol1owon thc suzace. This is the consequence of having the same randornness in Sf and in F(St, f).
J
'i
.)I
70 60
'
B
50 4 30
vt
!
1! 4A
20 1!(:, o 140
**.r-
......
.
a
'
'
'j -
;:t ..:
,
f
>-.. ... --
**G*--........
()
1' ------
120
-
1O(k)
-
80
60
.
:.2
0.4
.'. :
-x.--
n
'*''''e
08 .
q
j:) '
RL
t
.
.'
E :
2If T '1 means yearp'' the interes rate and thc volatility will be yearly ratcs. But .. WIIiCII T 1 may verjr well nlcan six' months, thrcc ment h s. or any time inerval during lion. stlc.'lt 1 used normali7m t i! e: tinandal instrument will ext merely T Under 1 We as a conditins, numben the interGst rattr Or the Volatilily must be adjllslcd tt thc relevant timr y Perild. . =
dirandom
lsonc
=
.
=
:
3
PDES
The patial diffcrential equation
in Asset Pricing obtained
by Black and Scholes is relevant
tmder some specitic assumptions. These are (1) the underlying asset is a stock,(2) thc stock does not pay any dividends, (3) the derivative asset is a European style call option that cannot be exercised bcfore the expiration datea(4) the risk-free rate is constant, and (5) there are no indivisibilitieg or transaction costs such as commissions and bid-ask spreads.
!
3O0
C H A PT ER
*
13 The Black-schotes PDE
4 Exotic Options i.:
ln most applications of pricing, one or morc of thcse assumptions will be V iO la ted lf so in general, the Black-scholes PDE will not apply aad a new PDE should be found. One exception is the violation of assuhlption (3).If style, the PDE will remain the snme, theoption is American The relevant PDES under these more complicated circumstances fall into o f a few general classes of applications. We discuss a simple case next. .
' ,
.
4 Yxotic Optlons
';
one
3.1
There is now a constant tcrm &.Hence stocks paying dividends at a constut rate do not present a major problem.
,
,
;.
:j
'
In thc previous sectionx a complication to the Black-scholes framework was discusse d ae pps satisned by thc arbitrage-free pricc of the derivative asset did change as the assumptions concerning dividend payments changed, This Section discusses another complication. Supposc the derivative asset is an option with a possibly random expiration date. For example. there are some and options that are known as banier derivativesl Unlike options, the pamff of these instruments also dcpends on whether or not the spot PZCC Of the underlying asset crossed a ccrtin barrier during tbe life of the option. If such a crossing has occurred, thc payoff of the option changes. aexotic'' options, we Ixjeny review some of these
E
Consttmt Ditpaxanzl.s
(
., J
,
If one is trying to pricc a call option, and if the option is writtcn on a stockthat pays dividends at a constant ratc of units per timc, thc rcsulting 1 PDE will change only slightly. Suppose we changc onc of thc B1ac k-scholes assumptions and introdu 5 constant rate of dividcnds, &paid by the tmderlying asset a Again, risk-free the portfolio form we can try to same bycombining thc underlying asset and the call option written on it: Pt plF(,%, f) +. %St. . (12) (i.' .
'down-and-out''
.'
''
=
,
% can be selected as #1
so that thc unpredictable is formed:
1
=
02
,
-F
=
,:
;t
.
(13)
x,
1
dPt
Ftdt +
=
Iyj the standard Black-scholes case, the call option payoff is equal to s K if the option expircs in the money. ln this payoff Sv is the price of the underlying asset at avm/on alid K is the constant strike price, In the case of a jloatinglookback call option, the payoff is the difference whcre sm..n is the minimum price of the underlying asset observed %. during the life of the option.4 A ftxedlookback call option, on the other hand, pays the difference positive) betwecn a flxed strike price K and S where the latter is (if max jj t e maximum reachcd by the underlying asset price during the life of the option, Thcse options have the charactelistic that some positive payoff is guaranteed if the option is in the money during some time over its life. Hence, everything else being the samev they are more expensive.
s
2
p
(14) l
jlsso't dt.
=
.
'
.
Up to this point there is no difference from the original Black-scholes approach discussed in Chapter 12. The time path of the Pt will aga in be completely predictable. ne difference occurs in deciding how much this portfolio shou ld am prcdictable) capital gains werr preciate in value. Before, the (completely exactly equal to earnings of a risk-free investmcnt. But now, the underl/ng stock pays a dividend that is predictablc at a rate of &. Hence, the cap ital gains plus the dividends received must equal the earnings of a risk-freo potfolio: dPt + &dt
4 1 Ixmkbuzk optiona
..
random component is eliminated and a hedge 1
;
-
,
-
E''' .; F
)
t
:y'
' .:
r ''
.; :
4
;
''
or J#, Putting this together with
(14)wc
r'F - rFstt
-
(16)
+ rptdt.
.-ndt
-
Fj -
1
jbst
ywuzjw.r
opjums
A ladder option bas several thresholda, such that if thc underlying price reaches these thresholds, the rcturn of thc option is in.''
'
:
''locked
get a slightly different PDE: -
kmin,
,
(15)
rptdt,
3'Tlzeseare also known as knock-out'' and options. jookback option is yoatingbecatzse the strike price is not lixcd. sknock-in''
a
=
fa
tJ-
f.1 k-
''up-and-out''
''standard''
k%.
''approximately''
Thc portfolio wcights ?:
3O1
vjz
j,
.
4n:
.
j
(
'
..
i
.
302
C HA P T E R
.
The Black-scholes P
13
.
4 Exk jo
;
; ;)
k-in Optvld
4.3 Trigger or
.. :
. f.
.
if the s barrier If is not'' life of the option. barriet during thc the rice falls below a P s rebate payoff. with expires hedv the option . as a somc reac ', A down-and-in option gives its holder a European option
'
o P( joyjs
Average orzjsian optiotts arc quite common and have payoffs depending on the average price of the underlying asset over the lifctime of the 7 op tion .
.
#.6 The Reletunf
:
4.4 Knoclt--t Option.s
It is clear from this brief list of exotic options that there are threc major differcnces between exotics and the standard Black-scholes case. First, th e e-wiration volue of the option may depend on some event hapening over the life of the option (e.g.,it may be a function of the maximum p underlying tiw asset price). Clearly, thesc make the boundary conditioas of complicated than the Black-scholes case. much more Second, derivative instruments may have random expiration dates. Third, the derivative may be written on more tltan one assct. A1l t hese may lead to changes in the basic PDE that we dcrived in the Black-scholcs case, Not all examples can be discussed here. But consider the case of knock-outoption Wc discuss the case of a call. Let the Kt be the banier at time f. Let and Fvh f, Ktj, respectively. be the price of thc underlying asset and the price of the knock-out option. If thc St reaches thc Kt during the life of the option, the option holdcr rece jves a rebate Rt and the option suddenly expires. Otherwise, it is a stan d ar d European option. In derjving the relevant PDE, the main difference from the standard case is in the boundary conditions, As long as thc underlying asset price is above the barrier Kt during the life of the option, / (0,T1, the same pos as in tjx standard case prevails:
. ,
options arc European options that exp ire immediatcly if, fl . example,the un derlyingasset price falls below a barrier during the life , the option. The option pays a rebate if the barrier is reachcd- Otherwises l Europcan option. Such an option is called is a Knock-out
'
'down-and-out.
'istandard''
..'
'5:
:
)
many different ways one can structure an exoti cases include thc following:
obviously
nere are optitm, Stme
t
E
4.5 Ofhcr F-4ptics commtn
',
( '
;..5.,.
:
dfdown-and-otlf'
,:
1
WIRiCh are derivatives where the underlying ysset * Baslt options, a basket of various financial instrumcnts. Such baskcts dampen volatility of the individual securitics. Bas ket options become more ai. t fordable in thc case o f emeyng rlztzr/zl derivatives. Multi-assct fyr/t/rtc have Payoffs depending tm the underlying price FOr example, the Payoff of Such a Cax may be y more t han One agget. '
<;
'
,
.
1
,
.
:
:
FtuNjz,Szv, F) Another
maxlo, maxtlz,
m
(
lg
X1.
.z:') -
possibility is the spread call F(x%r
,
S2T
,
F)
=
maX(0,
or thc portfolio call
where p1,
0z are
have a dual
Nfrke
ssimilarly,tere
has an upcfossing
!
,d
j
tzlr-
Sz7' )
(
j
h
1
K1,
-
-2
th?' .'.CE
=2F 'V'V + rF S $' f t
- KF + F '
,
0 if
=
(22)
and
.!
.(
'
T :..
F)
szv,
=
maxlo,
knownportfolio
F(q$jr,
'
;
'
F(&w,
.1
.
: >
PDFS
call
Sz,
(#1&r
+
01.b11 . )
-
F(.w, T)K,.)
(
A'j,
But if the
weights. As a Enal example, one
:.
tprlitirl;
st falls below K,
F)
maxgo,
maxlo,
sv -
A-wi.
(23)
during the life of the option, we have
'.;
(qlr -
Kl),
(k2T
-
K 2 )1-
undcrlyi'ng are up-and-in optifaus tlaatcome into effect if the of a certain barrier.
: ne up-an d-out option expires immediately if the tlnderlying of a cerlain barrier.
auet
. .), (21.E
Fs
..'.
=
-
price has an
:
r
:,
.
alset
k.
*ill
j'
!'
'
t)t
:
27'.
''''
.
.:
: ).
.
.r
if
'
'
st s
Kt
,
(24)
'Fhe form of thc PDE is the same, but thc boundary is different. nis resu lt in a different solution for FS / t K : as was discusscd earlier. ) 7
.
.
'
t r, x.jr ) = Rt ,
j
'.
l
....
-
13/*-1'
jg jj
Ofte:l arithmelic averages are used, and the averge
monyhjybasjs.
:1
caa.nbe computed
on a daily, weekly,
C H A P T ER
Solving
*
1.3
The Black-scholes PDE
Solving PDL
in
Practice
in Pmctice
PDES
Once a trader obtains a PDE representing the behavior over nne of a derivative price F%, t4 there will be two ways to proceed in calculating tMs value in practice.
5.1 Closedovm Alxtitnu The first method is similar to the one used by Black and Scholcs, which involves solving the PDE for a closed-form formula, lt turns out that tbe PDES describing the behavior of derivative prices cannot in every case be solved for closed forms. ln general, either such PDES are not easy to solve, closed-form formulas. or they do not have solutions that one can express as closed forms and numerical between difference First, let ug discuss the f) if the appropriate solves PDE F(St hmction of PDE. The solutions a a such satisfy equality partial derivatives as an
IR
1G U R E
3
,
rtst - rF +. F/ +
+
1lsslr zStc
=
0,
0 S St
,
(25)
Nowa it is possible that one can fmd a continuous surface sucb that the partial derivatives do indeed satisfy tbe PDE. But it may still be impossible to represent this surface in tenns of an easy and convenient formula as in the case of Black-scholes. ln other words, although a solution may exist, th solution may not be representable as a convenient function of Ss and t. We will discuss this by using an analogy, Consider the function of tne F(I) shown ill Figure 2. rrhe way it is drawn, F(/) is clearly continuous and smooth, So. in thc region shown, F(f ) has derivatives with respect to time. But F(/) was drawn I7(t)
3 2.5 2 1
#(f)
=
azeab' +
(26)
tu,
where ai, i 1, 2, 3 are constants, cannot represent this cuwe. ln fact, for a general continuouj and smooth function, such closed-form formulas will not exist/'g The solutitms of PDES in the simple Black-scholes case are surfaces in the three dimensional spacc generated by and FSt, l), Given a smoth and continuous curve in three-dimensional space F x f x St, the patial derivatives may be well deNned and may satisfy a certain PDE, but the stlrface may not be representable by a compact formula. Hence, a solution to a PDE may exist, but a closed-form expression for the fonnula may not. ln fact, given that such formulas are very constrained in reprcsentittg smooth surfaces in three (or higher) dimensions, tls may often be the case rathcr than the cxception. =
ut,t
.5
1
i:
'On the other hand, if yhe curve s of a tyjx,. one may be able to identfy it as a polynomial and represent it wit,h a focnula. For example, the cunre l Figtlre 3 looks 1tk'e a parabola and has a simpie closed-form representation tza + aj I + atl. as F 9If a cuwe is smooth and continuous, it however, be expauded may, as au ininite 'l-aylor series expansion. Yct Taylor series oxpansions are not closed-form formulas. rInhvy are representationa of such F(.). <spe-cial''
Simple
cs
j
'
E -1
in some arbitrary fashion, and there is no reason to expect that this curve can be represented by a compact formula involving a few tcrms in t. For example, an exponential formula
1
0.5
-0.5 F IGU RE
2
1
=
.:
'( .E.(
ER
c H A PT
506
13 The Black-scholes PDB
*
5 Solving
.
.j
F1
)
.r
307
: '
n
To illustrate the last step, let
:
..
.K0
!
1(E
-1000 -1500
-20
.
S
-25* -2n
E1 E
. .
:''
q. : .
.
..
.
.n:
,
0
..
o
Z
i
....:.......
'
10/
as
''
xxx
c
... ':.
'j
10
4
i
!:
20
1,
4
rrhe
KGd
ut
Lxplugged
?j
'
0 S t S T:
'!''
(27)
To solve this PDE numerically, one assumes that the PDE is valid foz' linite increments in St and f 'Ih'o '' par titions'' are needed, 1. A grid size for LS must be selected as a minimum increment in the price of the underlying security. 2. Time f is the second variable in Fvh, f). Hence, a gri d se for Af is needed as well. Needless to say, f LS must be *'sma 11 How small is can be dccided by trial and error. 3. Next one has to dccide on the range of possible values for St. T0 and tjw be more precise, one Selects, a priori, the mzimum values should maximumsmax as possible values o f S/. nese extreme be selected so that observcd prices remain within the range
,
2'. :'LL. :'
:
boundary conditions
Enax
.
must be detcrmined.
sxjzs
30 2:
7
'
10
E
o
,
1
'.
'min
Boundary lbr
4:
7
.
S
50
5
''smallp''
.,s
60
:
''
umin S S:
70
' ,:
,
,
.
rj.o
.:
0 S St,
x
,,
w
''
0,
,
,
j
' When a closed-form solution does n0t exist, a market pmicipant is E PDES. A numerical solutitm is like ;; forced to obtain numerical solutiou to E . calculating the surface represented by F(St, t ) Jirecf/y, without srstobtnm- l.: t . ing a closed-form formula for F(St, t ). Consider again the PDE obtnmed . from tlle Black-scholes framework: =
xmax
,
.
' '5
!l. 2
(29) .;.
.,r -1
Nsol%ftfmu
+ -FnrlS;
tjj,
.l
?
rF + Ft + rls
Fsi,
where F,y is the value at time tj if the price of the underlying asset is at and of The limlts of i, j will be determined by the choice of k. ymin We want to approximate Fuh, t) at a fmite number of points Fq. This is shown in Figure 4 for an arbitrary surface and in Figure 5 for the Blackscholessurface. In either case, the dots represent tlze points at wltich sizes of the grids and t determine how F(.% f ) will be evaluated. close t jwse dots will be on the sufacc. Obviously, the closer these dots are, the bcttcr thc approximation of thc surfacc. and We lct Fjy dcnotc thc o t7>that rcprcsents the th value for thc yth value for f. nese values for & and t will be selected from their in'' to Fst, I). ne result is written respec tjve axes and then aS g.. carry on this calculation, we need to change the partial differential equat jon to a difference equation by replacing aIl differentials by appro-
';
.2o
0
Fl GU RE
5.2 Numericul
=
'
d
i
20
Fij
y
.
!
a
-1()
tj
4. The
Practice
*
! )
-
in
5 Assuming that for small but noninfmitesimal & and tt the same PDE is valid, the values of Fvh, t ) at the grid points should be d e.termined.
:
'
PDF.S
Q.8
,
1.1
(28)
Boundar.y
surm
; ::J
1.
=1
'
0.2
ror
c
etl
.. . x
100
B0
s FI G t; R E
:&
Boundae for t
'-----
o:
E
5
1x
lu(;
:
'.
:g
(2 H A PTE R
308
.
13
309
bnclusions
The Black-scholes PDE ::
Priate differences. nere are various methods of doing this, cach wit.h a differentdegrec of accuracy. Here, we usc the simplest methodlt' IF p. A.F' LF 1 f rF + - o-lSl + r. (30) LS2 2 / ; where the lirst-order partial derivatives arc approximated by the correspondingdifferences. For tirst partials we can use the backward differences !' : Fij - Fi,j-j LF
.
'
k
'
For St that is very low, we let St sjsy
=
and s'min jj
rnin'
tg6j
(j '
'
A.SY
,.
:
ht
f
LF
Fq rsj
rs AS
(31)
-
!.
'', (32) t t 5'f. '
F-L,)
=
r)
Fsv,
,
-,
AS which is : '1 or we can usc forwarddifferences, an cxample of F 1 ) Fq LF rs 15 = rvh .f-85 (aa) J For the second-order partials, we use the approximations Fjj tjIF Ft-j,j F2+.1,/- Fij 1. 1 (34) 2: 1, Au kN 15-2 'T'he wherc i and 1 N. parametel's N and n determinc .; n j 1, Of the which decided number Points at to calculate the sudace Ffut, fz..j g'k we . ' 5 and N 22. Hence. excludmg For example, in Figurc 5 we can 1et n calcula/ of dots the values, have total 80 the points on boundary to we a on the surface. Thesc values can be calculated by solving recursiveb the ) : (system 09 equations in (30). The recursivc nature of the problem is due to the existence of thoundazy conditions. next section deals with these.
l''
-
,
=
maxgm.
(37)
K, ()j.
-
nese give the boundary values for Fq. ln Figure 5, these boundary regjons arc shown explicitiy. Using these boktndary values in Eq. (30),we can solvc for the remaining unknown Fq.
.
,
.
-
is an extremely low price. ncrc is almost no chance In this case, t jaat the option will expire in the money. ne rcsulting call prcmium is cjose to zero. For ( r, we know exactly that qsmin
;
:.J:
6 Concluslons
-
.k
,
.
=
,
.
.
.
,
This chapter discussed some examples of PDES faced in pricing dezivativc assets. We illustrated the dlfhcu jtjes of introducing a second random dement jyj prkcjxg can options. Wc also discussed some exotic derivatives and tjac way poss would change, Onc important point was the geometry of the Black-sc jwjes surfaces. We saw t jaat random trajectories for the underlp'ng assets price led to random Paths on this surface. nis is shown in Figure 6.
,'
=
.
.
.
,
.
.,
=
=
,
.
rrhe
,
)
Conditinns ) 5.2.1 rtlpgrlff/r.p Now, some of the Fij are known because of endpoint conditions. FG example, we always krlow thc value of the option as a function of St at 7 . expiration. For extreme valucs of St, we can use some approxl 'mations that l r arc valid in the limit. In paticular'. t , and 'i St that is high, let we very g . For 5, Ke-fl-t) (35) Fv h
.
.i,jy
eo
-
*
11:1i:5 is a price chosen so that the ca 11 pltz IlzllLl;k Here, smax the expiration date payoff.
is 17y ;
..'.
.
j:
.
(St,') 'WN. -...
()
Epfll!d:p 14:,
jr,s.s
lj,y tftlyk.. g
oqjy
i
0
'
:
'
-.-'''
.-
().12
. .... ''''' --------
,
j4a
' :,
().z$
.....
st
...........
12n
(s
100
:
xts will be seen bclow, elem@n pj. are ignorilv i, j subscripts for notatonal convenience. $: exists each tberc o'ne equation such ms (>)< i, For j equation depe,n j. this digerence d on of
'Mwc
:
cv
..
. (.:'Efiy
2o l
'F
'k'
yyyyy . E,:
co
=
umax
,
4o
':.:':.:'1
'
.
,iytn
50
ks'm.x
max
...
yk: kktjjk.-
st
--..
ao
o.a
so
1
,
llwe can
L'
also use centered differences.
F 1G U R E
i(t j't :
...
.
6
rj..1
:
C H A P T ER
.
The Black-scholes PDE
13
LLL :.
References
(b) More precisely, remember
,
E
.:
lngersoll (1987) prtwides several examplcs of PDES from asset pricing, Otzr of this topic is clearly intended to provide examples for a simple treatment introduction to PDES. An interested readcr should ccmsult other sources if information beyond a simple introdudion is needed. One good introdtzction This bok illustrates thc basics using MAPLE. to PDES is Betounes (1998),
:
l
J@
s
/!
<
vke--rteiz-qzbt-s)
=
nv J t; --r5
=
E
or, again,
.
E
'
) .
8 Exercises
(e-rlx1.,Y s t
WMch selection of
t
/j
<
J>
#,
t;
%*
l (7-2 )(/-.1
z
F
x
z
.
*
l
a martingale? Would
=
r +
g.
2
r? .
(d) Now try:
.)i L' .1
#t
=
.,
'
Note that each one of these selections
E.
.
where
-(u+
<
xve-net-slkt-bbl-rq
=
would make e-rtx
't
'
h
work? (c) How about
'')
eK,
-rf l-s'
M=r
;,
Let Xt be a geometric Wiener process,
.-
t;
.
EE
'l'hc exercises in this section prepare the reader for the next three chapters instead of dealing with the PDES. An interested reader will lind severl uscful problems in Betounes (1998).
=
Ec-rrla.I.''
from the previous derivation that
,:
;':;
Xt
31 1
8 Exercises
7
the e-nx
1 r - - CFa, 2 .
dehnes a different distribution for
1.
'
y)
c.2/).
Ngt,
-
r
FI.A-/1X,, s
fl
<
f'le1'' IY',
=
.S' < f
1.
s
<
t1
=
A'(t?F'-F,''f'i')
IFx,
s
<
..
. : :j
Using these, calculate the expectation:
L j
E (.Yt lX
J' p
J <
1)
l
.
dealg With obtzt'ining martingales. geometric process with drift #, and difusion parametcr 2.
WiS
SUPPOSe
CXWCiSC
(c-&A- I-Y s #>
<
E . .:
r.
That is, when would thet
(a) When wcmld the e-rtx t be a mmingale? following equality hold: EP
j
.
Xt is
',
IJ
=
e-rsx
'
S
; :;
.
L. )..'
P-T'X
=
17
Wiener process: l
=
cvpi '
(a) Calculatc the expected value of the increment dzt). (b) Is Zt a martingale? (c) Calculate .Eg&!. How would you change the defmition of Xt to make Zt a martingale? (d) How would S(4) then change?
'?.
/J.
)
X
Lj
'r
,,
where Xt is an exponential
.
And the trivial equality
r' (
Z(f
E
(a) Consider the deeition
Flc
Consider
j
.
.. 1.'
.
('
!;
.
l Translations of Probabilities
J.;
7rwxsL
.
yx77ww
'','','N o..v
w
.
The probability density
z
/(z,)
knownexpression
. 2. EE #
#
*
*
erlvatlve
rlcm E qailmlent
artingule
ro xlcts
(2r)
l .-jzj e 2=
=
(z,;
.
suppose we are interested iri thc probability that zt falls ttear a specific valuc J. Then, titis probability can be expressed by first choosing a small > 0, and nex't by calculating the integral of tlle normal dens inten'al region in question: over the
?. 5' 'f
c/uares
of this random variable is given by the well-
E
P J
'z
'! .E
1 2
-
-.
z;
<
<
,
'i+lA
1
+. -A 2
1
=
2=
z-ja
e -)z/ dz ,.
':
will not change very much is small, then Now, if the region around 11 to + !A. varies from means we can approximatc /'(z/) 2 nis as zt z the intcmal and write integral during this on the right-hand side of by f (:)
' '
1 Translations of Ptobabilities .
Reccnt methods ot derivative asset pricing do not necessarily exploit PDES implicd by arbitrage-frec portfolios. ney rcst on converting prices of such is done through transforming the underlying assets into martingales,
,:i
; !. .
.
(3)as
1 ,&.
z+ z
'
?
'l>
:-
E
rfhis
i,
probability distributions using the tools providcd by the Girsanov theorem. nis approach is quite diffcrent from the method of PDES. The tools involved exploit tle existence of arbitrage-free portfolios indimectly, and hence are morc diflicult to visualizc. A student of fmance or economio is likely to be even less familiar with this new set of tools than with, say, tho
-
,
.:
;.
.
.f(z;)
,i
J.7'
l,
x
1
.
2*
c-
)zzr Jz 2
=
.
t
:.:
1 2*
C
z+
.-.zz
al,
dzt
-
z-lAZ
-z
z2k
.
.$T
q
that is assoVisualized this way, probability corresponds to a ciatcdwith possible values of zt in small intervals. Probabilities tre called meanuresbecause they are mappings from arbitrary sets to nonnegativc rcal ''measure''
:
.
This chapter discusses these tools. We adopt a step-by-stcp approach. First we review some simple concepts and set the notation. As motivation, WC Show some Simple cxamples of the way the Girsanov theorem is used. The full theorem is stated next, We follow this with a sedion dealing with tl3e intuitive eVlanatim tf various concepts utilizcd in the theorcm. Fiapplied nally, thc theorem is in examples of increasing complex. Overall, few examples are provided from hnancial markets. ne next chapter deals W ith that. ne purpose tf thc present chapter is to clarify the notion of transforming underlying probability distributions.
2*
c
nis construction is shown in Figure 1. The probability in (5)is a by a redangle with basc l and beight /'(:). represented(approximately)
EL
PDES.
1
.
EE
t
0.4
*
l .'.
Q.3
'.!
:i '
G.2
:;. '.
Pxobubility ua ffMezture''
'
Consider a normally distributed random variable z: at a flxed time t, with zero mean and unit variance. Formally, Zf
%=
N'(0 1). >
3l 2
'
.
0.1
!
E
i
1
(1) )' t >
.' '
.
...
1ist''b
-')
ex p ( 5 z z )
-
;
F 1GU R E
'
.:.
tE
1
Z
.
!'
:.
C H APT ER
.
Pricing Derivacive Products
14
.1
! qunslartons of Probabilities
E1
numbers R + Fo r infinitesimal A, which we write as dzt, these measures are denoted by the symbol ##(z;), or simply dp when there is no confusion about the underlying random variable'.
Q.8
.
dpkj
=
# :
1
-
j
dzt
,i
<
zt
<
+
1
j
''
E. '
dzt
(6)
.
k'
:
This can be read as the probability that the random variable zt will fall within a small interval centered on and of inlinitegimal length #z;. ne sum of al1 such probabilities will then be given by adding these dpzj for various values of Formally, this is expressed by the use of the integral
'.
:5
,
6
.
'
.
X
..
d#(z,)
=
-X
1.
(7) ,
,)
A similar approach is used for calculating the expected value of zt,
.E'Ez,1
o.4
.
k. '
-.x
zt dpzt),
=
-X
'.;.
@)
:
value of zt. Geometrically, this deterwhich can be seen as an mines the center of the probability mass. The variance is another weighted ftaverage''
L'
;
avcage:
i'
o.2
'.
-x
- .7Ez,1!2 -x gz,-
E Ez'
=
.E'(z,112dpzt).
.?
(9)
'.'.
i ..S
ne variance has a geometric interpretation as well. It gives an indication of how the probability mass spreads around the center. Accordingly, when we talk about a certain probability measure, dP, we always have in mind a shape and a location for the density of the random
',,,
j
is transformed into another
p. = I
tion
,'
In this botlk we always assume tlla his density exists. In tj , thc underlying random variables may not exist.
.. l .
at zero:
(5
(11) E.
CE :
Other
settings, thc density fun
s
';
1o
z
2
'
$
(10)
0.
FIGU RE
:
.b.
normal density, this time centered
()
.':.;'
at
l. = -5,
-5
i
-
move thc . we can Ieave the shape of the distribution the same, but where the densityto a different location. Figure 2 illustrates a case
that was centered
.
-:()
'L
Under these conditions, we can subiect a orobability distlibution to two tmes of transformations. .
normaldensity
.
a
'E.
-
I
;
variable-l
'
'
j. ) .J.
-
We can also change the shape of the distribution. One way to do this is to increase or decrease te variance of the distribution. nis can be accomplished by scaling the original random variable. Figure 3 displays a case where the variance of the random variable zt is reduced from 4 to 1.
Modern metllods for pricing dcrivative assets utilize a novel way (f transforming the probability measure dp so that the mean of a random process transformation permitg treating an asset tliat carries a poszt changes. premium'' as if it were risk-free. This chapter dcals with this ve com Plicated idea. 'rhe
Rrisk
316
ClH A P T E R
.
14
Pricing Derivative Producl
.317
2 Changing Means
Both operations lead to a change in the original mean, while preserving other charactezistics of the original random variable, However, while the first method cannot, in general, be used in asset pricing, the second method becomes a very useful tool. We discuss these methods in detail next. The discussion proceeds within the context of a single random variable, rather than a stochastic process. The more complicated case of a continuous-time process is treated in the sedion on the Girsanov theorem.
0.B
.6
2.1 Method 1: OpcgOng
on
Po44ible Vltlues
The first and standard metbod for changing te mcan of a random variablc is used routinely in econometrics and statistics. One simply adds a #t.2 zt + constant Jz' to zj in order to obtain a new random variable h The h defmed tis way will have a new mean. For example, if originally =
0.4
(12)
.E'(z/1 = 0,
ll
then the
new random variable :1 will be such that
FA1
0.2
=
E'(z/1 +
v,
=
>.
(13)
2.1.1 Fwrzlpz/o7 In spite of the simplicity of this transformation, it is important for Iater discussions to look at a precise example. Supposc the random variable Z is dened as follows. A die is rolled and the values of Z are set according to the rule -10
-5
5
0
FIGU RE
10
roli of 1 or 2
-3
roll of 3 or 4 roll of 5 or 6
10
3
Z
=
-1
ln the following section, we discuss two different methods of s'witciting means of random variables.
Msuming that the probability of getting a particuiar nllmber is 1/6. we can easilycalculate the mean of Z as a weighted average of its possible values: .E'lz!
2 Changlng Means
=
1 1 1 j(10j + jg-3J + jg-ll
(15)
(16) = 2 Now, suppose we would like to change the mean of Z using the method Outlined earlier. More precisely, suppose we would like to calculate a new .
Now, flx f and Iet zt be a univuiate random variable. There are twcl WayS valu'q one c,an change the mean of z/. In the lirst case. we operate on the values countcrintuitive, assumcd by zt. In the second, and case, we leave te assumed by zt unchanged, but instead operate on eprobabilities associated with zt.
(14)
lg
can be negative.
F :
3 18
C H A P T E)R
@
'.
Pricing Derivative Products
14
Changing Means *4 ''
random variable with the lame variance but with a mean of one. We call this new random variable Z and let
2
z-
=
Using the formula for the mcan, we calculate the E 1:1: 1 1 1 'g/l 11 11 + j(-1 jg10 11+ j(-3 =
-
-
-
.
1 f.
(18)
',
'
('EE
As can be seen from this transformation, in order to change the mean of Z we operated on the values assumcd by Z. Namely, we subtracted 1 from each possible value, The probabilitics were not changed.
rt + Flrisk premiuml,
the world. Let a be the
(constant)expected .S'(.R1 !
=
E
,.'
obtain
.:',
ERt
+ #'1
rt + # +' a.
(1 +
. i. .
: I..
j
tE
,/
';
=
&(1+
#, + gmt
#t5).
(25)
can be interpreted as a risk
= (r(.j. r t )(j o.s),
(26)
The term on the left-hand side of this equation. .E-;gz%+1 jutj, represents means that expectedgross rettzrn, .E)ll + RJI.
):
'rhis
E :.
In the case of normally distributed random variables, this is equivalent r to presen'ing the shape of the dcnsity, while sliding the center of the d1X- qb tribution to a new location. Figure 2 displays an example. E lf #, is selected as -a, thcn such a transformation would eliminate the risk premium from Rt. Note that in order to use this method for chang ing meaflsj We need to k'Jt;)i' the Usk premium a. Only under thcse cfmditiong could we anive at subtracting t he r ight'' quantity f'rom Rt and obtain th* equivalent risk-free yicld. f This example is simple and does not illustrate w'/zysomebody might want !' next Such to go tltrough a transformation of means to begin with. The example is more illustrative in this respect.
f(.%+l!
.E'/(.5',+11
E
(22)
r.,)
Note that the positive constants p. or prcmium.Transforming (25),
,
=
=
1
.;'
.
(24)
tlle risk-free
k
nen Rf is a random variable with mean rt + The tirst metbod to change the mean of Rt is to add a constant and a new random variable k, Rt + v.. nis random variable will liave the mean
.
Here the left-hand side represents thc expected future price discountcd at rate. For some Jz > 0,
%
(21)
r/ + a.
.
(23)
hl-h.
.%-
:
risk premium:
.
,
1 > (1 + rt ) .E'dE>%+11
'!
where rt is the known risk-free ratc of treasury bonds with comparable maturity,and where E g.1cxpresses the expectation over possible states of
,
=
That is, on the average, thc risky asset will appreciate faster than the growth of a risk-free investment, nis equality can be rewqitten as
E
(20)
.
.E'/E.%+1 1> (1+
;
2.1.2 Example 2 We cao illustrate this method for changing the means of random varil'elevant GICS Using 2 example from nance. mtfc WC Of tfiple-Arrated corporate bond Rt with fkxcd f will have the a /cld expected value =
,
:
(19)
E'gS;1
vt,
'.
(17)
1.
J 2.1.3 .hxtlznJ7/tr be 1, 2, 'rhe example is dismzssed in discrete time 'Iirst. Let t (bserved the price of some financial asset that pays no dividends. The St is over discrete times r, t -I- 1, Lct rt be the rate of risk-free return, A typical risky asset & must offer a rate of return Rt greater than rt, since othemise there will be no reason to hold it, This means that, using F;g.1, the expectation operator conditional on information available as of time / satisses
'.(.
F,(1 + R,1
=
whichsays that the expected remrn free return approximately by Jt:
:'
(1+
r,)(1 + p),
of the risky asset must exceed the risk-
f,EA,1 21 r/ + y.,
:
&
(27)
(28)
F:
7
'
:
:
E .
Di
..:
. .
'k
.)
'@
in the case where rt and p. arc small enough that the cross-product term tlan be ignored. Under thcsc conditions, #, is the risk premium for htlding the asset for tme period, and (-f.)rt ) is the risk-free discount factor.
C H A PT E R
Pricing Derivative Products
*
analyst who wants to obtain Now consider the problcm of a snancial market valuc of this asset today. That is, the analyst would like to the fair calcmlate.St. Onc way to do this is to exploit the relation
1
Et
(1+
2 Changing Means
;
task at the outset. the second method for changing means does precisely this,
:
5.
2.2 Method 2: Operutfng crt Probtbiitcs
'
'b't
(29)
-
-'r+1
Rtj
''
second way of changing the mean of a random variable is to leave but transform the corresponding probabilit.v the random variablc neasure that governs z/. We introduce this method using a series of examples that gct more and more complicated. At the end we provide the Girsantw theorcm, which extends the method to continuous-time stochastic processes. nc idea may be counterinmitive, but is in fact quite simple, as Example 1 will show. 'rhe
bycalculating tl,e expectation on the left-hand sidel But doing this requires a knowledge of the distribution of R,. which rebefore Quiresknowing the risk premium >.4 Yet knowing the risk pre market value will nowhere the fair is Utilization of rare. knowing (29) go in ternas of calculating the mean of Rt without on the other hand, if one could work.s method might the the Another to use #,, way of transforming having of distribution Rt be found. must thc If a new expectation using a different probability distribution / vields an such as exp ression 1 E9 5'/+1 t (31) jj. -I- %j (
i'intacq''
'. ,'
'mium
,'',
'
.,
:'
-,.5
)
''transtbrm''
2.2.1 F-mmple 1 Consider the rst exnmplc of the previous section, with Z dehned as a f'unction of rolling a dic:
',
-
=
k
'
')
'
(zaverage
'onlybyknowingscan
down.
aa,
:E. :'
'L .'::
a-his
srther cxymplitxates
fBecause
1
(1 +
.571
R()
#
1 (1 + -SrA,)
''
1z
: .'.
'tgctting
')
k. .
5
k':.
E , (Z1 :
r
1
-->
=
3 1
-
-
--->
6)
=
-
1 3
--->
(r
3
? (Gettine *
or 4)
122
,.
.L.
= '
=
%'
1c2
a 1 or 2b =
429
'
tceuing:! or 4) / (Getting 5
r 6)
-
=
-
22 39 5 33
-
-.
(36) (3,)
.
(38)
Xote that the new probabilities are designated by 13. Now calculate the mean under thcse new probabilities!
-'
.
1
-g1()-2j
=
,3
Ftgctting
.1:*
the oucuutions.
the mzan of the distribution of R( could be made equal to r
ygz-sjzjjz
*c
'..'.
(30)
.
=
fatzettinz c 1 or 2
,
and the disvibution of R, .epinnea
(34)
2,
=
1 1 98 a+ z z +.-(-3-21 -(-1-21 3 (35) 3 3 3 Suppose we want to transfonm this random variable so that its mean becomes one, while Ieaving the variamce unchanged. considcrthc following transformation of the original probabilities associatcd with rolling thc die:
vartz)
j
mriod'sP11,
(33)
,
and a variance
,
srl-horc s an additional difliculty. The term tm the left-hand sidc of (29) is a zwalz-'vzr . of R(. Hencc, wc cannot simply move the expectation operator i.u front of Rt: function E,
E'lzj
.
3nis relafion is just th< desnititm of the yield Rt. If we discoun! the next by '1 + Rt, wc naturally recover today's value. becakuuted,
-
.k
The trick here is to accomplish this transformation in the mean without having to use the value of Jt cxplicitly. Even though ths seems an impossble
1
rOl1 of 1 tr 2 roll of 3 or 4 rojj of 5 or 6
with a prcvioesly calculated mean of
(32) ,
%.
-3
=
'
v,,
ele meau
2
.,
of rt. (kllown) wouldprovide an estimate of St. what would F,z:g.1and rt represent in this particular case? r/ will be the risk-freerate. ne cxpcctation operator would be given by the risk-neutral probabilities.By making these transformations, we would be eliminating the risk premium from Rt'. =
z
..j.
be veo, useful for calculating In fact, one could exploit this thiscan by using a modd that describes the dynamics &+.j, ''forecasting'' equality and then discounting the forecast'' by the This
I't
10
;.E J
! ,
,s,.
X/ -
321
.
':
=
..
42N
22
5
1101+ jj (-31 + jj L-11 1. =
(39)
:'
E
C H A PT E R
322
*
The mean is indeed one. Calculate the variance'. 5 22 122 2 11z + 11z + E >gz) (10 39 33 429 -
-
f ''
-
2
11
=
98 3
-.
..;
(40)
transformation of probabilities shown in ne variance has not changcd. (38) accomplishes exactly what the Nrst method did, Yet this second method operated on the probability measure P(Z), rather than on the values of z itself. lt is worth emphasizing that tbese new probabilities do not relate to the true odds of the expcriment. The true pro jlatljjjtjjx agsoctatcd WZ rolling the die are still given by the original numbers, P. ne reader may havc noticed the notatkn wc adopted. In fact, we need FPg.1, rather than F(.l. The probto write the new expectation operator as abilities used in calculating the averages are no longer the same as #, and the use of F(.1 will be misleading. When tbis method is used, special care should be given to designating the probability distribution utilized in calculating expectations under consideration.? 'rhe
4:
((
$>
EE
)
.
'q :
11
p@
'
) '
E :
'.,
Girsanov Theorem
323
?
3.1 A Normully Distrilnzted Rundtmz Wdcb!e
'
i. .
3
ne
Accordingly, if we have to calculate an expectation, and if this expectation is easier tta calculate with an cquivalent measure, then it may be worth switching probabilities, althoug,h the new measure may not be the one that governs tbe tnle states of nature. After all, the purpose is not to make a statement about the odds of vuilus states of nature. nc purpose is to calculatc a quantity in a convenient fashion. The general method can be summn rized as follows: (1) We have an ex-pectation to calculate. (2)We transform the original probability measure so that the expectation becomes easier to calculate. (3)We calculate the expectation under the new probability. (4)Once tlle result is calculated and if desired, we transform this probability back to the original distribution. We now discuss such probability transformations in more complex settings. The Girsanov theorem will bc introduced using special cascs with growing complexity. Then we provide the general theorem and discuss its assumptions and implications.
.:.
-(-3
-(-1
=
3
Pricing Derivative Products
14
The Glrsanov Theotem
Fix t and consider a normally distributed random
,T
zt
; 1.
,
.:
##(z;)
!''
dequivalent''
.:
1
=
f(zd)= When we multiply (zt) by dpztj, be seen from the following:
'
:
'..
=
1.
Csome
'
g/(z,)
'
;
(43)
.
1 2*
e-
jsz yyz ,. )(z/)+.sz,-
After grouping the terms in the exponent, we obtain the exprcssion !. .sjz .. ) dpzt e- z gz tz 2* Clearly dlhztj is a Fzew probability measure, desned by t
=
new
2
(
2 11
Zr-
&
we obtain a new probability. This can
E##(z,)lE;(z,)1
;
nc
(4l )
.N0, 1),
e- jjolz dz J. (42) 2= ln this example, the state spacc is continuous, although we are still working with a single random variable, instead of a random process, Next, dclinc thc function
'
''
partitllr readers may wonder how we found hc, new probabilities /(Z). In tbis conclitions to used unknowns and threc probabilities the as casz,i: was easy. We copsidered condition is tbat the probabilities sum to one, nc second is that the ftrst them. for ne so1vc third is that the variance equals 985. mean is one.
zl'.
Denote the density function of zt l3yfzt) and thc implied probability measurc by # such that
Th e examples just discussed werc clearly simplifed, Fir'st we dealt with random variables that were allowed to assume a fmitc number of valuesE. thc state space was fmite, Sccond, we dealt with a single random variable instead of using a random process. The Girsanov theorcm provides the general framework for transforming measure in more com- i one probability mcasure into another plicated cages. The theorem covers the case of Brownian motion. Hencc, the state space is continuous, and the trangformations are extended to continuous-time stochastic processes. t because, as we The probabilities so trarisformed are called VII see in more detail later in this chapter, they assign positive probabili- J ties to the same domains. nus, although the two probability distributions are diferent, with appzopriate transformations one can always recover one t measure from the other. Since such recoverics are always possible, we may and then, if :, Wan t to I1se the Oonvenient'' distributitm for Our calculations, 1 >itch desircd, back to the original distribution. idequivalent''
'x.
variable
.
=
dpztjqzt).
,
(44)
(45) (46)
324
C H A P T ER
*
14
Pricing Derivative Products
3 The Girsanov Tlaeorem
325
1
By simply reading from the density in (45),we see that #(zz) is the probability associated with a normally distributed random variable mean t and variance 1, lt turns out that by multiplying dpzf) by the function ;(zr), and then in changmg the mean of zt. Note that in switching to #, we sueeded this particular casc, thc multiplication by 4(z;) presen'ed the shape of tbe probability mcasure, In fact, (45)is still a bell-shaped, Gaussian curve with the sme variancc, But Pztj and #(z/) are different memsures. They have diffcrent means and they assign t/s/fcrcnf weigbts to inteaals on the z-axis. Undcr thc measure Pzt), the random variable zt has mean zero, EP EP Ez/) 0, and variancc EF j 1. However, tmder the new probability measure #(zr), zt has mean gzfq la,. ne variance is unchanged. What we have just shown is that there exists a function kzt ) such that if we multiply a probability measure by this function, we get a new probability. The resulting random variable is again nonnal but has a differcnt me-anFinally, the transformation of measures,
(z2, =
=
=
##(z,) f(z,) J#(z,),
(47)
=
which changed the mean of the random variable z?, is revenible:
(ztl-3dlhzt)
(48)
dpzt).
=
Thc transformation leaves the variance of zr uncbanged, and is unique, given p, and e. We can now summarize the two methods of changing means:8 * Method 1: Subtraction of means. Given a random variable z
Z-g =
-
1
z
#
=
Z 8We simplify the nomtion
glhtly.
'w
#
=
(50) variable
Z with
(51)
Nm, 1),
N(t), 1).
,
-
-
-
>
.
3.2 A Nornztlly Dlstdbuted Vector The previous example showed how the mean of a normally distributed random variable could be changed by multiplying the corresponding probability measure by a function zt ). The transfonued measure was shown another that be probability assigned to a different mean to z/, although the variance remained the same. Can we proceed in a similar way if we are given a vector of nonaally distributed variables? The answer is yes. For simplicity we show the bivariate case. Extension to an rl-variate Gaussian vector is analogous. With fxed 1, suppose we are givcn the random variables z:;, ht, jointb distributed as normal. The corresponding densiT will be
fzt,
z2,)
1
=
2.
g.2 9 ) (zz,-?z.z)1 - i E(zl,--pz., tr12 cr..ie 1
where fl is the variancc covariance matrx of D with ojl, i zI,, zcr. The
by f(Z) and
0b-
(52)
g.
'
If1!
2 tl'j
=
tn 2G2
2
=
(z1: (z'a'-
,,1 ) #.cl
c'12 Gl 2
Gl 2
r
-1
-
(53)
(zjr,zazl, (54)
,
la 2 denoting thc variances and lf1lrepresents the determinant: .61
N(0, 1).
transform the probabilities dP through multiplication tain a new probability # such that
Zl
-2-2
>
I
Will
.w.
Z1
=
(49)
1),
have a zero mean, Method 2: USZg equivalent measureg. Given a random Wobability #, WCII
*
Ng,
k l)y transforming Z
dehne a new random variable
2'
ew
Thc next question is whether we can accomplish the same transformations if we are given a sequence of nonually distributed random variables,
g'la
the covariance
between
2
n z, .
Finally, y.j, tz are the means corresponding to zl, and zzt. The joint probability measure can be dehned using .Ptzl/, z2r)
J(z1r,z2/) dz,, dzzt.
=
(56)
This expression is the probability mass associated with a small rectangle dzlzq centered at a particular value for the pair zlt zzt. lt gives tlle Probabllity that zLt, z2; will fall in that particular rectangle jointlj'. Hence the term joint density functbn. Suppose we want to changc the means of zy), ht from tj tn to zero, while leaving the variances unchanged. Can we accomplish this by transforming thc probabilty dpht, ntj just as in the previous example, namely, by multiplying by a fundion f(zl/ zzj )? ,
,
,
r'
.:I
.<
'g:.
326
C H A PT ER
.
E''.
Pricing Derivative Product.s
11
-Ez1J
(zll,
z2l)
=
,i
e
usingthis, we can define
arl
'':jj-'
s
h
j gsj j-'g?,.l j sa j
vl.y
jn,
sag
+
fwk
l .t
jj ,:1 Ju
J(zi,, z2,) d#(z:,, zz,).
=
dlhzjt,zz,)
=
1 e 2.,, Iflp
zplgij
-:s
Iyj order to convet this expression into
)
tut
-z,s,+. -.J
(5o 'j
PO
We recognize this as the bivariate norrnal probability distribution for a ran- h rt matrix n. ne . ; domvector go, zu,1withzc,)mean zero and variance-covariance accomplishcd the stated objedive. The nonzero by ;(z.,, multiplication ') mean of the bivariate vedor was eliminated through a transformation of tlze 2.
,
3.2. 1
.
.
.
jn j
.
3.3 The Rzzzlnn-Nltodym Derivatit'e
').
consideragain
E
underlying probabilities. 'rhis example dealt with a bivariate random vector. Exactly the same transformation can be applied if instead we have a random scquence of k z:,1. Only the orders normally dstributed random variables, gzj/,zu corresponding and matrices need in of the to be changed, with vectors (53) adjustmcnts in similar (57).
the function
t
We used the dljzt):
,
':t
4(zJ
) in
d?lzt)
j
j.
(())
e
-./+1
.w
L
r
a
c. ;
.
(61)
.: .i
..
.
E
.
We will now discuss where this fuoct ional fonn comes from. In normal tml. distributions,the parameter #t, wbich represents the mean. shows up an exponent of c. What is morea this exponent is in the form of a square:
as
1(zf -
p,) 2
(6c)
(69)
.,
:
=
f(Cl1'
=
This expression can be regarded as a dcrivative. lt reads as if the tive oj oe measure > with respect to P is given by (zt). Such derivatives ae called Itadon-Nikodym derivatives, and fztj can be regarded as the denaity of the probability measure # with respect to the measure P. According to this, if the Radon-Nikodym derivative of # with respect to P exists, tben we can use the resulting density ijztj to transform the mean of zt by leaving its variance structtzre unchanged.
(.
J(zt)
(68)
''deriva-
h'
which in the scalar casc became
/(z;) from
apzt)
,;
,
(oyl
.
(ztjdpztj.
=
#(o)
'
e
,,2
Or, dividing both sides by dpzt),
with future discussion in mind, we would like to emphasize one regu- Ikir : laritythat the readcr may already have obsenred. E:'E univariate of length k, or simply as a Thirik of z/ as represent ing a vector t randomvariable. In transforming the probability measures Pzt) into tzt ), ( the fundion f(zl) was utilized. This function had the following structtlre, .) 'i' -z;f)-.s+.lJz'!)-1s =
1:9
=
obtaining the new probability measure
,
' ',
f(z,)
c'
e -sz,+ 4
=
:
.WNote
Kzt) with
(z/)
':t
,
(64)
.
is what determines the functional form of (zt). Multiplying the original probability measure by (z;) accomplishes this transformation in the exponcnt of e. Given this, a reader may wonder if we could attach a deeper interpretat oyjto what the f(z,) really represents. Thc next section discusses this
,.
.
lyzsa
:(?
nis
:
.b
dzgj.
(6a)
,
.n
we need to add
.'L
a
1 (z,)2
j
-
(58) t
j j j.l)',
'z
327
...,
>tzjt, zct) can bc obtained by multiplying expression (53) by jht, zw), shownin (57).Thc product of these two expessions gives -)(z,,
neorem
.''..
.
l'szkt. zo) by a new probabuity measure
zz,) dl-hzkt,
Gjlsanov
'yhe
'(
The answer is yes. Consider the function dcfined by z2,lg
3
'F
Nlncidentally,
,,
thc function
4(z,)
a
subtrvgctv a mean from z;, whereas
E
'
.!
tlle hmction -1
J'(z,)
;
:. .. '.. .
e-wzt,t
=
-
e
.z,-
.)
.2
(6,6)
CHA PT ER
*
14
Pricing Derivative Products
Clearly, such a trlmsformation is very useful for a financial markct particwltile ipant, because the risk premiums of asset prices can be leaving the volatility structure intact, In the case of options, for exmple, the option price does not depcnd on the mean growth of the underlying asset price, whereas the volatility of the latter is a fundamental determinant. In such circumstances, transforming original probability distributions using 4(z;) would be very convenicnt, ln Figure 4, we show onc cxample of this function J(zy).
4 Statement
CONDITION:
dielintinated''
3.4 Eqltlulcnt
Girsanov Theorem
Given an interval dzt, the probabilities P and 13 satisfy /(#z)
>
if and only if
0
dlhzt) =
(zfldpztt
(71)
note that in order to write the ratio
J#(zr)
(72)
##lz,)
meaningfully, we need the probability mass in the denominator to be differTo perform the inverse transformatitm, we need the numen from zero, ent different f'rom zero. But the numerator and the denominator are be to ator probabilitiesassigned to infinitesimal intcrvals dz. Hence, in order for the Radon-Nikodym derivative to exist, whcn # assigns a nonzero probability to #z, so must #, and vice versa. In other words:
25
(z) z?bz 1
l'(z/) drzt)
=
(74)
I(z/)-' dlhztj.
4 Statement of the Girsanov Theorem In applications of continuous-time hnance, the cxnmples provided thus far will bc of limited use. Contittuous-time Enance dcals wit.h continuous or right continutms stochastic processes, whereas the transformations thus far involved only a jinite sequence of random variables, ne Girsanov theorem provides the conditions under which the Radon-Nikodym derivative J(z;) exists for cases whcre zt is a continuous stochastic process. Transfonuations of probability measures in continuous hnancc use this theorem. We first state the formal version of the Oirsantw theorem. A motivating discussion follows afterwards. The setting of the Girsanov theorem is the following. We are given a family of information sets (32).over a period I0,T1. T is linite.'o Ch'cr this intenral, we deline a random process (t? (/(; dBi- J j,' A du) (t t (E e xu =
jtl wj, ,
,
(,y(p
where Xt is an ft-measurablc process.l 1 The )P) is a Wiener process with probability distribution #. Wc impose an additional condition on Xt. should not vary
=
=
(73)
(70)
able to perform transformations such as
30
0.
nis means that for all practical purposes, the two measures are equivalent. Hcnce, they arc called equivalent probability measures.
derivative,
dlhzt) f(z,), d#(z,) = =
>
and apzt)
exist?That is, whcn would we be dlhzt)
#(Jz)
If this condititm is satisfied, then (z/) would exist, and we can always go back and forth bctween the two measures / and P using the relations
Metlles
When would the Radon-Nikodym
ln hcuristic terms,
of the
1
iltoo
20
mt1Ch''*
15
sgcl', zaj .< ..-5
1() 5 .3
-2
-1
0
FlG U R E
1
4
2
! zy
3
x
t s
p. zj.
ltdNote,that this is not a ven' serious restricton in the case of Enancial derivatives.A1l'ntst all linancial derivatives have Ilnite cxpiration dates. Oflzn, the maturity of the dcrivative ilqrumznt is less than one yeaz. Clnat
is, giveu the informafon sct
6, the
value of X is known euctly.
j'
:'; .j'
C H A P T ER
330
Pricing Derivative Products
14
*
5 A Discussion of
;
Gtnew''
.(i
get-l
JWFx
=
-
dh
,.: ;
=
3 #!
d(t
Jl'P).
/xr
=
of t
Also, we see by simple substitution
(79)
;i '
t) in (76).
=
(79),we
r';
Q
1+
(81)
Jl1Z;.
-l
t)
.'
3
). 1.
: (82) ..:5
#H$
f-
('t
:
is a stochastic integral with respect to a Wiener process. Also, the term : Lxs is f/-adapted and does not move rapidly, All these imply, as shown i.q Chapter 6, that tlw integral ig a (squareintegrable) martingale, r: E where u < martingale.
f
.
u
fX .1p
t)
.$
t/l4$ lu
I
(81),this
Due to
=
0
(sxs
implies that
fTT;,
(83)
(1 is a (squareintegrable)
Z WiCRCF
RWZSWC
PFOCCSS
f'n
UVCS
Witll
XSMCCt
to
bjF
h
ZRd
Wi $11XSPCCt
F1'I1a z 1 dctermined by zz and Pz.( ) .?4
an event with being of tlae event. funetion ..4
4/
whichis similar to importantpoints;
.j
:
'
:.
()
=
,
(88)
Js.
Then, taking the integrals in the exponent in a straightforward fashion, and emembering that P?() %
p
/
(0,T1,
=
p,
=
If the process (t dclined by (76)is a martingale witb resped information sets It, and the probability #, then C, defined by to
te
(g..y.)
,
Xu
...)
THEOREM:
=
-!zrj;xwdw;.-!.h'.-,1 zdlzq
Suppose the Xu was constant and equaled
.l
We are now ready to state the Girsanov thcorem.
11, H,: l
e
.7,
-
vkdu,
=
where we explicitly factored out the (constant)c2 term from the integrals, Alternatively, this term can be incorporated in Xlg.
:!.
l
,
Il1 this section, we go over the notation and assumptions used in the Girsanov theorem systenzatically, and rclate them to previously discussed examples. We also show their relevance to concepts in financial models. We begin with thc function 6
E.
But the term l
(86)
5 A Dlscussion of the Girsanov Theorem
obtain
l =
Xt dt.
-
''
180)
by taking the stochastic integral of
t/1,:
=
That is, I'I; is obtained by subtracting an J/-adaptcd drift from H(. Thc main condition for performing such transformations is that (t is a martingalc with E L'G 1 l We now discuss the notation and the assumptions of the Girsanov theorem in detail. The proof of the theroem can be found in Liptser and Shilyayev (1977).
;
which reduces to
is
Girsanov Theorem
ln heuristic termsa this theorcm states that if wc are given a Wiener HJ;, then, multiplying the probability distributitm of this process by proccss #t, wc can obtain a A?'w Wicncr process 12 with probability distribution #. The two processes are relatcd to each other through
:1
'rhis means that Xt cannot incrcse (ordecrease) rapidly over time. Equa- E1' tion (77)is known as the Novikov condition. property that turns out ) In continuous time, the density (t has a condition is satisficd, Novikov important. It if the that out turns very be to ( then t will be a square integrable matingale. We ftrst show this explicitly. ( Using Ito's Lemma, calculate the differential ll zllj d ' Xu A'J l d: l j.t r d gjjj (7g)
nus,
the
1
(84)
E
;
;i1 '
..
lx being the indicator
-1sI!sB'',-
.z
E
:
(jwj
,
Hzt) discussed
earlier. This shows the following
=
.:
)s2l1
(dmean''
.:l.
(85)
e
1. The symbol Xt used in thc Girsanov theorem plays the same role Jt played in simpler settings. It measures how much the original will be changed. 2, In earlier examples, g was time indepcndent. Herc, Xt may depend on any random quantity, as long as this random quantity is known by time r. nat is the meaning of making Xt Ji-adapted. Hence, much more complicated drift transformations are allowed for. 3, The f/ is a martingale with E (lj) 1,
'3.
to tilc Wpbzbiliv
the
=
.
.
F
' .3 t
.( .'l
C HA PT ER
Pr ic ing Derivative Products
11
*
E
J h Discussion of
the
Girsanov Theorem
333
.k '
Next consider the Wiener process l There is something counterintuitive about this process. lt t'urns out that both )'P2and Hz;are standard Wiener processes. Thus, they do not have any drift. Yet hey relate to each other by
JI'; J)); - Xt #f, whichmeans that at least one of these processes
..: ..;
E y
What is the meaning of lx ? How can we motivate this relation? lx is simply a function that has value 1 if occurs. In fact, we the preceding equation as .4
In the case where
./gl
=
TEla(w1
=
is an infinitesimal g/z
which is similar simpler settings.
to
=
:;
5.1 Applieation to
p,
tEr
a
0, Or,
(q5)
.
Jz dt
is norlzero. Recall
:rF;,
y,t +
=
t c
(0,x).
(96)
O.p;.
(qg)
write lgksrt+s
t .
jusij
+ x) + vsjplos
.stf
=
St +
-
u,ijyj .j. tpp;
(,:)
(99)
Jtx,
,'r
since IB(+x
j
-
Wzij is unpredictable
given
v%t.
Thus, for #,
,i.
:,
,
?
,
,j
'
SDES
0:
t.j(sj
=
st -
(j()1)
gt.
h,
,.
,,,
.)
>
Then l will be a martingale. One disadvantage of this transformation is that in order to obtain one would need to kriow Jt. But y, incomorates any Hsk prernium that the risky stock return has. In general, such Hsk premiums are not known before one filds the fair market value of the asset. The second method to convert St into a martingale is much more promising. Usn the Girsanov theorem, we could easily switch to y. an equivalent measure #, so that the drift of S is zero. do tus, wc have to come up with a function (St4, and multiply it bY the original probability measure associated with S S 2 may be a t CYmartingale under P,
t
in mtlch
l
-
i ')
.$
q%
',
:;
0,
Sf is not a martingale. Yet we can easily convert into a martingale tly eliminating its drift. one method, discussed earlier, was to subtract an appropriate mean from st aad detine
.j'
f---tn
>
> sg,sy.j.yj.rj s:.
:
(93) seen earlier
=
5
inten/al, this means
the probability transformations
WC Can
L
y, dy
Jjji
'
,
ds +
st
:
(92)
4z dl a4
&
t =
',
rewrite
l-Al
With
.'
(91)
fwl.
St
.
t
g-j(p;):
Clearly, St cannot bc a martingale if the drift term that
,
,-zkdt,
1
.
2'Jr/
i
.
The point 1, 11: has zero drift under #, whereas JKhas zero drift under P. Hence, 'j can be used to represent unpredictable enors in dynnmic systemsgiven that we switch the probability measures from P to #. using it as an error term in Iieu Also !' because V, contains a term of W4wolzld reduce the drift of the original SDE under consideration exactly by -xtdt. If the Xt is interpreted as the time-dependent risk premium, the transformation would make all risky assets grow at a risk-free rate. Finally, consider the relation FFlla
tjpqyj
j
'nft
point?
=
H( is assumed to havc the probability distribution P
j
.:
if must have nonzero d Xt is not identical to zero. How can we explain this seemingly contradictory
#w(.d)
ne
CE
(90)
=
0.12
with H() = with
.
i ,''
'ro
We give a heuristic example. i Let dSt denote incremental changes in a stock price. Msume t hat theR c hanges are driven by infnitesimal shocks that have a nlrmal distributiony g equation drien so that we can represent St using the stochastic differential by the Wiener process 'F; t ' x), tfp;, f (94) + e I0, d.% v-at
.
'
.'
E zy
IJ/j gkroy
lznis
=
: ( .
:
(1():)
formulation agaiu perruits ncgative prices a: positive probability, we use it because it is no:ationally comrenient. In any casu, tllo geometric sD'E will tye dealt witla in the ne.xt chapter.
' '.
L
.%t,
>
.
. ' ' ..
'y
JE C HA P T ER
334
j!Ei 1-
'
'
I
,-.
ara
L/+.vI/j
-1
zr cl
zra
yr,k
..1
qu
tltv)
'f.
=
j
1
-) s: -.w)l e2,w0-2t nis defncs the pzobability measure P. We would like to s'witch to a new probability / such that under becomes a martingale. Dehne Ez-f 1211 e- b - ) =
.
t
,
(&)
=
Multiply
/ .$ by this
d#st)
(S =
f
.
y
l )!
:
(jjjj,j /, St
(105)
.
t-
,''
J
.
the exponents =
i
(
:
1
.TE(-',)z dsf.
-
2,:rc2/
e
C
(107)
E
l .
But this is a probability measure associated with a normaliy distributed write the increprocess with zero drift and difzsion (m. 'Ihat means we can 11,/: t:f ments of Sf in terms a new driving term
ds/
=
cdbr
:.
(108) .
.
Such an St process was shown to be a martingale. is defined with respect to probability
'rhe
Weiner process
.j
Vf
t
.
'q: t'
6 Which Probabilities?
? 'v!
The role Played by the synthetic probabilities / appears central to pricin: disOf linancial securitics even at this level of discussion. According to the and discrete no-arbitrage in of cussion in Chapte 2, under thc condition a liquid markets in that trades security of pricc setting, the any timc willbe given by the martingale equality:
d'straightforward''
.),
:
::
.J '.
z')! ,
A market practtioner would then need to take the folsteps in order to price the derivative contractz
* First, the probability distribution # necds to be selected. This is, in general, done indirectly by selecting the Erst- and second-order moments of the underlying processes, as implied by the fundamental theorem of finance. For example, in case the security does not have any payouts and there is no foreign cun-enc'y involved, we lct for a small l > 0: F Sg5 d+a
S -.
'
('
t
with known F( lowing
,
E 'oc! ( t v j
' (.
(106)
=
= F / (D Fs z,
'.
:..
skt st-lu? f /,s e-
=
Ct Ej
get
ip..%-18,.2211 lnvlt
where t < F and tl:e Dt is a discount factor, known or random, depending on the normalization adopted. ln case tlzere are no foreipl currcncies or payouts, and in case sangs accoutt normalization is utilized, Dt will be a function of t-lzerisk-free rate rt. If rs r is constant, thc Dt will be known and will factor out of the expectation operator. The fad that the Dt and the probability # are known makes Eq. (109)a very useful analytical tool, because for all derivative assets there will exist an expiration time F, such that the dependence tf the derivative asset's price, Cr, on is contractually specihed. Hence. using Ct FCS: f ) we write.: can uw
:
t#(.,)
(stt
= c-J
Or, rearranging
) to
.
335
=
As usual, the superscript of the .E'(.1.1operator represents the probability expectation. measurcused to evaluate the transformation, this perform order a (Sg) function nceds to be to ln calculatcd. First recall that the density of St is givcn by f,
6 q/hich Probabilities?
J
#:
but it will bc a martingalc under E
)
Pricing Derivative Products
14
*
12
rth.
(110)
nis determines the arbitrage-free dynamics of the postulated Stochastic Differential Equation. * Second, the market practitioner needs to calibrate the SDES volatility parameterts). This nontrival task is often based on the existenc.e of liquid options, or caps/ioors markets, that provide direct volatility quotes. But, even then calibration needs to bc done carefully, . Once the underlying synthetic probability and the dynamic are determined, the task reduces to one of calculating in (109) thc expedation itself. This can be done either by calculating tl'ke implied closedform solution, or by numerical evaleation of the cxpectation. In case of closed-form solution, one would atake'' the integral, which gives the EPIFCSP, F)1: cxpectat ion
''fair''
sf
=
ElhDtsvj,
j''
(109)1, .
Smox
.Ft&., Nzxia
vldlhsv),
(1 l 1)
k'
E IJ J
C H A P T ER
.
14
Pricing Derivative Product.s
'i
A Method for Generating Equivakent Probabilities J
.
where the # Sv) is the martingalc probability associated with that particular insnitesimal variation in ,Sv. The Smin uLsxis the range of possible movements in In case of GiMonte Carlo'' evaluation, one would use the approximatitlxi:
i will be obtained probabilities pi
'fhat is,
J
truc used the
,
,sz.
.
F
vf
by multiplying the possible values by the that con-espond to the possible states.ls Clearly, if one in place of pt', the resulting forecast
.
f
=
1 E /' FlS r T)j 2 N -
,
j (F(Sv, F)),
j= 1
(112)
=
,
.
,
y IE $
) :.
t
:g
) .
j'
=
1
C
sw + '
j=l
.
.
# s.w.
.
+
PM qy%T
.
+
psfgb'kv
(115) (116)
(117)
=
:: .
and t
0 S Zt,
:
p
(J 13)
(118)
for a1l 1, under a probability P. We show that such Z can be very useful in generating new probabilities. Consider a set in the real line R, and define its indicator function as 1a: -4
kr,
p
.
As secn in Girsaaov Theorem, there is an lteresting way one can use martingales to generate probabilities. For example, assumc that we deEne a random process Zt that assumes tmly nonncgative values. Suppose wc select a random process Z that has the following properties: E'#(z,j l
..: .
rrhe
=
.
7 A Method for Generating Equlvalent Probabilities
IE
'rhen,
s, t
+
would be quite an inaccurate rellection of whcre St would be within an intcn'al A, because under # the Sf would grow at the rate (inacmzrate) rti rather tllan the trtte expected growth ((r; + p,)) that incomorates the risk prcmium g. Having misrepresented the possible growth in St, the mmingale probability # could certally not gencrate satisfactory forecasts. Yct, the # is useful in the process of pricing. For forecasting exercises, a dccision maker should clearly use thc real-world probability and apply thc operator F#g.1.
,
The role played by the ? in these calculations is clearly very important. Thanks to the use of the madingale probability, the pricing carl proceed without having to model the true probability distribution P of the process Stb or for that mattcr without haviog to model the rk premiam. Both of 14 which require difficult and subjective modeling decisions. 'lajs brings us to the main question that we want to discuss. Martingale probability P appeal-s to be an imporant tool to a market praditioner. ls it also as important to, say, an econometrician? In general, not at all. Suppose the econometrician's objectivc is to obtain the use of # would yield miscrable results. the best prediction of Sv. In order to see this, suppose the world at time T has M possible states. denoted by St,will then bc given by'. ubest'' forecast of
T
T
.E...
k
,1
XV V y=1
=
,
E.:
1, N is an index that represcnts trajectories of Sp where the j randomly selected from the arbitrage-free distribution #. This and similar proccdures aTe called Monte Carlo methods. ne law of largc numbers guarantee that, if the randomness ig correctlv modeled in the se/ lection of Sp, and if the number of paths N goes to infinity, the above average will converge to the true expectation. Hence, thc approximation can be made arbitrarily good.i3 . ne last step is simple. In case the discount factor Dt is known, one divides by D/ to express thc value in current dollars. If the Dt is itself random, then its random behavior needs to be taken into accotmt jointly, with the Sp within the expectation operator.
1
M
<
.N
#
:
l
ta
(1.14) . :
1
if Zt e
()
othenvise
=
..d
(119)
.
That is, 1v4is one if Zt assumes a value that falls in otherwise it is zero, We would like to investigate the meaning of the, expression: -d.,
E
E'CEZJIaI, 15Tobc more exact, here the p/ would 6e condilional ' ;. j
'E '.
(120) probabilities.
E'
.'j(
.
r'.!.
2H A P T E R
338
Pricing Derivative Products
14
*
A Method for Generating F-quivalent Probabilities
q
339
j
where represents a set of possible values that Zt can adopt. ln particular, would like to show how this expression dehnes a rlcw probability / for we thC Z t PTOCCSS, First, gomc heuristics. The cxpected value of Zt is one. By multiplying out'' the this process by tbe indicator function lx we are in fact Also recall values assumcd by Zt other than those that fall in the set that Zt cannot be negative. we must have)
1
./4
nus, in this special case, starting with the truc probability, #, and the expectation,
;.
('
EPLZ/ 1a4q,
;
(Gzeroing
:
,
we could gcnerate
..d.
E'
:J; .
'rhus
p
Second, suppose fl represents al1 possible values of this set into n mutually exclusivc sets, Wj, such that ztj +
.
.
.
+ zla
4i5
('
).i
Zt
'.
?.. ;(<
.( '
lxt + regardless of tc
.
.
+ l.u
.
nus
value assumed by Zt. E
or, after replacing, P ,1 E Ez
WEZ,1
1
=
=
(123)
'!
1(!
we can write
'
(124) /
E (.Z',1(l,
.
A'f'gz t 1
.E'#lz 1zfz + t q a: 1+
.
.
.
+ f'gz
t
1
z:a
Probzt
r'
(17.5):;. . (126) .''.
1
= 1. Thus each J,is positive and together they sum to one. If we denote these terms by
by definition,
.14.)
(E
't
(127).;
a ncw probability associated wit
f'(z,l
T.
Zt for se1
z/(z) dz
Js = Jzjf
,x;1
:
(128) '
iqccc: IL
Note that the values of #(,4f)
/3(z4i )
=
(129).
1.
.'
may be quite different from the ori
Pro b(Zt RSSOYZWU
'
ssnainiml,ml
)
e-'E.(,)1
.
..4
6
i
)
=
(130) 2'
P (A J),
Jl
se'
Of
positive rational
numbem
I
(135)
dz
(136) (137)
an expectation
of some function
Suppose also that we found a way of writing this g(x) using a Zt on Xt4 as above:
'
gj
. .r
1
dx. jnxtfxt
=
:
(
the fl will be a collntable
(134)
pAij,
Now, suppose we need to calculate .'(A',) under the probabilie #;
'
,
.
lickr''
as before.
-'
.:
.
(
is
=
..J
n
In cas tere
t '
and
#z.
zflz) dz
E '
0
=
lx/z/tzl
=
.
kzli)
(133)
/'(z) J.4j
Jjj
=
.;
tllC
=
have
and
:
.E'#IZII..4I/(ad.;),
.'e
T(z,I
j
=
we can claim to have obtained ..4f.That is, we have obtaxd:
nen,
,
EPLZtLLA,
Witll
a:0.
!'
=
1
=
ditrue''
.7
f
P
=
I
1f, in addititan, the Zt process is a martingale, then the consistency conditions for the new set of probabilities over different time periods will also be satislied. Without discussing this technical point we instead look at another way of exprcssing these concepts. Suppose the probability P has a densit.y function (z):
(122) t
nen
l
ayjd
.q
and that we split
f.
=
Jxlz
'r
(121)
0 s E (z, lx 1 .
a new probability distribution, if
'.
'
'
. .
=
Zthxt).
(138) (depending
.) (
C HA PT ER
nen
Pricing Derivative Product.s
14
.
('
note the following useful transformations:
7 A Metiitd for Generating Equivalent Probabilities @'W:-l''2'l
is one convenient candidate to the nonnegative proHence, e that discussed Zt in the previous section, lt can be used to generate we cess equivalent probabilities because it is positive and it.s expectation is cqual to
'
7
SE#(-Y,)1
(139) E?
gxtfxjdx
-
34l
onc.
In fact, givcn a set e R, we can defne a ncw probability ?Aj original probability the # by calculating the expectation: from ..4
$)
..
.::
''e hlxlfx)
=
t
( l 40)
(x) dx
zhxjf
=
dx
=
F
/
()
(#(x)).
=
.
'
Ct
.'j .:
7. 1 An Example Consider the random vaziable Zt defined by
z,
=
e
ErH':-
)
t
:
:
mz/1
.f
(1.41j
,
where Wz)is a Wiener process with respect to a probability, #, baving zero J mean and variance /. The 0 < cr is a known constant. Note that by dehrtition 0 S Zt. Now considcr its hrst moment. Taking the expectation directly: :.L
xt
.
:'
Thus, sulutituting for Sw.,in
.l.
c
'
Cfk
sgz
j
t
1
eGW)-fo/
-
=
-x
'
2t
1#)z
e-
Jj#).
: L
..
,
:
((
'
This simplihes to sgz
/
j
=
':,'' E1L. e- ) (R -qx) *
=
1 21
-X
)#JF
2
/gmaxgus
(143)
a - (p;-rwz) Jpi.
K (jj
-
'
(15j)
.
K (jj
-
(J52j
7.
ctz'lW$.-R$l-
t
)fr2(F-/)cr(T'-/)
K (jj.
-
l
(15g)
,
Or, after factoring outz
,t
(144), 7.
Lv = s'rs,frtl'l'l
-B$)-1.cr2(T-0
#
movr.
t
:
r'k. k
. di/ts But te hmction under the integral Sign is the density of a normallyWholl tributed random variable witb mean o-t and variance 1. Consequently, .: it is integrated from minus to plus infinity, we should obtain:
(15:)
)G2(T-/))
er(F-l)+dW/r-16)-J6r2(F-/)
sjijmaxg,
=
.4
,
d
jj4gl
,
(148)the option price will be given by;
t
t
.'
'
,
.
t
..1,
:! ,!;:
K ()j!
-
etrr+trlK-lfz2/l
s1)
=
c-r(T-/)s/gm%jy,e(r(T-2)+<(
=
I
.. ;: -2trr16+G
gmax t,yw
Note an interesting occurrence, A version of tle variable Zt introduced in ( 141) is imbedded in this expression. In fact, splitting the exponential term into two we can writel
(142) :
e -rlv-tlEi5
=
Now, we know that the solution of tMs SDE will givc an St such as:
,
j'
:'
(147)
.
=
k
E
J
where, according to Black-scholcs framework, the St is a geometric process obeying, under the risk-neutral probability /, the SDE; rutdt + frk/tfl'p). dst (149)
t '1
A
How could this function be used in pricing Iinartcal securities? Consider thc arbitrage-free price of a call option Ct with strike price K, written under the Black-scholes assumptions:
.
It turns out that this last integral could be easier to deal with than the original one in (139)-Wc now see an application of these concepts.
.E'rg/','FP;-ltWlll
/(.,4)
;
stazing
.=
h''*''x'
/,a-rw(H'z.-I'F;)+(:r2(7'-l)Lv (jj. ,
,rtz'-rl LK.'@
I
-
*''
$, <'
./
..'h'
(154)
Now, as before, let
'
wy
'':'
-= *111
iS nleans
1
2/
e-q (p;-,',)
z
zls,i
=
1.
'f.@
t (145). )
t
' '' vlfS
E za(zjl
=
1.
'
.L
....
'K-K'ka..rh'
= I
.'j ..
*'
=
146/..
(
.
.E'>I-,cr(p-r-'l&;)-l(r'(r-') movr.t J
;
:=:q.
ll1Zt
cGtB$ -l#)-lf'2tW-J)
=
We obtain;
.
'.x
ZT
%'*'''
.'
:
' h.
F/gz t
'r
maxgksycrlF-'l)
A'/gmaxtserlr-t) t
.rv-tb I.*'..A
-
l
'
-
/.-f,;#-r-p:)+lg2(r-z)h
(c-G(W$-W'?)+
!T2(F-'))x' >
(y-t'r(Bfr-W;)+1tF(T-I)j.&' -
y
(jj
j
(jj
v 0jj
.l'n
,
(155) (156)
E' .2
c H A PT
ER
14
*
.
'
!
>
Pricing Derivative Produet.s
10 Exercises
.
i
343
,
.;
for some proba bility# delined by: /(.d)
=
10 Exercises
i E
1, Consider a random variable correspondingprobabilities:
:'
(zj j, ,1
:
Notc that. by switching to the probability l t h e term represented by z r : and the expectation is easier to calculate. ln the has simply optionsa transformations that use this method turn s exotic of case pr icing Essentially, we j out to be convcnient ways of obtaining pricing formulas. expectations involving geometric proccsses will contain implicitly that see by Slzch Zv. lt then immediately becomes terms that can be represented tbis section. ne E Possible to change measures using the trick discussed in resulting expectations may be easier to evaluatc, ,
(ax l2lr
,
fddisappeared''
,
'
jax
,;
=
=
1, ptax
1)
=
-0.5, px ,2,
=
with the following values and the
.x
ptax
.?j,
.
-0.5)
= .2)
.2J,
=
,5/.
=
=
(a) Calculate the mean and the variance of this (b) Change the mean of this random variable to
,
,
appropriate
.
constant from Ax,
rrhat
random variable. .05
is, calculate
by subtracting an
':L m
..
8 conclusions
! '.'
.s
such that ay l'as mean Has tlie variance claanged? (c) do the same transformation using a change in probabilities, Now (d) that again the variance remains constant. so Have the values of k;r changed? (e)
,
.
As conclusions, we review some of the important steps of transforming tlwr martingale process. ) 2. into a
s
a
.05.
,
was done by switching the distribution of St fromj: - The transformation # by using a new crror term (, P to nis was accomplished '; still had thc same variance. . Thjs new error term representation (108)from (94) is that the me . What distinguisbes while the altered, presen'ing O f l is zero mcan propcrty of the err j accomplished cbanging the distributions, rather t by terms. This was ' random variable. underlying subtracting a constant from the example, the transformation was used to conr .j,4/..1 . More importantly, in this martingale. ln linancial models, one may want to appv' vert St into a tl/ the transformation to e '-ru t rather than St. e-rtst would reprcsent discounted valuc of the assct price, where th e discount is done with redefm spect to the (risk-free) rate r. The jstj function has to be l : in order to accomp lish this (.
g, Assume that the return Rt of a stock has the following log-normal distribution for fixed t:
,
'
,
'
.
logtlr)
.
x(Jt,
.x.
a. c ).
.
Suppose we let the derlsity of 1og(&) be denoted 1nyfRt) that Jt We further estimate thc variance as G2 ,17.
,
=
(a) yud
a function kRt) such that under the density, (RtjfRtj, jjas a mean equal to the risk-free rate r (,b)Find a (Rt) such that Rf has mean zero, (c) u n der wltich probability is it feasier'' to calculate
,
sjacj( ?
L . . r:
Transform jng stochastic processes into martingales through the use calculus. The so the Girsanov tbeorem is a deeper topic in stochastic method will all be at an that prov ide the technical background of this onc of thc more intui vancedlevcl. Karatzas and Shrevc (1991)provides comprehensive refereno. Shiryayev (1977)is a Liptser
discussions.
an d
Rt
.05.
=
':
9 R efetences
and hypothesize
,09.
=
(d) Is the
variance different under these probabilities?
.
3, The long rate R and the short rate r are known to have a jointly normal distribution with vaxiance-covariance matrix 5: and mean y,. moments are given by
E ...
'rhese
'
.
.
v,
Z
r
,E r,
'j .( .)E
.
' %. '.
' ':
=
.5
.1
.1
.9
.%
:
344
C H A P T E IR
*
14
E)
Pricing Derivative Products
E:
and
' .q'
M=
07 .05
. .,
,)
joint density be denoted by f (R, r), (a) Using Mathematica or Maple plot this joint density.
Let the corrcsponding
'
Equivalent Martingale 4easures
'
(b) Find a function QCR, r) such that the interest rates have zero mean under thc probability'. rldRdr. (R, rlfl, dP
'
'
't
..w
=
(c) Plot the 4.(A, r) and the new density. matrix of interest (d) Has the variance-covariance changed?
Appliccton.s
:
rate
vector
,
t :: .,: :
..
1
lntroductlon
In this chapter, wc show how the method of equivalent matingale measures can bc applied. We usc option pricing to do this. We know that there are two ways of calculating thc arbitrage-free pricc of a European call option Ct written on a stock that does not pay any dividends.
..%
J
:'jk
vh
....( j'
1. The originat Black-scholes approach, where; (1) a riskless portfolio is formcd, (2) a partial differential equation in F(St, f) is obtained, and (3) the PDE is stalved cither directly or numcrically. 2. The martingale methods, wherc one linds a probability # under which St becomes a martingale. One then calculates
E
'j
i. )
usynthetic''
' j
Ei.
.
-
c/
.j .'
.t.
=
s/r-rtr-/lgmaxtyw -
K, ()))
(1)
again, either analytically or numerically.
:1
The first major topic of this chapter is a step-by-step treatment of the mar-
tingaleapproach. we begin with the assumptions set by Black and Scholes and show how to convert the (discounted) a-sset prices into martingales. This is done by Ending an equivalent martingale measure #. nis application does not use the Girsanov theorem directly.
$.' . .1 ..'j '.?:
.
The Girsanov theorem is applied explicitly in tlte second half of the con-espondence between two approaches to asset pricIn particular, we show that converting call (discounted)
:.'
fhapter,where the ingis also discussed.
.
)
.L
J:
i
. .
. '.!
345
j'
346
!H A PT ER
.
:'
Equivalent Martingale Measures
15
.
2 A Martingale Measure
prices into martingalcs is equivalent to forcing the Ftus'j t) to satisfy a particular partial differential equation, which turns out to be the Black-scholes PDE introduced earlier. We conclude that the PDE and the martingale approacbes are closely related.
2./../ cakulation usingthe distribution
,
C
in (2),FgcFrj can be calculated explicitly. Substituting from thc definition in (4),we can mite * eF'A 1 1 jlkusj l'r (/yl ?. Sgc
: 2
A Martingale Measure
, The method of forming risk-free portfolios and using the resulting PDES ! was discussed in Chapter 12, although a step-by-step derivation of the , Black-scholes formula was not provided there. adopts matingale diferent method of equivalent ne measures way i a tedious but is points, derivation at obtaining the formula. 'I'he rests of same on straightfomard mathematics and consequently is conceptually very simple. we will provide a step-bpstep derivation of the Black-scholcs fonnula ; using this approach. , First, some intermediate results need to be discussed. These results are .( important in their own right, since they occur routinely in asset pridng
sjeyrj
c
'
nen
:
s'gdyj
X
!f.
process, j Nll't' tO/),
.
(2) :;
with Fo given. We defme St as the geometric process
X
x
i.
'
1
yypf.
oj
.j.
qy)
,
s) e - )
(.r.c-.3.12-2 +u-(.4m+!
x.
A: r)
.
g.zr'l
;
1 e 2./&22
-;
(j-, . (y,?+...z,)j2 J
dj.r;.
(j?)
=
esr-Fjtr
,
2
r z
(1c)
.
The moment-generating function is a useful tool in statistics. If its kth delivative with respect to is calculated and evaluated at 0, one linds the #th moment of te random variable in question. For exampje, the first moment of F by taking the J can be calculated derrative of with : respect to (10) =
:
:
q
#A,f J
!
FJ.
,.z
e(A#z+1.r
xgta+l(r
, :
obep
z-avzt
-cx)
xv(A)
.
called a generawea wicnerprocess, becase it a lxhasis asome-es and has a variance not neceoaruy equal to one. meau, nonzero ion, 24)may be random, as long ms it ia ndemndent of
luy.,
cy,,
-lsi+rscjzlgtasrq.js/z)
,
is an arbitrary parameter. The cxplicit form of this moment-t' irnportantlp' genera ting function is useful in asset pricing formulas. More momentz? t he types of calculations one has to go over to obtain the stochastii Operations in generat Z g fundion illustrate some standard , Calculus. The following section is useful in this respect as well.
(5)
.
But the integr:zl on the right-hand side of this cxpression is the area under the density of a normally distributed random variable. Hence, it sums to oae. We obtain
xsflis the initial point of St and is given exogenously.z We wcmid like to i obtain the moment-generating fundion of X. The moment-geoerating function denoted by M() is a specihc expecta- ; tion involving Fr, : ljer/j, oj.(A) (4) B' here
(n.s,g
:-J
-
-x
7
=
,
(6)becomes
=
sgrrrj
'
(S)
.
j
=
E:'
SLdY'
=
z
dv t. (a) -c.o g.n.a.c t ne cxptynent of the second exponential function can now be completed nto a square. The terms that do not depcnd on 1$ can be factorcd out. Dolg this, wc get
':.
''-
the equality in .
g
Now let F2 be a continuous-time
2,pv2/
ln this expression, the exponent is not a perfect square, but can be completcd into one by multiplying the right-hand side by
:.
Functm
e-
=
thc exponents:
:'
2..1 The Memenf-oeneraffng
j
..x exprcssion inside the integral can be simplified by grouping together
rne
formulas.
347
Now substitute
normal dism-bun
0 for
=
lgt +
g.
z tkje
jzj-yjga/
c
(1j)
in thig formula to get
.
j.
'. .):' .(
ou
'
:
1E. ..
J
=0
=
#t-
(12)
:'
!
C H A PTE R
348
Equivalenc Martingale Measures
)5
@
.5 Converting Asset Prices into Martingales
349
.1
For the second moment, we take the second dezivative and set
equal to
Using these, we can calculate the conditional expectation of a geomctric Brownian motion. Begin with
).
Ztz.1'O :
t?2
u
o
A=()
JA 2
(13)
(rkt.
=
'
Thesc are useful propcrties. But they az'c of secondary importance in asset formula in asset pricing pricing. The usefulness of the moment-gcnerating relatinship the cxploit Eq. tied We is to (10). FJ
.E'(c
j
tr/z/e.l fWl2
=
'
s'ae'''
=
f (E (O, x),
,
y)
2
tr f
'j
:
t,
Or, multiplying both sides by
):
E (-, Isu u
t
,
it
(15)...(, I
(21)
I
xs'al
=
F(cAF'1.
%,
cM.('-.)+l
(22)
(10)evaluated
fpt-s)
-
<
/.(1=
g) j
,Lj
at
=
1.
(23)
=
s
l''l is
.
-';z,Mt'-',)+z1t,'2(J-')
(24)
(7-5)
3 Converting Asset Prices into Martingales
!. '
.
(17)
dn.
F, +
Suppose we have as before St
.
AFI by
=
SoeI',
,
/ q (0,ca),
(26)
where 5$ is a Wicncr process whose distribution we label tnyP. Here, P probability measure that is behind the infmitesimal shocks is the affeetingthe asset price
.
''true''
''
j
,'1')
(1O
d.
=
l
r
t
-
Nmt
-
,)-
g'ztf
-
l
x)).
to calculations of the previous section, its moment-genetating given by .%()
=;
e>(l-X+
l t'f z l z(:-:)
.t.
Observed values of st will occur according to the probabilities given by P. But this does not mean that a hnancial analyst wouid lind this distribution most convenient to work with. Io fact, according to the discussion in Chapter 14, one may be able to obtain an equivalent probability # under which pricing assets becomes much easier. nis will especially be the case if we work witll probability measures that convert asset prices into
S.
S
Note that, by the dehnition of gencralized Wiener processes,
'l'
.$.1,
I
nis formula gives the conditional expedation of a geometric process. lt is routinely used in asset pricing theoly and will be utilized durirlg the following discussion.
J
(16)
).
#
Defme
'
J
$ =
e-gc-rfl
.
By dehnition, it is always true that F,
F(t, ay
=
.7
Nmt,
'w
1 .'
whcre F; again had the distribution
t
<
'leYij is the moment-generating fundion in But substituting this value of A in (10),we get
: '?
ln picing linancial derivatives using madingale methods, one cxpression that needs to be evaluated is tbe conditional expectation A'gi/ iSu, u < 6, where St is the geometric process discussed earlier. This is the second 1ntermediate result that wc need before proceeding with martingale melods. We use the same assumptions as in the previous section and assume that
st
Fle'l''
;..
'
Sv, u
Su
.
(14)
Proccsse; Qeometric
of
S --(
because X can be treated as nonrandom at time u. Recall that independent of L, u < f, This mcans that
) )
we have to take expectations as a result by itself. At several points Iater, of geometric proccsses, ne foregoing result is vely convenient, ill that it , pves an explicit formula for expectations involving gcometric processes.
2.2 Conditerutl Eoectation
E
:
.
(19) .h '.
function is
.
martingales. ln this section
! ,'
.L
(20)
?
E E: .J
measure.
,
..
'
we discuss an example of how to lind such a probability
:
350
(2 H A PTE R
Equivalent Martingale Measures
15
*
distribution of St is determined by the distribution Recall that the of F;. Hencc, the probability P s given by
:
.3 Converting Asset Prices into Martingales
E
Thc step-by-step derivation that follows will answer this question. Wc kpow that
Wtrue''
F
l
'w
(z2/)
Nmt
,
QE
'
(27)
jt) x).
t
'l'
7.
Now, assume that S, rcpresents the value of an underlying assct at time t, and let Su u < t be a price observed at an earlier date u. First of alla we krlow tat bccause the asset & is risky when discounted by the risk-free rate, it cannot be a martingale. In other words, undcr the true probability measure #, wc cannot have
wherc
'')
t
u
/7
Ij
<
e-r'gs
=
F/
Under the b
Rtrue''
,
''j
.
7
probability measure P, the discotmted process Zt, dehned
y
cOf),
j su
..
z
l
*
7
carmot be a martingale. Yet, thc ideas introduced in Chapter 14 can be used to change the drm of Zt and convert it into a martingalc. Under some conditions, we might bc able to find an equivalent probability measure #, such that the equality F.F
is satisEed.
nis
=
I u ge-rlqs-u,
Iz u,
fj
=
using
can also be expresscd E f' E zt
<
u
<
fl
=
c-ruy
tt
z
.
Egfe-rt-uts
E
:
r)
,L
(31) s
l
I
I
1= r
.'
& ,
M''
u
.-
*''*'
*'
'''
1
='
td
.-r(J-a)sap(l-u)+Jc2(r-u)
='
The parsameter p is now Ikxed in terms of the volatility J' and the riskfree interest rate r. ne important aspect of this choice for p is that the exponential on the right-hand side of (38)will equal one, since with this value of p,
,
'
t )
-r(.t - u) + p(/
'
.' ;
Substituting this in
Our problem is thc following. We need to find a probability measure 1-1 such that expectations calculated with it have the property
1
/
Ll ,
u
<
fl
=
(?a)
(39) '
#
E pLe-r'S I
.
(32)
j'
.5'
*''
'g .
ne drift in dz t will be zero as one switches the driving error term from with distribution P. the Wiener process H( to a new process 1V2 ne question is hfaw to lind such a probability measure I. We do this exp licitly in the nex't section.
3 1 Dete-ining
(g-/j
u
,
Note that because the expectation is takcn with respect to the probability #, the right-hand side of the formula depends on p instead of y.. Recall that the parameter p in (36)is arbitray. We can seled it as desired, as long as the cxpedation under > satislies the martingale condition. Define p as
.:
&:
(36)
tsing the probability given in (36).ln fact, the formktla for such a conditional expectation was derived earlier in .Eq. (25).We have
E9.
(3n)
e-rt $27
=
(35)
w'here the drift parameter p is arbitrary and is the only difference between the two measures P and P. Both probabilities have the same variance parameter. Now we can evaluate the conditional expectation < /), E #le-rt-us ,
;.
ln fact, because of the existence of a risk premium, in general, we have Epfe-rt,S t ISu u < fl > e-r'ls u (29)
N(Jtf, a2t).
-x.
Nlpt,
.:
.
,
# by
Now, deline a ncw probability .
(28)
&'
(34)
ct has the distribution denoted by P:
,
EPLe-''sl I
ljc''
=
.
351
e-ru.
:
-
(33)
/
j
x
Ll .
1
tr
r. u
.<
Transferring eru to the right,
.
That is, & becomes a martingale-3 How can we find such a >? whatis its form?
(38)): Elktt-rqt-s
,L E,
a) +
-
1
uj
(f -
fj
=
=
so .
0.
(40) (41)
'
E'
i .
L.
This is the martingale
tingaletmder /. :,j
2: )
':.
'-'ts
(c
,
condition.
Isu,
u
.<
/j
=
e-rvy
u.
It implies that e-rtst
(42) has become a mar-
7
' EE
.) C H A P T ER
:
Equivalent Martingale Measure
15
*
4 Application: The Black-scholes Formula
By determining a particular value for p, we werc able to lind a probability : distribution undc.r which expectations of asset prices had the martingale property. 'I-his distribution is normal in this particular case, and its form is given by
1
r - .j-a' 2 t
N
This probability is diffcrent from the (35). The difference is in the mean.
fz2/ ,
: : ;
L'
=
The previous section discussed how to determine pn equivalent mari j tingale measure #, when the distribution of asset prices was gov- : erncd by the probability measure #. It is instructive to compare the implied stchastic differential equations (SDE) under tlze tw'o probability measures. G<true''
:ri
dbb
'.
=
'''
, t was given by
.)
sl
=
s'acr'
I0.x),
=
p. dt +
JC,
g.
/ q
..
o-zt.
In other
''
,.
.:
...)
=
s'otrx I/sdt +
f:r
Jl4(l +
''
.f i!
. (;' '.j
1 (ksncr' 1j; tz' z dt,
(46). !
,
or, aftcr substituting St and grouping,
d.% =
Undef the
Zitrtte''
g.% + -1g'
z
'::
.k
.%
dt +
u'st
probability #j tlze asset Price Si
(47)
(CBI .
Satistvs
a SDE
.
With
(48)
.
-
.
1
1
z z jtw & + jtr St dt
(1/2)tr2cancel =
+ a''t d r,j,r /
(50)
,
out and we obtain the SDE;
rSt dt
+
U'SI
tl'l''d
(51)
.
(dsyntheticn
.'
2
l'l
This is an interesting result. Tc probability thal makes s'r a martingalc, interest rate r. switchcs the Jz';(#parametcr of the tar/ga/! vDE to the Wx&/c-hrd that contained is in general not known before St is nc p, a risk premium calculatcd. Thc r, on the other hand. is the risk-free rate and is known by assumption. Note the second difference between thc two SDES. The SDE in (51)is 1#RJ, which has the distribution #. This # drivcn by a new Wiener process has nothing to do with the adual occurrence of various states of the world. The probability mcasure # dctcrmines that. On the other hand, # is a very convenient mesure to work with, Under this measure, (discounted) asset prices are martingales, and this is a very handy property to havc in valuing derivative assets. Also, we know from hnance theory that under appropriate Onditions, probability # under which the existence of such a asset prices arc martingales is guaranteed if therc is no arbitrage.
E
(45)
((),x).
wolving
r
dst
)
'?
To get the SDE satisficd by St, we need to obtain thc cxpression for stochastic differentials dst. Bccause St is a function of F,, and becatuse we havc a SDE for the latter, 1to ys Lemma can be used!
ds :
:
(44)t
where F/ was normally distributed with mean p,t and variance words, the increments dYt have thc representation dY :
The terms
'q!
f G
'
:
o-vt
.
.
Thc lmplfed
1
z jtr Si dt +
pSt +
=
the same steps, we
Here we emphasizc, in passing, a critical step that may have gone unnoticed. By substituting r'; in place of )#;, we are implicitly switching the tmdcrlying probability mcasures from P to nis is the case because only undcr # will the error term in Eq. (48)be a standard Wiener process. lf we continue to use #, the crror terms f/'Prwill have a nonzero drift. In Eq. (48),p can now be replaced by its value 1 z (49) p r - -6r 2 Substituting this in (48),
;
SDES
r1)in (47).By following
and H( with dSt
:;
probability measure P given in
Jz with p
'
(4gj
.
Rtrue''
l'el'llace obtain
5''
:E '
1, a drift coeflicient (#,+ (1/2)tr2)q%, 2. a difhlsion coeflicient 3. Jmd a driving Wiener process 11$.
4 Application: The Black-scholes Formula
.f
.
$. '
ut,
The Black-scholes formula gives the price of a call option, Fvb'f the following conditions apply:
1: ;,
'nw SDE under thc martingale measure # is calculated in a simill. SiXrI fashion, but the drift coemcient ig ntw different. To get this SDE.we
, '
; E< :.
t. The risk-frce interest rate ig constant over the optionsg lifc.
*' :
,
/), when
...
;
l
.!
ClHA P T ER
354
*
)
Equivalent Martingale Measures
15
j
.
4
ApP
lication: Ihe Black-sclacles Fonuula
1,I .
I
L
2. ne underlying security pays no dl'vidends before the option matuas. 3. The call option is of the European type, and thus cannot be exercised ! before the expiration date. ' of the underlying security is a geometric Brownian mo4. The pzice tion with drift and diffalsion terms proportional to St. 5. Finally, there are no transadion costs, and assets are infinitely divisi- ' 'Jf ble.
property for e-rc
and the martingale
.
L
st
Ct
.
,
'
0
=
-rF + F3 + rkl
1tEraStzFss,
'
0
+ -
the bouadary condition is Flsw, z') where b; 'rheresulting formula is given Ke-rLI'-tlsdk F(-,.
f)
=
stNdqj
-
'
'
0
St,
::
r-v.-
max
-
-
-
-
a.
:s
t
:s
T)
(52)
.
*
;
-
,
v
-
/),
+
.g; . $? ',
-
K, (j)j.
-
maxt-sz
Ee
E
=
using the probability measure #. ne probability / is the equivalent jyj tjw pxvjous section,
.
K,
-
0J1,
(59)
measure and was derivcd
martingale
'
1
d?
zgrtwz F
'
sv
.
'
jj
N(J1
)
=
1
l
e-
l J.r. X
2
e
-.w
'
(60)
with
.'
'
jr,alrlzdY.s,
-plmtyy-tr-
=
.
utjg'r =
(61)
.
Using this density we can directly evaluate the expression
(55) k''
'
-c.o 2x underlying Let St be an asset, and Ct the price of a European call op- f. titm written on this assct. Assume the standard Black-scholes framewor k, : with no dividends, a constant risk-free rate, and no transaction costs. 0< objective in this section is to derive the Black-choies formula directly by j: using the equivalent martingale measure #. The basic relation is the martingale property that the e'-rtc t must satisfy the probabilfe lh
o
=
E
which can be written as
Co
>
-rr
gg
x
e
=
-rv
-.
maxt
sz .K
mauxls w .K
(jyg
(taj
t)jvjp,
(()g;
,
,
,
''
where we also have
'
..
Sv
't
under
, (e
-r(r-;)c
Ct
=
Et
r. ),
(s:)
these substituting
?,
=
'
-
K , 01
-x
y.r.
(64) e-iz!-ftrr-tr-ltr
2,77-G27*
)' ) dv z.
(65)
'Ib eliminate the max function from inside the integrai, we change tl'le limits of intcgration. we note that, after taking logarithms, the condition XCY-T
'
=
...
:
he
(63),
c-rl' maxl-ei'r
C'u
of thc call option. llerer > t is the expiration date option's payoff will be Sv -K if Sp > K. expiration, tlle that at we know tl,e call option expires with zero value. This permits one to write otherwise' theboundary condition C T maxlsz - K. 01. (5X W
in
=
t K
j
(5a)
.rz
Ctl
r(F - /) + fO(T tj dk (54) E' T r tz : Jn these expressions, T is the expiration date of the call option, r ks the g. risk-free interest rate, K is the strike pricc, and f;r is the volatility. ne f'unction N(x) is the probability that a standard normal rarldom variable (, :k N(Jll by given example, is less than x. For ln(u%/r)
=
maxtyz
=
,.
ith
-r(z-j)
(d
Let t 0 and calculate the option price as of time zero. Accordingly, the current information set It becomes &. nis way, instead of using conditional expectations, we can use the unconditional cxpectation operator E p (.1.
'
(53)
I
j!
implies
We now proceed with the step-by-step dcrivation of the Black-scholes formula by directly evaluating
F'
K, 01,
>
Et
=
1.
Irl order to dcrivc thc Black-scholes formula, this expectation will be calculated explicitly, nc derivation is straightfomard, yet involves lengthy cxpressitms. lt is bcst to simplify the notation. We make the following simplilicat j ons;
'
Under these conditions, the Black-scholes formula can bc obtained by solvJ ing the following PDE analytically:
W
355
i11 I ! j g I
(66)
j
'
Ej : ! '
11
!
I
I
p
1 '
1
J.
'
.
:
356
C H A P T ER
Equivalent Martingalc Measure.s
15
.
'u
is equivalent to
.''
-
Using this in
K
=
-
mcjry
-
'
') y,
(6g)
,
-(r-l/)r)2
Using the transformations in
:
x
Ke-rT
L
(69)
separately.
g
!
First we apply a transformation tlzat simpliEes the notation further. We define a new variable Z by z
=
:'
E
'X
1
) 2 .w1:4.%
2
'n'r
tN:p
=Q'
r
Ke-rT 1n(
)-(r-
3-
e-
-rcz . /
J
Jr
t,
( y'z -(r-)cr2)z)2
1 c-lz 277-
2
dz.
lnlv/ulj
) lntS /X') =
,
(
-
a,v
e
-rr <
1
e%
()
2=f.ra F
(r-,7lw
tr
(r
.-r'r,y
e-
a,)wt j,
*
(:
()
o.zx?''.r
w .(r.js)y.)a
1
(, -
2,27-
-dz
dz
'
zzzv-rznvfr-qwzlvs k
t)
E
5
a..v r
jz2 dz
side, using properties of the
1
g-jlzaxugzxfo
Jz
2=
-co
(g9)
-
tr2r
(80)
'
..., .
:n
m
cv-gy
gfj- z 0
gr..j
.
-x
Jg g--tz-ytsTj
.
s
H
E
=
Z +
:
;$
jjy .
2.
'Fhe terms in front of the integral cancel out except for Finally, we make the substitution
-
'nfiriy
.
lz
jszj'j-
,,
l
'
z
,.,u- Sf-es-
?;
(72)
('a)
.
Next, we complete the square in the exponent by adding and subtracting
'''
(73)
(76)
.
'. .
.%
ln(J;) (r JWIF twv'r
c-l z j dZ
Ke-rvgvja
we transform the intep-al on the right-hand Fmrmal densi'ty:
j,
4To see why llle lirnis of tlae integration change, note tlaat when l'w goes from 1n(,-) to tse transformed z deued by (70)w4ll be between
x,
)
ccn
'
:
i
I
(75) j
..x
.
d y.v
da
Kc-rT
=
';1j
But the towerlimit of the integral is closely related to the parameter #.2 in the Black-scholes formula. 4 Letting -
o
J
,y1.)
r,2
lnl.,f
,
rlnhisrequires an adjustment of the lower integration limit and the second integral on the right-hand side of (69)becomes Ke - r z
x
t
(70) )
.
j
(,p) z). Hence, we derived thc second part of the Black-scholes formula, as well as the value of the parameter #z. We are left to derive the ftrst part, SoNld, ), and show the connection between #1 and #2. nis rcquires manipulating the first integral on the right-hand side of (69).As a hrst step, we again usc the variable z delined in (70)..
;'
(r.- 2 O'.V-T
Jz
=
i
side of this ex-
1v2)w
c-lZ
aw.
.+
4.1 Clrulniimk
yvT -
.w
KV'/
jyj.j
%. (82)
t .
'
' (
t .
.
'
j
(74)and (75),we write
,i
.
We can now evaluate the two integrals on the right-hand
(74)
fLxjdx.
=
.k
, ,
-#a.
=
-L
fLx)dx
:
.5
)
2 tr
x
J
t.r
=
pression
'
i
Wc recall that the normal distribution has various symmety properties. One of these states that with flxj standard normal density, we can write
:
1
(r -
ty J-T
'
-rF(x$'
ln( K ) +
.
-(r-) e-zvn ijy ey! K) () e z. -&) 2*c.27ln(.% The integral can be split into two pieces: C* 1 z T z r :1 y' f ( l'rr - (r - J ) ) yw Cp V e - --i.c e- e r a r ln(.N) 2,/&277 & x ( -z7r(F? Ke -rr d.v w e l=r1T 1n()-) a
C':
357
j
(67) .
.
(65),
oa
4 Applicarion: The Black-scholes Formula we obtain the dz paramctcr of the Black-scholcs formula:
y:z zzln S p
and
3
!
'
:
E
'
(. :
. :..
358
C H A P T ER
.
15
lr
Equivalent Martingale Measures
Comparing Martingale
'y
to obtain
=
utl
2*
-x
'.
c
-H2dH
=
st)x(#:),
(g3)
'
z
where #c +
=
G
V'-/
(84)
.
i '(
This gives the first part t:f the Black-scholes fonnula and completes the derivation, We emphasize that during this derivation, no PDE was solved.
i :,, .j. .' . ..':13'
We have seen two contrasting approaches that can be used to calculate the fair market value of a derivative asset price. The lirst approach obtained the price of the derivative instrument by forming risk-free portfolios. Infinitesimal adjustments in portfolio weights and changes in the opticm price were used to replicate unexpeded movements in the underlying asset, St. This elminated all the risk from the portfolio, at the same time imposing restzictions on the way Fvb , f ), St and the risk-free asset could jointly move ovcr timc, nc assumption that wc could makc innitcsimal changes in positions played an important role here and showed the advantage of contineous-time asset pricing models. ne second method for pricing a derivative asset rested on the claim that we ceuld find a probability measure # such that under this probability, e r/F(.%, /) becomes a martingale. This means that
c-rFst,
f)
E
=
-rT
ge
Ftzz. F)
or, hcuristically, that the drift of thc stochstic dLe-rtFvh t ).1, ,
I.61,
We show the correspondence between the PDE and martingale approaches in two stages. The il'st stage uses the symbolic form of Ito's Lemma. lt is concise and intuitive, yet many important mathematical questions are not cxplicitly dealt with. rI'hc emphasis is put on the application of the Girsanov theorem. ln the second stage, the integral form of lto's Lemma is used. In the following, Ito's Lemma will be applicd to processes of the form e-rrF(.;. /). nis requires that F(.) be twice diffcrcntiable with respect to J,, and once differentiablc with respect to /. These assumptions will not be repeated in the following discussion.
.
5 Comparing Martlngale and PDE Approaches
( '
359
. application of differential and integral forms of Ito's Lemma, * the martingalc property of 1to integrals, . an important use t)f the Girsanov theorem.
.
1.
dk
PDE Approaches
'T'hcdiscussion is a good oppotunity to apply some of the more advanced mathematical tools introduced thus far. ln particular, the discussion will be another example of the following:
(
dz-'vs.'''lz 1
ankl
)
E r .?
.i,
5.1 Eqzatltzlence of the TuhoAp/votzches
i?
)
ln order to show how the two approachcs are related, we procecd in steps. ln the hrst step, we show htaw e-rts l can be converted into a martingale by switching thc driving Wiener process, and the associated probability measure. In the second step, we do the same for thc dcrivative asset e-rilj.s f), These conversions are done by a direct application of the Girsanov theorem, (The switching of probabilities from # to / during the derivation of the Black-scholes formula did not use the Girsanov theorem explicitly.)
iE .'
,
:
13
'
.'
(85)
E
differential
5. C'fprlvcr/jn,e-rf into a Martingale We begin with the basic model that determines the dynamics of the tmderlying asset price Suppose the undcrlying asset price follows the stochastic differential equation .1
,:
(86)
.'.'.
,i.
WaS Zero.
:
Black-scholes formula can bc obtained from either approach. One either solve the fundamcntal PDE of Black and Schoks, or, as we did could ear lier one could calculate the expectation Epfe-rrFsv, w)I1tj explicitly using thc cquivalcnt measure #. In their original aticle, Black and ScbolO chls the first path, The previous section derived the same formula using thC martingale approach. involved somewhat tedious manipulations, but waS straigbtfo-ard in terms of mathematical opcrations concerned. Obviously, these two mctho d s should be related in some way. ln this SCC tion between the two approachcs. we show the correspondence ne
'
dSt
',i
=
p,(,,)
dt + fr(.T;)#J#),
f G
,'
,
l .'
LL
'
: E'
y
d.%
.
.!
.
. '
'
'
.
.(
(87)
the drift and the difhlsion terms only depend on the obsen'cd underlying asset price St. lt is assumed that thcsc coefficients satisfy the usual regularity conditions. )P) is the usual Wiener process wit.h probability measure #. Wc simplify tls SDE to keep tNe notation clear. We write it as
E
'rhis
g0.x),
Where
.
=
p,t dt + uj #r1,: .
(88)
r .? .;
(J H A P T E R
360
E
Equivalent Marringale Measures
15
*
:
ln the first sectitm of this chapter, e-ru .1was convertcd into a martingale by directly finding a probability measure #. Next, we do the same using the Girsanov theorem, Wc can calculate thc SDE followed by e-'tSt, the pricc discounted by rrk thc risk-free rate. Applying Ito's Lemma to e- fr we obtain fte-rzj + c-rr ds /*
dLe-rtsJ st =
I
'.
;
1
-rlfr t dw./ (x) - rx l j dt + e will -nS and will have be drift, not In general, this equation e a zero not a t martingale, /
E/z,, ruvl
>
-
sinceSt is a
dXt
(89) . :;
e-rtf;
.
?
:1J :)
But, we can use the Girsanov theorem to convert e-rS t into a martingale. . Wc go over various stcps in detail, because this is a fundamcntal application E' of the Girsanov theorem in fmance, 2 'T'he Girsanov theorem says that we can find an f/-adapted process Xt h and a new Wicner process ( such that .i tvf dXf + #F). (92) ) $
=
=
where the f, is defined
)
F.
(93) 7
,
'
as
:
.
(-,
e
=
jt()
.:t-
.
Jyp
-
;
ay'j
du
(q4)
.
Jgc-rl,rl
=
. i y''
r
:
k$t 1
=
e-rlgpwl
-
rsj dt
5Here r5'; dt is an incrcmental eaming if
ut
-
e-rtn
tfa't-, + e-rtc
g)'fzt l
.
dollars were kept in the riskdree
aruset, a nd
#@c-r;1F+ e-rl #E
=
(99)
=
e-rlf-rFdtj + F ds i -h .<
1 z dt j F xx o
.
(100)
.
Choiccs.
.
L'.
/';
dLe-r's1 l
''
=
.
Or we can use tlze original SDF, in :
d.S,
system arc well-behaved. (
into a martingale.
'Fhe important question now is what to substitute for dSt We have two and #, e-Nst Under is a mamtingale. We can usc
i
(96)
I)j
+ e-rt F t dt
1
.
#(e-
dLe-rtl-st,I)j
j
Group ing t he te r'rrl S
(98)
.
,
,:
(95) li
dA-rl. rks'rj dt -I- e-r/tz'zlll1'f e-rkg.t -
rthdh
Note that on the right-hand side we abbreviated F(St /) as F. Substittzting for dF using lto's Lemma :ves the SDE that governs the differential dbe-rlyqst, r)j:
S
-
r/
de-rtFLst,f))
:
We alstlmc that t h e process X / Satisses the rcmaining integrability conditions Of the Oil-sanov theorcm, 6 The important equation for our purposcs is the one io (92).We use this to eliminate t'he J'M''',in (90).Remiting, after su bstitution of d'. t
''
e
=
,
.
is given by
Gt
5. l.2 Converting e' rt Ffst 1) into a Martingale The dcrivation of the previous section gave the precise form of the process Xt needcd to apply the Girsanov theorem to derivative assets, To price a derivative asset, we need to show that e-rtF(5'/, J) has the martingale property under A ln this section, the Girsanov thcorem will be used to do this, We go tlrough similar steps. First we use the differential form of lto's Lemma to obtain a stochastic differential equation for e-rtFut, tj, and thcn apply the Girsanov transformation to the driving Wiener process. Taking dcrivatives in a straightforward manncr, we obtain
;
'j
(97)
-
We use these in converting Le-rtFvt,
....
Thc probability measurc associated with dr #,J/,
if wc define this SDE under the new Wiener process. l1 addition, # will the drift term to zero. nis can be dXt as
/*/ - rxsl dt.
=
de-rtvhj
'
risky asset-s
361
We assume that the integrability conditions required by the Girsanov theorcm are satised by tls dXt equaling tlle term in the brackets. This concludes the first step of our derivation, We now havc a martingale measure #, a new Wiener process J1'), and the corresponding drift adjustment Xt such that e-rtst is a mftrtingale and obcys the SDE
)
(91)
0,
PDE Approaches
:
. Substimting for dvh and grouping slmliar terms, =
and
According to the Girsanov theorem, will be a standard probability #, 11/-2 be a martingale mtzurfrc if we equate accomplished by picking the value of
j y
.
dLe-rts/ J
5 Comparing Martingalc
'
.
=
e
'rlo
#72.
(101) .
(87)..
v,tdt +
trf
JH,:.
(102)
C H A P T ER
362
Equivalent Marringale Measures
1$
*
Wc choose the second step to illustrate once again at what point the Girsanov theorem is exploited. Eliminating the dSt from (100)using (102),
dLe-rtljst,f)1 =
Jl1 + e-rt Ft dt + Fgp,,
tff
+ h #W(l +
1
.j
:2
Fsarh dt
,
(103)
Rearranging, I)j
dke-rsyjst,
and
PDE Approaches
363
But, in order for e-nFit, t) to be a martingale under the pair the drif't term of this SDE must be zeo,; This is the desired result: 1 2 FI + Fss.rJ-/ + Fsrst 0. - rF + jy
Now we apply the Girsanov theorem for a second time. We again consider the Wicner process JP/,defincd by: + dXt /JP, JT,PI
(105)
=
andtransfozm
(104)using
the SDE in
d (e '-rtFs
t>
f))
the Girsanov transformation;
(7.2 e-rl -rF + F t + F gt + S.F 2 xx I dt
=
.j.
-
e -r/(Tkw.dx
e-rty / y.s #s
-j.
t
t
(106)
,
#vy Again, note tlze critical argument here. We know that the error term that drives Eq. (106)is a standard Wiener process only under tbe probability probability. measure#. Hence, # becomes the relevant in Eq. (97): derived already been value of dXt has ne dXt
We substitute this in
de-rtyst,
=
#'l -
rSI
dt
M
(107)
.
-'t
-rF
+ F, c
1 + F ! + Fsla.t + -Fsxt7l 2
-rl (J'J
Glf.
v.t - -
r-h
rt
dt
(108)
#Hz-/
nere were some critical steps in this derivation that are worth further discussion. First note the way the Girsanov theorem was used, We are given a Wiener process-driven SDE for thc price of a Enancial asset discounted by the risk-free ratc. lnitially, the proccss is net a martiltgale. The objctive is to convert it into onc. To do this we use the Girsanov theorem and find a new Wiener process and a ncw probability # such that the discounted asset price becomes a martingale. The probability measure / is called an equivalent martingale measure. This operation gives the drift adjustmcnt term Xt required by the Girsanov theorem. In the prececding derivation this was used twice, in (95) and in (106): This brings us to the second critical point of the derivation. We go back to Eq. (106): dfe-rtFst,
r)j
=
1 a dt -rF + Ft + Fsv,t + jjFsso.t
e-rt
-
e-ra
l F dx t + e-rtc d F u'
(112)
(ff,lfr y
l
-
the valuc of dXt means adding dXt
=
#,
-
-
rSt
dt
(113)
to the drift term. Note the subtle role played by this transformation, dXt is defined such that the term Fsg.t dt in Eq. (104)will be eliminated and will be replaccd by Fxr dt.
.
rrhe
,
f))
e-rt -rF
(111)
*'
Sim pltfy'in g
dbe-rtljst,
a'tlh t/11:,
5.2 Cdfcul stepsof the Deritufion
Here, substituting
r)1
(110)
The drift parameter is zero.
(106): J,
=
,
(104)
/Jr/,#,
This expression is identical to the mdnmental PDE of Black and Scholes. With this choice of Jj, the derivativc price discounted at the risk-free rate obeys the SDE dle-rkmvh t )1 e-rl
1 e-rt -rF + F/ + Fsgt + jFxxc c dt
=
+ c-rte l F.5' JH(.
=
Comparing Marringale
=
e-rtf-rF
=e
7
1 + Ft + ,-lsc't z + Fsrst 2
dt
+ c
-rt
rt y.,
#jj.zt,
(109)
?We know that if there are n.o arbittuge possibilities, the same f wll conve.rt all asset prices into martingales.
(
364
(2 H A P T E R
.
15
Equivalcnt Martingale Measures
.
6 Conclusions
$ ''
In other wor ds, th e app lication of the Girsanov theorem amounts to transformingthe #n/ term p,, into rst, the risk-free rate. Often, books on ..!, derivatives do this mechanically, by replacing all drift parameters with the risk-free rate. The Girsanov theorem is provided as the basis for such transformations, Here, wc see this exp jujjjy. Finally, a third point. How do we know that the pair ll; # that converts .' e-rtst into a martingalc will also convert e-rtkmvt, t4 into a martingale? t. This question is impotant, because a function of a martingalc nced not . itself J be a martingalc. 7 This step is related to equilibrium and arbitrage valuation of hnancial assets. It is in the domain of dynamic asset pricing theory. We brie:y mention a rationale. As was discussed heuristically in Chapter 2, under proper relations among asset prices will yield a unique man EjE conditions, lkrbitrage tingale measure that will convert a1l asset prices, discounted by the risk-free rate, into martingales. Hence, the use o f the same pair J#;, # in Girsanov transformations is a E' consequenceo f asset pricing theozy If arbitrage oppotuoities existed, we ?, couldnot have done this.
We assume
fz'/
is such that ,
# s jtyjq(y
Thjs is thc Noviktw condition
k;
t
.
T
tl
g . . ctt fpu
)2dg j
xj
.cj
(yj5j
.
thc Girsanov theorcm and implies that the
tf
jntcgral
,
,
365
!
e-rucr Ll F$
t/r
(j16)
N
f
js a martingale
'
:
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under
#.
But the derivative asset pric,e discounted by e-rt is also a martingale. makes the lirst integral on the right-hand side of (114), l
,?
:.
(,
e
-,u
-r y, +
s t .j.
.1
2
y ss
g.c u
oy
.y
ss u
su,
( j j-y;
.
Jt (trivial) martingale as well. But this is an integral taken with respect to time, and martingales are not supposed to have nonzero drift coemcients. Thus, the integral must cqual zero, This gives the pmial differcntial equa-
,
'
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5 3 Integml Fonn *'
.
tion
,
x
The relationship betwcen the PDE and martingale approachcs was ds- l cussedusing the symbolic form of Ito 's Lemma witich deals with stochatic @ ,
-
1 z + Fsrk Ft + -F..tn 2
=
0
t
2E
0,
st
:
0.
(118)
This is again tbe fundamental PDE of Black and Scholes.
,
differentials.
rf +
$t. .
As emphasized several times earlier, the stochastic differentials under gtand for intcgral equations in thc S: cons ideration are symbolic terms, which :. background. ne basic concept behind all SDES is the Ito integral. We :,1 usedstochastic differentials because thcy are convenient. and because the already involved tedious cquations. ) calculations !h' using lto's Lemma. analysis the integaral form of bc done same can ne withoutgoing over a11thc details, we repeat the basic stcps. representcd .l The value of a call option discounted by the risk-free rate is ) usual by e '-rws t, f), Applying the integral form of lto's Lemma, .
as
6 Conclusions Tllis chapter dealt with applications of the Girsanov theorem. we discussed Several important technical points. In terms of broad conclusions, we retain
the jollowug.
nere is a certain equivalence betwcen the martingale approach to pricing derivative assets and the PDES. one that uses In the martingale approach, we work with conditional expectations taken wit.h respect to an equivalent martingale measure that converts a1l assets discounted by the rigk-free rate into martingales. 'Thcse expectations are very easy to conceptualize once the deep idcas involving the Girsanov theorem are understood. Also, in the case where the derivative asset is of the European type. thcse expectations provide an easy way of numecally Obtaijling
;;
e-F(Sf, =
tj
.,
t
F(S t)r 0) + t +
d-F&
-rF + F/ +
0
e-ru (p y l
1
i.'
+ Fsrsu g-fxxcJ
du
(j14) ,t,
N
j
t
,z
N
.
t)
2
11Z2
Note that we uSe interest
the risk-free
arbitrage-free
.'.
in Placc of
F), and consequcntly
x replace
,,
p.t by r : 4
'
rate.
.
;
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agset prices.
It was shown that the martingale approach implies the PDES utisame lized diference t)y the PDE methodolor. is tat, in the martingale ne
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.
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C FI A P T E R
366
.
Equivalent Martingale Measures
15
8 References
$
:
(d) Now
comcs the point where you use the Girsanov theorem. How can you exploit the Girsanov theorem and evaluate the expectation in the above formula easily?
approach, the PDE is a consequence of risk-neutral asset pricing, wheres in the PDE method, tane bcgins with the PDES to obtain risk-free prices.
(e) Write the fmal formula for the chooser option.
i
7 References The section where we obtain the Black-scholes formula follows the treatCox and Huang (1989)is arl cxcellent summary of ment tf Ross (1993). reselts. martingale The same is true, of course, of thc treatment the main of Dufhe (1996).
2, In this exercise we work with the Black-scholes setting applied to foreir currency denominated assets. We will see a different use of Girsanov theorem. (For more details see Musiela and Rutkowski (1997).1 Let r, f denote the domestic and the foreign risk-frec rates. Let St be the exchange rate, that is, the price of 1 unit of foreign currency in terms of domestic currency. Assume a geometric process for the dynamics of St:
E .
r ,
;
8 Exercises
e '-rl-
-/).7
) ; J:
--(r.-,).s(max(x. est, the chooser option is worth:
an d thus at time ,
;!
,,.,())!,,!,
-
HS
t
7
f)
max (C(k%,f).
=
/)j
Pbht
C(f
,5'
11
l
)
-
Pt,
&)
=
i
S:
Jz,
(. A :
=
Does tliis remind you of a wcll-known parity condition? ; (b) Next, show that thc value of the chooser option at time t is given by Hft, &,) max stl, clt, + e-rtr-rlm ) (c) Consequently, show tltat the option price at time zero will be ,
;
:.
-
(c(r,
,stj
'
-s)
-
.
given by 'prretrpi-trz'
z;(0,
.s)
=
where S is the
c(0,5-)+ e-r''E underlying
gmax gr time -
price obgerved at
*
zero.
0jj
,
.
r
.
2
,
=e
rr;,p
-
zlm2 r
J'FW'T-
=
l CPT6IP
=
zt Lf
-
rzjdt r +
-
tzJp;j
-
rt,
,
where Zt 1/u%. (e) Under which probability is the process Zten/eft a martingale? (9 Can we say that # is the arbitrage-free measure of the foreir economy?
.
:
z
a What does Girsanov theorem imply about tlle process, W; under #? (d) Show using Ito formula that
t
.
c-'tF-''A'
-
/(W)
.r
show tbat:
(a) Using thcse,
.
$
f,
1G)l+g.p;
a martingale under measttre #? (c) Let / be the probability
.t
.:(. ,,'
Soe (r-y-
S eft t S er
,.
E5'l'
,
=
where H( is a Wiener process under probability P. (b) Is the process
'':
-
c
-
St
.; i
Imaxt-syK, ()) l1(.j
o'stdh.
-
.2k(.
whereas the value ol. the put is: ?)
(r fjstdt +
=
(a) Show that
3
rrhe
=
dSt
;.
1. ln this excrcise we use the Girsanov theorem to price tlw chooser chooser option is an cxotic option tlat gives the holder the option. right to choosc, at some fumre date, betwecn a call and a put written on the same undcrlying assct. Let the T be the cxpiration date, St bc the stock price, K the stnke price. If wc buy tlie chooser option at time 1, we can chofasc between call value of thc call is or put with strike K, written on St. At time f the
clst, z)
367
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zMerton (1973)was an early attempt to introduce sthastic interest rates. Yet, this was in a world where the underlng asset was agairl a stock. Such a complication can, by and ! vge, still be haadkd by using classical tools. New tools start being molc prwcticat wtien tiie derivative is interest a'ezjy'e, in the sense that the payoff depends on tlle value azld/or path tjy iuterest rates. oljowed
.
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In this cllapter we brietly outline the basic idtas behind the new tools. ne j sslles discussed in the following chapters are somewhat more advanced,
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' bg l'Robustness Of Black-scholes assumptions is ontt reastm wby te frmula contnues to i fcGsionals. For e=mple, one still obta kks reasouably accurate 2 vely popular with market pro sive ; prices w hen volatility is stochastico tr when intercs t rates move randomiy. A comprcben . a1. Karoui E1 is t'f formula ; e.: (1998) Black-scholes xmrce on this aspect
t ' 1
of American-style derivative securitics were
However, especially for interest rate derivatives, such an assumption cannot be maintained. It is precisely the risk associated with the interest ratc movements that makes these derivatives so popular. lntroduciag unpreoctabjewjenercomponents into risk-free interest rate models leads to some furoer complications in terms of mathematical tools. srjajly,sotice that Blackucholes assumptions can be maintained as long as derivativcs are short-dated, whereas the considcration of longer dated instruments may, by itself, be suficient relaxing for assumptions a reason on constant interest rates and vtlatility. second part of the book discusses new tools required by such modificatioas and introduces the important new results applicable to term stnzc-
ne Erst part of this book dealt wit h an introduction to quantitative tools '( that are useful for Classical Black-scholes J#rm(JcJl, where underlying sccurity St was a nondividend-pay ing stock, the risk-frce interest rate r anux the underlying volatility r were con: tanq the option was European, and wherc t here were no transactions costs or indivisibilities. The types of derivative securities traded in hnancial markets are mucb i more complicated than such '(p la in vanilia'' call or put options that may s.t this simplifed framcwork reasona bl y wdl. In fact, some of the assumptions 4 :,y; j. used by Black-scholes, a lthough often quite robust, may fall signmcantaz E ities.l Ncw assumptions irltro- ;$ short in the case of interest-sensitive secur duced in t hejr place require morc complicated tools. ( These new instruments may be similar i.n some ways to the plain-vanill? derivatives already dismzssed. Yet, there are some non trivial complkationsMore importantly, some new resu its have rccently been obta ined in dealing with interest-sensitive instruments an d terrn strttcture of interst rates. These powerful results require a different set o f quantitative tools in timir . .
$
First, a majtrity of flnarlcial derivatives are American style, containing early exercise clauses. A purchaser of snancialderivatives often does not have to wa it until the expiration date to exercise options that he or she has purchased. This complicates dcrivative asset pricing signicantly. New mathematicaltools need to be introduced. Second, it ig obvious that risk- fr ee nterest rates are not constant. They arc subjec-t to unpredictablc, infinitesimal shocks just like any other price. For somc financial derivatives, such as options on stocks, the assumption of constant risk-frec rate may be incorrect, but still is a reasonable approxmation.
,
1 lntroduction
are serious
derivatives.z
'
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!.
2. The risk-frce interest rate r was always kept constant.
t
q:
369
Recall tllat the examples discussed in previous chapters were by and large in linc with the basic Black-scholes assumptions. In particular, two aspccts of Black-scholes framework were always presen'cd.
=
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370
C i l A P T ER
.
16
Results
and
.
Tools for Interest-sensitive Securities
3 lnterest R.ate Derivatives
i:. .'J(
but they all have practical implications in terms of pricing highly liquid dcrivative structures. Chapter 17 wiil reintroduce the simplc two-state framcwork that moti- $ 'E vate d th e first part of this book. But in thc new version of models used ia Chaptcr 2, we will complicate the simple sct-up by allowing for stochastic r way, we short rates and by considering interest-sensitive instruments. such as normalization and tools such motivate important concepts as can the forwardmeasure. ') The major topic of Chapter 18 is the foundations for modcling tlle term ' strudure of interest rates. ne deEnitions of a forward rate, spot rate, and term structure are given here formally. More important, Chapter 18 1- t.' troduces the two broad approaches to modeling term structure of interegt (' rates, namely, the classical and the Heath-larrow-Morton tmproach. Lennw: ing the differences between the assumptions, the basic philosophies, and the t practical implementations that one can adtpt in each case, is an important the valtlation of interest-sensitive instruments. step for undentandjng Chapter 19 discusses classical PDE analysis for interest-sensitive securities. This approach can be regardcd as an attempt to follow steps similar PDES satised to those used with Black-scholes PDE, and then obtaining :J by default-free zero-coupon bond prices and derivatives written on them. The main difhculty is to :nd ways of adjusting tbe drift of the shorbrate process. Short-rate is not an anet, so tMs dzift cannot be zeplaced with the ' risk-free spot rate, r, as in the case of Black-scholes. A more complicated operation is needed, Tbis leads to the introduction of the notion of a market price of interes't rate risk. The corresponding PDES will now incorporate E this additional (unobsend) variable. i Chapter 20 is a discussion of the so-called classical PDE approach to uollmed in. fzxed income. Chapter 21 deals with the recent tools that are '.J. pricing, hedging, and arbitraging interest rate sensitive securities, srst .it topic here consists of the ftmdamental relationship that exists between a. '.''' Qlass o condititmal expectations of stochastic proccsseg and some partial differential equations. Once this correspondencc is established, fmancinl market participants gain a very important tool with pracical implications. i T his tcol is rclated tfl the Feynman-lac formula and jt ig dealt with in this chapter. Using this one can work either with condi- f'E: tional expectations taken with respect to martingale measures, or wif.h t.11:* corresgonding PDES. The analyst could take the direction which pro
process, a property wltich complicates the utilization of Feynman-Knc typc correspondences. Finally, Chapter 22 discusses stopping times, which are essential in dealing with American stjrle derivativcs. This concept is introduced along with a certain algorithm called dynamic programming that is very important in its own right. In this chapter we also show thc correspondencc between using binomial trees for American-style securitics and stopping times. We sec that the pricing is based on applications of dynamic programming. Stopping times are random variables whose outcomes are some particu1ar points in time where a certain process is being For example, option American-stylc exercised call expiration before be the date. an can Initially, such exccution times are unknown. Hencc, the execution date of an option can bc rcgarded as a random variable, Stopping times provide thc mathematical tools to incorporate in pricing the effects of such random variables. These mathematical tools are patticularly usefnl in case of interest sensitive derivatives. Hence, before we procecd with the discussion of the tools, we nced to discuss bclly some of thesc instruments. nis is done in thc following section.
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371
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3
lnterest Rate Derivatives
One of the most important classcs of derivative instruments that violate the assltmptions of Black-scholes cnvironrnent are dcrivatives written on interest-sensitive securities. some wcll-known intcrest rate derivatives are the following/
:'
.':
,
..
'
* lnterest rate futures and fo-ards. Lct Lti represent the annualized simple interest rate on a loan that begins at time q and cnds at time /j.-:. Suppose there are no bid-ask spreads or default risks involved. Then, at time t, wherc f < fj < /F.j.1 we can write futures and forward contracts on these ''Libor rates,'' L lj 4 For example, forward loans for the period gf/,r;.+jlcan be contacted at time f with an interest rate Ft The buyer of the forward will reccive, as
.
'rhe
',
t
itcorrespondence,''
.
Simpler
Cheaper) (()r Of thC Other
numerical
. .:
,
calculations.
Skbscript
'
Cmccpt;
t'official''
:
j
j
,
51n te following, the reader will notice a slight changc in notalion. In particular, the lme will be deaolcd by zi. nis is required by the new instruments. 'll-ibor is the London Interbank Ofrcd Ikate. lt is an intcrbaak rale asked by sellers tf fuods. It is obtained by polllng selected banks in London and then averaging the qutcs. l'Iznco, depending on the seiection ot banks, there may be several Libor rates on the same maturity. ne Brtjsh Bankcrs Msociation mtlculates an Libor that forms the basis Of most of thtse Li%l Instrumonts.
:
t
6
01 introduced in Chapter 21 are the generator implications a StoC hastic process, Ktlmkgorcv's backward equation, and the f0r':' especially go-called important Markov property, The latter is O f the MarkA Short shown behave is ) not to as a rate, because the latter mo del9 tf
Sorlle
,
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C HA PT E R
*
3 lntcrest Rate Derivatives
'E
Results and Tools for Interest-sensitive Securities
16
..r,(
.
373
1:
FM iuHems
a loan, a ccrtain sum N at time ti and will pay back at Stime lj+1 the sum N1 + Ft8), where the is the days adjustment factor, rate agreements (FRA.). Already discussed in Chapter 1. thcse * Fo-ard instruments provide a more convertient way of hedging interest rate rist ) Depending on the outcome of Ff > Z/; or Ft < Lq, the buyer of a i y # aid-in-arrears receives, at tne (+1, the sum ;
t
'',
Receipt
=
''
- Ltj)5
N-'t
Q
.j
,
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j
1 i
:J
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if it is positivea or pays
t
L
Conact
N (Fi - Lt J, if it is negative. 'The F.RA rate Ft is selected so that the time l price ot':h !J the FRA contract equals zero, This situation is shown in Figure 1. ln case'..t umelt of FRAS traded in adual markets, often the payment is made at the time the Lt is observed. Hence, it has to be discounted bv tl+.Z,f :). n<' is also shown in Rgure 1. . Caps and floors. Caps and floors are amoog some of te most Ikui: interest rate derivatives. Caps can be uscd to hedge the risk of increasinp' interest rates, Floors do the same for decreasing rates. are essen . . baskets of options written on Libor rates. Suppose t denotes the prescnt and Iet /, t : fn be the starting JJlzr o%t e#',. an interest rate cap. Let ln be the ending date of the cap for some fix j 'Fhen fa-j be reses dates. for evely ca E /z, a, l < fa < tn. Let the *11.3l that applies to the period ti /f+j, the buyer of the cap will receive, at : '...) li- 1, the sum ? .
t2
Set
Paymcnt
l
Da*
Date
Da*
A Standa
Xme
11
1
I
''
1
(Ft)
E
't,
Lt, :,
-
(Ft - Q)
FRA
Receipt
=
NIFt-LtX (1+Ln)
.:
'/:::;
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'lnhey
!E @
,,
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,
.
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Set Date Settlement Date
Date
I
Time
t2
l
t
,
,
tl
t
,
F'
,
t
!
!.;
F 1G U R E
....
; E4 'g'
N max
g:(f,, -
dI
Rcap), 01 ,
.
..
.
This formulation is shown in Figure 2. At time tt t-he option expires. + Rcap&j, then the holder
(gj,
lf the price of the bond is lower than 1/41 receivesthe difference:
kst: Lt6 is the underlying Libor rate obscrved at time ti, the . '! time days adjustment, and the N is a notionltl amount to be decided at / G The Rcap is the cap ratc whch plays the role of a strike price. ln a sense, a caplet will compensate the buyer for any zcrease in future Libor rates beyond tlle level Rcap. Thus, it is equivalent to awj fption With expiration date mitte.n on a default-free discount bond . maturity date ti-vk, with a strike price obtained from Rcap. ln parti Wlm the strike price that applies to this option is the 100/(1 + Rcap&), Rcap is the cap rate, and the is, as usual, the days adjustment. :
where
.
%
Payoff
,
=
100
(1+ Rcapnj
1* -
(1+ fw/;:)
,
..
Othemise, the holder receives nothing. nat the caplet is equivalent to a option on Ltj with a strike Rcap is also seen in Figtlre 2. Here, if we viewthe caplet as a call with expiration date fs+.l written on the Libor rate L, then it should be kept in mind that the settlement will be done at time fj.j.) rather than at time ti An interest rate lloorlet, can similarly be shown to be equivalent to a call option with expiration t, on a discount bond with maturity ti-v,
'
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Results ana
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Elti-itcap)gj
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These are some tf the basic interest rate derivatives, 'T'he scope of this book pyevents us from going into more exotic products. Instcad, we now would jike to Summarize &OmC key elements of these instrumcnts and see what types of new tools would be required.
.
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a,
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Equivalenuy, it can be viewcd as a put option on tl'ie Libor rate Luk,tbat :, at time ff+.:. cxmires 1. Chapter also in discussed instruments Thesc rate Interest were maps. E) lE' an exchangc of lainvanilla interest rate swaps paid in-arrears involve generated by a nxedpre-sct swap rate x against cash qows gen- .' cashnows qoating lbor rates 1.s. ne cash nowsare based on a notional eratedby /j+1. clearly, t,' a swap is a more comamountN and are settledof atFRAS. timo l rate of ne swap s set so that the plexform a sequence p equals of the zero. swap contract t pr ice Bond options. A call option written on a bon d g ives its holder the rigbt :t' Strike Price K. SinC,Q the Price 0 f A to buy a bond with price Bt at the bond depends on the cun-ent and future spot rates, bOn d optjons wl %': ts in I-a-lxlry SensitiA to movementg in rt Or for that matter, to movemen '' rates L t ' Itre options writtcn on swap contracts. Depen swaptions. swaptions they are very liq uid. At time t a practitioner may buy maturity on amouut N' option on a swap contract with strike price K an d sotional 'i
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Introducing interest rate derivativcs leads to several complications. is best seen by looking at bond optfons, and then compaing these with the Black-scholes framework. The price of a bond B( depends on the stochastic behavior of the current and f'uture spot rates in the economy. Hence, at the outset, two new assumption: are requircd. (1) Bond price Bt must be a function Of the current and future Spot ratesy and (2) the spot rate r; cannot be assumed constant because this would amount to saying that B/ would be completely prcdictable, which in turn would mean that the volatility of the underlying security is zero. Hence, there would be no demand for any call or put options written on thc bond. Thus, the very first rcquirement is that we work with stochastic interest rates, But then the resulting discount factors and the implied payoffs would be dependent on interest rates. Clearly this would make arbitrage-free pricing more complicated. The second complication is that most interest rate derivatives may be American style and any explicit or implicit options may bc exercised before their respective expiration date, if desired. Third, the payouts of the underlgng security may be different for intereSt rate derivatives. For example, in case of mark-to-market adjustments the fact that spot rates are stochastic witl, in generala make a difference in evaluatng an azoitrage-free futtzres price compared to forward prices. This is the case becausc with mark-to-market adjustments the holder of the contracts makes/receives periodic payments that Euchtuate as interest rates change, But tlzese mark-to-market cash flows will also be discounted t.jy stochastic discount factors that are afected by the same interest rate movements. The resulting futures prices may be different from the price of a forward contrad that has no mark-to-markct requirement. 'rhis
j T;
Bond Formulation
a75
The option expfres at tfme F and the swap will start at some date rj, F :j Fj and end at time Fa, F1 < Fa. The buyer of the swaption contract will, at expiration, have the right to get in a faed-payer swap contract with swap rate &, notional nmount N, start date Fj, and end date Fa. Hence, the value of the swaption will be positive if actual sw'ap rates at time Fy rjamely the Rv ltave moved above x.
.'j
Libor Formulatifm
complications
,
:
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5 376
C H A P T ER
.
Results
16
Tools for Interest-sensitive Securities
and
:
' k.
6 References
.
.
Similarly, if a bond makes coupon payments, the underlying security, Bt, will also be differcnt than a no-dividend paying stock These are some of the obvious modilkations that are required to deal with interest rate derivatives. There are also some morc technical implications that may not be as obvious at the outset. One of these was mentitme,d above.
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Intcrest rates are not assets, they are more like on assets. rns means that the arbitrage-free restr ictitm that consists of removing tlte unknown drift, /t, of an asset price Si, in the dynam ics!
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$ i.. . f ' .
I
and then replacing it with the risk-frec rate, r, is not a valid procedure :: anymore. In the dynamics of rt writtcn as, t. . i L' drt alrt f jdt + (rrt, !)tW'), ' other means. This makes tlm the drift art, tj has to be risk-adjusted bymuch t application theorem of Girsanov more complicated. In f'ax' j. ractical 15' risk-frei the switch from 116tf a new Wiener process, 11:, dehned under the ? ' measure#, cannot bc done in a straightfomard way. Given the Glrunov ? correspondencebetween the two Wiener processes'. :
7k
7 72' ''
.
=
(t?(G, 1)
telrt
-
,
'
Amd it is not clcar at the outset how A? can be determined. nj , c, x This is a signihcant complication comparec to r-ne Diacx-xacnchua tut ion: r
=
ap.
-
,.
i
'
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'
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9
5
.j
!
:
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: yj
r
...j.:?
f is a traded asset. Fundamental theorem o wllich possible when equality. Yet rt not being an uasset, a si nancewould then imply this substitutionis not valid.
p
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,
:
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,
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7
'
:
coucjusions
'I'he book of readinp published by Risk and Beyond is highly recommended as an cxcellent collection of readings concerning interest rate
,>
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I
'
6 References
.%t
.
l
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jE
hto'j,
1
j '
chapter is simply a brief summary and cannot be considered an introduction to interest-sensitive sccurities. It has, however, the bare minimum necessary for understanding the tools discusscd in the remaining chapters.
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;
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i j
.
r))#f + a'rt. t)d J1't ,
qk
.
; :
''tricking''
:',
.
' requ ire: of interest rate dynnmlc.s
The modications dh
t dt
! i
ii
someof these complications can be handled within a Black-scholes fzamework by either making small modillcations in the assltrnptions or by them in some ingenious way, But the early exercise possibility of interest rate derivat-ives and stochastic intercst rates are tw'o modicatitans that have to be incorporated in derivative assct pricing using new mathel-lft iCa l tools ne following chapters are intendcd to do this
,,
=
!
,
Clcarly, this very broad class of interest rate derivatives cannot bc treated using the assumptions of the Black-scholes environment,
!
W?; d Jjzr -
i
j
I
stocks.
,,!'r
d
! E
,
,
.'
j
r
rrhe
,i,
=
I
'rhe
?E
+ eutdkk,
j.
i '
p
Another complication is the coexistence of many interest rates, Note that within the simple Black-scholes world. there is one underlying asset St. Yet within the ixcd-income sector, there arc many interest rates implied by diferent maturities. Moreover, these intcrest rates cannot follow very diferent dynamics from each othcr because they relate, after all, to similar instruments. Thus, in contrast to the Black-scholes casc for interest rates, one would deal with a vector of random proccsses that must obey complex interrelaresulting k-dimensional dynamics tions due to arbitrage possibilities. arc bound to be more complicated. Note th a t in case of a classical Black-scholcs environment, modeling the risk-free dynamics of the underlying asset mclmt modeling a single SDE, where over-time arbitrage rcstrictions on a single variablc had to he taken into account. But in the case of interest rates, the samc overtime restrictions need to be modeled for l-variables. There is more. Now, arbitragc restrictions across variables nccd to be specihcd as well. Last but not least, therc is the modeling of volatilities. volatility of a b0n d has to vary over time. After alI thc bond matures at some specific date. Hence, these volatilities carmot be assumed constant as in te case ot
) ,'
4.1 Dri# Adfxstvnent
gvhdt
j
4.2 Tenn Strucfuze
g.
'.
=
;
.%.
dSt
!
377
. ! :
.
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,
:
E. F,
C HA P T ER
16
Results and Tool.sfor Interesr-serusitive Securities
) t'
alld their Pricing. The
derivatives
and the
.
Ceader
exlensive treatment in Rebonato
Shtuld
also consult Hull
(1998).
(20() I
t
.!!
j.
7 Exerclses
i
5.
Arbitrage Theorem in a New Setting
Lq
1. Plot the payoff diagrams for the following instruments: (a) A caplet with cap rate Rcap 6.75% written on 3-month Libv t Lt that is about to cxpire. .t (b) A forward contract written on a default-free discount bond with .l( 2 years. The forward contract cxpires in 3 months. ne matur contracted price is 89.5. / (c) A 3 by 6 FRA contract that pays the ftxed 3-month rate, F, agaiast . ! J
=
'
Nomwliwtion und Rtmdom Intercst Rutes
'J '
,
Libor. = 7.5%. Tho E (d) A flxed payer ltcrest rate swap with swap rate zeceivcs Libor. Staz ato : and 6-mont maturity 2 has years swap months 6 ago. was cxactly Exed payer swap .',' (e) A swaption that expires in 6 months on a z-year .(.
.;.
(.
1
,..'
with swap rate x
.6%.
2. Which onets) of the following are assets tradcd in hnancial markets: (2)
(b) (c) (d) (e) (9 (g)
Libf) A s-yearTreasury bond A FRA ccmtract A caplet Rettzrns on 3o-year German Bonds Volatility of Federal Funds rate Arl interest rate swap 6-mO11t17
.'..
=
.!'..
The motivation foT tlle main tools in derivatives pricing wms introduced in the simple model of Chapter 2. nere we discussed a simple construdion of synthetic (martingale) probabilities that playcd an essential role in thc ftrst Because of this book. tl:e setting was very simple, it was well-suited part for motivating complex notions such as risk-neutral probabilities and the crucial role played by martingale tools. Chapter 2 ctmsidered a modcl where lending and borrowing at a constant risk-frec rate was one of the three possible ways of westing, the tther tw'o being stocks and options written on these stocks. Chapter 2 interest rates were assumed to be constant and a discussion of imerest-sensitive hnancial derivatives was deliberately omitted. Yct, in linancial markets a largc majority of the instruments tlzat trade are interestsensitive products. These are used to bedge, to arbitrage the interest rate risk, and to speculate on it. Relnxlng the assumptitm of constant intercst rates is, thus, essential. As mentioned in Chapter 16, rebxing the assumption of constant interest rates and then introducing complex interest ratc derivatives creates a aeed for new mathematical tools, most of which were discovered only lately. nis chapter attempts to motivate these notions and introduces the new tools by using a simple discrete-state approach gimilar to the one utilized in Chapter 2. However, the model is extended in new directions so that these new tools and concepts can also be easily understood. By expanding
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379
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c HA P
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ER
.
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:: 17 Arbitrage Theorem '
A Model for New Instrumenrs
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381
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. framework of Chapter 2. one can discuss at least three malor the sirnpled TJ . additional results. F under normalizationof the concept The lirst sct of issues can be grouped '< This is the technique of obtaining pricing equations for ratios of asset prices r7 haS a numerator and a denominator. instead of prices themselves. A ratio In a dynsam1c ' setting both of tbese c hange. ne expected rate of change of cach element may be unknowo, but under some conditions, tbe expected numbcr, For example, the ratio of the tw'o, may be a Arlpwzz rateof change ofand . ratio of denonnator may grow at tlze .; a detcrministic thenumerator the will ratio if the itself stay same. same unknown rate. But the numerators and denominators in pricing formulas are carefully seleaedv and if the Girsanov theorem is skillfully exploited, modeling of asset price ; 2 dynamics can be greatly simplified. In order to start discussing the issue of normalization, we first let tlim anotlwr and then try to randomly from one perio d to short rate uctuate .2 remain tbe same. Clearly, /, see whethcr basic results obtained in Chapter 2 applicable derivatives, interest-sensitive to this makes the discussion directly securities of needs such to asgume stoc hastic interest y pricing iven that g main pfaint. rates, But this is not the ? It turns out that once interest rates become stochastic, we have now .: ; W ays of searching for synthetic probabilities, especially when we deal wi th . philosophy of the gcneral apz interest-sensitivc instruments. Although the 17roaclt introdtfced in Chaptcr 2 remain.s the same, tlie mcchanics change); jxin a dramatic way. ln fad, one can show that llsing dterent synuuet; Prou 5. a bilities will be more Practical for differcnt classes of financial delivativex. Obviously, the final arbitrage-free pricc that one obtains will be identical in synthetic probabilityv but te 5. eac h case. After all, what matters is not the Yet unique state-price vector, some synthetic prbabilities ma/ un derlying be more practical than others. Thig gimple step, which appears at the outsct inccmsequential, turns Out ' . to bc vel'y important for the practical utilization of synthetic probabilities, chooslo, that discover hnance. ln fact, we ddmcasuresp''as a pricing tool in Probability can simplify th' , another equa 1Iy over measure onc chaptcr is to explnm' : Pricingefort dramatically, The second objective o f this ( setting.l simple this comp lex idea in a numbtk It is also t.he case that earlicr chapters dealt with a very limited centered on p1a in vanilla opfoe' of derivative instruments. Most discussion withinthe Black-c holesenvironment, Occasionally, some forward contra q
! l
was discussed. The present chapter is a ncw step in this respect as well. Forward contracts and options mitten on Libor rates or bonds are the most liquid of a11 derivative instruments, yet. their treatment witliin the simple setting of Chapter 2 was not possible with constant spot rates. In this chapter, we incorporatc thcsc important instruments in the context of Finance and show that thcir treatment of the Fundamental requires additional tools.
.
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A Model for New lnstruments
j
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we nced
rst the simplitied setting of Chapter 2. A nonto remembcr dividcnd paying stock a European call option Ct, and risk-free borrowasd lending considered in a two-state, one-period setting. ne ing were yundamental neorem of Finance then gives the following lillear relation between the possible future valucs and the current arbitrage-free prices of thc three assets under consideration'.
l
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ut,
;
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'f
st
=
Ct
:j
(j..j. rhj
(j u d st-vh stya C'& Cr+.,s ..j. rhj
j
..
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J+a
,
j
;
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:
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is the time that elapses hetween the two time periods, the u and thc thc two states under consideration, and tbe llz:. > (), lid > 9) are state prices, Thc hrst row represents the payoffs of risk-free lending the payoffs of the stock St, and and yoaowing, the second row fepresents t jae tjajrlj row remesents payoffs of the option Ct.2 According to the Fundamental neorem of Finance, the (#u # J oxist there wgl be positive if and w are no arbitragc possibilities given d). exist and arc positive, reverse is also tnle. If the (r, qj. then there will be no arbitrage opportunity at the prices shown on the Iefshand sjde. e r jsk-free probability # was obtained from the lirst row of this matrix,
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prices
lone, can also ask tho following question, Given that we want to convcrt IISK t choose martngalesby modifying the true pro babilitydistribution, is there a way we c.arl way? synthelicmeasure irl some xyyesl',
inW
2wz
;
:
+. (1 + .L4d
,
(jejyrjjug =
=
( 1 + ralyyu (1 +. rhl
ty
make slight mtxlificatiorls in the notation compared to the simple model used in Chapter 2. In particular we introduce iudcxing by u and d, which stand for the two-statcs.
:
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(1 +
pd
.
-
J
js
=
.
a, '! : i
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382
C H A PT ER
Arbitrage Theorem
17
'
2 A Model for New Instrtlments
r
383
2,E.
nus, the probability P expected returns of the risky ssets so that all expected returns became equal to the risk-free rate r. Hence, the term isk-neutral measure or probability. In the next section we extend this framework in two ways. First, we add another time period so that the effects of random Puctuations in the spot Second, we change the types of instruments rate can be taken into aount. considered and introduce interest-sensitive securities.
gave
iTmodified''
l'bd
1 #? + =
The conditions
prices t) u ,
,
#
0
/u, 0
<
'
l-id are satisfied given the positiveness of state
<
,
E
.
with l'bu7 z-;d we had tw'o numbers that were positivc and that summed distributo one. These satisfy the requircmcnts of a probability :d titm within t his simple setting, and hence, we called the /&, synthetic, probabilities. These probabilitics, which or mtre precisely, risk-neutral said nothing abut the rcal-world odds of the states u, d, were called
',
nus
R
L
'
-neu tral'' due to the following. Consider the second and tltirtl rows of the systcm above in isolation: d u u -F'sd S/ St+h* /+.,: g)
!
I'iS k
(.
=
Ct
=
I)d
c',ia v
a
Multiply the tx, by (1+rA)/(1+r) the pricing cquations'. L
t
=
j
Sv /+..
car+a
+
/B
(1 + rhj
,
11 E js.i U/+a1 r)
=
Pu,#;' to obtain ..k
=
.
tfol
.
(1 + rLt
'd
2
..
@)
r-
=
down, down
.
=
,
,
2
down, up
=
a
=
up, up
-4
,
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an
.
'rhen,
Ct
=
f7+s
1 hjlsu
(1 + r
1
= (1 +
?.a)
ji r
E
(j
.j.
.
.F
1
d
+ Ctu
sa)
r
#?
:;
(5)
a
s
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rt 2U
i
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ull
..-* .
l
expectation operator here the .E'3(.j denotcs, j.s uj-ual, the (conditional) that uses the probabilities P Pd Note that we ltre omitting the t subscript in F3(.1 to simplify the notation in tMs chapter. l According to these pricing cquations, expected future payoff of the risky price. rate give the currcnt arbitrage-ce assctsdiscotmted by the riak-free IM prices though market Even lt is in this sense that /&, are 'e u St Ct, contain risk prcmia, they are nevertheless obtained using thc P as if they come from a risk-neutral world, There was a second important resu lt that was obtained from these pricing equations. Rearranging (4) and (5),we gct S/+a 1 + rh E >
W
l
N
,
.
U
tl 17
.:
.
j
l
E' I
h
v#d
,
,
=
1+ r
=
,
;g
T'11
h'j:
ru
tz
rd
wx
utl
-.
du
..--
d
%
rd '
tl
C 2+A
Ct
l 1
--
:
,%
E>
--'-
I
tz
la
time B I G t) R E
.
r
.(
:
'
dd
-'..
tz
.
.
=
up, down.
It turns out that a minimum of two time periods is necessary to factor in the effects of the random spot ratc rt ne situation is shown in Figures 1 and 2. An investor who would like to lend his or her money between h and does this at tbe risk-free rate contracted at time no matter which state occurs in the immediate futalre, his or her ret'urn is not risky,
.
= (1 +
.
,
'
1
ud
24.1
< tz < la, but keep the We consider two periods described by dates each assumption of two possible states in time period the same. Adding number of possibilities. This way, one more time period still increases the 1 there will be fourpossible in looking at time ts 1 + 21 from time (OPJ, i 1, states, 4J,dcscribing the possible paths the prices can follow at time-nodes (fl fz fa ):
,
(a) t
,),
and introduce the +
2.1 The New Envitonmenl
.
1
states
1% ' .
.(j
(J i l A P T E R
384
Arbitrage 'Theorem
17
.
spo Rates
Libor v-.
l
/
j A Modcl for New Insrruments
J'
cxtends arbitrage pricing to more interesting assets and, at the same time, gjvcs us an easy way of shoping why changing normalization is a useful tool for the linancial market practitioner, In particular, within this framework measure and compare its we will be able to introduce the scalled fonvard properties with the tisk-neutral measure seen earlier. Hence, we assume that there are Iiquid market.s for the following instruments 6
'?
rtI
rt
=
=
KrltAwn
Rarldom to be observed at tl # ! i !
( ..
..
i
!
J
Ltl
; ! E
! i : .
-
uckuklwn,
'
3 period Libor
#
E
E
t1
t
.
. .t .
:ts
.
. A savings account with no defaktlt risk. At time f cme can contract the rate rt, and after an intewal one is paid (1+ rt). lf the investor with short-term stay irwestment, he or she will have to wants to this Altrw f + A. time spot rate rt-vh at contract a terest rate is default. A forwardcontract on an interest rate Lt. free. lt is aso a spot rate that can be contracted for more than just one period. For example, it could represent the simplc interest rate at which pmonth Libor. Hence the a business borrows for 6 months, such as the choice of Lt as the symbol.7 @A s/lorf-maturity default-free discount bond with time-f pricc #(f, /3(), t < la. 'Fhis bond pays one dollar at maturity /a, and nothing else at other times.
rlme
t2
tg
385
:
u
,
A
'
'rhis
because the payoff is known/ Regardless of the state up or down that nmy the westor will receive the same incomc (1+r,, Because the occur at riskless borrowing and lending yields the samc rcturn whether the up and down state occurs, in a model where there are two time-nodes, 1, lz. it will be as if the spot rate does not Euctuate. So thc effect of any randomness in rt cannot bc analyzed. But as we add one more time period this changes. Looked at from time 1+ 11 1, the spot rate that the wcstor will be offered at timc Q of with the risks. hzvestor this investmcnt, staying will present some By type thiglf' rate of interest rf may end up lending his or hcr money either at a will available at fm ls; rtu 'Which spot of these rates be rate or at a tz /3,4 the randomness not known at timc fl Thus, with three time-nodes although with the onewill be an important tctor, rates of the relevant. randtmness of Chapter 2 this period framework was not situation was presented in Figurc 1 For time-node t, wc have Iwfz possible 5 d spot rates, rtzu and rtzn and hencc tbe value of rz is random. that we introduce to Chapter 2 is in the se'rhe second modcation lnstcad of dealing with stocks and options written lection of instruments. securities and forwards. This stocks, interest-sensitive we considcr o'n these
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=
=
. A apmaturity default-free discount bond with time-l price #(f, F). This bond pays 1 at mattzrity date F, Because we want this bond to have longer maturity, we let f < /a < F, Choing a numerical value for F is not necessary in our model. * A FRA contrad written on the Ltz that results in payoff N(Ft3 - Ltz) at time %.Here the Ft is a fonvard rate contracted at time h If Ftk > Ll, the buycr of the FRA (Fonvard Rate Agreement) pays the net amount. If Ft 1 < Ltg thc buyer receives the net amount. Note that the payment depends on the rate Ltg that becomes krtown at time tz. but the proceeds from the FRA are paid (received) at time l3. This is what makes the FRA. in-arrears. is the days adjustment. . Alld. finally, we consider an intcrest rate derivative, say a call option written on BtL fg) or a caplet that involves the Ltg The derivative expires at time f r3 and has the current pricc Gj.
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.
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lhat
7
there s no default risk in this settjng.
4This means t-w0 time-periods.
J '.'
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l(3r
'Gmarktt''
tt:
:
,
fluctuationsin
so
the payofs
if this investmen!
,
=
invcstment is no! entirely approprihte for a savinp s'nis shows tha! thc term WW money markct) account. The investment is risk-free in the sense that what WZW risk. 11.*price will n0t :hC Cfld Of the Ctmtrac! is known. The contracl has no tbat the contract period because the r, is constant. In addition, we assume 'ucuateduring the payoff at tlw end of the contract period is ocmsfnnt. ' is no default-risk eiter. 'risk-fzee''
there
,
I
'E.
is rolled-over.
'.' .t
.
6In this conlcxt, the liquidity of markets means that the assets can bc instantaneously bought and sold at te quoted prices. 7.A.:dermed in the pevious chapter, Libor is the London Interbunk Ofered lkate, an Zterest rate at which banks can borrow money in London. Libor rates arc tse,d as benchrnarks 4nd Libor-based nstruments form an imporlant proportion of assets in bank balance sheets.
386
cu
S'TER
,.
Arbitrage Theorem
17
.
2 A Model for New lnstruments
t ('
'
the right-hand side consists of the product of possible payoffs at time i/. matrix equation is similar to the multiplied by t.he states prices t5, used complications, several comments in 2, given the one new chapter yet, order. in are ne nrstrow of this system describes what happens to an investment in savings account. If one dollar is invested here, it will return tbe known a. a, (1+ rt, ) at time /a and an unknown A,a (1+ rt, ) at time ra. Timc t return is random because, in ctmtrast to Rtj thc Rtz is unknown at there are fw'o possibilities, and this is indicated by the time l:. At time superscripts a, d on the Rdt Rt't This row is similar to the hrst row in Eq. z z (1),CXCCPthere the elements are not constant. Ncxt, consider tbe second row of this matrix equation. The Ftt is a forward rate contracted at time 4 on the random Libor rate Lt,, which will Hence, we have here a Forward Rate Agreement. be obsen'ed at time jt turns out that FRAS have the arbitrage-free value zero at ctpnpuc/-time, because no up-front payment is required for sigriing these contracts. This cxplains the seccmd element of the vector on the left-hand side. Also, according to this FRA ctmtract, the difference between the known Fl 1 and the unknown Lt, will be paid (received) at timc f3, and this cxplans thc second row of the matrix. Clearly, thcrc arc four possibilities bere,8 nc third and fourth rows of the matrix equation deal with the two bonds WC inelttded ila the system. ne Bst and Bh denote the time- arbitragefree prices of the two zero-coupon bonds, the first maturing at tne l3, the sccond at some Rlmre date T, respectively. Note that the value of the short bond is constant and equal to one at ts because tls happens to be thc maturity date. On the other hand, the plice t)f the long bond docs not have tlais propcrty. are four possible values that Bt? can assttme, The last row of the matrtx cquation rcprcscnts the price, Ct of a derivative sccurity written on one or more of these assets. Finally, the ( fj j j u, J) are the 4 state prices for time ts. They exist and they are positive if and only if there are no arbitrage opportunities. As Whre
we now need to stac k these assets and the corresponding payoffs in a f? matrixequation similar to the one used in Chapter 2. But, lirst we make i' somenotational simpli juations. ' we denne the gvoss risk-free rcturns for periods and as follows; '
'lhis
,
''risk-free''
')
Rt
=
(1+'
Rtz
=
(1 + ru,l.
%
387
:.
J
I
=
=
,
.
,
EE
Althoug,h we will revert back to t he original notation in later chapteu, for the sake of simpllfy' ing the matrix equation discussed below, we simplify tlaenotation for bonds as wel j we 1et
,
:) '.
.
B'I 1
=
which represents the price of the tbe
S(f1
fa),
,
.
,p !
bond at current period
dishort''
,
,J
and 1et
,
.'E
B
l
Bts
=
#(f1 F)
=
Bt3'
,
,
).
.';
FJ
'
(
j-
fg
bond at times h and fa, respectively. We represent the price of the of the long bond for the interim period lz. do not need to consider the price nen, notional amount of thc FRA contract denoted by N we set the equal to 1, because this parametcr plays an inconsequential role j n otlr model. Finally, we assume that all interest rates are expresscd as rates over eriods Of length rate This way we dt not need to multiply an P ZSO Obtain Corregponding Of intender Wis is a A to returfls, by a fador equaling Alternatively A 1 simpllfy' one the notation. as to one can takc year. Now we can write the matrix equation implicd by t he Fun damenlal diKusse d rem of Finance. Stacking the current prices of the Iive instmments relation: obtain the the side, 1 left-hand above in a (f x ) vector on we t
: : '
.
'lhere
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=
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Fl-lnder
p
time t,. tx givca by -rhen,
:
.?
,
=
at somc conditiens the fonvard contracts may involve an immediate settlement the payment (receipt)will equal the present value of tile difference, and will
Fr - .Iaj,
:7
'.
1
0 B.% fl
js
/1
C'
R /1 R 6J R srI R& R rl R& R /1 Rdtz 11 lz 4 (F/I a L 2 ) (Ft, Z/ 2 ) Ft I L'4 tz ) (Ft L - L12 ) 1 1 1 1 aa au llul dy 14, luu /Jlu t cx c14 f7/3 ?.,! '.$ cats
=
.
.
vdu
v
mza
(6) r'. '.,
,
,
%(.
(je yjvatjvas
.
'
.-. E
J
f-J2
.
nis s tl'le case for example, for most lJolward Ikate Agreements, tradcd because this eylye' of settlemen! OII involve a rtzljo of two random variables, called the FRA-adjustrnent. By assuming that somv furlher complcations, tjelj at limu avoid sllcil nonlirlearities at tilis inital stage. Irl addition, we t;
:
ud
r//
1+
.
)
#NN
-
-
-
.
(
settle in-a#ars
anyway.
in thc, markct. But, they w111introduce the contract is semost interest-rate
'
g1
:
388
C H A PT ER
Arbitl-age Theorem
17
*
'
two periods and fotzr states of the world is the smallest systcm in which the issue of normalization can be discussed.
..;
,
.
E
Consider now how tis set'up differs from that of Chapter 2. First, in this matrix equation, the risks of investing in a savings account be explicitly. investment is risk-free only for one period. An seen can investor may be certain about the payoff at time fc t + A, the immediate future. But, one pcriod down the line, spot ratcs may yield a higher or Iower return depending on the state of the wodd i j. Hence, in general terms, the current spot ratc rt is known, but rz.j-a is still random. It may be (thc Rlt j state). down Conscquently, d carries up state), or u, (the a superscript indicating this dependenct on thc realized state at time tz. Second, note that the securities included in this model are quite different from those of Chapter 2. ne forward contract and the bonds considercd here are interest-scnsitive instruments and thc pricing of them is likely to be more delicate than the assets selected for the simpler model of Chapter 2. The same is true for the option Ct. The option is written on interestsensitive securities. Finally, note a straightforward aspect of the model. Because one of the bonds matures at time ts, its payoff is known and is coutant at that time. nis simple point will have important implications for choosing a synthetic measure) that is more convenient for pricing the probability (martingale new assets introduced here. We can now consider the important issue of normalization that determines the choicc of measure for the instrumcnts under consideration. But lirst we need the following caveat.
i.Fr'
2.2
rt'
Grisk-free''
izatiot
Again we bcgin with a review of the framework in Chapter 2. Consider how the risk-neutral probabilities were obtained given the Ftmdeztmental Theorem (f Finance. More precisely, take the cquation for given by Equation (1) and earlier in Chapter 2:
y
E,
=
?, ,'h
Rlow''
G
=
.
uf
,
st
..
su y,u.j- sd
=
J+a
:'
('.
' ,'.
St
=
(1+
u u S/1a'#
(j
'y
''
an d then recognized
E
.,4 L
.
., ' ).
y)
-
a
s,+x
(1 +
rhj
./+a/
+
d
ts
/ in this equation,
(1 + r) (j o ra)
we
(8)
ra) are in fac.t the Jsf This resultcd .
in
d
y-js +
1',+a y-js . 1 + ra) (
(qj
o
.8
ftl Now, during this operation, when the + arc substituted for the rl, an extra factor is left in the denominator of each z l+J term. This factor is ( 1 + rla and represcnts the return of the isk-free lnvestment in that pamticulr state. But, recall that one-period-abead risk-fyee retmrn is eonstant, and hence this factor was successfully factored out to give:
.
? i t '. b L.
..
-
., (j
1
,.a)
+
g-l.j-s/u .$,+..sp''q
+
d
.
(10)
Howcver. notc that in the new model with two periods, this return will be a random variable and a similar operation will not be possible. N evert h e less the point to remember here is that the process of introducing the risk-ncutral probabilities in the pricing equations resulted in the nonualization of each state s return by the corresponding rcturri of riskft'ee lending, ln fact, the substitution of #'' for 4i is equivalent to dividing every possible cntry in tbe matrx Eq. (l) by the corresponding enty of the ftrst ow, which repregents the payoff of the risk-free investment. As mentioned earlier, under the risk-neutral mcasure #, asset prices will have trends. lndeed, under # a11 expected rettzrns are converted to r and this means that they will drift upwards. Thus. the prices themselves will not be marthlgales under #. But, by with risk-free lending,
t:
,
'
cc
?
jj
y
5
Rnormalizing''
':
;
'. :
4,'41+
that the S/
rl
.j. xhj
u
.
,.
,4
4J
2+,i
In order to introduce thc risk-neutral probability multiplied each ') by thc ratio (1 + r')/(l + r):
.
k. Remark In this chapter the Rt and the Lt denotc the short rate and the Libor process, respectively. ln principle, these two are different processes, with Lt having a differcnt mattzrity than the spot rate, which is by de:nition the ' rate on the shortest possible tenor. But, because we want to keep the instnz- ! J ments and the model at a minimum level of complication, we assume tllat , 1. the Lt is the one-period Libor, This would make R( and Lt the same, but : even with this consideration a1l results of this chapter still hold, ne chapter ' treats these as if they are different notationally because with longer mato ritjr Libor rates this equivalenc,e will disappear. Using diffcrent notation for E. Lt and Rt will help us better undergtand the Libor instruments and their relationship to spot rates in these more general cases. t ' Libor. But this would reque Arl alternative is to considcr a fpwperiod '' model which will lead to a much more complicated matr a frcc-period of cquation than the one considered here. In this sense, the compromise
2. 1. l
No
jz.''
'rhe
,
389
:;
was the case in Chapter 2, it is important that for aIl i, j'. ud > ()
,
2 A Model for New lnstruments
.:
.. ..t
.
e'
(2 112t P T E R
17
*
3trbitrage
R7aeorenA
the expected return (i.e.,the trend) of the ratio becomes zero. In other upwards by the words, the numerator and the denominator will normalized variable becomcs aod drift expected the a martingale. It r samc will have no discerrtibk trend. Note one additional characteristic of this normalization, ne division by (1+ r) amounts to discounting a future cash flow to present. But, in normalizing by risk-free lending, one/rul discounts, and then averages using the probability # to get: 'trend''
St
=
St-vh
J3
E
four states. Thus, proceeding in a way similar to Chapter 2, we can exploit the remaining equations of system (6)in order to obtain the corresponding martingale equalities. For example, the third row of the sptem gives the short bond's arbitragefree price under the measure #: jyx
/1
(11)
,
( 1 + rj
2 A Model for New lnstruments
=
1 (1+ % )(1+ +
The r bcing constant, tis simplifies to
=
/2
r7
tz
witb i, j
=
=
(1+
r,,
(
)
B& f1
)(1 + d'g),#/,
(13)bccomes: 1 #'zu+ l'bud+ ?du+ #dd.
(14)
u, d. Then Equation =
If there are no arbitrage opportunities, and we will have 15Y >
the state prices
0.
# ud
+
1
(1+
rjj
)(1+
r()
jsdu
J3dst ,
=
1
E#
(1+
wherc the rt, is random, and hence, tional) cxpectation sign. By mtwing to continuous time and we can generalize this formula for arbitrage-fzee price of a default-free by: Bt, T)
)
where the state subscript is applied only to Rlg becatlse at time h, the Rt is known wjth certainty, We can detine the four risk-neutral probabilities in a similar fashion:
#ii
rt
1 + r,, )(1 +
k/N
?d
RN
/1
1
(1+ rt, )(1+ z'i )
(12)
.
nis is the pricing equation used several times in the lirs! part of this book. EP is the expedation As with all expectation operators uscd in this chapter, conditional on time t information unless indicated othenvige. Would thc same steps work in the two-period model that incorporates interest ratc derivatives? The aoswer is no. Ctmsidcr trying the same strater in the new model shown in (6).Sup#P by starting pose wc dtcide to detezmine the risk-neutral probabilities again with the ftrst row of the system which corresponds to the savings account: + R /1 Rd/2 424 R + R /1 Rd 4uu+ R 11 Nw (13)
l
rll )
1
+
Of
p 5'/ (1 .j-1rh)E E'%+a1 -
# uu
-
(15)
fsill will be positive
Clearly, as in Chapter 2, we can use the Iii as (( they are probabilties assoeiated with the states-of-the-world, cven though they do not have any probabilistic implications concerning the actual realization of any of the
cannot be moved out of the
h
=
Bfu %
(1 + +
rtl )(
1 + r: 2
)
I-ttu +
then assuming a ctntinuum of states, maturity T t < F. ne bond will then be given
rudu Etp e J'-.
'
=
Bdd ts (1 +
rt,
)(1 +
hd 2
)
#?d :5
-1
(
(condi-
an arbitrary zero-coupon
,
This equation will be used extensively in later chapters. Wc now get similar pricing formulas for thc long bond in system going over the same steps. nc fourth row of (6) givesl B
(16)
(18)
'
rq )(1+ rtg)
+ r/ 1 )(1+ r?
-
.pud+
)
pedy
(6) by
Bdu t?
(1+
#JEt
r, I )(1+ rtdz )
(z4.))
Ora
Bt
1
=
E f.
Bt
s
j- + r: j( j ..j.. tj L j
(
j
.
(21)
:'
Here, % is again a random variable. But so is Bt because the time fa is not a maturity date for this bond. For this reasm, the equation in tbis form willnot be very useful in practical pricing situations. Finally, using the second and the Efth rows of (6) yields thc pricing equationsfor the hvo Libor instruments, the FRA and the caplet dcrivative Ct, respectively! ,
0 Ct i
=
=
E z3
,
E
. (1 + %)(1+ %) gz r
.,
.1 )(1+ rtgj c
j- ,;
j
;
%
(22) ..j..
-
(
?it
=
12
x
A
to obtain
k
()
(Ft) =
..
.,.
(2a)
.
r,.)(1 + ni l'
(1 +
.'
E
side by the correspond-
*2
A
E'
c
)(1+
)
rt j
.j
yt, - Lt,d )
r:,
y
(Fd: -
puu+ p cv
(.1+. r ,1 )(1+.,.,a),
E
.
L/u )
-
(1 +
.
.
.
.
/y
l
v'
.
'
1
(1 +
Multiply and divide each term on the right-hand ilm (1 + rs441 + r.i ) and relabel using
!.
(1 +
rt ,
(Fjj - .I,J/, ) LL) d c pu + y . )(1 + r;z) z ( 1 + o )(1.j. tjd)z
u
,
(gy
-
Factoring out the F/j which is independent of thc realization of any future
) f..
,
:.
statc:
Thus, proceeding in a way similar tf that in Chapter 2, and using the savings account to determine the #, does lead to pricing formulas similar to the .,; ones in (18)or (19),Yet, within the context of the instruments ctnsidered l here, and w1t.11stoc hastic spot rates, the use tf risk-neutral probabilities # y turns out to be less convenient and, at times. even inappropriate. lt forces a ; marketpractitioner into handling unnecessary dilculties. ne next section :, illustratcs some of these. '
1
=
Rt Rt,u '/'
uu
+. R ,1 Rutz
I
whichafter relabeling
jjeff
.j.
p ': Rdt,
1
(1+
zt
rt, )(1 + rtg )
.L.)?
.
).
(
1 + rtt j 1 + r?a
)
,
.!.
du
+ R 2, Rdtz pdd,
(24)
+ rjyjs
..,
=
(J,5)
(j s
rt, )(.j+.
ptu 2
)
1
pud +
(j
.j. rt,
)(yo rta) z
pau
psu
cv r/j )(1 + la
(1 +
.
r;; )
pud.j.
(1 +. rt, )( ! +
tj,s
)
?dd
Ld/z
(1 +
rtj
)(1 + oj)
pdu
(29)
.
we can wrjte as2 .1
Ft.E
LLt
#'JJ.
-
puu+
y /2
t
,
gave:
((
+.
d
t
savings account equation:
L
1 ?uu + >s''z+ #du+
uzd
+. rj 1 )(1+ r)g 2)
(j +
.
2.3 StymeUndesfruble P'ropeufes The probabilities f' were genera tcd using the
1
y
y,
.
'
'
(.1
+ rt
x
.-
E
=
--
24.1-f- rtg)
1
#
(1 +
x
g
GJ(1 +
%)
Lt 2
.
(30)
Rearranging. we obtain a pricing fonnula which gives the arbitrage-free
5.
FRA rate Ft : 1
..
Now consider thc details of how these probabilities were used in pricing the FRA contract. First, note that pricing the FRA. means determining an F Jausesuch that the time 1: value of the contract is zero, This is the case lj, all F-'lAs are traded at a price of zero and this we consider as the be arbitrage-free price. The task is to determine the arbitrage-free F implie by this price. From the second row of the system in (6)we have ,
'
Ft
1
=
-
E>
1
-
j )(I+-r,a
(1+rq
)1
E
1
.
(1
.j.
r'
)(1+ o ) z,
tg
2
.
(3j)
'.
rjju expression clds a formula to deternne the contractual rate Ft using the risk-free probability 1. But, urilike the case of option valuation with constant interest rates, we immediately see some undesirable properties of the representation.
.
0
=
(F/I
-
Lt,u )4 uu + (F 2,
-
Lwt, lal
)#JJ. + (Fj , - Ldral''z + Ft 1 - Ldtz
(26) .,' .
E
(
'
;.
.!
cHh
394
p TE R
17
.
2 A Model ftarNew Instruments
Arbitrage Theorem
.395
.'
First, in general F, is not an usbiased Ft,
#
estimate of Lt, :
lf-,al
E
'
(32)
.
'
1
1 =
y;
After canceling we
jr
1
EI 3
;jj
-
(1 + rr)(1 + rtg)
jj
( yoy (.s
E
tfsjnj .
cv
.
The only time this will be the casc is when the rl and the Lt are statiatically separately: nerl, tlae expectations can be taken indepertdent. Ft
Proceeding in a similar fashion for the Libor derivative same argument. The pricing equation will be given by:
:
wjth :
:
.
?
t
g:c
EF
jsyj a
.
t!
jg4;
' Under this extreme assumption the fonvard rate becomes an unbiased es- 'E timator of the corresponding Libor process. Blt, il practice, can we really maturity lbor rates are statistically g say that the short rates and the longer independent? This will be a difhcult assumption to maintain. As ; Consider the sectmd drawback of using the risk-neutral measure with taken expectations we noticed earlier, the spot-rate trms inside the )'' dt'l nt factor Ollt. ln contrast to thc simplc model of Chapter respcct to 2, where r was copstant across states, wc now have an rt, that depen on :' stcchastica thc state u, #. Hence, the denominator terms in Eq. (31) are and stay inside the expectation, i Third, the pricing formula for the FRA in (31)is not linear. This prop- :') sight, can be quite a damaging aspect of the crty, althoug,h harmless at first major inconvenicnces for the mar- .. use of risk-ncutral measure. It creates tlle ket practitioner. In fact, when we tr.!rto dctermine the FIIA rate F,j or namely model fwf.l need processes, to pricc of the derivative Ct, we now . the rt and Lt, instcad of one, the Lr. Worse, thcse two processes are correevaluating y te of task The complicated way. lated with cach other in some expressions. corresponding expectations can be arduous with nlzmlinear .. in a denminated Ftt is the dehnition, A linal ctmmcnt. Note that, by Iiskvalue that will be settled in period la. Now consider how the Curre n jy l Pricing Eq. (31).The Pricing formula wit neutral mcasure / tperate: WW works by Iirst discounting to present a value that belongs to time f3. L: tlies to ; after taking the averagc via the expcctation operator, the formula that is ( this discounted term in tfme la dollars, simply because rcevress ; eventually how the contract is settlcd. freo; arbitrageClearly, this is not a ver.y efhcient way of calculating the altoget jwr discounting the with fotward rate. In fact, one can dispensc d because btlth the Ftt and Ltg are measured in time ts o gars!
ts.
'lahe
we now consider an alternativc way of obtaining martingale probabil tfes, pthin the same setup as in (6)and with the same 4/, we can utilize thc thiy.dequation to write; B;
'
Dividing by B; 1
'
'd
=
=
1
puu+
s) .;
1
ud
sf
+
z1
1
j
v
(36)
1
sjr,
$
du
+
1 RqI:,
.....
/
aa .
;;q
.
(37) (38)
,
.
1
this equaton becomes j
nuu + npd +
=
'
Because thc
'y
:
fj
yyu
o nad.
arc positive under the ctndition of no-arbitrage,
.
zr
fj
>
() '
/> y
.s,
g
(yq) we have
jyj)l
This means that the a new set of synthetic marprobabilitieg. ney yeld a new set of martingale relationships, We call the Wj (hc fonvara measurc. Before wc consider the advantages of the fomard measure, over thc risk-neutral measure, #, wc make a few Comments on the new normalization.
:,
m-i.i could Iyc used as
tingale
u
'n'il
.
,
.
.
E. :
ffd
+
and labeling,
.'
:
du
+
,
.
E:
4*
+
1
1
.
:
.35)
2.4 A New Nonntynwjfx
.:t
.
(
'
Clearly, using the moncy market account to define the probabilities # as done Z Eq, (24.), crcates complicatitms Which were not prcsent in Chapter g. jujow we wjll see that a judciolls chok.e of syntlletic probabjlitieg can get around these problems in a very convenient and elegant way.
.
y
c t,
securitics.
;
gktt
- + r) (1+ rt, )(1
make the
A ain jj the ct is an jnterest-sensitive derivative, the same problems g #s will be random discount factor cannot be factored present. otlt of the expectation and the spot rate wll in all Iikelihood be correjated wjth the optio'n payoff Cts if the latter s written on interest-sensitive
:
(33)
1
E
=
'
'.
G, we
'
.
'*'
396
C H A PT E R
@
Arbitrage Theorem
17
2 A Model for New lnstroments
j
i: .
nil,
First, to move from cquations written in terms of state plices to those expressed in terms of =, we need to multiply a11state-dependent values by which is a value determined at time Hence, this term is independent B'% I of the states at future dates, and will not carry a state superscript. means that it will factor out of expectations evaluated under Second, note that we can define a new forward measure for cvery default-frce zero-coupon bond with different maturity. Thus, it may be more appropriate to put a timc subscript on the mcasure, say, wy, indicating te maturity, 71 associated with that particular bond. Given a delivative written on interest-sensitive sccurities, it is clearly more appropriate to work with a fotward measure that is obtained from a bond that matures at the same time that the derivative expires. Fla, note how normalization is done here. To introduce te prob.. abilities in pricing equationsl we multiply and divide each (J' by the Bt /B; as x'iib this n motmts to multiplyina, i'n the maAfter relabeling the ! each trtx Eq. (6), assct price by the corrcsponding entry of thc short bond by the BtL.s Bst Hence. we say that we are (: normalizing rrhese and relatcd issues will be discussed in more detail below.
:! ';
.
!'
'nis
',
t) A;j =
E '.
.n..
!
a*
*'
Frj ;; i
$'
.
evaluating the similar cxpectation undcr the measure m To do this, we take the second row in system (6) and multiply every element by the ratio B;1 //.D1 which obviouslv equals one;
.
.j ;
;'
.'
'i
)
0
=
(F,k
-
'b
,
u Zlz)
+(F ,, -
-'
'
Bs11
sj1
$ uu + (Frl
B5h
LdJa )
sj
-
l
&11
sr f
1
/
UJ
-1-(y /1
-
kd
f)
)
Bq;1
(44)
+
(45)
of the Libor process Lt, 'This means that we
qvii.
gfsraj
E=
=
,
(46)
,
=
(f,,+zaJ
Et=-a
(47)
,
-
.
8-,''r(.1
. .?,
.
)x
sr
1,
with the lt being the information set available at time /. ln this particular case, it consists of the current and past prices of a1l assets under consideration. Nexq we recall the recursive property of conditional expectation operators that was used earlier:
:
dV
Ar$Tg.lf,
=
E E:
g,
*1
Et*
(42) .:
ldd
.
where the subscript of thc E7f operator indicatcs that the expectation is now takcn with respect to information available at time t + Tlmt is,
,
.
xdljj + Adra
.1
''
'*'
.J.u/2 )
+
F, h
:
Now consider
'
m'du
Thus we obtained an important result. Although the F/( is, in general, a biased estimator of Ltj, under the classical risk-neutral measure it becomes arl unbiased estimator tf Lt, under thc new forwardmeasure m Why is tbis rclevant? How can it bc used in pradice? Consider the following gcneral case and revert back to using the k instead of the ti notation. Let the lbor rate for time t + 21 be Sven by Ldq-c, thc current forward rate be Ft, and consider its future value Ft-a with s > 0,9 we can utilize tlAe mcasure = and write:
.
.'
t,
+
FtL
:
(z, j
I'WJ,
+
Lut,m-ud Ldttavdd z-d/a-rrfl'zj gz-/z-rri'u
,,
'''
#
=
(
2.4. 1 J'zo/pdr/d. of the Normalation We now discuss some of the important results of using thc new probab ility measure = instead of #. We proceed in steps. First, recall that within the setup in this chapter, the use of te risk-neutral measurc, #, leads to an equation where the F; is a biased estimator of the Libor process Lt. In fact, we had
(41)
fa'r'2
w'here the right-hand side is clearly the expectation evaluated using the new martingale probabilities now have:
*
-
'n>$
.
,
.
E#
'n'ud
gfvu/a+
11
5
.
and that tlley sum to one, we
*1
/
ry
Ft,
g>) -
'zri/,
Note that here the B( has conveniently factored out becatzse it is constant givcn thc observed, arbitrage-free price Bf Canceling aad rearranging:
;.
'
397
are in fact the corrcsponding elements of obtainafter factoring out the Ft, :
,'
whichsays that the castsnow.lo
t
Recognizing that the ratios
abest''
EfI''QE11 EI E1 '
=
'
,
(48)
forecasts of future forecasts, are simply tbe fore-
:
gnus F is the FRA rate ebsenrod bsewedwithin a short interval of time
,1'
fk'
# 71
.'
(4g)
Lytgan,
.
;
J
'.
) jk
'.
the
x
t<now,''
A.
whereas
the Ft..s is t14eFRA rate that will be
is used here in the sense of mean square urror.
% :
.;
C H A PT E R
398
17
*
est imator of L t +2a,
New 18 because Ft is an unbiased
under
=
We
;
According to this last eluality, the conditional cxpectation of function of a z, is multiplied by the arbitrage-free price of the short bond. That is, the problem of modeling and calibrating a bivariate process has completely disappeared. Inside the expectation sign there is a singlc random variablc Lt,. Here, we see the following convenient property of the new measure. vjje fomard mcasure, =, first calculates the ex-pectation in time /a (i.e., forwad) doilars and then does the discounting using an obseaed arbitragefrec price B; in contrast, the risk-neutral measure hrst applies a random discount fact'or to a random payoff, and then does the averaging. Note that in proceeding this way, thc risk-neutral measure misses the opportunity of using the discount factor implied by the markets, i.e., the #; during thc Pricing process. Instead, thc risk-neutral measure is tlying to recalculate the discount factor from scratch, as if it is part of the pricing problem, leading to the complicatcd bivariate dynamics. We will see anothcr eumplc of this in the next section.
't
C,an
'jzr
..
write:
''
Ft
.
., .'
r
Fr
=
.
'
'.
.
'
: '.
r,
t1
becomes:
(47),this
Nowa su bstituting from relation
f5o %
,,
z, L.l-'f+2l1 E+.,s
El
=
;
7 <
place'':
iright
F/+a
Et'r gA-r+ a)
.
' ,
,,
)
T'CE .
.
.
/ .
'
pays the
is the price of a cap Iet. At expiration, t13e
.
';.
(51)
,
measure whicbsays that the process ( pt ) is a martingale under theftlrwwrd will be ( forward vcry prices convowill this property f see later, we 'w.As when preliminary A instruments. interest sen sitive pricing some rate nient already be Scen by looking at the similar conditional this of cam le examp expectationfor the derivative Ct. caplet Suppose the
CE
'.
to introduce an
conditional expectations
and use the recursivc property f operator at thc
(49)
Etvrjz,2+ aaj
=
t
E=
2 A Model for New lnstruments
1
Arbitrtge Theorem
@ ;
Sum:
c s someImpnctzzifms .
'.
Ct,
N max
=
(z, ,z
K, 01
The proccdure followed in previous scction chose a bond which maturcd at time t = rs in order to obtain a synthetic probability (measure) under whicb the martingale equalitics turned out to be more convenient jor pricing purposes. ne choice of B';j as the normalizing fador was dictated partly by this desire for convenience. In fact, any other asset can be chosen as the normalizing variable. Yet, the fact that B8;!:L matured at timc tz made the time f(!: value of this bond constant. ne convenience of the conditional expectatlons obtaincd under is the result of this vcry simple fact. lt is tltis last property that makes jj ij tjae coetticients of the &, d-dependent terms Ltz or Ct constant relative to the infonuation set availablc at time f: in equations such Because tlAe
'.(
(52)
,
'
:E:
).
whereN is a notional amount t hat we set equal to one, the K is the cap-rate fz. The selectedat time and the Lfz is the Libor rate re alized at time the provides purc uaser Payment is made in-arrears at time t5, and hcnce, borrowing costs in increascs against insurance sort caplet the f o some of
:E
,
f
: ,
beyond the level K. (f the clasHOW S hould one plice such an instrllment 9 Consider thc use risk-neutal sical isk-neutral measure ? Using standard arguments and tjje probabiiity 15 we bave
..
.
.
,
'5'
r,
.
6'
,
as (28). they were constants, these cocflkients could be factored out of the expectation operators. This is an important result becausc it eliminated the need to calculate complex correlations between spot rates md future values of interest rate dependent prices. Also, due to this we avoided working with raadom discount factors. sut the choice of normalization was important for another reason as wcll. Under carefully chosen normalization, fonvard rates such as Ft, beeome martingales, and were unbiased estimators for future values of spot rates such as Lt-. implies that one can, laeuristically speaking, replace the future of a spot rate by the corresponding fonvard rates to tind the current arbitrage-free price of various interest rate dependent securities.
7
Ct
=
1
E
S
(1+
1
maxz'/a
r,1)(1 + rljt
v A,
-
a, k'l
fqa '-
'
i;: '
:
As discussed earlier, in this pr ic ing equation, the (random)spot rate % Libor ra te Lt , and hence the witb tlle (random) is likcly to be correlated and calibrate a bivmiate proc ess will be forced to mo deI mr ketpractitionerprice tlw caplet. oruer to f''' measurc in the last equat ion of sptem (6) pve s uszg t forwara
'
,
r''
in vet,
)
.'
: .
.
he
c,,
=
,
/:4: .
E(kI
B;,E%r
%-
,
which means
c 1!
=
BstvEv max
g1,u -
K, 0)
.
'nis
'
(55)
-value
,
.k :
'
.
.
.; .j
C H A P T ER
400
.
Arbitrage
17
q.
'l-heoreal
said in this chapter to two
we close this sect ion by apygng what was time setling. pricingexamples in a cont inuous
2 A Mtxlel for New Instnlments
:'
.
risk. Obviously, assuming deterministic spot rates would not be very approPriate here. But, if the assumption of deterministic rt is droppedx then the discount factor does not factor out and we cannot use Eq. (59)under A The fomard measure can provide a convenient solution, Using the arbitrage-free price of the discount bond Bt, r + j, we can instead write the pricing equation under the forbvard measure: p; ej gl(. r + slLp( Lpjxsj
,
j :. .
.
.:
.
7
Wr itC
r
,,
-
'
>
U;
k
2
Blt, T + 8)Et
.
=
'
.
p;
ge-
V+'rudulyt
wsj?s
zzwlxj
-
.'
:
=
.
1.v.)Nj
(62)
.
=
.
.
rjnja
,
:(
gp
()
g(/$-
f;
Et gy v j (,y; is an equation that one can exploit convcniently to find the arbitragefree value of Fr. 'I'he critical point is to make sure that in calculating this average one uses the fomard measure = and not the risk-neutral prob-
'
'171us,
6
0:
=
G
;
exchange of xow, we know that forward ctmtracts do not involvc any We have initiatim iation.li contract at cash at the time of init #(
Now, usc the fact tllat
(56) ,
.
(6j.)
,
s () js the tenor of Lv. Here Bt, r) is a value obsen'ed at time sence.it factors out of the expedation operator'. wjyere
.
*
.
=
J,
,
.
E E
(
J, 5. l Fe FRA Contract -Z,r)AT, at some Suppose we have a FRA contract t hat pays the sum Ft and Ft is the forward notional amount is where < N T, a futuredate F+, t variable, L The F t is obsened at contract-tne f. z. Price of the random T + &, the Bccause this is a cash llow that belongs to a futurc datc the casb flow will be pven by usu ap, current valtle denote d tyy#: of tbe using by the is discounted martingaleequality, wbere t he future cash tlow risk-neutral measure we can thc Under &. + risk-free rate rs, t S s S r
4gl
,;
ability
! LIAC formula in (56)7Because Ft is k How can this price be zet givcn Vailishes. If thc spot rates chosen so that the r ight-hand side expedatin dO. A Value for Ft will be to ;: js eagy th very determinis%tic, assumed to be ar e fador, liscount the out factoring obtained by c-an be easi jy i: VO t r dv F:3 Zzl N&, ) (58) J'l 1 t (F t
p
.
.
2.5.2 cplct As a second example to the power of the fomard measure discussed abovc we consider pricing issues involping caplet, Let Ct be the current a price of a caplet written on some Libor rate Lt with tenor &and with cap rate K. suppose the notional amount is N 1 and that the caplet expires 1, except for the notation on Lt. at time F. We 1et According to this, t-liebayer of the caplet will receive the payoff ,4
,
,'
=
then setting t he
JZ)
Id
M
-
=
.
eqtiai to zero and 0
=
Canceling:
Eti' IFt
-
f-rl
The Fj1. that ma kes the currcnt Price one where Fr Et? EZrl =
',
(59)
,
the forward cmtract
t:f
=
tEt
zero
iS
the
cr
'
(601i
price is equal to the That is, when spot rates are determinist jc the forward risk-neutr al meaure #. Blackthe frecagt of the futufe Zz under Of constant interest rates at varscholes framework exploits the assumpt ion usable the stock same assumption is not opt ions. But, ious points in pricing important The most when one is dealing with interest-sensitive secur ities. hedge interest rate d gucb to is the traded nee securities are reason t hat ,
Kbest''
: : FE
Ct .
L
K, ()j rz-w..: this instrument
=
Etp - /,r r, as maxzw-,
U
-
K, (jjj.
(64)
We also know that at time r - the Fp-s will coincidc with Lw-.12 so we may dccide to use F t as the After all as time passes,
'
E k
Sunderlying.''
,
,
cash exchan#e's, but are Prtw ided as a guar- ; ' llAny margins that may bc retltl ired are not . .(. Settlements future. n tlle E an tCe toward
SPOt
.
.(
max
at time r. As mentioned earlier, will protect the buyer against increases in Lp-s beyond the level K. Normally 0 < & < 1 and in the above the rigbt-hand side will be proportional to 8. How does one price this caplet? supposewe decide to use the riskneutal probability #, We know that the arbitrage-free pricc will be given by2
j ,
.
=
I2At axy time, the fomard rate for an immediate rate for that period.
loan of tenor &'will be the same as tbe
'
:'
:
.'
(2 i4 A P T E R
402
*
l7
Arbitrage Theem
t;:
2 A Model for New Insrruments
403
. ;;
D
Thks this variablc will eventually coincide with the futtzre spot rate Lr-n. process is called the forwardLibor process. with a This suggcsts that wc model log-normal folward rate dynamics probability #, original under the Wiener process W( defmed
2.5.3 Ntprmljzc/)n as a Tool Above we discussed the important implications of normalization and measure choicc from the point of view of asset pricing, with particular emphasis on interest rate sensitive securities. Are there any implications for the mathcmatics of snancial derivatives? We see from the above discussion that the fundamental variables are in fact tbe state prices (f/J. When thcre are no arbitrage oppormnitics, these prices will exist, they will be positive and will be unique. Once this is dctermined, the hnancial analpt has a great dcal of llexibility concerning the martingalc measure that he or she can choose, ne Fnthetic probability can be selccted as the classical risk-neutral measure # or thc fomard measure m dcpending on the instruments one is working witb. Hcnce, the issue of which measure to work with becomes another tool for the analyst. In fact, as suggested by Girsanov therem. one can go back and forth between various probabilities depending on the requirements of thc pzicing problem. Jn fact, consider a normalization with respcct to BI and the corresponding mcasure that we just used. Clearly we could also have normalized with the longer maturity bond Bt and obtained a new probability, say given by:
f. .
:,
..L'
;; ,
,
J
dFt
=
:,
pF, dt + o.Ft dls
'S
enviroluuent. and then apply tlic Black-scholes 1ogic as in a Black-scholes to determine the Ct. riskIf we proceed this way, thc fint step will be to switch to /, the mazingale The is not Ft a neutral probability, But this creates a problem. under /. So, as we switch probabilities and use thc Wiener process, V,, deEned undcr >, the forwazd rate dynamics will become
q:,
E .
,'
: :
1
:) .
''
dt
=
g.F/tl1,r,, g F, dt + s
,.
;.
'
':
theoremwherc the p,* is the new riskmdjuated drift implied by the Girsanov E' Black-scboles unlike the Under / this drift is not known at the outset. So, replaced by the known stock price is l casc where the drift of the underlying unknown to . difqcult with a (and constant) spot rate r, we now end up dctermine. Consider what happcns to the forward rate dynamics if we use the k w1t11 instead. Under the forward measure obtained forward measure will be a & T time for Tl-normalization, the folward rate Ft delined l BLtb 13 martingale. Hence w: can write.
.
E
'lv
'-
,'
'
'/ tj =
,,
';r
1
#
Bt
i)
(65)
.
:i
-
A11prices that maturc at time F would then be matingales oncc they are nonualized by the Bt. Note that the ratio
,
.
';
#F t
=
eF f #H(c
,
:
f'
A very convenient property () is a Wiener process under where the unbiased of this SDE is that tlie drift is equal to zero and the Ft is an I'F;A
=.
1
i'
7r /
,'?
--J B
1.
ia' N--
cs timator of Lr-s:
) Ff
E'
=
,!
can be uscd to write:
::'
(fvz-al ,
E.
f*lm diffictty of determining an unknown drm. We nere in Mt' Black-scholes type argument and price this caplet J go ahead with afashion.l4 ) straightfonvard
'J
.,
is no additional
?
13Itis important to rcaliz, that under Will not be a martingale.
it
(66)
7
-Bl
'
fomard ro'? differgrlt normalization thb particular
?. .'''.'
kt
: Elo need to discllznt tile OP Black-k,lei . 14Aremazing difference is in the units used hre. 'There is unlike the namics. rate forward nis the if use we present the payoffto cxpressed in time dollars. ' whcre the stock price dynamics dS( are i' envirorlment
:
'
fji =
-,
'&
i.l
-
B/
&
.
l
This way one can go from one measure to another. Would such adjustmeats be any use to us in pricing interest rate sensitive Securites'? The answer is again yes. When we deal with an instrument that depends on more than onc Lv with different tenors F, we can Erst start with one forward measure, but then by taking the derivative with respect to the other. we can obtain the proper terms'' that need to bc introduced. tlcorrection
-'
(67)
'
.
:LL ::%.
C H A PT E R
404
*
17
r
Arbitrage Thcorem
3,
. The annual volatility is
!
and forward In this chapter we introduced the notions of normalization role in priclg delivative securities measure. nese tools play an important in a convenient fashion, More than just theoretical conccpts, they should be regarded as impolant tools in pricing assets in real world markets. ney are especially useful fgr any derivative whose settlement is done at a hzture
g'
!
i
.
, '
g
,
;
yt
iTup''
E
(Tdown''
2. Suppose at time t 0, you are given four default-free zcro-coupon bond prices #tf, T) with maturities from 1 to 4 yeal's:
:',
=
2 '
'
/40, 1)
:%
.94,
P(0, 2)
=
.8(9, 3)
.92,
=
.P(0, 4)
,87,
=
,80
=
.
(a) How can you a spot-rate tree to these bond prices? Discuss. (b) Obtain a tree consistent with the term stnzcture given above, (c) What are the differences, if any, betwecn the tree approaches in Ouestions (a) and (b)? dt''
y .')-
''
r:( E.
4 References
3, Select tcn standard, normal random numbers using Mathematica, Maplc, or Matlab. Suppose interest rates follow the SDE:
)
ne book by Musiela and Rutkowski (1997)is an excellent source for a reader with a strong quantitative background. Although it is much more demanding mathematically than the present tcxt, the results are well worth the efforts, Another possible source is the last chapter in Pliska (19W). Pliska trcats these notions in discrete time, but our treatment was also in discrete time.
h
drt
:'t';
dt +
.02r/
=
.06r/
:11s.
ytssume that the current spot rate is 6%.
. ,.
(a) Discretizc the SDE given above. (b) Calcmlate all estimate for the followirlg expectation interval ,04,
'.(
5 Exerclses
E
j-
'
csdsmaxyj
(ljj
.06, -
,
and the random numbers you selected. Assume that thc expectation is taken with respect to the fme probability. (c) Calculate the sample average fo
1E
on the spot rate rt'. I ,.!
..) .' T
#rt + e'% J?l';.
E
gc-V
''daj
$
* Thc annual drift is
and then multiply this by the sample average for:
r
E
gmaxlr.
Do we obtain the same result? (d) Which approach is correct? :
.t j
using a time
=
'
* 'T'he rt follows: =
.
(a) Suppose instruments are to be priced over a year, Determinc an appropriate time interval A, sth that binomial trees have five steps. (b) Wlaat would be the implied u and d in this case? (c) Determine the tree for the spot rate rt. (d) What arc the and probabilitics implied by the tree?
1
t
s'
normalized by tho * The price of all assets considered here, once of becomes F), bond Bt, arbitrage-free price of a zero-coupon a martingale under
12%
=
The current spot rate is assumed to be 6%.
r;
date, in future dollars. Some of tbe main results were the following. When we use the fon ward measure n'v obtained from a default-free discount bond Blt, T), three things happcn:
drt
405
I
,.
3 Conclusions
Suppose you are given the following lformation
Exercises
k '
J
' .
-
.06,
0))
.
406
C H APT E R
.
17
Arbitrage Theorem
(e) Can you use this result in calculating bond prices? dynamic (9 ln particular, how do we know that thc interest rate arbitrage-free? displayed in thc above SDE are dynamics if we (g) What would happen to the above interest rate switched to risk-neutral mcasurc #? arbitrage-free bond prices, How (h) Suppose you arc given a series of framcwork in obtaining the can you exploit this within the above arbitrage-fzee dynamics for rt
Modeling Term Structure and Related Concepts
l lntroductlon ne previous chapter was important because it discussed the Fundamental Theorem of Finance when interest-sensitive securities are included in the pidure. We obtained new Tesults. The issue of normalization, the use of foward measures within Libtr instruments, and ways of handling the simultaneous existence of bonds with differing maturities was introduced using a simple model. Now it is time to take somc steps backwamdand (11.scuss the basic concepts in morc detail before we utilize the results obtained in Chapter 17. In particular, we need to do hvo things. The new concepts fl'om Exed inCome are much more fragile and somehow less intuitive than the straightforward notions used in the standard Black-scholes world. flxed inCome concepts need to be de:ned first, and carefully motivated second. Othemise, some of the reasoning behind the well-known bond pricing formulas may lAe difEcult to grasp. Next, at this point we need to introduce somc. important arbitrage relationships that are used repeatcdly in pricing interest-sensitive securities. The next cbapter will consider two ftmdamentally different methodologies llsed irl pricing interest-scnsitive securities. nesc are the so-called classical approach and the Heath-larrow-Morton approach, respectively. Our main pupose will be to show the basic reasoning behind these fundamentally different methodologies and highlight their similarities and differences. But 'rhese
407
.' .'.('
408
: H A P T ER
18
Modeling Term Structure
and
:
Related Concepts
Main Concepts
to do this wc must Erst introduce a number of new arbitrage relations that exist between the spot rates, bond prices, and fonvard rates, ne til'st arbitrage relation that we need to smdy is the one behveen invcstmcnt in very short-term savings accounts and bonds. Suppose both of these are default-free. How would the long-term bond prices relate to depositing money in a shol-term savings account and thcn rolling this over continuously'?l It is clear that when one buys a Ionger term bond, the commitment is for more than one night, or one month. During this period, scvcral risky events may occur, and these may affect the price of the bond adversely. Yet, the overnight wcstmcnt will be mostly immune to the risky events because the investor's money is returned the next and hence can be reinvested at a higher overnight rate. Thus, it appears that long-tcnn bonds should pay a premium relative to overnight money. in order to be held by risk-averse investors. ln thc Black-scholes world the switch to the risk-neutral metsure eliminated these risk premia and gavc us a pricing equation. Can the same be done with interest-sensitive securities and random spot rates'? We will see that the answer is yes. ln fact, the clas- ' sical approach to pricing interest-sensitive securities exploits this particular arbitrage relation cxtcnsively, The second arbitragc rdation is specilic to Exed income. Fixed incomo ? markets prode many liquid instrumcnts that arc almost identical except j for their maturil. For example, we have a spectrum of discount bonds ; that are differentiated only by their maturitjr. Similarly, we have forward )( asped : rates of different maturities. It turns out that this multidimensional of interest-sensitive instruments permits writing down complex arbitrage rclations between a set of zero-coupon bonds and a set of forward rates. I ln fact, if we havc a k-dimensional vector of bond prices, we can relate . this to a vector of fonvard ratcs using arbitrage arguments. nese arbitrage relations form the basis of the Heath-larrow-Morton approach to pricing interest-sensitive securities. should 7 Th us, one way or ano ther the material in the present chapter bc regarded as a necessary background to discussing pricing of interestsensitivesecuritics, :'
idlong''
).
:
?
'
',
'
.:
'
)
au overnight interes
.
'rhcse
B( l F)
=-
e-'lft'
r)(F-J)
(1)
It is the rate of return that corresponds to an westment of #(f, F) dollars which returns one dollar after a period of lcngth LF /1. Herc, the use of ah exponential function justifiesthe term continuoualy compoundcd. Note that there is a one-tonc relationship betwcen the bond pricc and the yield. Given one we know the other. arc also indexed by the same indices T and t. Next, we need to define a continuously compoundcd forwardmfc F(/, X U). Tbis concept repzesents the interest rate on a loan that begins at time r and matures at time U > F, The rate is contracted at time t, although cash transactitms will take place at future dates F and U. The fact that the ratc is continuously compounded implies that the actual interest calculation will be made using thc exponential hmction. In fact, if one dollar is loaned at time T, the money returned at time U will be given by: rrhey
e #(l,:&)(&-r)
(a)
.
Note that the F(l, T) &) has (hree time indices whereas discount bond prices each came with two indiccs. nis suggests that to obtain a relation between fonvard mates and bond prices, we may hav to use fu,tl differcnt bonds, Bt, Tj and #(f, &'), with maturities F and U, respedively. Between them, these two bond prices will have the same time indices (/, 6 &).
.:
2.1 lrhxee Cumwes '
ne basic concepts defned in the previous scction can be used to dehne used routinely by market professionals. These three artz the yield eunge,the diseount curve, and tlze credit-spread cun. ne so-called swap ''curvcs''
E
We begin with some de fmitions, some of which werc introduced carlier. r The price of a discount bond maturing at t ime F obscrved at time / < T ; will earn
'rhe
'
2 Main Concepts
investment
will be represented by the symbol #(r. r), rt will again denote the :1riskless borrowing. rate stantaneous spot The spot ratc is instantaneous on Jn the sense that the loan is made at time t and is repaid after an infinitesimal period dt The spot rate is also riskless in the sense tbat there is no dfault risk, and the return to this instantaneous investment is known with certainty,z two desnitions were seen earlier. Tbe Iirst new concept that we now dcsne is the contirmously compounded yield, Rt, F), of thc discount bond, Bt, F). Given t.he current pricc of the bond #(/, F) and witlz par value $1athe #(1, F) is deined by the equation:
,
1In practcc, the slaortest-erm
409
rate.
...,
L
l'Bay
kBut, as we saw earlier, the sfxt rate itself carl be random vaable a well have a market rsk it rollcd-over.
and the
westmcat
410
CHAPTt R
.
Modeling Tem Stntcture
18
and
Related Concepts
in hxed income marc'urve, which is perhaps tbe most widely used culve kets, is omitted. nis is due to the limited scopc of this beok. Wc do not consider instnzments. We only deal with mathematical tools to study them. The forwardcurve which consists of a spcctrum of interest rates on for!he ward lcans contracted fcr various future dates will be discussed later in chapter. 2.1.1 The Ke/# Ct4rve ne yicld curvc is obtained from the rclationship betwcen the yield Rt, T), and the discount bond price Bt, F). We have: Bt, T)
=
e -a(l,z)(r-2)
(g)
,
whcre #(f, F) is the arbitrage-free price of thc F-maturity discount bond. Thtus, to obtain the yield R(f, W) of a bonda wc lirst necd to obtain its price. Thcn. Eq, (3)is uscd to get the continuously compounded yield: 1og(1) - log B(t, T) Rt, F) (4) T- t =
=
logltf,
-
F) .
T-I
wil bo Here, we have 0 < Bt, T) < 1 as long as t < T. Thus Iogg#tf, F)1 will posjtiveR(t, T) be the a negative number, and hence Now, assume that at time / there exist zero-coupon bonds wit.h a 111 where T'''dx is the longest maturity spectrum of maturities r (E gl,T'MA'I of these bonds be given by the set price the available in the market. Let B(t, F) in this set wc can use Eq. (4) For each (#(/, F), T l (f, T). nen we have the following A(f, corresponding and obtain thc Seld definition. (I, i.s ON: The spectrum of yields (R(f, F), F DE FrrJ&'r1).
T'Mfll
called the yield curve.
between the yields of the bonds ne yield curve is a corrcspondence respective maturities. and their risk celain class belonging to a is an extension of the yield given above of the yield The dehnition curve yield curves provide the spco curve notion used by practitioners, Observcd Trcasuries, at a Jinitenumber of maturities, Hero, trum of yields on, say, that at any time t, there is we assume not only that time is coptinuous, but bonds. An invcstor can always buy and sell a a continlum of pure discount mamrities extend liquid F-maturity bond, for any value of F < T'''&'r.These from the immediate tenor, F
=
f+
#f,
(5)
2 Main Concepts Tmax prtwiding a continuous yield to the longest possible matuzity F there curve. According to this assumption. given an arbitrary T < the corrcsponding yield because it will be will be no need to directly obsencd in the markets. =
F'''J2
i'interpolate''
2.1.2 The Discount C'urkz In spite of thc popularity of the term cations instcad use the discount canze.
curve,'' most market appli-
'dyield
DEFINITION: 'The spectam of default-free zero-coupon bond priczes jlt, T), T EE gf,FIJ,with a continullm of maturities that belong to the samc risk class, is called the discouns curpc. ne discount culve is more convenicnt to use in valuing general cash cfvwJ represent a general cash fltw to bc flows. In fad, let the fch < reccived at arbitrary times rj < Fz < L F. The present valuc CF/ of this gcneral cash flow can be obtained by simply multiplying the amotmt to bc received at time T by the corresponding Bt, Tf). In fact, the discounted value can easily be obtained by using arbitrage-free zerocoupon bond prices with maturities falling to the corresponding T;.. nis present value is ,
.
.
.
,
=
ch J-ll at, z'jlcy,i =
(6)
,
izzz
The reason why this works is simple. ne price S(/, 7l) is simply the current arbitrage-free value of $1 to be paid at time T). The discount is dircctly quoted by the market. Hence, thc discount curve will play an essential rolc in the daily work of a market practitioncr. 2.l.3 The Credit Spread Curve Yield cuwes and the discount cun'es are obviously valid for bonds of a given risk class, When we look at the spectrum of bonds fB(t, T), T i (q T''=1l, we implicitly assume that the default risk oxt these bonds is the same. Othenvise, thc difference between yields would not just be due to differenccs in the corresponding maturities. Hence, for each risk class we obtain a different yicld curve. (discount) rrhe difference between these yield (discount) will ctkdit indicate the cuncs mrctztf:, the supplcmental amount riskier credits havc to pay to borrow money at the same maturity. The cocxistence of different yicld cun'es that represent different risk clagges leads to the so-called credit-spread curp'e. DEFINITION: Given two yield cun'es (R(f, F), F i correspond to dcfault-frce bonds and the fRtb,T e
gf,T'N/IJJthat rrrliullthat
tI,
7
' L.
CH A PT ER
18
.
Modeling Term Stnlcturc
and
Related Concepts
correspondto bonds with a given default probability, the spectrum of the spreads, (s(/,F) fktv Rt, Tjj, T d II. F rcvx jJ, is calvd the =
credit
,pzrftf
curve.
2 Main Concepts
?
.'
-
/,,
.
Yield Curve
.
?
.
Indeed,
besome practitioncrs prefer to work wit.h a correspondence tween the crcdit spread and the maturity, instead of dealing with the yield . use of the credit spread cun'e will be more practical if the curve itself, traded instruments are written on the spreads rathcr than the underlyipg interest rates. In this book, we omit a discussion of credit instruments and' E.l assumethroughout that thcrc is no default risk. Hence, there is only one yE risk class and there the default risk s assumed to be zero. i:..
'rhe
''
.
l
,,.'
.
1
.
2.2 Mxemeats
(n4
t
.'j. f..
Yicld Cuae
Malurily
')
'
Before we deal with more substantial issues we also would like to dixazu the comparison between a shjt in the yicld cuve and a movement along it: TE Given a yield curve, qRt, Tj, T l g/,F ''MZIJ continuous in F, note that J? at time /, we can considcr fww different incremental changes. First, at atly instant f we can ask what happens to a particular RLt,T) as ?- changes by 'J a small amount dcnoted by JF. Here, we are modifying the nmturity of A particular bond under ctmsideration, namely the one that has maturity F, by' dT. In othcr words, we are moving along the same yield curve. Accor y to this, if the yield cun'e is continuous and smooth, we can obtain the dcrivative; '1
changes, time will pass, new Wiener incrcments are drawn, random shockts) affect the spot ratc and the yield cun'e shijts. It is important to realize that as f increases by dt, thc cnfre spectrum of yields will, in generalv change. nus, the dynamics of Exed income instruments are esscntially the dynamics of a cunzc rather than the dynamics of a single stochastic process. The implied arbitragc restrictions will be much more complicated than the case of the Black-scholes environment. Af-ter al1, wc need to make sure that the movements of an entire cul'vc occurs in a fashion that rules out arbitrage.
,
t
'(;.
p.
.f(
'
.
:
y,
J
1
't'1
Ij.
.
dRlt, Tj #(F). LLIT =
'
Ej '
)8
tr E
(7)p,z
!g'..k
T), F G Il, F'''&21).Tlm-''. . nis is simply the slopc of the yield cunre (.&(/, quantitics are shown in Figure 1. The g(F) is the slope of the tangent to tlm'! continuous yield curve at maturity T. Figure 2 displays the correspon . '. situationwith the discount curve. Yield curves are generally classilied as negatively sloped, positivel/is As the shape of the cttr@xi sloped, and flat. ney can also exhibit .$ changes,the slope cbanges s well, lt is important to realize that an increrqi EI
'j
.
,
.
Gthumps.''
1
.
.'
l'lutaq
involving bonds with different mattlrities at the t and, at time r, every Rft, T) is known. Also, because there are no Wien oj 1,: increments wolved in these movements, the derivative emn be taken in l.' (E standard fashion without having recourse to Ito's Lemma, vlns contemplatc is second change A that we can a type of incremental ation in the time parameter /'. ne incremental change in the spednlm Shocks. M y yields Rlt, T) due to a change in time f, will involve rarldom .:...r t an
Cvcrimcnt
kt?o;d
Discount Ctu-ve
:
i
j
I
.*'*
'
' . .
:
..-....--...--....
s
: :: ::
E E E
u
u+l
slj)jle n
-
B(t, u)
'
:
Fl GUR E '
B(,u+)
Marity ,r
2
!
.
.'
!:
414
(2 H A PT E R
*
Modeling Tenn Structure
18
and
..
3 A Bond Pricing Equation
Related Concep/zr
415
)
Othemise, there will be arbitrage opportunities. It is inst-ctive to review tbe underlying arbitrage argument within this setting, c-'(T-') First, suppose #(/, F) > Then one can short the bond during :hc pcriod gJ, F), and invest e -r(T-,) of the proceeds to risk-free lending. At t ime T the short (bond)position is worth -$1. But the risk-free lending will return +$1. Hence, at time T the net cash flow will be zero. But, at t the investor is still left with some cash in the pocket because
Also, note that for stochastic differential' s such as dRt, T), Ito's LentmaE needsto be used due to involved wienercomponents. we are now ready a fundamental pricing equation that wiIl be uied tbroughou'z Et to introducc second part of this book, namely, the bond pricing equation. the '
.
'
;
:;hi' ':
sj/
ln tbis section we start discussing thc rst substantial issue of this chapter. wc derive an equation that gives the arbitrage-free price taf a default-free zcro-coupon bond #(f, F), maturing at timc F. We go in steps. We flst bcgin with a simplifed case where the instantaneous spkt rate is constanta and then move to a stochastic risk-free rate. This way of proceeding makes . . it easicr to undel-stand the underlying arbitragc arguments. !,
,
gn)
''
.
:
-
-,(w-,)
e
'.
-E.
. s
Thus we 'hrst let the spot rate rt be constant:
r, 5
Bt, F)
e.
r(r-t) .
yqt p)
:
'( .
.,.
r
Consider the rationale behind this formula. ne r is the continuously coml r c-r(T-/) pounded instantaneous interest ratc, ne ftmc tion p lays the role L.i exponential f function F the of a discount factor at time t. At eqlmj s a1I times. the otlter bond, At same as the maturity value of the ,1 whicb is Side of right-hand 1. the < F, the exponential Hence, fador is lcss than t f ; discounted to time-r dollar, Eq. (9) represents the present va 1u e of one :( at a constant, continuously compounded rate r, Now an investor who faces thcse instruments has the following cboices. , can invest c-r(F-0 dollars in a risk-frce savings account now, He or sbe worth will T be $1 Or the investor can buy tbe r-mattllitr and at time this reium 1 and F) dollars now. This investment will also discountbond pay B 'f i risk, default anu T. Clearly we havc two instruments, w ith no $1 at time with the same payoff at time F, nere are no interim payouts either. lf interest rates are constant and if there is no default risk, any bond that promisesto pay one dollar at t ime T will have to have the game prke as the initial investment of et-r('r-') to risk- fr ee lending. That is. we must have.. y
=
i
I
!
! 1
.().
such arbitrage
j
opporttmities is
'
=
r-r('r-))
'
(1))
q
'
i
.
I
3'2 Stochzzstic Spot Rztcs
:t'
','
earned during the innnitesimal inten'al gf,l + #fl. Thus, rt is known at t, but its future values :uctuatc randomly as time passes. The Fundamental of Finance can bc applied to obtain an arbitrage relation between and 7.) tjw stochastic spot rates, rt, t l gI. r1. s(/ tilizc the methodology introduced in Chapter 17. We take the curwe u rent bond price Bt, Tj and normalize by the current value of the savings account, which is $1. Next, we take the maturity value of the bond, which is $1 aud normalize it by the value of the savings account. 'Fhis value is cqual
:
,1
:'
.
,
.1'
.
r
#(f,
F)
=
e
-r(T-l) .
(10)Ek' .
': ..
.'.;.
'
! : ' '
.
;
i:
h
.
.
1
erheorem
,
,
:
whenthe instantaneous spot rate rs becomes stochastic, the pzicing formula in (9)wili have to change, Suppose rt represents the risk-free rate
.
,
!
'
Hcncc, this relationship is not a dehnititn ., or an assumption. It is a restriction imposed on bond prices and savings accounts by the requircment that therc are no arbitrage opportunities. Notice that n obtaining this equation we did not use the Fundamental Thetarem of Finance, but doing this would havc given exactly the same result.S
E
(9)
,
'
''bond
>
.
=
,
(
Then the price of a dcfault-free pure discount bond paying $1 at time F will be given by:
I
()
>
sjj vj
-
The only condition that would eliminate when thc pricing equation'' holds:
:
(8)
n
=
e-r(r-/)
rjxex
'
3.1 eonsfunf
-
.
.
x
Raze
E
1 !
The other possibility is Bt, F) < e-rl r-f) at time t, one would -r(z-J) dojjars, and buy a bond at a price of #(1, F). When the borrow e (jate arrives the net cash Qow will again be zero. The +$1 received ma (uyjty from t jw bond can be used to pay the loan off. But at time /, there will be a nct gain!
,
'pot
,
,
i.
3 A Bond Priclng Equation
E '
il ,
. .
(
3see
F-xercise 2 at tlx end of tbis chapter.
.' :
.j
!
j
j
I
T 416
CHA PT ER
*
18
Modeling Tenn Saucture and Related Concepts
.3 A Bond Pricing Equation
'
'
.r
rsds because it is the rcturn at time T' to $1 rolled-over at the instanto eh Dividing $1 by this value of rate rs for tbe entire period s e (f, taneous r. a e-lk ' savings account, we get the
.
i
.
According to chapter17, this normalized bond price must be a martingalc under the risk-neutra! mcasure /. Thus, we must havc Bt, T) e-f
wherethe term appliedto the par
Gd,
value
EI) =
gr-.lfrszj
E'
t
(1a)
,
ij
('
.
)
i,
''.
I''
can also be interpretcd as a random discount factor
I
'
.
$1.
.
Some further comments about this formula are in order. First, the bond ?.: priceformula givcn in (12)has another important implication. Bond prices dependon the whole spectrum of future short rates rsmt < < r. In other words,we can Iook at it this way: the yield curve at time t contains a11the information concerning future short rates.4 relevant Second, therc is the issue of wliich probability measure is used to calc'u- t. latethese expectations. One may think that with the class of Treasury bonds . being risk-free assets, there is no tikpremium to eliminate, and hencc, there ! is no need to use the equivalent martingalc measure. This is, in general, incorrect. As interest rates become stochastic, prices of Treasury bonds will i ; risk.'' contain amarket ney depend on the future behavior of spot rates ). E and this behavior is stochatic. To eliminate thc risk-premium associated r. With Cisksy eqtlivalent martingale WC nCCQI SIICU to tlse measures in evaluatill CXPIVSSiODS 2S ill 12) ( We now discuss this formula using discrete intervals of size 0 < 'This ' will show the passage to continuous time, and explain thc mechanics of the )( : bond pricing formula bettcr.
lI
('
-
',
WC n0W
':.
,
Sctting.
(
.
'
(1+
'
j
'
,J
1
(1+ G)(1
+ r,+aA)(1 + rr.j-zaA)
,
:
' :'. t
l t
.':.. ;
(13)
y :
;'
.
: .
'
L.''
e
+r,+-aa)
-->.
(14)separatcly,
.-rr,
e
-r!
I
,SA
.e .'
-r; ,aaA
to
t.16)
e-
p
..
tl
g-rl-roa...-r.aja j
0:
.
'
provide the optimal forecasls in the sense of minin um mean square error givcn an irdbrmation set It
(15)
zzu
or as
.
exmctations
+ r-+.aa)...(1
. ;
)
,.
Now, recall
'
j
(1 +r)(1t
( j4)
'
.
,
;
.,
.
Next, apply this to each ratio on thc right-hand sidc of obtain the approximation .g,= '
:
r, is the known current spot rate on loans that begin at time t and and r,+a, at timc f + are unknown spot rates for thc two future o-yca peods, urilikethe case of continuous timc, tbese are simple interest an by market convention, are multiplied by a.
e-rj
(1 V rjA)
t
3.2.1 Diacrete Tzr/e Consider the special case of a three-period bond in discrctc time. lf A represents some time intezval less than one year and if f is the
.
t
,
!
conditional
-
=
,
..
4lkemembertlaat
1
.s,p
r;A)(1 + raal (1+ rt ruas) W ith the condition that the I + n. is selected so that F the approximation that when rj is small one can write:
s
wbere end time rates, d
'
:
,
Et
thc hcuristics of mtwing to continuous time in the present setting, the formula
we move from three periods to an n-period
-
:
=
Show
s( t r;
.')
:
AS
to Contvvl4m:sTime
(13)becomes:
':!
.
'irisk-free''
3.3 Mxing
,1
u
Bt, t + 3a)
According to Eq. (13),the bond's price is equal to the discounted value The discount factor is random and an (condithe payof at maturi. t ional) cxpcctation operator needs to be used. The expedation is taken With regpect to the risk-neutral probability #. We normalize the valuc of the bond at times t and t + 31 using the saving and borrowing. As mentioned earliera for time t we divide the bond price by 1, the amount invested in risk-free lending and borrowing. For time f + 3 wc divide the value of the bond at maturity, which is $1,by the value of rolling the investment over at future spot rates raa, r/+.2,. ne expectation is conditional on the information set available at time t. nis information sct contains the current value of rt. Of
'r1.
)a ? to-w.,s =1
--..
e-
Ljo )-r t
y
(jyj
given that a1l technical conditions are satisfed. Thusa as z --y 0 we move from discrete-time discounting, toward continuous-tue discounting with Yariable spot rates. As a result, discrete-time discount factors get replaced by the exponential function. Bccause interest rates are continuously chanxconti-ning, an integral has to be used in the exponent. we obtain thc nus uous time bond pricing formula; Bt,
'.r)
=
El)
gc-
JJ rxtsj
4l8
C H A P T ER
+
18
Modeling Tenn Structure and Relatcd Concepu
4 FolavardRates
419
4 1 Diacref c Tmc
3.4 Yields (ZDZI spotRltes
.
We can also derive a relation between the yields S(l, 73 and the short rate %. We can relate fumre short rates to the yield curve of time l using the hvo equations in (12) and (3).Equating the right-hand sidcs:
e-rt(,,z)(z'-r) spt =
Taking logarithms:
rsd:'j
gg.1k''
(19)
.
'frfrslsl
Rt, F)
Bond Prices
and
-
=
loc E? eE T- t
1
'
-'
'
(20)
,
be visualized as some sort of We sec that the yield of a bond can (roughly) the life of the bond. In expected during prevail to average spot rate that is of fact, in the special case a constant spot rate, t < S < T (21) yve obtain;
To motivate the discussion, we begin with two periods, If represents small but nonintinitcsimal timc intcwal, and if t is the then a the price of a hvo-period bond will be given by Bft, t + 2A). nis bond will yield a cash flow of $1 at maturity date t + 2A. nus, one pays #(1, f + 2.), at time tn and the investment pays off $1 at timc f + 2,'. Now suppose liquid markets are available in fomard loans and consider the following alternative investment at time 1. We make a fomard loan that begins at timc t + which pays an interest of F(f, t-, t -#-2.)k& at I + 2A. I-et the total amount loaned be such that, at time f + 21 we receive $1. and denoted by Thus, the amount of the loan contracted for time I +
B*
is/
B*I+u&
1
=
(j +
yLt t .j.
yj ,
,
(24)
'
t .j. gjyjj
Now this is an amount that bclongs to timc f + Wc necd to discount the B*1:- a to timc f using the currcnt spot rate, This gives the time f value of the fonvard loan, which we call B? : .
j1' jg. . F -
#
Rt, Tt
1ogEt =
-
f
rs.g.j
jog =
-
-
g.y
z.;;
F - t
(22) (23)
Hence, tlae yield equals the spot zate, if the spot rate is indeed constant-
4 Forward Rates
and Bond
1
=
(1 +
rtz)
g
1
(1+ F(/, t + A, t +
2k),I
j
(25)
.
Finally, after rccognizing that for pcriod t, rt is also thc trivially dctincd becomesiC' forward rate F (f, f, t + fl,', the #/#
Prices
In this section we obtain another arbitrage relation that shows how/brw'W cnlcial rates relate to bond prices. It turns out that this relationshipplays a role in the moderzl theory of ftxed income. Let F(l. 7) U) be the cuncnt forward rate, contracted at time t, on a loan that begins at time T and matures at time U > F. and As mentioncd earlier, to derive a relationship between the B(t, T) Tls time U. F(/, T) Uj we need a second bond, B(t, U), that matures at is easy to see, The Ft, T; U) is a market price that corporates time f 111period bctween the times F and U. We formation concerning the (future) cxpect the bond Bt, F) to incorporate all rclevant information up to timo will F, ne longer maturity bond Bt, t./), on the otller hand, is a price tat be principle should in incorporate al1 information up to time U. Hence, we information concern&) all necessary able to extract from Blt, T) and #(/, ing the F(f. 7; &). As before, we obtain tltis relationship ftrst in discrete limit. and thcn take the continuous-time time. using intervals of length .,
.
B* t
=
1
(1+
F(f,
1
1, t +
)J gl+
F(f, t
+
,
f+
2A)1
.
(26)
According tta this if, at time /, we invest the amount B; at a rate rt, and then at t +/. roll this investment at a predetermined rate F(/, / +,', t +2), we get a payoff of $l at time t + 2/. But this is exactly the same payoff given by the strategy of buying the bond Bt, t + 2A). nus, if the credit risk involved in the two strategics is the same, wc must have #(f,
l +
2)
=
#,#
,
5ln the following expressions the reader may notice that the folward rates, say, Flt, F T + dre milltiplied by niq is ueeded becalsse the F(.) are assumed :6.4 be armual rates, Wllerets the A is suplxsedly a small arbitrary intenral. By market convenlion. the lbrward interest eamed during is not F(, T; T + ), but Ft, 7; F + z) timcxs For examplc, if the anlklal forward rate i!l 6%, and if onc year is madc ot' 3* duys, then a three-month loan VII earrt 6 times 1/4 percent. ),
.
,'
,.
sAny loan tbat begins now can be called a trivial fotward loan.
!
.#'
.j' .::
420
CHAPTER
Modeling Term Structure and Related Concepts
18
*
#
4 Forward Rates and Bond Prices
.
.
or
we move from discrete time toward continuous-time discounting. As a result, lw'tp things happen. First, the discrete-time discount factors need to be
j .'y
Bt, t + 2)
1
1
(27) : t + 2A)) + A)) g1+ F(f, f + '(: Note that sincc alI the quantities on the right-hand side of this equation are known at timc f, thee is no need to use any expectation operators in th fonuula. 'The relationship between the bond prices and the current forward ,) rates is cxad. ) What happcns when this arbitrage relation does not hold? One would simply short sell the expensive investment and buy the cheaper one. 'The . paymcnts and receipts of two positions will cancel each other at time l + 2, whilc leaving some prolit at time 1. Hence, there will be an arbitrage ;:2 opportunity. . 5? # 4.2 Metring to Celltinutv Time 1. 'i. Suppose now we consider n discrete time periods, each of length A, so ) that F t + n. The formula becomes =
fl + F(f,
4.
.
f, l
,
..
>
xplaced by the exponential function. Second, instead of discrete forward
. E :E .:
''
s E
.
rates we need to use instantaneous fonvard rates, Bccause instantaneous fomard rates may be different, an integral has to bc used in the exponent. Again, note that there is no expectation operator in this equation because y) a jj s(/, are quantities known at time t.
ne
formula:
:
B(t. %.
'
'
.
i
g1 +
F( t, t t +
lXl
,
Now use the approximation has
.
(1 +
.
(n
F( t, t +
F(I,
-
1)
,
t +
nt
T U) and
)1
(2g)
T U),J
z
,.r,4
x =
:
--F(/,d,J+AJA '
s(/, z)
.
..
Bt
'
-
--F(l,/+rI-1)A,/+.r;AJA
.
-
-
r LC-
-
-
'
'
yjvslds
,.(,
=
Take logarithms of these equations
log Bt,
;
(,34)
r) -
log Blt, T + 1)
v =
-
=
1*
F(J, sjds.
(37)
T
l' OW, suppose is small so that the F(/, F) can be considered during the small time intewal (z:z'+ aj. we can write:
:
',',:
,
,t
'r
s(/
=
,y,)
,
xjjm
a-+0
log Bt, F) - log#(/, F + A) .&.
'
.
,L
7nis
will facilitate the derivations of HJM arbitrage conditions
latcr.
yy
(38)
.
That is, the instantaneous fomard rate Flr, F) is closely related derivative of the logarithm of thc discount cun'c.
.)
:
constant
#4 jj F + ) F( t, r)A. - log This equation becomes exact, aftcr takng the limit:
:
-'>'
u:
BLjj F)
jog
.
,
'rhus,
(J6)
.
:.
which means that we can let 0 and increase the number of intorvals to obtain the continuous version of the relation between instamaneou: E forward rates and bond prices, . e-L , #(;'yV# Blt, T) (32);( , is given that the recurring technical conditions are all satished. The F(f, l
Ft, sjds
/
7+
...
.->
woa
F(/, sjds +
t
;) '
'
and subtract:
.F
z'qnN
1
z+a sjas ) e-S syj,
T+
.
:p'
'
'
.;'
2 ar ILC'
e-.L
=
and
:.
(a9)
,
--Ft/,/+.&,l+2A-I
'
-r
i
.
1 1t--'u/ , ? I.t:r But products of exponential terms can be simpliEed by adding the exNnents. So l,,+a)a-,(/,,+a,,+za)a...-,(,,,+(u-1)a.,+sa)a Bt, t + nhl z e(31) Fqt, /+(@- 1)A7t-izlt = (t - E2.1 1
ncous fomar tj yates. We can also go in the opposite diredion and write F(l, 71U) as a function of bond prices. We prefer to do this for the maturities r and U = F + L.' Thus, consider two bonds, #(f, T) and Bt, T + ), whose matu. rities differ only by a small timc inten'al > 0. Thcn writing the formula l
:.
are small, one
fr(,,z;s,)a
g-
prices as a function of instanta-
(t!2)twice:
(2
1
(3:3)
5
..'j'
the Bt, Tj as
Write
s(/
.
that when thc F(/,
(1- + and
.
F(l,')W:
.
g
L .
?'
-
.
';
1
r
e-
=
gives prices of default-free zero-coupon
=
=
F
''
.
.
Bt, Tj
42 1
'
:':g
(39) to the
C 422
c H A PT
ER
18
*
Modeling Term Structure
and
Related Concepts
By going through a similar argument, we can derive a similar expression for the noninstantaneous, but continuously compounded fomard rate f7);S #(/, ZJ logStf, F) logAtl, U) U-F
5 Conclusions: Relevance of
Relationships
the
once discotmted by the instantaneous interest rate rt. The bond Blt, T) paid $1 at matuty, and tbe discounted value of this was
(d-
-
F(1, T) Uj
=
,
(40)
where F(J, 6 U) is the continuously compounded fonvard rate on a loan that begins at time F < U and ends at time U. The contract is written at time t. & we get thc instantaneots folward rate F(f, F); Clearly, by letting F ljm F(l, T; U).
=
I-U
(41)
It is obvious from these arguments that the existencc of F(l, F) assumes that the discount cun'e, that is, the continuum of bond prices. is differentiable with respcct to F, te maturity date. Using Eq, (39) and assuming that some teclmical conditions are satished, we see that F(f, 1)
=
rt
(42)
.
Bt, T)
5 Conclusions: Relevance
It is time to review what we have obtained so far. We have basically derived three relationships between the bond prices Bt, F), the bond yields #(/, F), the fomard rates F(, 7; &), and the spot ratcs rt. The tirst relation was simply delinitional. Given thc bond pricc, we delincd the continuously compounded yicld to maturity Rt, T) as; Rt, T)
=
-
log Bt, Tj F - t
,
(43)
The second relationship was the result of applying the same principle that was used in the fist part of the book to bond prices; namely, that the expectation under thc risk-neutral measure 13 of payoffs of a linancial derivative would equal the current arbitrage-frce price of the instnzment, Izsed as a SEarlicr in this seuion whcn dixussing the discretu time case, F(l, E &) was t)f symlml for simple forward rates. In moving to contilzuous time. and swilcbing to tt) use compounded A continuouslgv rates. symbo! function, te same now denotcs the exmnential symbols for Ihe two differcnt would perlzaps be of prgceeding to use appropriate way gre concepts. But the notation of lhis ehapter is already too ctmplicaled.
j.
El)
=
gc-ftd'j
op-
(44)
.
Thus, this second relationship is based on thc no-arbitrage condition and as such is a pricing equation. That is, given a proper model for r/, it can be used to obtain the acorrect'' market price for the bond #(f, T). The tlrd relationship was derived in the prcvious section. Using again prices of the bonds an arbitrage argument we saw that thc (arbitrage-free) B(t, F), #(f, U) with & > F, and continuously compounded forward rate F(l, 71U) were related according to!
nat is, thc instantaneous fomard rate for a loan that begins at the current time f is simply the spot rate r/. of the Relationships
7* ds rk
The spot rate rt being random, we apply the (conditional) expectation under risk-neutral the relation: obtain #, the to measurc erator
->
Ft, F)
423
Ft,
T; U4
1ogBlt, F) - log Bt, Uj
=
U-T
(45)
,
This can also be used as a pricing equation, except that if we are given a Ft, T; U) we will have one equation and fwo unknowns to determin here, namely the Bs, F), #(1, &). nus, before we can use this as a pricing equation we need to know at least onc of the Bt. F), Blt, Uj. ne addition of other forward rates wottld not help much because each forward rate equation would come with an additional unknown bond price.g To sum up, thc hrst relation is simply a dehnition. It cannot be used for picing. But the other two arc based on arbitrage principles and would hold in liquid and well-functiorting markets. ncy form the basis of the two broad approaches to pricing interest-sensitive instruments. Thc so-called classica1 approach uses the second relation, whereas the recent Heath-larrowMbrton, HJM approach, uses the third. We will study these in the nex't
ehapter.
gsuppose wc brought in unother equation
contairlintr
Blt. U):
1ogBtn U) - log Blt, F(/, U, 5 U We will have two equations, but three uaknowrjs, namely the Bt, Again, an additional piece of information is needed. -)
-%)
=
(46)
.
,
-
7N),
z?(/,&)
and B((,
.).
424
C HA PT ER
and
Modeling Term Srmcrure
18
.
Related Concepa
6 References
Exercist!s 2, Consider a world with two time periods and tw'o possible states at each time t 0, 1, 2. Tlnere are only two assets to invest. One is risk-free borrowing and lcnding at the risk-free rate r,., i 0, 1 The other is to buy a two period bond with current price Bv. ne bond pays $1 at time t 2 whcn it matures, =
can consult this exccllent book for discrete-time lkxed income is one source that contains a good Rebonato (1998) models, Jarrow (1996). of interest rate models, thc other good source i and comprehensivc review For a sunrey of recent issues, see Iegadeesh the publication by Risk (1996). and 'Tbckman (2000). The reader
7 Exercises 1. Considcr the SDE for the spot drt Suppose the parametcrs Wiencr process.
a,
=
a
(/z-
z,
tEr
rate rl rtj dt +
(47)
trtM.
are known, and that, as usual, I'I'; is a
(a) Show that E
Erlrdl
p, + (G
=
pr (r,.Ir,l
=
#,) c-tzt'-o
-
c zf'-
(1-
(48)
c-M@-'))
(49)
,
(b) What do thesc two equations imply for the ctmditional mean and x? variance of spot rate as s (c) Suppose the market price of interest rate risk is constant at A G). Using (i.e.. the Girsanov transformation adjusts the drift by thift and show the that the bond price function given in the text, given diffusion parameters for a bond that matures at time s are by -->
Jt
B
=
U.B
VA 1 r , + -a
(
=
z
-
a
1
-
-
e
e-as-
)
-a(,-,)
,)
)
(50) (51)
.
(d) What happens to bond price volatility as maturity this expected? What happens to the drift coeficient as maturity (e)
approaches?
Is
approaches?
Ls
tbis txpected? (9 Finally, what is te drift and diffusion parameter for a bond with --+ oo'? very long maturity, s
=
.
=
(a) Set up a 2 x 4 system with state prices /. i, j u, d that gives tite arbitrage-free priccs of a savings account and of the bond S. (b) Show how one can get risk-neutral probabilities, #, in this setting. (c) Show that if one adopts a savings account normalization, tllc arbitrage-frcc price of the bond will be givcn by =
B
=
E i
1 (1 + rf))(- 1 +
n)
.
2 The Classical Approach
427
That rtb or for the instantaneous fonvard rates, will also bc is. they will be valid under the risk-neutral measure /. The so-callcd classical approach uses the first arbitrage relation and tries to extract from the 1A(!, F)) a risk-adjusted model for the spot rate rt. This will involve modeling the dri.ft of the spot rate dynamics, as well as calibration to observed vtlatilitics. An assumption on the Markovness of rt is used along the way. The Heath-larrow-Morton (HJM) approach, on the othcr hand, uses the second arbitrage condition and obtains arbitrage-free dynamics of k-dimensional instantaneous fprward rates F(f, F). It involvcs no drift modeling, but volatilities need to be calibrated. lt is more gencral, and, usually, less pactical to use in practice. ne HJM approach does not need spot-rate modcling. Yet, it alst demonstrates t'hat the spot rate rt is in general not Markov. ln this chaptcr we provide a discussitm of these methods used by practitioners in pricing interest-sensitivc securities. Ciiven our lrnited scope, numerical issues and details of the pricing computations will be omitted. lnterested readers can consult several cxcellent texts on thcsc, Our focus is on the understanding of these tw'o fundamcntally different approaches. Glrisk-adjusted.''
Classical and HJM Approaches to Fixed lncome
1 lntroduction Market
practice in pricing intcrcst-sensitive securities can proceed in two
differentwap depending on which of the two arbitrage relations dcveloped in the previous chapter is taken as a starting point. ln fact, Chapter 18 discussedin detail the bond pricing equation Bt, F)
X gd-
?lF rstdj
=
,
which gave arbitrage-free prices of default-free discount bonds #(/, F) under the risk-neutral measurc #, nis was a relation between spot rates rt and bond prices #(f, F) that hcld only when there were no arbitrage possibilities. The second arbitrage relation of Chapter 18 was between instantaneous fomard rates F(f, F) and bond prices: .B(t, F)
:=
!''-
-F
?r
F(! .lJ. '
-
Obviously, both relations can be exploited to calculate arbitrage-free prices of intcrest-sensitive securities. nc market practice is to start with a set of bond priccs qBLt, T)J that above of eithcr the argued arbitrage-free. one to be can rcasonably be nen mcdel for determine rt oz' used and a relations can be to go rclations taf Because F1). thc i two forward rates (#(/, for the set s (f. conditions, the modcl that one obtains for above hold under no-arbtrage Ibacltwardss'
'),
426
2 The Classical Approach The relationship
behveen bond prices and instantaneous spot rates, #(/, F)
=
y'J
jg-
Et/3
cs,'j ,
can bc exploited in (atlcast) two different ways by market practitioners. First, if an accurate and arbitrage-free discount curve (Bt, T) exists, and try to obtain onc can use these in Equation (1), go an arbitragc-free model for the spot rate rr. Onc can then exploit the arbitage-frcc characteristic of this spot-rate modcl to price interest ratc derivatives other than bonds. Second, one may go the othcr way around. lf thcrc are no reliable data on the discount cul've Bs, T), one may first posit an appropriate arbitage-free model for the spot ratc r/, estimate it using historical data on interest ratcs, and thcn use Equation market prices ftar (1)in getting illiquid bonds and other interest-sensitive dcrivatives. Both of thcse will be called thc classical tzp/zmlc to pricing interest rate derivatives. Wc will see that. one way or another, the classical approach is based on modeling tlze instantancous interest rate r!. in the firsl case, by starting from a Set of bond prices jBt. F))., and in the sccond case, from data available tm rt process itsclf. Gtbackwardsj''
Rfair''
l
'
; .%
(: .J.
428
C H A PTE R
*
Classical and HJM Approaches
19
J.
The Classical Approach
Fixed lncome...
to
:!
'.
so we start by looking at some simple
None of these are straightfoward, examples.
i:' ..
,
'g'
'
' (
First, consider the case in which we prefer to model rt directiy. Suppoge, in an economy where discount bonds do not trade actively, we' is, is constant at r. have reasons to believe that
; :.
),
'lnhat
can write:
y'
q
) '
=
Et
r jd
.''
j
the expectation T)
=
E;
(1 .a,g
.
.
y p
,
trivially and obtaz:
:'. l
e-'T-V%
y
Thusj starting from a posited model for rt we obtained a bond pricing eqpzl- 91 tion namely a Closed-form formula that depends on the krlown quantities j (Jl Iy and r. i . Using this equatifm, we Can PZCC illiquid bonds. To givc an exnmple, j,
We then have the following prices for 1 2, 3, 4 year t.: Lj discfmnt bonds; ,05.
maturity
r
=
,
r
X(l , t V 1)
=
#(f
3)
=
t + 4)
=
#(f,
l +
!q
'
Blt, t + 2) ,
i
.95,
=
.;;!
j
.90, .i!'
.
86
?
Eh
=
#(/, t + 2)
=
Bt, t + 3)
=
Bt, t + 4)
=
'
yps
-
.82.
a (r: , (j !
ttdt + bfrt, fltul
at,
=
wth
yj
j.jy
=
,
jj
(j
m
,
Tj,at js, the rt process is in fact constant at r.1 usug tjajs jasrmationv we can price interest-sensitive derivatives written y ourjce' or on Bqt, F). For example, a bond option will have an arbitrage-free equal to zero because rt is constant. P 2.3 Tlke qenevalCuae
supposeone obtains a reasonably accurate obsewation on the discount cun'c jBft, a), t < u s w) that one can assume to be arbitrage-free. Then, Equation (1)says that the same spot rate process rs must satisfy the following set of equationslz
l
.82.
pfjty ro)
:'
t
st jj
py;
=
a
.
j- )? o,j yjj jty-jyrl
and hence its d.riR f ..'
L.'
#(f,
L)
''Jhese priccs are identical to tose zwe
=
in
rstoj
ggl-i
'
g,
'
'
above, the r, was constant,
.86,
then infer from these prices that the r j process is in fact following d%
*13.d k our original assertion about the ctnstancy of rt is correct, these prices will be arbitragc-free. However, note that if we had posited a nondeterministic model for rj, tlttN he application of the same procedure would be problematic. In fad, .
.90,
E
,
lf
neutralmeasure .J'5 In the under # was zero.
.95,
'..
.''
SUPPOSC
we can
jjyo
..'
:
Bft, t +. 1)
:
.:
Rtake''
Bt
,
.
X( 1, F) Because r is constant, we
: .
.1 '
(1)we
:
Now suppose we do not know what type of stochastic process ri follows in reality. ln fact, suppose otzr purpose is to determine this process from observations on liquid bonds that trade in the market. ln particular, suppose we observe thc following discount curve:
:i
c
Using relation
j
(L
'
2.1 Example 1
2.2 F-awmple 2
Ei E:
..
429
s
EJ
j-
sxample1.
are assllmng that bonas lwwc a par value of 1.
o;
,
(4)
,
(5)
rn Gfj
S
gj
.
)'
430
CH A P T ER
*
Classical and HJM Approaches
19
to
Fixed lncome
2 The Classical Approach
: '
. .g
wjxro
To < F)
<
.
.
<
.
J
k
I
have reasonably accurate and
which
maturities
1 at we arc the n + bond prices. j arbitrage-free discuss these equatonj. Gven the martingale mcasure that deus t-et expectation operator El) the right-hand sides of al1 these equa(.1, fincsthe depend on the same spot rate process r,, albeit with different 6.. The tions detcrmine the left-hand side of these n+ 1 equations. The prob) marut faced by the oractitioner is to determine one model for thc interest rate lem r, such t an these equations are satislied simultaneously. How at process consiswe guarantee that thc specific model selected tbr the rt wfll be J canwith thesc n + 1 set o equations? is indeed no straightforward
'
.
E
:
'
mo
t
t t
E.
.,
vog
,
drt
(
(7)
alrt, tldt + blrt, /'lJF;,
=
j)
f/trr, tj, (r,, /) an d the probabilistic be- j' and second, one has to seled the havior of the driving process K, such that the system of equations shown ;' in (5) are satisnedl we consider two examples. ?
Geometric SDE j complications, To see that the system in (2)-(5)comes with 'E we select for the spot ratc r,, a geometr icsDE driven by a wiener k .,4
2.3.1
suppose under process
(fhiddcn''
the
/:
dr/ sr,(// + =
Hence we have postnlated that in are given by:
art
,
,
f
)
=
y,rt
,
:
(8)
g,.y#wz;.
(7)the drift and
,
t
the diffusion coeffcicnts .
or 2,
tj
w
pz = p;.
m.,,
(9) ?, ...
will immediately be some headaches. A spot rate process obeyitv this model would eventually go to plus or minus infinity dcpending on the of the rt will be con- ? (x7. A1sO, the Pcrcentagc vlatility sign tf #,, as / represent the ;. stapt. Clearly, these do not seems to be ideal properties to First rates do not r , interest bchavior t)f overnight rates obscrved in rea Iity Wtrends.'' volatility seems to ; Second, in reality percentage interest rate have
nere
->
.
.
:
u omly n the 'Nte the implicit assumption bere. Thc incremcnt iu tlle smt rate depends ? 3 be will As wtl sec. below, this current r/, and hence the spot rate has a Marktwian character. . conditions are sat isjed. where arbtrage markets flxed-income in ial case spec .
::
E:
.;
.
;
=
;
'
.
fof
;
a spot m/d
.
-x.
''-lle-nl
-will
'rhis
.,?),
=
jr.
tentt-et us illustrate somc fof the difculties wolved. task. Jn fad consider what this requires. First, onc has to postulate
ye a complicated nonlzear function of the level of spot rate rt, rather than just a constant. sut putting these two difliculties aside, considcr the problem mentioned above: namely, how to select the g. and (z. such that all the equations in the system in (5) are satisued simultaneously'?4 This is no smple task. In fct, gven the reasonablv accurate obscwations on the bond prices, jBt, 6.), i (), a (a-)-(5) wc have n + 1 equations with kmown left-hand sides. But the free narametem t'la,- Mtovest rate model that we can choose arc onlv the lz 'and the a. unkn-o--w to satisfy a system of n + 1 cquations b cho hvo not going to be possible unpess therc are strong interdepcndencies among obsen'ed bond prices fBt, 6.), i 0, nj, so that n - 1 of these equations arc in fact redundant, Then, tlie sastem would in fact reducc to two equations in hvo unknowns and a set of /s, (z. that fits the obsen'ed arbitrage-free djScount culve fBt, F)) can be found. sut jyow attractjve is it to postulate such strong dependencies among the n + 1 bond prices that one obsen'es in liquid markcts? Obviously, tle spot-ratc process postulated in Equation (8)is quite inadequate for practiCal Pricing pumoses. Other modcls must be sought. being
(6)
Ts
4jl
g
S
UJMkW.
Js.-- Us '-ni;
Jsing
,,,
JJJ Meanmeverting ajoy/a The geometric SDE may be inappropriate for describing the dynamics of tlae spot rate, but fzom the above arguments we Iearned something, nrst, tn appropriate sDE should be seleded for rt, and then the parameters of this model should be determined (calibrated) so that the spot-rate model jits,, tjje (jjscount cuwe (A(f, 6.)Jgiven by liquid markets. If this can be done, and if the obsewed discount bond prices jpt, T.4,i (), n are arutrage-free, then the resulting model for the spot-ratc process r, woeld also be arbitrage-free. It could be used to price interest-sensitive dcrvativesThus, one may ask if one can postulate a sDE more realistic than tjw geomutric proccss discussed in the first example. In fact, consider the mearsreverting spot-rate process with variable f?y and a square-root diffusion component'. .4
'
'
=
..,
.mean''
dr
u ere,
=
tdy- rtldt f the
+ o.ypidy
.
( t(p
for each time period parametcr ot is allowed to assume a dtfferent known value. augments the number of free parameters that nis one has in systcm (5).For examplc, in a discrete setting with: ,
to
<
j.!
.<
.,z .
.
.
u
jjjj
,
4Tjae sc j ect jqm oy y; as a Wlener process i: already made. hecause real-world spot-rate may contain jumps. presses
nis
alst may not be appropriate
432
(--H A P T E R
+
Classical and HJM Approaches
19
tta
Fixed Inctame
there will be m + 3 free parametcrs to select in the interest-tate model, namcly the
't'#r-8t, ,
,
.
.
-
,
0t,- A, ,
(12)
z.1'-
This gives more flexibility inhtting thc interest rate process to the obsen'ed discount cun'e. f#(f, Fj), i 0, nJ,5ln fact, wc can not only Iit the rt volatilities as we1l.6 See Hull and bond but fit it bond pricts, to process to for example. Whitc (1990) Bcsides, unlike geometric processes, mean-reverting proccsses are x. Also, given inknown tmder the right conditions not to explode as t fmitcsimal steps, thc r/ process that will be generatcd by the mean-reverting modcl will not become negative given the difusion component adopted here, =
,
,
The ClassicalApproach
433
f) has where tlte drift because it is assumed to be adjusted for a the interest rate risk and consequently the H( is Wiener a process under the risk-neutral measure #, We consider two cases,
tr,
GKtilde''
2.4.1 zl One-Factor Model Suppose we want to price a derivative instrument that is sensitive to onlylts price is denoted by Crt, f). The expiration date is T and thc r? expiration payoff is given by the known function Glrv, F): Crvs T)
-->
2.4 Using tte Spot Rate Mtxle! Supposeonc successfully completes the project to extract an arbitrage-free
model for thc spot rate rt from the pricing cquation: j' rsds Bt, F) E:j3 =
ge- j
n
fT /)z
+ blrt, r)z/I#;,
5The. parameters n and m need not be, the same. lhat the t'n-hat is, we can calibrate the free parameters of the SDE shown in (10)so liqzlid volatilities of B(t, ir') obtained lom Equatftm (1) rrzatch lhe volatilities obsezved in optiorls markets n these bonds. TWecan mention at least three specific uses, but there arc many others that we do not go instrtzmnt into bechduseof the limited scopc of this book: ('l) It may bc that thcre is a traded 'trr, 1) fhat can be smthetically replicatcd using thc traded Ynds #(/, T). (2) The Cjrt, ) be some suspicion t,1*1 mhty bc a new instrument that does not yet tradc. (3) There may model for r. one call C.-(rr,?) is mispriced by the markcts. Then, by using an arbitragc-free arbitrage, takc lhe hedging, or speclllatim and irlstrumcnt price for proper calculate a positionq. Or, tme could simply use tbe price in invcstment banking opcrations. 'slir'*
Grv, T).
One could immediately use the pricing equation: ctr/
,
7-)
=
g,-hr
e-/'
z'j
r-d,ctrz.,
.
This expcctation can be evaluated using Montc Carlo metods; or it can be solved for a closed-form solution if one exists', or it can be converted imo a PDE, as will be seen in Chapter 21', or it can be cvaluated in a u'ee model. 'Ils will bc possible because we would already have a dynamics tbr rt under thc 13:
.
How would this model be uscd? The answcr to tlis qucstion was bziclly mentioned at the beginning of this chapter. Thc bond pricing equation is uscd to extzac.t an arbitrage-free spot-rate model from the existing term structure because using this model one can then pric.e other interest-scnsitive securities and obtain arbitrage-free prices without having to look at the markets for these secuities.7 To see the use of the spot-rate model, considcr the following setup. A reliable term structure (#(f,FJ is given and is exploited to extract the arbitrage-free model for rt =
=
drt
=
J(r; tjdt + blrt,
1)t:/16.
The rest is just computation.
2.4.2 Second Factor Things can get somewhat more complicated if we want to price derivaa tive instrument that is sensitive to rt and, say, to Rt, long rate, which is a not perfectly correlated with rt. Supptse the pricc of this new iostnzment is denoted by Crt, Rt /). The expiration date is again F, and the expiration payotf is given by the known function Gtrr, Rv, F): ..4
,
ctzv,r) one could
=
Gtrz., Ry, Tj.
again write the pricing equation..
ctr/, F)
=
llr,dsGrv, ge-
Etl5
Rv,
z')1 .
But, the model would tot be complete. In fact, we do not yet have an arbitrage-fre model given the factory'' Rt. Before we can proceed and calculatc the price, we need to obtain a risk-adjusted SDE for Rv as well. For these issues we refer thc reader to Brennan and Schwarz (1979) and the related litcrature. lt must be realized that the two rt and Ar may lmve complex time-varying correlation properties processes and computationally the problcm may get much more dicult tltan the case of a single factor. fsecond
C H A PT ER
434
.
19 Classical and
HJM Approaclws
to
Fixed Income
2.4.3 F/ze Importance of Calibration (ne obtains the lt is important to understand the process by wbich and esmcthods ectmomctric spot-rate model in (2.#.J), If me used onlj' f), trtrr, rcsulting 1) the and diffusion timated a continuous-time drift art, modcl written as #r/
=
f7tr,
t 4dt+ o'rt
,
f )#F;*
would not be called arbitrage free. Econometric metbods yield estimates valid under the for the rcal-world parameters, and the model would be estimated dircctly be 11.7 can real-world probability #. The Wiener process residtlals. repession from the data as continuous-timc It is the backward extraction of the r/ process using Bt, F)
=
El)
j-
2.5 Cmnpurison u/ith thc Bluck-cholcs World securities We see that the classical approach to pricing interest-sensitive calibratioa tis that also We modeling. sec amounts, essentially, to spot-rate effort is not trivial, especially when discount bond prices arc not perfectly related to each other across maturities. arbitrage reMore importantly, if one pursues the classicai approach, fitting to through indirectly, model the into strictions will be incorporated priccs, bond discount with of set starts a the initial yield cun'e. One tii-st model for find rt that tries a then to and onc or the corresponding yields that obscrved structure term ulits'' the so ?1F Gd''j Et? Bt, F)
ge-'
is satisfied for cvery T. Black-sclzoles 'This is quite differcnt from the philosophy used in the restricarbitrage the 'There, book. of this world discussed in the tirst part model by repladng tions were directly and explicitly incomorated into the drifl. M notation for tbe rsk-adjusted thc following chaptcrs we will have a different risk-adjusted drift as wrie the able will to be we devclop new concepts that we can use, we market the itz this adjustment, case, or, ctrr, r) hrbqrt s l), where the / is he Girsanov drift risk. price of inlerest ralc SIIl
-
Term Structure
to
the unknown drift of the underlying process by thc known spot rate. was no need to model the drift term of the stock price process. nc latter was simply replaced by the (constant) spot ratc r. As a result, Blackucholes approach reduced the problem to one Iaf volatility modeling. The assumption of a geometric process for the underlying prtcess St simplilied this further and percentage volatility was assumed to be constant. Thus, in this sense, the spot-ratc modeling that forms the basis of the classical approach appears to be a fundamentallk different methodology from the arbitrage-frec pricing as scen until now. This leads to tlze following question: Is thel'e another approach that one can use, which will be more in line with the philosopby of Black-scholes? The answer is yes and it is the Heath-larrow-Morton (HJM) Model. 'rhere
/''r'Jj
that yields an arbitrage-free model because the probability used in this modeling is more pricing cquation is the /, Hence, arbitrage-free spot-zate also based is problem. lt calibration on judicious than just an estimation or models,8 choice of pricing
=
(J The iIJM Approach
The HJM Approach to Term Structure ne arbitrage restrictions that wc have been studng are the result of common random processes that inqucnce discount bonds that are identical except for their maturity, lf the liquid bonds that determine thc term structure jBt, T)) are all inlluenced by the same unpredictable Wiencr process 7, the respective prices must somehow be related to each other as suggested by the pricing relation: 'i-r.r'j
#(f, T')
=
Et'
g-
.
Thc classical approach to pricing intcrest-sensitive securities is an attempt to extract these arbitrage relations from the Bt, T) and then summarize them within an arbitrage-frec spot-rate model'. drt
=
rt, tjdt + brt, /)#P(.
This is indeed a complicated task Iaf indirect accounting tor a complex set arbitrage relations between mrket prices. The Heath-lanow-Morton, or as known in the markct, HJM, approach attacks these arbitrage restrictions directly by b'ringing the forward'ratcs to the foefront. The idea is based on the second arbitrage relation developcd extensively in Chaptcr 18. As mentioned there, there are dircct relations between discount bonds that arc identical except for their maturity and forward rates. lt is sufhcient to review a simple case. Let #(/, F) and #(f, U) be two default-frec zero-cotlpon bonds that are identical except for their maturity U > F. Let F(f, T) &) be the interest rate contracted at time f on a default-free forward toan that starts at F and ends at U. Herc, F(.) is a perccntage rate for period &-F. Thus, no days Of
436
(2 H A P T E R
.
19
Classical and HJM Approachesto Fixed Income
the discussitm in Chaptcr 18 permits adjustment factor is needed. condition:g no-arbitrage tbe writing rrhcn,
(1 + F(l, T &lj
=
Bt, F) Bt, s)
.
)
or for that to calibrate and/or estimate any additional drift coecients, automatical;y will be risk. adjust A11these coeocients for tesc matter to incomorated in the forward rate dynamics. ln other words, if we decided to modei the fomrard rates Ft, T; Uj instead of the spot rate G, the arbitrage relations can be dircctly built into thc fomard ratc dynamics similar to the case of Black-scholes. The development of the HJM approach is based on this idea. Of course, in this framework we still have to calibrate the volatilities. Also, we need to seled the exad fonvard rates that the pricing will be based on,
nich
to
Term Structure
437
developed in Chapter 19: Bt, T)
=
e-
J' f
Ft,sbd, .
where #(f, is the rate on a forward Ioan that begins at time s and ends after an ininitesimal timc pcriod ds. Writing (hc arbitrage rclation as .)
We thus have two bonds with different maturities in a single expression that contains Flt, T) &). Now consider the joint dynamics of these variables. SDES Because bonds are traded assets. in te corresponding we can rcplace the drift parameters by the risk-free ratc h. Thus, up to this point evcrything derivation, But, note that according to the is identical to Black-scholes arbitrage relation above, the ratio of the tw'o risk-ncutral bond dynamics will be captured by the movements of a single forward rate F(I, T; U). ln other words, once risk-neutral dynamics of the bonds are written, the SDE for the folward rate F(f, T; U will bc determined. There will be no need
3.1
3 The HJM Approach
Fonvuwd Rate?
Here we have several options because the arbitrage relation can be written in several different ways, The original approach used by HJM is to rnodel the continuotlsly comis, use the relation pounded instantaneous fomard rates F(/, Fl-that S(l, (J) : 9Lct us repeat the, arbitrage condition using somewhat difbren Ianguage. 1he time U thc present value of a sure dollar to be received at a later datq U. 11s wersc iz LT) value of $1 that we have now. Dividing the inversc by 1 + F(/, -C brings a tirlx U value to time F: 1 (1 + f' t, T; U)1l( 1, Uj started Multplying this by Blt. 7-) should bring it back to $1, the amount that we orignally with: 1 1. #(/, 'J') j 1 + F(l, T; U))A(t, U) 'rjac
=
This s the case sincc Blt, F) is the prescnt value of $1 to be received at time F.
Bt, F) Bt, ./)
=e
y; z,(,,,)a.
we can obtain an arbitrage restriction on the dynamics for continuously compounded instantaneous rates F(/, T), as will be done in the next section. But this is only one way HJM models can proceed. Another option is to is, we can use use forward rates for discrete, nonin:nitesimal periods, models that arc based on the F(f, T; U). Letting & F + A, we can modcl arbitrage-free dynarnics using the relationship: rnlat
=
g1 + F(/, T; F +
ll
#(f, T) Bt, T +
=
)
.
Here, we can kecp the l > 0 favd and consider the joint dynamics of thc #(f, T), Bt. T + ) as l changes. ne joint dynamics can bc modeled with the risk-neutral measure, or depending on the instrument to bc priced, with thc fomard measure introduced in Chapter 17. Proceeding this way leads to thc so-called BGM models, after the work in Bracea Gatarek and Musiela (1996).ne remaining part of this chapter will proceed along the lines of original HJM approach by using the instantaneous fonvard rate F(/, F). 3.2
Arbitrugc-lvcc
mics in HJM
From the relationship bctween the default-frce pure discount bond prices Bt, T;.), Fj < Frnax, with maturity rj and forward rates F(/, F) dcrived in Chapter 19, we have: Bt, Tj
=
e-t
Frt '
uldu .
(13)
Recall that there is no expedation operator involved in this exprcssion, betause the F(l, u) are a11forward rates obsen'ed at fime 1. They are rates on fomard loans that will begin at future dates u > t and last an insnitcsimal period du. For the ncxt section adopt the notation Bt Bt, T) alld assume that for a typical bond with maturity T we are given the following stochastic differential equation: =
C H A PT ER
dBt
*
19
p-lt,
=
Classicaland HJM Approaches ro Fixed Income
F BtlBtdt
+
tw(f,
T; BtlBtdjl'b
(14)
is a Wiener process with respect to the real-world probabilwhere thc this SDE. First, the ity P. Wc need to emphasize three points concenng volatility, but is bond diffusion parameter is written in terms of percentage VIT
the SDE is driven by a Wiener not necessarily of geomctric form.ltl sccond, wit.h differprocess indexed by F. This means tahat,in principle, every bond shock. Later, we ent maturity is allowed to be inEuenccd by some difcrent P;W will required be to be the will see the single factor case where a1l the same. And third. note the new way wc write the diffusion parameter, rt. 71Bt4 is explicitly made a function of the maturity F. This is nccded in the dezivation below, but will be abandoned in latcr chapters. Now, bonds are traded assets. ln a risk-neutral world with application of the Girsanov theorem, tbe drift coeftlcient can be modilied as in the case of the Black-scholes framework; (15) dl t t't B dt + a't, T Bt)B,d%l',
3 The HJM Approach
t
B''J
is the ncw Wiener where rt is the risk-frce instantaneous spot rate, and switching from P to That is, by risk-neutral #. under the measure 2rocess bond eliminated unknown dynamics. the in drift thc have P, we Given these SDES for bonds, we can get thc dynamics of the F(/, F) fl'om Equation (13).Begin with the arbitrage rclation introduccd in Chapter 19, and discussed above: logltf, F) logltl, F + ) F(f, T T + ) (16) (F + 1) T is used to deline thc non-instanwhere a noninfmitesimal intenral 0 < and taneous fonvard rate, F(l. 71F + l), for a loan that begs at time F identical that are ends at time F+ A. nis is done by considering twt bonds in a1l aspects, except for their maturity. which are apart, Now, to get the arbitragc-free dynamics of forwaz'd rates, apply lto's Lemma to the right-hand sidc of (16).and use the risk-adjusted drifts whencver needed,l Apply Ito's Lemma lirst to log B($, T4 to get: -
=
,
# (log#(I,
a
Bt, F)1 d Elog
=
.j vldBt,
sjf,
1
tz'tf, F) gstj, g-)c
T W/)zBt, zla,j. (j.g)
txmsfnnt19A gcometric SDE would have the diffusion parnmeter written as O'B, witb r volatility not conis bond well. I'len, percerltae Hcre we have lt, 7-, B,j depend on 1.: as
stanl here.
mistakcnly
11Here, applyng Ito's Lemrna memls varyiug hc t parameter. ne reader may 21 0. Ths will bc done, but for tim lhink at this pont that wz are trying to take the limit as time beng the is kept constant.
439
r)1
=
Lrtdt
jl fwtf, y) s,)2) dt
-
Now agply Ito's Lemma to fflog Blt, T + sion wlth T replaced by T + A:12 d
(logS(I,
=
F+
) and
+
(w(j,
get the equivalent
j)(wtr,
z'+
Btjl)
h,
dt +
cLt,w+
expres-
,
sjltfuz.
(19)
It is important to realizc that the first tcrms in drift of the SDES for #(f, F) and BT F+) are the same because the djnamics under consideration are arbitrage-free, Under ?, discount bonds wlth different maturities will have expected rates of returns that equal the risk-free rate r This is essentially the same argument used in switching to the (constant) nsk-free rate r in the drift of the SDE for a stock price St utilized in Blackucholes derivation. Now substitute the stochastic differentials (18)atd (19)in the definition of F(/, T) r + A) given in (16)and cancel the common rtdt terms: .
dF(t, F; T +
1
= ga +
1
)
(c(/,F + (541,T +
u.
2
)) -
#(f, F +
,
,
Blt, T + A))
-
o't,
T Bt, T)) zj dt
ajt, T)Bt,
r))j t/r,6
.
(20)
This is the final result of applying Ito's Lemma to (16).This equation gives the arbitrage-free dynnrnics of a forward rate on a loan that begins at time F and ends period later. Now, we can 1et -->. 0, This will gjivethe dynamics of the instantaneoua foward rate. To do this, note that the way expression (20)is written. On the right-hand side, we have two terms that are of the form:
)-
y(x +
glx)
ln expressions like these. Ietting 0 means taking the (standard)derivative of g(.) with respect to x. Writing these tcrms in brackets separately and --
-.+
,
z;yjdy.
)1
Lrtdt -
-
k
Term Structure
Simplifying and then substituting from the SDE for the risk-adjusted bond dynamics in (15):
'rhe
=
to
12Af1oya.ll, 1h0 tw'o honda are identical, except for their matklrities.
440
C H A P T ER
then letting the rilt-hand
19 Claoical and HJM Approaches to Fixed lncome
*
0 amounts to takiog the derivative of the tw'o terms side with respect to T. Doing tMs gives
->
1 lim 21 ,&-..a
gc(/,T +
lim
A->t1
1
z
))2
#(1, T +
T)Bt, T))
tzf,
=
,
-
ptr(/, T;Bt,
g
op ))
gtrtf,F + A, #(f, F +
a
/9t7'41, T; Blt,
T)S(f, T))2j
rlt,
tr(/,
6
bFjt,
Bltb F)))
get the corresponding SDE for the
lim JF(r, T; T + A)
=
--yt)
tF(f, F).
Or, dF(t, T)
ct,
=
z?tEr(/,
+
o.t. TJBt, F)) d( pr
T; BLt,F))
T)Bt, JT
r))
tjs
y
(21)
3.3 Interpretafa'en The HJM approach is based on imposing the no-arbitragc restridions directly on the fomard ratcs. Fil-st, a relation between forward rates and bond prices is obtained using an arbitrage argument. Then arbitrage-free dynrlmics are written for Blt, T). Given thc SDES foT bond prices, a SDE that an instantaneous forward rate should satisfy is obtained. To sec the real meaning of this, suppose we postulate a general SDE for the instantaneus fozward rate F(f, F); =
J(F(f, F),
/)#f +
:(F(l, F), f)JH(,
tw(/,
T Bt, Tjj
ty(7'tf,T;#(f, T)) JT
.
(23)
F),
/)
Jfr(I,
T Bt, F))
(24) JF Hence, the previous section derived the amcf no-arbitrage restrictions on the drift coefscient for instantaneous forward rate dynamics. This is similar to the Black-scholes approach that was seen several times in the first part of tbe book. There, the drift term p. of the SDE for a stock price Sf was replaced by thc risk-free interest rate r under the condition that there were no-arbitrage possibilities. Here, the drifl is replaced not by r, but by a somewhat more complicated term that depends on the volatilities of thc bonds under considcration. But, in principle, the drift is detcrmined by arbitrage arguments and will hold only under the condition that there are no-arbitrage possibilities between the forward loan markets and bond prices. nrouglzout this process no rate modeling'' was done. It is worth emphasizing that the risk-adjusted drift of instantaneous forward rates depends only on the volatility para' meters. This is again similar to the Black-scholes cnvironment where there was no need to model the expected rate tf return on the tlnderlying stock, but modeling or calibrating thc volatility wwu needed. lt is in this sense that the HJM approach can be regarded as a true extension of the Black-scholcs methodolor to flxed income sector. =
.
Iforward
where the (r(.) are the bond price volatilities. Wc have several cornmcnts to make on tMs result,
#F(l, F)
-->
The difhlsion parameter will be given by:
.
(20)we
44l
A rcader may wonder how one would obtain these risk-adjusted parameters that are valid under the condition of no-arbitrage. Well, the prcvious section just established that under narbitrage, risk-adjusted drift can be replaced by:
.r))
j
Term Stnzctuze
to
4(F(f, F), f)
F))
JF
Putting these together in iytstantaneoua fomard ratez
-
(m
3 The HJM Approach
(22)
where the aFjt, F), !) and h(F(f, F), f) are supposed to bc the riskadjusted drift and tlle diffusion parameters, and the J#: is the zisk-neutral probability.
3.4 Tl
rt fn the
Approuch
Further, note that in the HJM approach there is no need to model any short-rate process. In pmicular, an exact model for tlle spot rate r; is not needed. Yet, suppose there is a spot ratc in the market. What would the Sbp,s obtained for the forward rates F(l, T) imply for tbis spot rate? The question is relevant becaesc the smt rate corrcsponds to the nearest insnitesimal forward Ioan, the one that starts at time f R-hus, realizing tlmt ,
F(/, t) (25) for a1l f, we can in fact derive an equation for the spot rate starting from the sDEs for fozward rateg. Before wc start, we simplify the notation and rt
=
C H A PTFR
*2
*
Classical and HJM Approaches rta Fixed lncome
19
bs, t) in (24).Then, write the integral equation for write: bFls, F), f) F(l, F) using the new !z(.) notation! =
s(/,w)
-
F'(o,
w)+
jjTbs,
' bs,
whcrewe used (23)and tation for the spot rate r/: =
F(0,
1) +
Next, select F
bls, ujdu
l
=
l
#.$+
0
y
bs, /)#Hrt,
(26)
where the bls, tj is the volatility of the Fs, tj. The srst important result that we obtain from this equation is that tlle fomard rates are biased estimators of the future spot rates under the riskfree measure. ln fact, consider taking the conditional expcctation of some future spot rate rv with initial point t < 'r:
Et/
Erzl =
.E? l j/-tj
+,
glj
,
bs,
Et
,-)
1.
bs,
T
bs
.
/
F(f,
Et>(z'zl
'r)
(28)
-
The second major implication of the SDE for r/ has to do with tho non-Markovness of the spot rate. To see this, note tbat the rt given l>y Equation (2$ depends on te term: J
n
1) !7(-v,
pt
j
bs, ujdu
ds,
(29)
all past forward rate volatilithat, in general, will be a complex function of aaccumulation'' simply of past chanjes not is particular, this term an ties. In the way a typical drift or diffusion term would lead to
.2',
M(G, ds
t
t)
brs, -)J1K.-
J(j
(30) (31)
>(J,
uldu
ds,
(32)
$
and would not be captred by a state variable. The difference between (29) and (32)will depend on interest rates obscrk'ed before t would make the intercst rate non-Markov in general. Next we see an example. rrhis
-
.
3.4.1 Constant Forward f'latilities Suppose al1 fonvard rates F(/, T) have volatilities that are constant at b. Then for each one of thcse fomard rates the equation under no-arbitrage will be given by:
The dynet'mics
Here, the fomard rate in the first expedation is known at tue /; bence it light-hand comes out of the expectation sign. The third expectation on thc side is zero because it is taken with respect to a Wiener process. But the second term is in general positive and does not vanish. Hence we have:
r-- ?k
bls, t 1) -
r)
#F(f,
(27)
'rlp:
,
--2
t
uldujsj
'
Et>
+
g -
'
Term Struc:ure
to
In fact, the new term in the equation for rt is more like a cross product. Hence. the similar term for an interest rate obselved period before the rt would be
a to get a represen-
t
t
bs, tj
rldws, j'bls,
+
x
in
/
rt
ult/ajts
.J, (24) (21). z')
3 The HJM Approach
=
blT
of the bond pricc will be
tjdt + 5#46.
rtBlt, F)Jl + bT tlBt, F)#F). From thesc wc can derive the equation for the spot rate by taking
dBt, T)
integrals in
=
(26):
rj
=
F(0, t) +
1
jb a,t
2
+ /)'r'6,
(33) (34) thc
(35)
which gives the SDE drt where the F/(0,
f
=
(G((),f ) + bldt
+ JJJF;.
(36)
) is given by Ft (0, f )
=
pF(0, t) .
t?f
Note that according to this model, the spot rate has a time-dcpendent drift and a constant volatility.
3.5 Anotltcv Advuntuge oj the HJM Appwouch The I'IJM approach exploited rates and bond prices to impose stantaneous forward rates directly. model the expeded rate of change
the arbitrage
relation betsvccn fonvard restrictiorls on the dynamics of thc inBy doing this it eliminatcd the need to of the spot rate.
:'
:.; :
.
444
C H A P T ER
*
19
Classical and HJM Approaches
to
.:1
Fixed income
'i .
.
i
tt
But the approach has other advantages as well. As was seen in ean lier chapters, a k-dimensional Markov process would in general yield nonMarkov univariate models. Hence, within the HJM framework one could in principle impose Markovness on the bchavior of a set of forward rateg and in a multivariate sense this would bc a reasonable approximation. Yet, in a univariate sense when we modci the spot rate, the latter would still behave in a non-Markovian fasition, nis point is important because currcnt empirical work indicates that spot rate behavior in reality may fail to be Markovian. Hence. from this angle, the HJM approach provides an important :exibility to market prac-
,'
''
.:
r
.
r'
I
.
7 i EE:
E !
i
to
lnitial Term Structure
,
,
.
,
,
drt
E !
'S;
't
''
'
E
The HJM approach is clearly the more appropriate philosophy to adopt from te point of view of arbitrage-free pricing. lt incorporates arbitrage restrictions directly into the model and is more iexiblc. Howcver, it appears that market practice still prcfers the classical ap- C'. proach and continues to use spot-rate modeling one way or another. How y' can we explain this discrepany? As discussed in Musiela and Rutkowski (1997),modeling the irlstanta- 1 / . neous spot rate has its own difculties. W'hen one imposes a Gaussian struc- ?j ture to SDES that govern thc dynamics of the JFtla F) and when one useg t, constant percentagc volatilities, tlle processes under consideration explodo .' in hnite timc. nis is clearly not a very desirable propcrty of a dynamic j model. It can introduce major instabilities in the pricing effort. 'F lt is also true that there are signmcant resources investe d in spot-rate is, again, a great dcal of famodels both tinancially and time-wise. miliarity with the spot-rate models, and it may be that they provide good g'l t.', alnproximatiml to arbitrage-free pricts anmay. 0The recent models that exploit the fonvardmeasure secm tt be an SWCC to Problems of instantaneous fomard-rate modeling, and should lx considered as a promising alternative.
.
drt
,
( t j ''i-. ? ,
'
:
.
.,
l .
4 !
.
'
.)
'
(
tE:
to Initial Term Structure
:2
rt
=
rt-x +
(ztx
-
his hOw this could be dcme in practice. 'IYY WC IWVCI' ShoWed QDSCLkSSiOIV t numerical issues book tries to keep to a minimltm, but there arc some casG of pricing metho ds facilitateg the understanding practical where a dscussion
../
trdl
Euler scheme:l3
rtnjz
+
t7.lllzl
-
H/;-a1,
(38)
where is the discretization intcrval. nc rcmaining part of the calibration excrcisc dcpcnds on thc method adopted. We discuss some simple examples, ,
1
Suppose we know that incrcmcnts IH'I Wt-al are independent and are normally distributcd with mean zero and variance A. Suppose we have also calibrated the volatility parameter tr and the speed of mean reversion a. Hence, there ig tnly one unknown parameter x. Finally, we also have the initial spot rate ra. Consider the following exercise. Seled M standard normal random variables using some random number generator. Multiply each random number by V&Start with a historical estimate of ar and obtain the firxt Monte Carlo trajectory for rrl starting with rll and using Equation (38) recursivcly. Repeat this N times to obtain N such spot-ratc trajectorics:
l frltl'Illr,' ,
:E.
)dt+
- rt
4. 1 Mtmte Cflo
'p
;.
a(x
and then discretize this using the straightforward
1i
r
'rhcre
=
C'..
k
rt
bLrt,fl#H(
to this term strtture. How can this be done in practice? Several methods arc open to us. ney a1l start by positing a class of plausible spot-rate models and then continue by discretizing it. Thus, we can let rt follow the Vasicek model:
''
.
Pgtzctice
4 How to Fit
tldt +
art,
=
')
.
3.6
Fit rt
to
of the conceptual issues. Some simple examples of how an arbitrage-free spot-rate model can be obtained fall into this category. We discuss this briedy at the end of the chapter. Suppose we are given an arbitrage-free family of a bond prices #(f, Tf), i 1 n. Supppose also that we decided to use the classical approach to price interest-sensitive securities. Assuming a one factor model, we first need to lit a risk-adjusted spot-rate model =
;
f
E
titifmers.
Mlrket
How
'''
-
.
-
,
11 lr''vr -
;
.
'f
J'
Ef
l3Euler scheme replas difrerentiaks by qrst differences. It is a first-order approximation that may end up causing signitleant cumulative errors.
r' 446
C HA PT ER
*
19
Classical and HJM Approaches
to
'rhen calculate the prices by using the sample equivalent pricing formula:
(t rj) ,
=
a'skr
1
p
6 References
of the bond
where the ri/ are t he f'th clement of the y'th tree trajectory and Nk is the number of tree trajectories for a bond that matures after Tk steps. These trajectories depend on the ui #j, and hence, these equations can be used to determine the latter. To do this we need to impose enough restrictions such that the total numbcr of unknown parameters in the tree becomes equal to the number of equations, The tree parameters can then be obtained from these equations. ne tree will fit the initial term structure exactly, An example to this way of proceeding is in Black, Derman, and Toy (1984).
g
r: . jy;?.s , a
j
,
where M may be diferent for cach bond, depending on the maturity. Now, because l was selected arbitrarily, the 6.) will not be arbitrage-free. But, we also have the obscn'ed term structure, which is known to be arbitrage-free. So, we can try to adjust the & in a way to minimize the distance:
tf,
'rmax e
i uuc 1
447
,
e-
j=1
Fixed lncome
2 I#(r,T') - Bt, Ff) I
4.3 Closed-Fbnn
'sollanu
Supposc we can analytically calculatc thc cxpcctation: csdxj Bt, Tj Etz3
This way we fmd a value for K such that the calculated term structure is as close as possible to thc obscrved term structure. Once such a K is arbitrage-free, in the determined, the rt dynnmics bccomes (approximately) sense that using the modcl parameters, and this new % one can obtain bond prices that come to the obsen'ed term structure. 'sclosc''
=
'rlhtlx
min
slodeu
K
.
E=
ne prcvious appzoach used a single paramctcr x to make calculated bond prices come as close as possible to an obsewed term structure. fit was not perfect because the distance bctween the two term stnzctures was not reduced to zero, although it was minimized. By adopting a general the it. tree approach one can Once we considcr a binomial model ftr movements in r, wc can choose the arbitrage-free the relevant paramctcrs so that the tree trajectories term structurc and the relevant volatilities. For examplc, we can assume tllat we have N arbitrage-free bond prices. Suppose wc also know the volatilities o'i of each bond S(f, L). Let the up and down movements in r; at stage i bc dcnoted by ui, di, such that: rrhe
K'improve''
'lit''
uidi
eumple
:n) I.a(r,
of obtaining
G(r/, Ff
,
an
x)
1 .
arbitrage-free (approximately)
5 Concluslons This chapter has brielly summarized the fw'o major approaches to pricing devative securities that depend on interest rates. The classical approach was shtawn to be an effort in spot-rate modeling. The arbitrage restrictions were incorporated indirectly through a process of an initial tlunre.'s The HJM approach on the other hand was ari extertsion of the Black-scholcs formula to interest-sensitive securities. ftfitting
.
,
1 Nk
This is another model for rt.
1
1
=
Given this restriction, the tree will be recombining and at evcry stage we pill have i unknown parameters, ne nex't task will be to dctermine thee ui #j by using the equality: S((), Tk ) =
.(''
and get a closed-form solution for the S(/, F), as will be discussed in the ncxt chapter. Suppose this results in the function: Bt, T) Grt, T; K4. nen, we can minimize tbe distance between tlle closed-form solution and the observed arbitrage-free yield curve by choosing K in some optimal sense: )
4.2 Trce
j-
=
-
n 1 rf'a e - )Z;
-.
=
1
,
6
References
'lhe best source on tliese issues is Musiela and Rutkowski (1998). Of course, this source is quite technical, btlt we recomrnend that readers who are scriously interested in lhcd-income sector put in the necessary effort and become more familiar with it, The excellent discrete time treatment, Jarrow (1996), should also be mentioned here.
448
C H A PT E R
*
Classical and HJM Approaches
19
to
Fixed Income
1. Consider the equation below that gives interest rate dynamics in a settingwhere the time axis (0,F1 is subdivided into n equal intervals, each of lengt.h 1: rl+a
where the random
r: +
+'
tr?
trltl#;ia
W?))-f- frztH'l
-
-
Wzkal,
r/-ha R !+A
(W$+a
=
-
awsx j,
M)
-(all .
=
Rt +
r,
tz 21
a 22
Rt
H''jia 1
+
,
a+2/+.(
Suppose
R'J lii6 #rl + *1(V6+.& ) + t6( -
3. Suppose at time t = 0. we are given four zero-coupon bond prices (Sj, Bz, #3 A4Jthat mature at times l = 1. 2, 3, 4. Titis forms the term strudure of interest rates. We also have one-period fomard rates (, , fi is h , ./'aJ,where each = f 0 on a loan that begins at timc f = i and the rate contracted at tne ends at time f = i + 1, ln other words, if a borrowcr borrows $1 at timc t = i, he or shc will pay back N(1 + hj at time f i + l ne spot rate is denoted by ri. By defnition we have ,
(a) Explain the structure of the enor terms in this cquation. In particular, do you lind it plausible that M-a may cnter the dynami of observed interest rates? (b) Can you write a stochastic differential equation that will bc the analog of this in continuous time? What is the difficulty? (c) Now suppose you krlow, in addition, tlmt long-term intcrest rates, Rt, move according to a dynamic givcn by A/+.a
a12
(a) Derive a univariate representation for the short rate rt. (b) According to this representation, is rt a Markov process? (c) Under what conditions, if any, would the urtivariate process rt be Markov?
are distributed normally as -
=
fz1 1
where the error term is jointlynormal and serially uncorrelated. rt is a short rate, while Rt is a Iong rate.
error tcrms
Ws
09
has the following dynamics,
7 Exercises
=
7 Exercises
l'V-A),
-
where we also know the covariance:
=
rtl The
.
all fomard ioans are default-free. each At time peziod there are fwc possiblc states of the world, denoted ,
=
a
(#JJand
by (?.f/di
A;IAIPZJI'Jp.
=
.
:
=
1, ?, 3, 4J.
(a) Looked at from time i 0. how many possible states of the world are there at time i 3? (b) Suppose =
=
for thc vector process
Can you write a represcntation Xt
(#1
rt
=
Rt
(vector)Markov X
process Xt, r/
=
R:
.87,
=
%
B4
.82,
=
.75)
=
and
(./)
=
8%,
/1
=
9%,
h
=
10%,
fs
=
18%1..
Forrn tliree arbitrage portfolios that will guarantee a net positive return at times i 1, 2, 3 with no risk. (c) Form three arbitrage portfolios that will guarantee a net return at time i 0 with no risk. (d) Given a default-free zero-coupon bond. Bn, that matures at time t n, and alI the forward rates (f, s-1J, obtain a formula that cxpresses Bn as a function of fi. =
=
=
,
Bz
,
such that X is a first-ordcr Markov? r (d) Can you wnte a continuous time equivalent of this system? @) Suppose short or long rates are individually non-Markov. ls it possible that they are jointlyso? 2. Suppose the
.9,
=
.
.
.
.
CHA PT FR
*
(e) Now consider to the system:
19
Classical and HJM Approaches
to
Fixed lncome
the Fundamental Theorem of Finance as applied
B1 B1
s' 2
/1
#3
sy
Bd3
42
B4
,2
ad4
Bz =
.
Can al1 Bi be determined indcpendently? (9 In the system above can aIl the (/fJ be dete=ined independcntly? (g) Can we claim that all h are normally distributed? Prove your
Classical PDE Analysis for lnterest Rate Derivatives
a'llsWe'r.
4. Consider again the setup of Ouestion 1. Supposc we want to price thrce European style call options written on one period (spot)Libor rates 0. 1, 2, 3, as in the above case. Let these option prices be L with i denoted by G. F-ach option has the payoff: =
c
where N is a notional general.
=
N mxgz.j
-
K, 0j,
amount that we set equal to one without loss of arly
price such an option? (a) How can mu (b) Suppose we assume the following: (i) Each h is a current observation on the future unknown value of Li. (ii) Eacb h is normally distributed with mean zero and constant variance cri (iii) We can use the Black formula to price the calls. .
(c) Would thesc assumpticms be appropriate under the risk-neutral measure obtained using money market normalization? Fwxplain. (d) How would the use of the forwardmeasure that corresponds to each Li improve the situation? (e) In fact, cao you obtain the forward measeres for times f 1, 2? (9 Price the call optitan for time t 2 using the fomard measure. =
=
1 lntroduction The reader is already fa rniliar with various derivations of the Black-scholes formula, one of which is thc partial differential equations (PDE) method. ln pmicular, Chaptcr 12 showed how risk-free borrowing and lcnding, the underlying instrumcnt, and the corresponding options can bc combined to obtain risk-free portfolios. Over time, these potfolios behaved in such a way that small random perturbations in the positions taken canceled each other, and the portfolio return became deterministic. As a result, with no dcfault risk the portfolio had to yield thc same return as the risk-free spot ratc r, which was assumed to bc constant. Otherwise, there would be arbitrage opportunities. The application of Ito's Lemma within this contcxt
rcsulted in the fundamcntal Black-scholes PDE. The Black-scholcs PDE was of the form: 1
rvtl - F'F + Ft +
2
+ - r.r 2
ut
zFss = 0,
(1)
with the boundary condition: Ftv
,
F)
=
max
(.% -
K, 01
.
'The r is the constant risk-free instantaneous spot rate, the S, is the price of a stock that paid no dividends. the F is the time f price of a European call option written on the stock. The K and the F are the strike price and the expiration date of the call, respectively. ln Chapter 15 it was also 45I
7 452
n-
H A P T ER
.
20
Classical PDE Analsis for lnterest Rate Derivatives
mentioned that the solution of this PDE corresponded to the conditional expectation Fst,
tj
.......-
E') gc-'fr-/)F(,,.,
r)j
(3)
,
calculated with the risk-neutral probability #. Given that we are now dealing with derivatives written on interestsensitive securities, we can now ask (at least) two questions; PDES in the casc of interest rate derivatives? For * Do we get similar example, considering the simplest case, what type of a PDE would the price of a default-free discount bond satisfy? @ Given a PDE involving an interest ratc derivativea can we obtain its solution as a conditional expectation similar to (3):?
ncse questions can be answcred in fww different ways. First, we can fol1ow the same approach as in Chapter 12 and obtain a PDE for discount bond prices along the lines similar to the derivation of tlle Black-scholes PDE. In particular, we can form a portfolio and equate its deterministic rcturn to that of a risk-free instantaneous investment in a savings account. Application of lto's Lemma should yield the desired PDE.I ne second way of obtairiing PDES for intcrest-sensitive securities is by exploiting the martingale equalities and the so-called Feynman-Knc results directly. In fact, when we westigate the relationship between a certnin class of expectations and PDES, we are led to an interesting mathematical regularity, It turns out that thcre is a very close connection between a representation such as:
Bt,
F)
EtI3 =
j- h'nrxd'%BT Flj
(4)
and a certain class of partial differential equations. In stochastic calculus, these topics come under the headings of RGenerators for 1to Difhlsions,'' aKolmogorov Backward F-quationy'' and more importantly, ''Feynman-lfac formula.'' Using these methodsa given a conditional expcctation such as in (4), we can directly obtain a PDE that corresponds to it and vice versa. Of coursea this correspondence depends on some additional conditions concerning the underlying random variables, but is clearly a very convenient tool for the Enancial market pradititmer. Yet, the discussion of thcse ern'' methods should wait until the next chapter. Ktmod-
Introduction
453
In this chapter we show that prices of interest rate derivatives will satisfy similar to the mdamental Black-scholes PDE using the steps.'' But, this derivation will still be fundamentally different thlm tlle one followed in Chapter 12 because the underlying variable will now be the spot rate rt. Spot rate is not an asset price, in contrast to the St which reprcsentcd the pricc of a tradcd asset in thc Black-scholes wor1d.2 Obviously, the difticulties associated with spot-rate modeling will be present here ttlso. Thc derivation of the fundamental PDE for interest-sensitive securities will follow steps similar to thc classic paper by Vasicek 'T'he essential (1977). idea is to incorporate in the dynamics of the returns the arbitrage conditions implied by a single Wiener process3 that determincs the random mtwcments observed in more than otle asset. ln the case of the Black-scholes approach, we workcd with two securities, the underlying stock and the call option written on it. An insnitcsimal randort movement in the price of the stock also affected the price of the option. Hence we had fww prices driven essentially by the same source of randomncss, These securities could be combined in a careful fashion with risk-free borrowing and lending so that the unpredictable random movemcnts canceled each othcr and the rcsulting portfolio became The same idea can be cxtended to interest-sensitive securities. For example, cxcept ftr their maturities, bonds are instruments. They are expected to be inlluenced by the similar ivnitesimal andom llttctuations. Hence, under some conditions, a portfolio formed using two (or more) bonds can be made risk-free if podfolio wcights are chosen carefully. Yet, tbere are differences when compared with the case of stocks. ln the classical Black-scholes derivation, the spot rate was assumed to be constant. This assumption did not appcar to be vcry severe. In the case of interest-sensitive securities, the assumption of a constant intercst rate canlot be maintained, On the contrary, tbe randomness that drives the system comes from infinitcsimal Wiener incremcnts that affcct instantancous spot rate rt. But, this latter is not an asset price as mentioned earlier, The unkmown drift of intercst rate dynamics cannot be simply made equal to the l'isk-frce rate by invoking arbitrage arguments. This introduces major comPlications in the derivation and numerical estimation of PDES for interest rate derivatives. ln fact, although the steps in the following derivation are mathematically straightfomard, they are somcwhat more convoluted than in the case of plain-vanilla call options written on stocks. Finally, we should reitcrate that the approach adopted here is heuristic just like the derivation of the Black-scholes PDE. A techniPDES
Gtclassical
Rriskless.''
idsimilar''
Eiclassical''
arld as a restllt tle 1We remind the rcader that risk-free portfolios are nOt Klf-financinp is not mathcmatically accurate in continuous time. Yet, tme stZ obtains tile PDE because the extra cash fow invested or withdrawn over tirne has an expecletl value ty.f this heurisdc ZGrt. Wis i%ue WaS discussed in Chapter 12 in more detail. We keep uilizing metht)d w'itil the condititln ha the reader kcps in mind this important pyint. melhd
Zlcorrtzd''
'ne
r is moze likr a percentage return,
3oy in case of
lwo-facter
a pure numbet.
modelsa two ndemndent
Wiener
,
i
! @
!
,
' ! .
1
i
;'
1
proceses. :!
E'l :
'.
454 c H A P T E R
.
Classical PDE Analpis for lnteresr Rate Derivatives
20
:
3 Market Price of Interest Rate Risk
. 2
' lk.,. . t:. ' :
condition cally correct derivation would incomorate in the argumcnt thc carlier, the discussed that the risk-free portfolios are also self-hnancing. approach below may not yield self-hnancing portfolios. x/ts
;
't
3 Market Price of lnterest Rate Risk
12
To derive a PDE for a discount bond's price, we first need to form a riskmade of the two bonds #1, Bl at time t.4 In particular, frce portfolio without any loss of generality, it is assumed that 0L units of J.ll are purchased, and ezunits of Bl are shorted, for a total portlblio value:
'
(
.:
.+,
'
..
.:.
F1
.
2 The Framework
.(
.k
,
' (
provided two Thc hrst step is to set the framework. We assume that we are bond prices, discount default-free SDES describing the dynamics of two ? < Thc bond .( Fc. that T1 such Tz #(f, E ) and S(f, Fa), with maturities F simplify the notation, prices are driven by the same Wiener process H( To write: subscript f and in this section we ignore the t'nc 7 1 Bf ?, Tj ), (5) B
.
.+
'..
':
9LB1 -
=
.
'
=
B
a
=
E
%
,
(6)
Blt, Fg).
dynamics) These bond prices are postulated to have the following dB1 ;&(B1, /)st#r + tn(At, z).atJJ#;, =
t. .
dB1
=
/J-
(B2 t4B2dt +.
,
,
j)#2Jg(.
.
(8)
i .: '
.:
:
'
'
21 :
)
dtg''
:
or after
'.
replacing
d.+
=
from the =
'
. '
(0
!
tn
(#1,flsldJ#;j
1
crjfl
-
%hBl)
Cl =
s
:
tfn -
(n
tn
)
B1
jlj-t/jjyj
szt(zz -
((/.t1S'
=
-
%yaB2) dt.
tl4j
,
G1
-
'nese
;
.+.
4e1'hetime subscript is ignored fir notational
L
.k
tjBkdt +
(13)
tn
)
nB2
becomes
z?.i:
(j5;
(16)
These increments do not have a Wiener component and are completely predidablc. These steps justifythe particular valttes chosen for the portfolio weights 8I, weights werc selectcd so that the J'l1 tcrm drops from %. the SDE of the portfolio nis is similar to the derivatioo of the
i
J,,
'.'
SDES
gp,(#l,
d'+
t
-
o,dB - ogdBl that give the dynamics of dB, #S2:
=
= 0. This gives the incrcmental changes in the portfolio value:
'
imposed on this smt-rate dynamics; the pa- ' assumed to depend only on the latest obser1) rameten ar, t), bLrt, are ' not affcd the drift and volatility < / d s'a tion %, st) that previous rs., this Markov We alrcady know from the previous chapter that parameters. model. Still, tbe t term-structure general of rt will bc violated in a property approximation. reasonable classical approach procecds assuming that it is a
0,
.
'.
pz'ob a bility P. Note the cbtical restriction
(12)
.3X,
Grouping the Wiener increment JJF;, we see that its coefhcient zero after rcplacing the values of 0L and 0z
,
,
)
(#,(S2y tlBzdt + o(Bl,
=
.
O'L
- eg
.:
'lclassical''
trl
n -
,
but W,'7. Note two points, First, the difhasion terms are a f'unction of the samevolatility ''J the Second, 2. depend on diffcrent diffusion parametcrg a';, i 1, volatility, but the bond #parameters are written in terms of pcrcentage tile drift namics are not neccssarily givep by geometric processes because 1, Bi. 2, and i and difftision parameters are also allowed to dcpend on SDE. ( geometric requircd by a are not constant as would be nccd approach to we now msit Because we arc adopting a by: given of bc dynamics the We rt let model. interest rate an t (9) . drt art, tjdt + brt f )(fH?), assumed where thc drift arf /) and the diffusion blrt, tj parameters are historical data, or as in the to be knobvn. ney are either cstimated from worth em- ?' also is It market pzices. practical approaches, calibrated using real 1,ko* : thc witb respect to phasizing that the P( herc is a Wiencr process =
B a(
=
:
(n4.!?2
=
where g'j, 1, 2 are the volatility parameters trj(Si, f ), (rz(B2, t ) of thc two bonds as described in Equations (7) and (8).As time passes, this portfolio's value will change. Acting as if the portfolio weights are constant, the implied innitesimal changes will be given byl
,
(7)
.97
,
#1((z'z - c)
,i
! .';
() ())
,
G2
0L
'
,,
,
;
ogs2
Suppose the portfolio weights are chosen as:
@
,
=
455
'.
.
simplcty.
K
..1
;
'
456 t'l H A
P T ER
*
20
Classical PDE Analysis for Interest Rate Derivatives
(tW2#'l
4 Derivacion of
,
d.q
=
(t7'2-
oj
(1p
)
ttrz#.l Simplifying the
,@,
=
zi )
dt and rearranging,
(/.z1 r/)
-
rt.pdt.
o.yg/
(t.n-
s .:
-
l#,z(7-2
frj
G) '
#S(f, F)
!
rk
;
dr
j,z
,:
4. '
': ;.
.,,...
(j ().
%--#
.
':
##(G, f)
: .
:; 'r
art
=
,
tjat + brt,
yjtfp)
g2)
wc get:
'.
.
(21)
a
Substimting for drt from
.;.
'
(1a)
1
Brdrt + Btdt + LBrrbrt, tjldt.
=
'
;
we obtain;
=
''
'
fhlhj
-
The third step of the PDE derivation for bond prices is to use the previous results in lto's expansion for Bt, F). Remembering that #(/, F) is also a function of r,, and applying Ito's rule:
: %
of nis sDE does not contain a diffl'sion term and the dynamic bebavior and claim standard riskless. argument thc Hence, use we can now d.@, is that this portfolio should not present a'ny arbitrage opportunities and its deterministic return should equal the rtAdt :
457
4 Derivation of the PDE
'
.
Lqq
PDE
!
:
.:
y
the
'
<
t'
by
Olhlydt.
-
j
'
t.
Black-scholes PDE. Indeed, replacing the 0i, dividing and multiplying .e, and arranging the d-+ can be written as:
.
.'
. ('
:: .
')
'rhat is, the risk premia offcred by bonds of differcnt maturities are equal, . ) ): Risk of volatility premia normalized corresponding by the parameter. once :..q! per unit volatility are the same across bonds. Bonds with higher volatility y risk-premia.s Pay PEOPO rtionately higher nis result is not vely uncxpected t because at the end, these bonds have the same source of risk given the :.,1: common JT'Kfactor. Obviously, if one of the bonds was a function of an j (' additional and different Wiener process, say Wz7,then even under a n y i arbitragc condition, risk premia per volatility unit could be differen t acrou negat*e. ! yij well risk be premia can vcy bonds,Note, in passing, that these Now, during this derivation the maturities of thc tmderlying bonds were ,L' E selectedarbitrarily. Thus, similar equalities should be true for al1 discotmt :E : ! 1 bonds as long as their dynamics are driven by the same Wiener process j ' ) nis gives a term Atrr, f ) that is relevant to al1bond prices, Bt, 6.): ,
=
1 Brart, t) + Bt + '-Brrbrt, 2
1) a
I)#r#l.
dt + brt,
(23)
where again the JP; is a Wiener process with respec.t to the real-world probability #. This SDE must be identical to the original cquation that drives thc bond price dynamics. Simplifying the notation, this SDE is;
dB
';;
't
#,tsa
=
tjBdt + rrCB, /)/.1(/H.),
(24)
'
..!
.
.
under the probability #. This means that we can equatc the drift and diffusion cocfhcicnts. Setting the two difhasion coeflicients in (23)and (24) equal to each othcr, we obtain;
b(r t t)B
:;
.
.
; '
;
* CZ-
i
G) =
f).
(r,,
(20) t'
.t. This term is called the market prce of interest rate tik. As citn be seen from (hC UCFWXiOS, followZg r! it iS ifl WRCFZIZ fuzctioll Of Tf aRd 1. Btlt il While Simply is Will dependence thig assuming that write WC SCC it aS , i' i of bond matur ty. independent the again that 1, is kept in mind, Note risk WaS It is worth menticming that a similar market J'ricd of dul./ In *nused explicitly. Black-scholes framework but was not Pr escnt in the Of securities interest-sensitive we ) with Black-scholes PDE, trast to the case PDE: here. deriving the do have to use the A/ cxplicitly in .; tllc
tiolx
i
:'
'
i
'
''
'
....
r $
I
'
!
'
1
: '
,
=
(25)
OB .
where ojB, t4 is abbreviated as o'. Equating the drifts in
(23)and
(24)gives:
1 Brarl, r) + Bt + '-Brrbrt, l)Z. (26) 2 Here wc have twro equations (25)and (26)that we can exploit in obtairting thc po's for bond prices. In facta this last Equation (26)is already a PDE except for thc fact that it contains the unknown p,B, t). Also, 'note that up to this point we did nothing that would incorporate the arbitrage restrictions tjaat we must have in this system,ti It turns out that thc way to eliminate the drift /z( B, tj from (26) js by using arbitragc arguments. Recall that in the case of Black-scholes PDE, one simply the y.B, tj by the constant spot rate r. But in the prcsent case this is not possible because we keep using the spot-rate drift alrt /) in (26). lf we replaccd the g,(B, 1) by rt. this woutd require adjusting the spot-rate drift art /) in (26)to its risk-neutral #(B,
f)#
-
=
(dunlnown''
E
'I
;.
,
I I
KGrcplacef'
,
,
.
sainoher way of wying this is that the Sharpe Ratios of the Nnds are equal.
.g
.'
L
.j'
6'rllere
will be arbitrage
restrictions
driven by the samc Wiener process !,#;.
because we have assumed
that all bond prces
arc
r,'' J ::
458 C H A P T E R
.
.2
E'
20 Classical PDE Analysis for lnterest Rate Dexivatives
4 Derivation
'
and it is not clear equivalent as wcll. But the rt is not t'he price of an asset resolved by utilhing bc problem can how this adlustment can be done. risk the market price of intcrest rate r. ln fact, Equation (20)gives the market price of risk kt as: :
J') /.z.(.8,
rt
-
A,
=
-
- 6X
(
459
;. ..'.t.
:
.?
'.(::
k ',.
....
.k '.!
,
,.
%
:
.
EE. ..)
(29)
rB + #(rj t)Bv.t. ,
(26)and
1) jn
side of this for By.B,
Now substitute thc right-hand
.'
'
!,
This gives:
lxBb r)5
The gencral strater in deriving the PDE was similar to thc case of Black-scholes, The maim difference arises f'rom thc fact that the driving process in the present case is not Stb the price of an asset, but is the spot rate rt which is a pure numbcr. Hence the no-arbitrage ccmditions have to be introduced in a differcnt way tltan just makit'tg the unknown drift coeflicient equal to the risk-frce rate. ne approach was to modify the drift of the bond dynamics using the market price of risk for rt. Thc reader should realize that letting
Z
'
...
=
4.1 A Ctmzprll'iafm
'.
'E....
...j.
(28)
=
'
''
(2):
-
;
.
.
W
shown in or, using thc equivalence of dlf'fusion parameters B(b4B, t ) G) ,. - brt,- tll)r .
'.
.'. ..
'(,
., 1
.
r.5
rear-
rj
Cangc, .
J1(S, /)S
,
.1 . '' ?
Note that thc be written as:
PDE
We now summarize the major aspects of this derivation and compare it with the approach takcn in the case of Black-scholes PDE,
7
%
(27)
,
t
r';'
'rhis
Bralrt,
of the
.
f)
1 + Bt + jBrrblrt, 5
L
Bv(J(G, r)
-
r
()Z
rtB
-
-
#trt tlBrf ,
drtft I.bB, tj is now eiiminated. '
1 brl, r)z/) + Bt + 2-Brrbrt,
=
0.
1.
(30)
.:
'i;J
:'.
This can inally
:E'
..'
t)
-
rtB
=
0.
. ...,
';
(31)
.
(32)
1.
i ;
..
'
:
.
.
.
.
.
j
'
?
''':
:?
B( T; F)
j
,
bond .L rrhis is a PDE for the price of a default-free pure discount ' Gmdition of thc simpler case than is B(t, F). ne associatcd boundary t i' Black-scholes. 'T'he bond is default-free and at maturity is guaranteed to (f spot rates at that time: c; have a value of 1, regardless of the levcl =
'
:;
.;t
2,
j
' '
,':,
alrt, /) and diffusion . lf tne had an inlercst ratc model with known drift still need an ts- ; coeflicicnt blh, ), to use this PDE in practice, one would usable. Also, it is worth re,,e t i mate for thc t. Othemise the equation is not ris k-adjusted f alizing that in this PDE the coefficient of Br is equivalent to a t. drift of the spot-rate dynamics. dynamim & In fact, it is as if we are using the drift from the spot-rate Girsanov tlleorem : writtell undcr the risk-neutral measure /. lnvoking the W( detlned tmd/r ) for Equation (9) and switching from the Wiener process obtain a new SDE foT rp P to the Wiener process *2 dcfined under /, we , . r)r,/I.::. ',,, (33) . .y (d?(r/,t4 - brt, /),) dt + brt, t rate rist'' 'Whene'er Th e dlift Of this SDE is now adjusted for the Switched Jt(.) from to G, one needs to switch the bond price driftg are l).r). #(G, J(r/ (J(G f) Spot-rate dynamics from ) to '
y t
(
.'.
'j
;
..
:
'
I
): I
L
j
: '
.
'
.
:
'
=
:
idinterest
;
-
,
,
,
.%
:J
?
riB + blrt, /)Sr)A/,
(34)
as was donc in Equation (29),introduces the no-arbitrage condition in thc equation implicitly. However, notice a rather important difference. In the case of the Black-scholes dcrivation, by using the no-arbitrage condition we succeeded in completel.y eliminating tbe nccd to model and calibrate the drift of the stock price process St. In fact, in thc Black-scholes derivation, expccted change in St did not matter at all. Thc option price dependcd on the rclevmt volatilities only. In case of the spot-rate approach to pricing intcrcst-sensitive securitics, the use of no-arbitrage conditions will again introduce thc spot rate rt in the PDE. Yet, along with the rf, fww ncw parameters enter, namely the spot-rate drift art, t4 and the market pricc of interest rate rigk. Thcse parameters need to be estimated or calibrated if the PDE is to be used in real-world pricing. As mentioned in the previotts chapter, this is a departure from the practicality of the Black-scholes approach, which required the modeling of volatilities only. But it is also a change in philosophy because, in a sense, a complete modeling of the rt process is now nceded. A second fundamental point of the abovc dcrivation is tbe assumption of a single driving process rt. Remember that the dynamics of al1 bond prices were assumed to bc driven by the same univariate Wiener process H( Because the same Wiener process is present in thc. SDE for the spot ratc r/, this assumption enabled us to obtain a convenient no-arbitrage condition that was a functitn of a single markct price of risk t. Clearly, this may not be the case. Making a single stock pricc a function of a single random process, H(, may bc an acceptable approximation', doing the samc thing for a set of discount-frcc bonds ranging from very short to very long maturities may be more questionable. ,
.
2
dr
=
.'
460 C H A P T E R
*
20
Classizal PDE Analysis for lnterest Rate Derivative
Nevertheless, our purpose in this book is to display the relcvant tools actual markets. The rather than obtaining satisfactory pricing metbods for end.7 useful this to assumption of a single factor is
5 Closed-Form Solutlons of the PDE
5 Closed-Form Solutions of
the
PDE
46l
with the boundaq condition
B$TT)
=
1.
But this is nothing other than the ordinary diferential equation ##(J, T) + rBt, T) 0. dt with terminal condition #(F) 1. Its solution will be given by =
solved for a The fundamental PDE for bond prices can sosnetinaes be Bt, F) to ties that formula explicit closed-form solution. This way, an relevant and the parameters spot rate rt, the maturity F, the tj and , can be obtained. tj, brt, art, Black-scholes arld the The analogy is with the fundamcntal PDE of and the S( assumptions enough process on Black-scholes formula. Given PDE able that solve to to the constanc'y of the interest rates, one was enough given framework, the ln present get the Black-schoies formula. assumptions about thc intercst rate process rt one cari do the same for the bond price PDE. We discuss some simplc examples.
=
#41 F)
C-FtZ'-J)
=
'current''
5.1 C-e
This bond pricing function will satisfy the boundary condition and the fundamental PDE. lt is the usual discount at constant instantaneous rate t. 5.2 Case 2: A Meun-Reveuing rt Suppose now thc market price of
(n!1)
at We begin with an extreme case. Supposc the spot rate is ctmstant for rl, rt r for alI t. nen tc SDE drt art, tjdt + brt, /)JM,
drt
=
=
(trivial)parameters: alrt, t)
bfn tj
=
=
(36)
,
=
0.
=
0.
which originally is given Thus, the fundamental PDE for a typical Bt, T),
by
a
=
(x -
rtj dt + hJI'P),
(37)
whcre B'' is a Wiener process under the real-world probability. Note that 6 the volatllity structure is restricted to be a constant absolute volatili deoted by 5. Suppose hlrtbcr that the parameters a, a7. b, and are known exactly. Then, the fundamental PDE for a typical #(f, T) will reduce to:
paid Further, bccause there is no interest rate risk no risk-premia sbould be for it: A
is constant:
but that the spot rate follows the mean-reverting SDE given by:
1) A Detenrdnfstic rl
will have the following
lisk
Br (tz(& rt4 -
1 h) + Bt + jBrrb z
-
-
rtB
0,
=
(38)
This setup is known as the Vasicek model, after the seminal work of Vasicek (1977). lt can bc shown that the solution of this PDE is the closed-form expression given by the bond pricing formula Bt, F), for time f 0, =
Br (t? :) -
1 + Bt + jBrrbz
-
rtB
=
0,
(35)
-ta.J
.)(1-e--)(A-r)-'rA=
e
,
(39)
where
willreduce to Bt + TB 7It should be remembcred well.
,(0, F)
(i-e-aF)2
lhat
this assurnption
=
0, is ofterl made in actual prking projects as
R
=
x
-
bl
h -
-
a
-y
RM
(40)
and is the current obsewalion on the spot rate. Given some plausible estimatesfoT tlle unknown paramcters we can then plot this function. r
462 C H A P T E R
.
20
Classical PDE Analysis for Interest Rate Derivatives
5 Closed-Form Solutions of the PDE
463
5.2.1 Ekample For cxample, consider an economy where the long-run mean of the spot rate is 5% and where the spot rate is pulled toward thc Iong-rn mean at We thus have a rate of .25.
and
,25
a
=
(41)
.05.
K
=
Further, suppose thc absolutc interest rate volatll' ity is b
.0
=
15
during one year..
.015
(42)
.
To apply the formula, wc need thc market Assume that we have
price of interest rate
(43)
.1cj
JL
-
rl
=
...
where the p,, .vare the unkn' own bond drif't md volatility parametcn. we know that
ncn, 1
(44)
-.10.
A,
isk.
=
Uging these parameters, we can calculate thc bond pricing fundion Blt, T) that will tlcpend on the initial interest rate r and on the mamrity curve'' discussed earlier. parametcr T. This is the so-called b a The graph of the jB(t, T), F l (0,Fmaxjl with (z 5%, r = r at three diffcrent levels for the spot rate r .J.5, x 15% are shown in Figure 1. Because thesc are discount bond priceg, the short maturitics have values closc to 1, whcreas longer maturities get progrcssively chcaper. ne corrcsponding yield curve is obtained by taking (minus)the logaThc yield rithm of thc discount cun'e and then dividing by the matur. of initial spot ratcs. cunre is sht)ma in Figure 2 foz the s.4'nnc set determinG Note that the mean-reverting aspect of the interest rate SDE well that the yicld cunre can have upward- or downward-sloping curvcs, as model currently, the lf as flat ones. This is becausc if the spot rate is % 5% itg toward will as wc consider the long back mean it that gt7 assumes priced automaticaliy by using ratcs on be bonds. nus long bonds would will by using sbort priccd be the average around 5%. whereas short bonds the long-rtm below rates closer to lf %. The case of a current short rate upward-slopirig yield curvc. mean is the revcse and gives an market price of risk Figure 3 shows the effect of changing the valuc of 5%. assuming r on the discount c'uzve, didiscount
.015,
-.10.
=
=
=
2
h/ 10 ' I121
14
'
'''.
.5%,
.05)
=
=
=
=
0
''''
12
'
01
' .
.
.
'
.
.
()()d$
---..
J
.....=,
0.06
zr
r
o
5%
=
.5%
C=
0 ()4 0.02 0
1
1
2
4
.
I
I
I
6
8
10
I
I
12 14
j
16
FIGUR E
j
18 2
j
j
20 22
j
j
24 26
j
j
28 30
M2UFi$
464 C H A P T E R S(0, T )
.
Classical PDE Analysis for Interest Rate Derivtives
20
8 Exercises
465
where the functions
1
.d(1,
xzltlar)
r), 2
=
C(f, F)
=
C'(f, T) are givcn by
y e 1/2(a++v)r
(cr+
+ y)
2(ty +
(e'yr
+ y)
MT
11 (49)
1) + 2 y
-
C
2
1
.-
F
(eT
1)- + 2 y
-
(50)
.
and the y is given by
'y
(cr+ )3 +
=
2 bz
.
(51)
One can act in a sinlilar fashion and plot the yield curves for this case.
Concluslons
6 FlG U RE
5.3 Csc 3: More Cemplex Fornks for nere are several other models that result in closed-form solutions bond prices. spot For cxamplc, in the casc of Cox-lngersoll-Ross, the f'undamental rate rt is asstlmed to obey the slightly different SDE
drt
=
crta
+ dprlap;,
rldt
-
(45)
volatility. which is known as the squarc-root specilication for imerest rate will by! be given The PDE that will correspond to this case
((z(x
-
r)
-
blrjB
+ Bt +
r
1
g
BT
=
(46)
(47)
1.
eqtlation. This PDE can again be solved for a closed-form bond-pricing of Vathe complex than case The resulting expressitm is somewhat more sicek. lt is given by
B(t, F)
=
7
References
The PDE solution for bond prices can be found in all major sources. ne reader may, however, prefcr to read lirst t.he original paper by Vasicek tbat can be found in and Beyond.'' Two other good sources are Cox-lngersoll-Ross and White (1990). and Hull (1985) tvasicck
Brrb z r - TB = 0,
with boundary condition F)
This chaptcr dealt with the classical approach to deriving PDES for interestsensitive securities. We see that although the majgr steps are similar to the cse of Black-scholes, there are some major differences in terms of practical applications and the underlying philosophy between thc two cases, The classical approach to pricing interest rate-sensitive seculities rests on modeling tle drifts of the underlying stochastic processes, whereas the Black-scholes approach was one where only the volatilities needed to be modeled and calibrated.
Wl6 F)d-C;'FX
@
(48)
8
Exerclses
1. Suppose you are given the following SDE for the instantaneous spot ratez drt
=
kn-td?i
,
(52)
where tlle J#) is a Wiener process under the real-world probability and the c. is a constant volatility. The initial spot rate r is known to be 5%.
2H A PT ER
466
.
ClassicalPDE Analysisfor Interest Rate Derivatives
20
(a) What does tMs spot rate dynamics imply? (b) Obtain a PDE for a default-free discount bond price Blt, Tj under these conditions. (c) Can you determine the solution to this PDE? (d) What is the market price of interest rate risk? Can you interpret its sign?
Relating Conditional Expectations to PDES
You are given the spot-rate model: ,#1.#7, (53) d% a (m rt4 dt + where the 1&)is a Wiener proccss under the real-world probability. Under this spot rate model, the solution to tlic PDE that corresponds the closed-fonn bond to a default-free pure discount bond B F) gives S(f, F); pricing formula tt-'-atT (l-e-(F-rJ)(l-r)-(r-l)A-oV ) (54) #(f, F) e =
-
.)j)2
=
where
?k
=
x
-
d,
-
bl -
y
(55)
.
of Now consider the following questions that deal with properties formula, this represented by bonds whose prices can be (a) Apply lto's Lemma to the bond formula that gives #(/, T) above and obtain thc SDE that gives bond dynamics. (b) What are te drift and diffusion components of bond dynamics? is Derive these expressions explicitly and show that the drift g given by:
and that thc
(c) (d) (e) (9 (g)
#.A
y diffusion parameter #
=
rt -
.g-a(z-l)j
,
equals!
(
--+
lntroduction
Throughout
this book we keep alternating between mathcmatical tools for tw'o major pricing methods. Using the Fundamental neorcm of Finance and normalizing by the mcmey market account, we often used the representation F(5,,, /)
=
.E:
je.j'' cd.ast5y,
wlj
(1)
to price a derivative with expiration payoff FSv, T), mitten on St. According to tltis, the conditional expectation under the risk-neutral measure, ?, of futurc payoffs would equal the current arbitrage-free price Ftkf J), once discounted by the random discount factor c-.( t rv-, When the h was conStant, as was the case under the Black-scholes assumptions, this formula simplilied to: ,
-a(z-l)j b -a 1 e of maz'ls it expected that the diffusion parameter is independent ket price of risk ? bond What is the relationship bctween the maturity of a discount and its volatility? risk-free rate ls the lisk prcmium, that is, the return, in excess of this imporrisk? Is of price proportional to volatility? To market tant? drft and diffusion parnmSuppose F x, what happens to the eters? What does the R represent? -
1
(liscolmt
-
.
.F% l tj k
'>
=
e-rT-tle'l '!'
gFty:r,. r)j
(2)
.
At other times, the pricing was discussed using PDE methods. For example, in the previous chapter, using the method of risk-free portfolios we derivcd the PDE that a default-frec discount-bond price #(f, T) must satisly' under the condition of no-arbitrage! Br (J(r,,
/)
-
,?,(r,>
r))
+
1 .%+ j.Brrbrt, 467
t) c
- z.,s
=
0,
(3)
e'
468
C H A PT E R
.
Relating Conditional Expccratiorts to PDL'
21
with the boundary condition
B(T T)
=
(4)
1.
Similarly, under the Biack-scholcs assumptions with constant spot rate r. Black-scholes PDE for a call option we earlicr obtained the fundamental writtcn on St: with strike price K and cxpiration T, 1 0. (5) bhlvsr+ Ft + jlqrlrt z - rF =
The boundary condition was
(6) max gts'z &), 01 approaches Thus, the pricing effort went back and forth between PDE and approaches that used conditional expectations. Yet, both of these methods are supposcd to givc the same arbitrage-free price Fut, t ). This sugbetwcen conditional gests that thezc may be some dccper correspondence PDES that are shown in (3)or expectations, such as in (1) or (2),and the rcspectively. (5), tj is given by ln fact, suppose wc sbowed that whcn a function FLS: FSy, T)
=
1)
s',
-
/)
-
469
Fvh, /). lf one could establish a PDE that corresponds to such expectations, this could give a fastcr, more accurate, or simply a more practical numerical method for obtaining the fair market price F(x%, 1) of a hnancial derivative written on Sf.3 Alternatively, a markct practitioner can be given a PDE that he or she does not know how to solvc. lf thc conditional expectation in (8) is shown to be a solution for this PDE, thcn this may yield a convenient way of for Fuh. t). Again, thc correspondence will be very uscful. In this chapter we discuss the mechanics of obtaining such correspondences and thc tools that are associated with them. Llsolving''
,
r)q Le-br-dsl-s-r, ,
(7)
1) would automatically whcre Ft5'j, f) is twice differcntiable, tbe same Flvt, satisfy a specific PDE. And supposc we derived the general form of this PDE. TMs would be very convenient. We discuss some examples. A1l interest rat derivatives have to assume that instantancous spot rates of Finance wotlid are random. At the same time, thc Fundamcntal Thcorem always permit one to write the dcrivatives' price Fvh, t ) as
Fls,
PDL
to
-
,
Fs,
2 From Conditional Expectations
wlj
ge-zrrsas-t-r,
zti'
(8)
expectations under the risk-neutral measure. As a result, such conditional especially thc case for interarise naturally in derivative pricing. nis is assumed constant, and bc cannot est rate dcrivatives, where tbe spot rate random. hcnce, thc discount factors will have to be twaluate. The But thcse conditionai expectations are not always easy to jndecd. Ofttn, task complex makc this a very stochastic behavior of rt can used. there is no closed-form solution and numerici methods need to bc numerically, speed and acEven when such eeedations ean be evaluatcd methods. Thus, it may l>e alternative necessitate curacy considerations may rcprescntation tliat avoids the quite uscful to have an alternative the arbitrage-free price calculating expectations conditional in cvaluation of corresponds to tlie conFt/, f ). ln particular, if we can obtain a PDE that numerical schemes to calctllato expedations ditional (1) or (2),we can use
tdirectl
2 From Conditional Expectations to
PDES
In this section we establish a correspondenc between a class of conditional expectations and PDES. Using simple examples, we illustrate that starting with a function detincd via a certain class of conditional expcctations, we can always obtain a corresponding PDE satislied by this fundion, as long as some nontrivial conditions are satished. ne main condition necessary for such a correspondence to exist is Markovness of the processcs under consideration. Our discussion will begin with a simple example that is not directly uscful to a market participant. But this will facilitate the understanding of thc derivations. Also, we gradually complicate these examples and show how thc methods discessed here can be utilized in practical derivatives pricing as well, 2.1 Cse
1: Cmutcnt Dlscount Ft-s
Consider the function Fxt) of a by the conditional expectationl F(x,)
=
l-andom X
E,P l
e-p''#(x
process xr e N
)ds
,
g0,x),
defned
(9)
where p > 0 represents a constant instantaneous discount rate, #(.) is some continuous payout that depends on the value assumed by the random proEPt cess xt. (.1is the expectation tmder the probability # and conditional on thc information set h, botb of which are left unspeciEed at this point. 'I'he process xt obeys the SDE: dx3 ydt + t7WH(, (10) where pz, c' are known constants. =
1For evample, in dealing wit American-style derivatives, it will in general be more convenient to work with ntlmerieal PDE metizod irlstead of evaluaizlg 1he conditional expectations through Monte Carlo.
..
pa . ,.,.y
(--H A P T E R
470
Relating Conditional Expectations
21
.
to PDF..S
2 From Conditional Expectations
PDL'
to
4-11
''
This FLxt)can be interpreted as thc expccted value of some discounted random variable future cash tlowg(.x,) that depends on an fy-measurable deterministic. < xs. The discount factor 0 p is markets will, in general, in snancial of interest the cash Clearly, tlows is especially the case for be discounted by random discount factors. thc momcnt. A1I we interest rate derivatives, but we will leave this asidc at to wdnt to accomplish at this point is to obtain a DE that will lead ltentl that the stcps study in detail We to in the expectation (9). this, random discount factors can to this PDE. Once we learn how to do introduced, easily bc expectation (9) in several We now obtain a PDE that corresponds to applied to more complicated stcps. These steps are general and can be expectations than the one in (9). We proceed in a mechanical way. to ilthat the initial iustrate the derivation. To simplify thc notaon wc assume point is given by t 0. and split the pcriod (0,x) First, consider a small time inten'al 0 < represented by the intenral i9, j, in two. One being the immediate future, and tlze othcr representcd by (A, x),
7.
: :
Itcorresponds''
Flxnj
c-*g(;r.)JJ
EP
=
+
A
0
:.
)('
F(x
,.
k
:L
,::'
-ki
6
.j J-
:
.
-
:
.'EE t.. : .
.
.!(i ) .fr' .(
;$L Er',
y,P 0
=
;
'7
.
jA'P t'
replacing the T hus, (.11.2 we get..
This ptrmits
EI
lF'Pa
Er 0
x -&
c
-*
# (xs )ds
2Re,cal1that at time
=
'
EP
(.1)
=
l
.
eJ(.) operator
.j
E;
e-J'AF# a
(12)
by the operato:
:' .
J
e-ptl-)
# (.':)ds
.
=
,
We
WZ have more irlformation than at time t
=
(I-
=
(17)
0.
divide all terms by
,
and
e-/k,gtxll,& +
(c-VA
-
1)F(-u) + (F(.u)
-
F(xo)j
(18)
0.
=
=
lim
-..
A
1
:
e-n'%
# (-x)ds
.
,.
=
gx
o
).
(20)
The third term, on thc other hand, involves the expectation of a stochastic differential and hence rcquires the application of Ito's Lemma. First, we approximate using Taylor series and m'ite;
.'
.
''
'zrk,-lxal ) -
c
'f
F(xs)
-
=
,
(1* ? '$''
.
.jt,--ofq
zzgz--txa -
x.)1 +
(zt)
.
'
B'Here lhe F(xz) is te value of F(.) observed at tirne z 0. It is condititmal on Thc otl the other hand, is the value lhat will be obsen'cd arter a time interval of length at t A. It *1 be ndililal on
E. ',
l
jo
,-fx-)1
co
+ e-nFxxj
--.
.
:.
jo
0 of each term on the left-hand As the Iast step, we take the limit as k side. ne second term is, in fact, a standard derivative of eex evaluated at x 0: 1 e-' a lim -p. - 1) (19) a-+pX The first term is the derivative with respect to the upper Iimit of a Riemann integral!
'j .'.
.j
...
a e-psgnlds
side and moving them inside the
.
(
'.
.
in
(16)
.
.
(131.'t'
(.1
E;
Qu
expectations. ' The third step will apply the recursive property of conditional :' nested, it is the expecAs seen earlier, when conditional expectations are Thus, ff wc ; tation with respect to the smallcr information set that matters. y have l't 1s. wc can write: EP
.
..
'
;.5
(12) t': $ 1
# (xs jds
+ c-#AF(.u)
tr-/gtxsldx
feafrangel
t..-..tjjjli,1
).'1
g-pl
E;
=
As the fourth step, we add and subtract F(.u),
4.Jz
.;!;k. ..
,'j
.
ds
E;
)).:..
.
.
...
C -ps #(A:)
)
Grouping all terms on the right-hand expectation operator we obtain:
$) !
t,.:
.n'
A
U
.
'
P
(11):
A
,
r
Z'
(15)
.
'
':
1nThe second step wolves some elemcntazy transformatitns that are b this right-hand of side tended to introduce a future value of #(.) to thc .. rewrit- J expression. In fact, note tbat te second term in the brackets can bc L. e-'# as! ten aftcr multiplying and dividing by g-ptx-hl
E; j-pFtaulj
=
a
r
,7:'
(11)
.
e-#'.(x,)Jx
i 11.
.5
'
e-pglxajds
:
.
:q
=
*
EoP
nis Iast expression can now be utilized in
t
1
X
. :
r, 1. . ..:.'.: g
But we can recognize the term inside the inner brackets on the right-hand side as the F(.u) and write/
/.)
't
rfhis
z
'
L,
=
,t
F(.v),
..
7! . :;
:
.ta,
=
'
.
.r:.
C H A PT ER
472 Then let
k
-+
*
Relating Conditional Expectations
21
0 and take the eoectation 1 p lim u.F: gF(.u)
-
F(x.)j
to
PDF.s
1 2 Fxp, + jFxxt:r
,
A*0
(22)
the formula as a where Jz is the drift of the random process x that enters xoj. result of applying the expcctation operator to (.r reach Replacing the Iimits obtained in (19)-(22)in expression (18),we the desired PDE:
#(f, F)
1Fxxe 2 -
j
JIF + #
=
(j
,
(23)
where the &, Fxx. 15 and g are all functions of x. behveen the condiOne may wonder what causes this correspondence conccpts scemed tional expectation (9)and this PDE? After all, these two A heuristic arkswer to this question is to be quite unrelated at the outset. the following. value'' of cash 'rhe PDE corresponds to the expectation of the conditional the F(.) by given is 0ow stream ('(.r,)). If this present value of xo and funetion arbitrary expectation shown above, then it cannot be an expected the constraints due to its behavior over time must satisfy somc PDE. the constraints lead to fmure behavior of x. These Fxo) is the result of an optimal forefunction the precisely, More projecting ways in which Fxt) may cast. nis optimal forecast requires variable x3, determinischange over time. Expected changes in the random and thc second order lto gxt), tic changcs in the time variable f, payouts F(.). The optimal pTechangcs in corredion all cause various predictable The PDE that corresponds diction should take these changcs into aount. obtained in sucb a way that the to the conditional expectation operators are and its variance expected value of the prediction error is set cqultl to zero, consideratioll. 4 taken ztt'l is minnized, once tese predictablc changes are Ktpresent
correspondence between a We now sec a more relevant example to the Sm PDES. fact, In we now apply the class of conditional expectations and prices. dgscount bond derivation to obtain a PDE for default-free pure wif.h default-free purc discount bond spot Consider the prjce Bt, T) of a that the instantaneous maturity T in a no-arbitrage setting, Msume 4In fact, note that in obtahling Zef 0.
the PDE we replaced
the Wiener complnent of the
-r
wit.b
Et>
473
(e-
jr czdaj
BT F)
(24)
,
1.
=
Here the expectation is taken w1t.11respect to the risk-neutral measure P and with respect to the conditioning set available at time f, namely the It. This is assumed to include the current observation on the spot rate rt. lf rt is a Markov process #(f, F) will depend only on thc latest observation of jJ, r;. Because we are in the risk-neutral world, as dictated by the use of will the follow the rt the dynamics given by SDE: drt
=
(tI(G, t)
r#trj,
-
f)1
dt
+
blrt, /)#H(.
where H( is a Wiener process under the risk-neutral thc market price of interegt rate risk delled by t
=
/L -
measure
rt
(25) A 'Ihe
A., is
(26)
with t, g' being the short-hand notation for the drift and diffusion components of the bond price dynamics: dB
y,B, tlBdt + =B,
=
/)z14/1,9).
Thus, we again have a conditional expectation and a process that is driving it, just as in the preous case. nis means that wc can apply the same steps used there and obtain a PDE that con-esponds to Blt, T), Yet. in the present case, this PDE may also have some practical use in pricing bonds. lt can be solved numerically, or if a closed-form solution exists, analytically. The snme steps will be applied in a mechanical way, without discussing the details. First, split the intewal if, F1 into two parts to wrile: Bt,
2.2 Cse 2: Bond Pricing
=
with
-
F g + x
PDES'
to
rate h is a Markov process and write the price of the bond with par value $1, using the familiar formula:
to obtain: =
2 From Conditiona Expectations
'r)
=
.E:
Second, try to introduce the
(2g) gte-.?7-rxd.l te-.?itrxl'lj future price .
of the bond, Bt +. ?, F), in
this expression. In fact, the second exponential on the right-hand sidc can easilybe recognized as B(t + F) once we use thc recursive property of conditionalcxpectations. Using .
E:
we can write Bt,
r)
=
(.jj gl,#+a Bt a, g(e-.$'-r,z&)
r.)
EI)
=
s,>
,
+
Flj
(28) (29)
474
2H A PT E R
Relating Conditional Expectatiolzs to
71
*
PDF-S
2.3 Ctzse 3: A
because Bt +
T)
,
Eil.-h
=
?-'''-
'-.:1.91
gc-
(30)
-
add and ln the third step, group a1l terms inside the expectaton sign, : subtract Bt +. T), and divide by ,
1s/ f
2 From Conditional Expectations
gj-./7+&r>J: -
1)
s(f +
r) + (stf +
,
,
r) -
T)11
Bt,
=
0.
(31)
T) S(f, T)1 to the leftNote that this introduces the increment (S(l + applying Ito's Lernrna. hand side. This will bc used for 0 of the first term in this equation:s Fourth, takc the limit as k
Qenexalfztlhtm
F(r/
,
--
I(
-
ts
s(, .y.a. z)j
j)
-.('-ar
'
-
Then, apply lto's Lernma to the second term in tion:
1 lim Et (#(1+ i -..()
F)
,
= Bt + lrl/tn,
1)
take the expecta-
(31)and
Afh(r/, f)1 + jBrrblrl,
/) z ,
(33)
1) where the drift and the difhlsion of the spot-rate process art, t), !7(rl, used.ti are obtain the ln the Nnal step, replace these limits in expression (31) to conditional expectation (24): PDE that corresponds to the
- rtB + Bt +
BrLart 1) ,
-
htbrt,
1
+ jBrrbrt t ()1
,
1) z
=
0,
5(71T)
1.
=
E
$
F f
(ch'
r'j
gruldu
.
(36)
This F(.) would represent the price of all instrument that makes interest rate dependent payments at times u (E (f,F1, and hence needs to bc cvaluated using the random discount factor Du at cach u: Ds
=
rd, e ju
(g,y)
,
time u.7
Vrious instruments and interest rate derivatives, such as coupcm bonds, financial futures that are markcd to market, and index-linkcd dcrivatives fall into this category where the arbitragc-free price will be given by conditional expectations such as in (36).nus thc methods that were digcussed in the last two sections can be applied to tind thc implied PDE if the processtes) that drive these cxpectations are Markov. Thc corresmnding PDES may bc exploited for real-lifc pricing of these complex instptrnents.
(34) 2.4 iome Cltyd/icutimz
with, of course. the usual boundary condition'. =
f)
lt is interesting to note that the expectation of this Du is notbing other than the time / pricc of a default-free pure discount bond that pays $1 at
1
-
(g2)
w)j
A(l,
-
z).
.-x,s(f,
475
We have scen in dctail two cases where the existence of a ccrtain typc of conditional expectation led to a corrcsponding PDE. In thc ftrst case there was a random cash llow stream dcpending on an underlying process xt but the discoum rate wms constant. ln the second case, the instrument paid a single, flxcd cash flow at maturity, yet the discount factor was random. Ckarly, one can combinc these tw'o basic examples to obtain the PDE that corresponds to instruments that make spot-rate dependent pamuts glrt) and that need to be discountcd by random discount factors:
,
1 iim-. A o
PDL
to
(35)
of a This is a PDE that must be satisEed by an arbitrage-free price risk. In Chapter 20, the same PDE was pure discotmt bond with no-default obtained using the method of risk-free portfolios, of limit sHere we are assuming that 111e,technical conditions permitting the iptcrehange and expcctation operators are satisfied. depends on t as well as on rt. ftrnlike the previous example, here the Blt, T) functionbefore. did not exst Hence the,re will be an addilonal B, term that
We need to comment on some issucs that may be confusing at the first reading. zl-lere we cannot directiy apply the frP(.1 operator to D. because the glru) will be corzelatcd with the D.. If such correlation did not exist, and if #(.) depended (m an independent random variable, say xu only, then we could take expectations scparately and simply multfply the payout by the t'orresponding discount bond price Bu ttl discount it: r r 15 hu fl Ez #(%r,)(y & s u g r l#(x x lz/uj ys..u
=
J
z
t
assumirlg tlzat the neccssary intercharlge of the olycrators is allowed. On t,he other hand, equation (38) can always be applied if we uscd the fonvard measure as discussed in Chaptcr 17.
.'
j
(-aH A P T E R
476
*
21
Relaring Conditimal Expecrations
to
PDES
2.4.1 F/zd Importance of Markovness The derivation used hcre in obtaining the PDE that corresponds to the class of conditional expectations is valid only if the undcrlyinj stochastic processes are Markov. lt may be worthwhilc to see exactly where this assumption of Markovness was used in the prcceding discussion. During th9 derivation of the PDE, we uscd the conditional expectation operators Z7rg.l that we now express in the expanded form, showing the conditioning information set cxplicitly: E
r
?
c-
)-*r 3 ds gxutdu '
t
Ifl
=
E
W
f>
v;d
r ! ds
-
c
gxujdu
t
-
Ft,
Irt
(39)
(40)
rt).
These cmerations are valid tmly when the rf process is Markov, lf thi/ assumption is not true, then thc conditional expcctations that we considerod would depend on more than just the rt. ln fact, pst spot rates (rs,s < would also be determining factors of the price of the instrument. In other words, the latter price could no longer be written as F(l, r,), a ftmction that depended on rt and t only. Thc rest of thc derivation would not follow itl general. Hencc, we see that the assumption of Markovness plays a central role in the choice of pricing methtads that one uses for interest rate dezivativcs.
z.5 Wlclx Dvift? One may also wonder which parameter should be used as tbe drift of straightforwaTd, the random process in such PDE derivations. answer is but it may be worthwhile to repeat it. The ccmditional expectations under study are obtaincd with respect to probability distribution. For examplc, when we write the some (conditional) arbitrage-frce plice of a bond as: 'lnhe
S(f, F)
=
E(
j-
?'rfGdsj ,
(41)
risk-neutral probability we take the expectation with respect to #, the consideration under is rt, this choice of Given that the random process risk-adjusted drift for rt the risk-neutral probabili rcquires that we use and write the corresponding SDE as (42) drt (/?(r;,1) - ktbrt, tj) dt + blrt, l)dM,
2 Fttam Conditionak Expeccations to PtlEs
LKreal
world'' SDE:
drt
=
art,
where the 1*7 is a Wiener pross
tjdt + brt, f)#H;7, with resped
to real-world
(43) probability #.
'
Hence, within the present ccmtext, while using Ito's Lemma, whenever a tbrt, drift substitetion for d% is needed wc have to use art, /) t)j and not art, l). This was the case, for exnmple, in obtaining the limit in (33). Will the nonadjusted drift ever be used? question is intercsting because it teaches us something about pricing approaches that use other than the risk-neutral measure; formulas that, in principle, should give the same answer, but may nevertheless not be very practical. In other words, the question will show the power of thc martingalc approach, lndeed, during tlle same derivation. instead ff using the risk-adjusted drift we can indeed use the original drift of thc spot-rate process. But this requires that the conditional expectatitm under consideration be evaluated using the rcal-world probability P, instead of the risk-neutral probability. However, we know that an expression such as
t( '
'rhe
Bt,
w)
je-
Jr r,j
.P
=
:J
(
(44)
I
cannot hold in gcneral if the Bt, T) is arbitrage-free, and if the expectation is taken with respect to real-world probability #. If one insists on using the real-world probability then the formula for the arbitrage-free pice will instead be given by:
ge..('r
#(f, F)
=
Et.p
rs6uej'
E(r,,.),p7 -
jztrsqsl?tslj ,
:
!
I ''
!
1,
(4s)
where a1l symbols are as in (42)and (43). One can in fact obtain thc same PDE as in (34) by departing from this conditional expectation and using exactly the samc steps as before. The only major diffcrence will be at the stage when one calculates the limit corrcsponding to (33).There, one would substitute the eal-world drift art, t) instcad of the risk-adjusted dn'ft.
j '
!
t(
(
2.6 Aaothe'r Btmd Pdce Forrnltlz ne main focus of this chapter is the correspondence between PDES and conditional expectations, But, in passing, it may be appropriate to discuss an application of equivalent martingale measures to bond pricing. The preceding section considered /:$,5 bond pricing formulas, One used the martingale measure l and gave the compact eApression: #(/, T)
=
instead of thc
l
477
=
Et9
j-
LC
G'j
1
! .
l
(46)
.
The other used the rcal-world probability P and resultcd in Bt, T)
=
E;
ge-f'
!
rsJ>cf'I2(r,.,J)J%*-l(r.,s)2J'1j .
g
(
C H A P T ER
478
Relating Conditionai Expectations
21
.
to
PDF.S
Of course, the two Bt, F) would be identical, except for the way they are charaderized and calculated. 'rhe question that we touch on brie*y here is how to go from one bond price formula to the other. nis provides a good example of the use ol Girsanov theorem. First, we remind the readcr that within the context of Chapter lf two probabilities # and # are equivalent if they are rclated by ,
dh
(48)
(tdptb where the Radon-Nikodym derivative (t was given by (Au#W'l - l sil ('t e L =
=
(49)
,
where A.t is an f/-measurable prccess,8 We now show how to get pricing formula (47)startg from (46),assuming that aIl technical conditions of Girsanov theorem are satisfied. Stazt with the bond pzicing equation: Bt, F)
Et>
=
?7*rsdaj
gd -
(50)
,
Write the same pxpression using thc definition of the conditional expectaEtP'. tion operator 'j3
j-
./;F
r,dzj
r,dy)
(g.('
=
(1
dp-z
(51)
range at which future rt will take values. Now, shown in (48)to substitute for d? in tfse the equivalence betwcen / and # this equation:
where the l is thc relevant
j
j-
.s> l
substitutng for (,r .(''
Et>
rspyj
(g/,F
rxaj .
(52)
j.vdp
we get the desired equivalcnce:
ge-
.
= Etz
rsdxj
(e.r
rsf/'j
ejo''la.dH?z
-
)
..'csJ;e?-,'E(r.v.,):Fe-)
je-
Jdj
2s
(53)
dp
lrs-.spda'lj .
(54)
probability ob-
This is, indeed, the bond pricing formula witb real-world tained earlier. of default-free Thus, the connection betwcen the two characterizations tbeorem simple once te pure discount btnd pyices becomes very that the show did not derivation, abovc thc utilized. in we Of coursc, is clearly is risk. Btlt it a drift term ; is the market pzice for interest rate equation. adjustment to the interest rate stochastic differential 'iirsanov
8ln tls partictllar
caqc,
AJ
wili be tize rnarket
price of spot interest rate risk.
3 From PDF.S
to
Conditional Expectations
479
2.7 Which Formulu? Expressions (46) and (54) give two different characterizatitms for #(/, F), But the second formula, derived with respect to real-world probability, seems to be messier because it is a function of r whereas characvterization (50) does not contain this vaziable. Hence one may be tempted to conclude that if one is utilizing Monte Carlo approach to calculate bond prices, or the prices of related derivatives, the formula in (50) is the one that should be used, lt does not require the knowledge of t. The appearances are unfortunately deceiving in tltis particular cse. Whether tme uses (46) or (54),as long as one stays within the boundaries of the classical approach, Monte Carlo pricing of bonds or other interest-sensitive securities would necessitate a calibration of 2. ln the casc of (54)this is obvious, the r is in the pricing formula. ln the case of (50),some numerical estimate of the A, will also be needed in generating random paths for the rt through the corresponding SDE under the zvz/rriag?leprobability A
drt
=
art,
/)
-
/#(r/,
/))
dt + bt,
f)#M.
(55)
Obviously, this cquation becomes usable only if some numerical estimate for is plugged in. Thus, in tane case, the intcgral contains the / but not the SDR In the other case. the 2 is in the SDE btd does not show up in the integral. But in Monte Carlo pricinga the market participant has to usc both the 'itegral and thc SDE. That is why t-le approacb outlined here is still thc approach and rcquires, one way or another, modeling underlying drifts. HJM approach avoids this difficul. i.t
'Kclassical''
'rhe
3
From
PDES
to Conditional Expectations
Up to this point we showed that if the underlying proccsses are Marktw and if some technical conditions are satished, then the arbitragc-free prices cbaracterized as conditional expectations witb respect to some appropriate measure would satisfy a PDE. That is. given a class of conditional cxpectatiens, we obtain a corresponding PDE. ln this section we investigate going in the opposite direction. Suppose by an asset price Fsi, tj. Can we go from we are givcn a PDE gatished there to conditional expcctations as a possible solution class? We discuss thig within a special case. We let the F(BI, tj be the price of derivative that is written on the Wiener process J#; dchned with a snancial
480
C HA PT ER
Relating Condititanal Expecrations
21
*
to
PDL
respect to probability. The choicc of a H') as the driving pross may not seem to be vezy realistic but it can easily bc generalized. Furhera it permits the use of a known PDE called the heat equation in cngineering literature, Suppose this price F(H( f) of thc derivative was known to satisfy the ftllowing PDE:
3 From PDES
Ff +
Fu';g
(56)
0
=
and that we have the following boundary condition at expiration, F( M$, F) ,
=
l
=
F(I#;
1 a2F
j g-sf zkp,j PF
dy
-(
+
Ft +
-
F(M':, 1)
JF dt + ppszz!,Tr
j asza jl dt
(58)
+ Fw,aI,#).
T
J,
Y
EG(114/.)1 -
JF
#u,
dl'l
.
(62)
Thus, if we can show that the sccond expectation on thc right-hand side is
function F(.) can be determined by t'elking the zero, then the (unknown) expectation of the known function G(.). But this rcquircs that: T
Et>
pz-
tuj
(63)
().
=
dllz-
To show that this is the case, we invoke an important propcrty of Ito integrals with respect to Wiener processes. From Chapter 10 we know that if (Ht) is a nonanticipative function with respect to an information sct h, and with respect to the probability #, then the expectation of integrals with respect to J;fziwill vanish: l
E;
lr(H$)dH(;
0.
=
(64)
Lct us repeat why this is so. The H( is a Wiener process, Its increments, #r#;, do not dcpcnd on the past, including the immediate past. But if (J#;) is nonanticipative, then /,(1) will not depend on the eithcr. So, in (56)we have the expectation of a product where the individual terms are independent of one another. Also, one of tbese, namely the #14t, has KGfuturen
1' JF
=
pl.F
1'
F! +.
/
z'
Jl1$ +
r(
F, +
ds.
EFP'P'
,
I
j'sx
ds
(59)
(60)
0.
=
Using this and taking the expectation with respcct to
/
of the two sides of
(59),we can write: Et?'(Ftp' z,,
X
=
(57)
Recall that the partial derivatives Ft and Fp.pr are themselves functions of k;' and s. Now, wc know something about the integrals on the right-hand sidc. As second integral a matter of fact, using the PDE in (56),we know that the equals zero:
Equation
f)
l
where we use the fact that the Wiener process has a drift parameter that cquals zero and a diffusion parameter that equals one. nis stochastic differential equation shows how F(F;, tj evolves over time. The next step is integrating hoth sides of this cquality from t to W: F(W'w,T)
,
(J(H''w)
,
481 =
F)
for some known function G(.). We show that the solution of this PDE can be represented as a ctanditional expectation. To do this, we hrst assume that all technical conditions f): arc satisied and start by applying lto's Lcmma to F(H( -
Conditional Expectations
Now. F(Hi, T) is the value of F(.) at the boundary f T, so we can repiace it by the known fundion G( l#'z). Doing this and rearranging:
,
1
to
niean
ZCrO.
Going back to equality namcly the
(62),we F
.E'/ #
see that the term we equate (JF
dpz
t
J'r,.'q
is exactly of this type. It is an integral of a nonanticipative f'unction with respect to the Wiener process. This means that its expectation is zerox given that F( satisfies some tecllnical conditions. .)
E:
gj'
l7 d.;j
=
F(J#:,
/) +
T
F,>
PF ,?sz
/
#l,F;
.
(61)
(66)
()
-
t
Thus wc obtai' ned:
#'(p;2) ,
=
EP l
lclp-wl! ,
(67)
of the price F( I/PI/) as a conditional expectawhich is a characterization of condition the G(J#y) and the probability #. This funcboundaly tion tion is algo the solution f the heat equation. ln fact, beginning with a ,
'r)1
(65)
,
.
to zero,
482
C H A PT E R
*
21
Relating Conditional Expectations to
PDF-S
PDE involving an unknown function F(/, W ), we determined the solution probability, w1f.14 as dn euectation of a known function with respect to a respect to which H( is a Wiener process. :
4.2 A4arkov Propcrty This property was seen before. Ltt SDE: dut
4 Generatots, Feynman-Kac Formula, and Other Tools Given the importance of the issues discussed above, it is not very surprising that the theory of stochastic processes developed some sptematic tools and concepts to facilitate tbe treatment of similar problems. Many of these tools simplify the notation and make the derivations mechanical. nis is the case with the notion of a generator, which is the formal equivalent of obtaining limits such as in (33),and the Feynman-lac theorem, which gives the probabilistic solution for a class of PDES, We complete this cbapter by formalizing these concepts utilizcd implicitly during the earlier discussion. 4.1 1to Diffuslo'ns
=
ast,
tldt + rvh,
/)#H(,
f (E
(0,x).
(68)
only.g We now assume that the drift and diffusion parameters depend on St The SDE cari bc written as: dst
=
=
alutjdt
xv
be an 1to diffusion satisfying the
+ tr(5'f)#H(,
t
(E
(70)
(0,cr).
Let /(.) be any bounded function, and suppose that the information set It contains a1l Su, u :i f until time t. 'Then we say that St satisfes the Marknv Jrcwilr/.pif ; E
I .J,1 If(.5+h)
=
E
1 E/'(-%+) .%1
,
>
0-
(71)
nat is, fumre movements in St, given what we obsen'ed until time 1, are likely to be the same as starting the proccss at time t. ln other words. te observations on St from the distant past do not belp to improve forecasts, given the St.
4.3 qenevatov /J tzn Ito Di/hlio?z I-et St be the lto diffusion given in (70).Let fvtj be a twice differentiable function of and suppose the process St has reached a pmicular value st as of time f We may wondcr how f (&) may move startirlg from the cunvnt state h. We dene an operator to represent this movement. We let the operator bc dcfined as the expected rate of c/ztlagc for fSf) as: ,%,
A continuous stochastic process St that has hnite first- and second-order moments was shown to follow the general SDE! dSt
483
4 Generators, Feynman-Kac Formula, and Other Tools
asflds
+ trt.5'rllT#;,
t
6
(0,x),
(69)
where the J(.) and c(.) are the drift and diffusion parameters. Processes that have thfs characteristic are called time-homogenous Ito ffu-orl-.'The and (11results below apply to those processes whose instantaneous drift (z4.) and fusion are not dependent on f diredly. Usual conditions apply to fast.'' tr(.), in that they arc not supposed to vary diflsions. of properties 1to discuss We can two tTtoo
SDES utilized in 9In almost all case,s of interest where there are no jumps involved, thc latter is especially popular practice arc either of geometrie, or of mean rcverting type. widely believcd to havc a mean reverting wilh intcrest rate derivatfvcs lxlatlse the shct't ratc is would be a functien of character. Under thesc conditions, the drift and diffusion parameers initial tenn stnlcture. rpatch ailowed the timc is to dopendencc on -%only. Howevcr, often rrhe
,
-.4
((&+a)If(&)) flstl
E
lim ( A a-a Hcrc thc small case letter st indicates an already obsen'ed value for The numerator of the expression on the right-hand side measures expected change in f(St). As we divide this by z', thc operator becomes a rate of change. ln the theory of stochastic proccsses zd is called the generator of the lto diffusion St. Some readers may wonder how we can desne a rate of change for fv), which indiredly is a function of a Wiener process. A rate of change is like a derivativc and we have shown that Wiener processes are not diffcrcntiable, So, how can we justly'thc existence of an operator such as one may ask, The answcr to this question is simple. does not deal with the actual represents an opected rate of change. rate of change in f(&). Instead, Although the Wiener process may be too erratic and nondifferentiable, note that expected changes in fst) will bc a smoothcr function and, under some conditions, a limit can be defined.lo Afh)
=
-u)
.
.%.
azl
.4,
..4
.,4
oEvery expectation titmlar values.
represents
an average.
B.ydefmiton, averagcs are smoothcr
than par-
3
484
C H A PT ER
4.4 A
*
for
RepT'cserlftztier
Relating Conditional Expectatiorus to PDL
21
4 Generators, Feynman-Kac Formula, and Othex Ttots
..4
The corrcsponding
First note that is an expected rate of change in the Iimit. That 1, we consider the immediate future with an infinitesimal change of tlme. nen, it is obvious that such a change would relate directly to lto's Lemrna. In fact, in the present case where St is a univariate stochastic process:
the operator
,4
astjdt
=
+ fr(-$,)#I,#),
t e' (0,txz),
(73)
is given by: Af
af
=
pf
1 u JV , o't M z
+
y
d.r(,sl
1./?2TjJ,+
gtk/,?.y M+
,
os,
Hence, thc difference between the operator Lemma is at two points;
(74)
ntld;. os
(75)
E=1
+
fyxj
J a,
1
f jtt7.rov )ij J-'ll''-q oyoyj
(78)
,
=1
/=1
where the term hafj represents the f/th clement of the matrx crtthln). 'rhe difference between the univariatc case and this multivariate formula is the existence of cross-product tcrms. Othenvise, the extension is immediate. In most advanced books on stochastic calculus, it is this multivariate form of that is introduced. The expressitm in (78)is known as the inhnitesimal generator of .(.).
4.5 Kolmogorx's Bcckwurd Eqwzzer Suppose we are given the lto diffalsion Also, assume that wc have a functlon of St denoted by f St). Consider the cxpectation: ut.
8%
and the application
..d
J'-!ap
k
k
JJ
..4
,
lt is worthwhile to compare this with what lto's Lemma would give. Applying Ito's Lemma tt futj with Si given by (73): -
--I.J =
will then be given by
operator
k
,4
dSt
.,zl
485
f (-
of Ito's
These two differences are consistent with the definition of A.smenticmed calculates an apected rate of change starting from the immediate above. state st. -d.
.p1.
,
(.(.%)I 1
(79)
--
.
,
1) represents
.-
..4
,
Jf
.4y.
(80)
=
-J/ Remembering the definition of
4.4.1 Multivariate Case For completion, we should provide the multivariate case for Let Xt be a k-dimensional lto diffusion given by the (vector)SDE',
E
-
where fSthe forecasted value and S- is the latest value observed before time t. Hcuristically speaking, is the immediate past, Then, using the t) may operator, wc can characterize how the fSchange ovcr time. This evolution of the forecast is given by Kolmogorov's backward rblftzlon: ,
1. The (fHz)term in lto's formula is replaced by its drift, which is zero. 2. Next, the remairting part of Ito's formula is divided by dt.
/)
-.
ad.'.
-.4.
dz'Lt :
a j, :
..=
dxkt
,.1
l
*'''t
d /' .j-.
*-'
vk 1 *'''t
akt
d HG
.lk
t
:
rrk k f
*'''
(76)
#W/r12
where the ait are the diffusion coeocients dcpending on Xt and the tp are the diffusion coecients possibly depending on Xt as well. nis equaticm is written in the symbolic form; dXt where al
.)
=
atdt +
tz'/##),
is a k x 1 vcctor and the
oj
/
is a k
e
I0,x),
x k matrix-
(77)
ga
m
1 z z? f jyts yj DSa It is easy to see that the equality in (81)is none other than the PDE: 1 c atfs + hrt J.u. Af -
=
at
lf
+
.
nx
a
=
(81)
(82)
nus, we again see the important correspondence betwcen conditional expectations such as and the PDE in different ways: @
rrhe
=
Effst)
I
,-1
(83)
(81).As before. this correspondence
h.%-f) >
s--/)
satisfies the PDE in Equation
can be statcd in two
(81).
k
.E ! 'k :L'.'. '.
C H A PT E R
486
.
*
Relating Con ditionalExpectacions to
21
eiventhe PDE in Equation (81)we can tind an
thePD E is satisflcd. ...s.
...
fs -
,
,'
h
.
* ' ',ii.:,
PDF-S
'
.'
8 Exercises
4
,j;j
i.. j..
t'' 'Y. 4(. (
5 Feynman-lfac Formula
X'.E
vkJp.
'j
t'i.
.r
y.y
uence,
.
ne
ykynman-xac formula is an extension of Kolmogorov's backward equation as well as being a formalization of the issues discussed earlier in this chapter. The formula provides a probabilistic solution f that corresponds to a given PDE.
1': :
j'j %L
,
(:
rt.
.%
,
4(tt,,
ijt
:,.
L
E (;!> k9-g (
;
Feynman-xhc Formula: Given
,lu.lsf.'
4.5.1 E'turzw/e
jtf, rt )
nl
'
Consider the function!
:;
.
t,
kl) ,
!)
1
=
e-
(.5.-&))?2:
,
liat
,.'-
=
:;
.
p
.( .
w jwre tjw operator
.1
ot
(85) ))j
dW6'
.,
know that
twice-
m
Af
=
=
'
;':;
?.;.
a
at
=
case, we have: 0
(r, j
ft-
-
-.fx..
=
2
:.
: ..
,g
,
4
',
:
'
''
()ne
.'
References
Several intercsting cases using this corrcspondence are found in Kushner (1995). nis sourcc also gives practical ways of calculating the implied poss,
E
:
is suc h ftmction . lt turns out that the conditional density #4-$,, &, witb resped to t and thc f.- To see this, take the hrst part ial derivative substitute in (89). The equation 111 sccondpar tial with respect to and E bc satislied. According to this result, thc condition al density fun-tion of a (generab PDE Kolmogorov 's backward equation, Ths ized) Wiener process satisies associated tf St will value with a pa rticular tells us how tllc probability . initial point the given &. passes, as t f)
Orrespindcncc between poss and some conditional expectations is useful in practical asset pricing. Given an instrument with special charvery acteristics,a market practitioner can use this correspondence and derive the implied PDES. These can then be numerically evaluated,
,
..
(89)
as
R''hc
'
(88)
substitutingthege. Kolmogorov's backward equation bccomcs: 1
expectations
6 Conclusions
. '
'
(8X
.
/
,E
'E
. '
=
(92)
.
oy
,
.
and
2
Hence, the Feynman-Kac formula provides conditional PDES. a solution that corresponds to a certain class of
q
7
-
&
(85),in this particular
or
a
c72 1 f + -% a :
'
,?
,
-
at
p /
'
density. Wc a we apply Kolmogorov 's formula to this satisfy backward Kolmogorov's would f S function f (.) o differentiable equation. . 1 2 (86) ftm f? / f. + 2 U't x.
(91)
is given by:
'
=
...
.:
A.f - qrljf,
=
-
,
.
ds /
(90)
,
a
..
'.'
eter
jd-tkuvtrsldxytrx j
we have
'
-
'
Bet according to
E;
=
.
... .....Jt. (g4) .L. 2rr/ .k . this is the conditional density function of a wiener inspection shows t f 0 and moves over time with zero Processthat starts from S at time #'E drift and variancc t. equation for this pro)' j If we wcre to write down a stochastic differential the drift parameter as zero an d tbe diffusion paramCe y, s we would choosewould satisfy; :: as one. The %
pt
axn
487
'E'.'..
that
,) such
.: /. j/ j Soltltitm fOr thc P,-'x-. ra..ja xn result means that f(S- 1) iS a Of correspondence bethe Kolmogorov 's backward equation is an example PDES in tlis earlier seen tween an expecta tion of a stochastic process and chapter. ryjs
..
'
:
8 Exerclses
ut
'
1
,
'
'
:
.
evolve
;
bond prie.c /J(/, T) satishes the following PDE!
supposethe -
ime
'
rt
s
+. lt +.
i
a srlt - xy s ) +. -s,,fy. 2
=
(j
(:9)
488
c
H A PT E R
.
21
Relating Conditiona: Expectations to A(T; T)
Deline the variable Pr(u) as )tmrr ds j'd( tr, P'(u) e e-
..):/8.k -
,
=
where
s
is tbe market price of intercst
(94)
1.
=
l'ate
PDF-S
)A(&sx)2J
(p.5;
,
,
lisk.
(a) Let #(f, F) be the bond price, Calculatc the #(#F). (b) Use thc PDE in (93)to get an cxpression for dB(t, F). lnd take expectations with (c) Integrate this expression from t to T respect to mmingale cquality to obtain the bond pricing formula: ,J)2J'lj j't' ruds ,.)J)K- l trs Etp (96) #(/, T).
gc
Stopping Times and American-Type Securities
gFltrr
-
=
where thc expectation is conditional on tbe currcnt assumed to be known.
,
rt which is
1
lntroduction
Options considered in this book can be dividcd into hvo catcgories. The hrst group was characterized using a pricing equation that depended on the currcnt value of thc underlying asscts St and on thc tne 1. For example, the price of a plain-vanilla call optitm at tne t was written as: C,
F(<;,
=
/).
(1)
Given the observed value of St and thc time t, tlAe option grice was detcrmined by the function F(.), Plain-vanilla European options, where the was a geometric process, scllinto tllis category.l The second category of options, although not dealt with cxtensively in this book, wcrc those that wcre classihed as path-dependent. The price of these options at time t depended not only on the current St, but possibly on observed before time t as well. An optitm's some or on all othcr values of payoff at expiration time 1' could dependa for example, on the average of the last N values observed at discrcte times: u$/
.,
t At expiration,
<
fj < t2
<
.
.
<
.
t,sv T =
a call option holder could for examplc be paid'.
Cw
S, + St- + ''
=
max
.
2
.
.
+
SI&
N
-
K, 0
.
where K is somc strike pricz. 'Adtlitional
also assumed
assumptions
concerning
to get a closcd-fo'rm
ncl dividend
formula for F(.). 489
payments
antl constant interest rates
were
49O
C H A PT ER
.
Sropping Xrnes
22
American-Type Securiries
and
Under these conditions the time T price of this call option could be written using:
Cp
=
F(Stj St,, ,
-
-
-
,
(4)
Sv-yF).
Clearly, this expression will look somewhat more complicated for time t, t < 7:2 Yet, pricing this sort of exotic option is not necessarily more diocult than the case of plain-vanilla exotic options. ln fact, according to what was said, the payof of this option occurs at expiration date F, and in this sense the option is still European. ne only complications expression. Thus, alare the additional h terms that show up in the though the option is path-dependent, and the payof depends on how one gets to an expiration value of the underlying asset, a Monte Carlo-type approach can give a reasonable approvimation to Ct once the dynamics of St is correctly postulated. Notice that for neither of these two categories of options the westor has to make another decision once the option is purchased. In both cases, one waits until expiration and exercises the right to buy if it is profhable to do so. Alternatively. the option holder can close te position and sell the option to somebody else. But no other decision has to be taken, Hence, no other variable enters the formula/ Now consider anAmerican-stuvleoption. These securities can be exercised at or before the expiration date F. Once the investor buys an Americanstyle option he or she will havc an additional decision to make. The time and to cxercle the option must now be chosen. ne investor cannot just sit where q # (0,F1. wait until expiration. At sme critical time denoted by 0, the gain, realize and it may be more prohtable to exercise the (call)option So
than hold on to the call until expiration maxgsz
(5)
K,
-
-
to get
L,% x'l
EP
gct:r-elr maxls'z
-
K, 01
I
21n fact, no closed-lbrm formula may exkst. 5We always assume tbat the intcrest rates and volatility are constant.
.
(7)
y Study Stopping Tlmes?
2
Even if thc notion of stopping times was limited to the class of Amcricanstyle securities, it would still bc necessary to stttdy stopping times, It is true that most Iinancial derivativcs are American-style and stopping timcs are necessae to price them. But there is more to stopping times than just American-style derivatives. We need to study stopping times not just because tbey are thcoretical notions uscful in theoretical formulas, but also because there are some very specilic numerical algorithms that tme needs to use in determining dates of early exerdse. That is, we study stopping timcs because of numerical considerations as well. ncre are properties of optimal stopping times that make some approachcs more convenient than others when it comes to pricing. By learning these properties we can reducc the time it takes to calculate whether, at a certain timc f*, an option should be exercised or not. Or, il terms of the 0, whether one has: e dtexercise
(6)
K>0J.
al,j
>
nis means that the discounted value of the expected payoff may be less than what one gains by simply exercising the option at timc 0. From this it should be clear that with American-style sectuities, the decision to exercise the option is equivalent to hnding such critical time periods 0. Note that under these conditions, thc pricing formula for the option may depend on the procedure used to select the 0's, as well as on the prcviously discussed variables. Such 0's are called stopping times. When the date to exercise is chosen in some optimal fashion, they are callcd optimal stopping times and play a crucial role in pricing American-style securities.
which means
martingale ln factn at some eritical time 0, the expectation under the what tme than Iess be mxgs'y K, the future payoff 0J may measure P of That is, witb K. and option received exercised the %may get if one have: constant spot rates, we may -
2 Why Study Stopping 'Times?
''
=
t*,
(8)
or (9)
which means not exercise.'' By doing these calculations faster or mome actmrately, one can reduce costs and capture arbitrage opportunities better. Hence, the properties of algorithms used to determine stopping times will be an important part tf the pricing cffort. nere are othcr reasons for studying stopping times. Optimal stopping times are in general obtained by using the so-called dynamic programming approach. Dynamic programming is a uscful tool in its own right and should be learncd whether one is interested n pricing derivatives or not. It just bappens tat the context of stopping times is a very natural setting for presenting the main ideas of dynamic programming.
C H A PT E R
492
*
Stopping Times and American--f'ype Securities
22
2.1 Amzrrictmt-y!,e Securlfe: explidt options, American-type derivative securities contain impiicit or which can bc exercised before the expiration date if desired, This causes has to charsigncant complications both at a t.hcoretical level where one where acterize the fair-market value of the security, and at a practical level one has to calculate this price. Bermudan-style options are a mixture of Amcrican and European optimes other than the extions, can be exercised at some prespccised F!. At the piration date. Yet, they cannot be exercised at a1l times during (0, < tx F < < tz date of issue the security species some specihc dates during which the option holder can exercise his or her option, stopping'' pcrspedivea the comFrom the point of view ef the options Bermudap are very similar to Amercan-style plications created by of stopping timcs and the rediscussion introductory securities. ne same Bermudan options. lated tools will bc suocient for Arnerican as well as for American opwith Hence, in the remaindcr of this chapter we work only tions when dealing with stopping times. rrhey
=
idoptimal
4 Uses of Scopping Times
493
been exercised or not. In other words, given lt we can differentiate between the possibilities:
which means that option llas already been exercised, or
(12) which means that the early exercise clausc of the contract has not yet been utilized. nis property of r is exactly what determines a stopping fzzzc. DEFINITION: A stopping time is an fi-measurable nonnegative dom variable such that;
1. Given It we can tell if (13) 2. We have Pr
Stopping Rmes Stopping times are special type random variables that assume as outcomes this random time periods, /. For example, Iet ' be a stopping timc. Thcn of its second. thc and that range random, means two things. First, that is obscrved, it is > When outcome 0. F an possible valueg is (0,FJ for somv will be in the form:
ran-
<
x)
=
1.
(14)
ln casc of derivative securities in general, we have a qnite expration period. So the options will either be exercised at a hnite time. or will expire unexercised, This means that tlze second requiremeot that r be finite with probability one is always satisfed.
'r
(10) period. That is, the outcome of the random variable is a particallar time bond. Thc option Now consider an American-style option written on a and the cxpiration f 0 can be exercised at any time between the present this option if iie or she date denoted by T. ne option holder will cxercise expiration date until the thinks that it is better to do so, rather than waiting of tbe contract. date,'' which is of grcat imporHence, we are dealing with a the asset. ln fact, the right to exez'tance from the point of view of pricing American security cise early may have some additional value and pricing an must take this into aceount. It is obvious that given 'rhus. we 1et r represent tbe early exercise date. whether the option has already the information set. It, we wiil be able to tell =
itrandom
4
Uses of Stopping Times
How can the stoppiag times, m, be utilzed in practice? The most obvious use of T is to 1et it denote the exercise date of an option, With European securities, there was no randomness in exerdse dates. The security could only be exercised at expiration, Hence, we can write: P'r
=
F)
=
1.
(15)
With American-tj'pe securities, is in general random.' Consider an American-style call option FCSL f) written on the underlying where St follows a SDE: security .'
,
ut,
dSt
=
ast
,
f)#/ + rS:,
1)4/1.9), t
4In some spedal eascs it is never worlh cxercising corresponding r will again equal F.
e' (0,x).
the American-style
(16) oplitm, and the
r
C H A P T ER
491
22
.
Stopping Times and Arnerican--hpe Securities
with thc drift and diffusion cocfticients satisfying the usual regularity conditions. The price of the dezivative sccurity can again be expressed using the equivalent martingale measure ?. But this time there is an additionai complication, The security holder docs not have to wait until time F to exercise the option. He or she will exercise the opton as soon as it is more prolitable to do so than wait until expiration. worth In other words, if one has to wait until expiration, the asset will be j-r(W-rJ mmxls'w K, 0)j FSt, 1) (17) =
X
5 A Simpliied Setting
495
where r is the constant instantaneous spot rate and the 0 < ts is a known dividend rate, The I#; is a Weiner process with respect to risk-neutral measure #. We let the Ct denote the price of an American-style call option, with stmike K amd expiraton date F, T < t that is written on St. Suppose we decide to price this call using a binomial tree approach, ne methodology was discussed tartier, but is summarized here for convenience. We ftrst choose the grid parameter and discretize the St in a staadard way',6
-
,
lz
.SL
this with, at time t. If the option can be cxercised early, we can comparc say, l)*
Fst,
=
sup
gF,z
rcpf r
Sid
'rljj
ge
-4T-''F(.%., t,
,
(18)
opportunitiess and the f' where the *2 w is the sct of all possible stopping represents a possible date where the is the optimal choice for r. Here, option holder decides to exercise the call option. Hcnce, at time /, we can calclzlate a spectrum of possible prices indexed by T using the possible values for the stopping time F(S % , f, among ail these T. To fintl the corrcct price we then pick tbe suprcmum ,r)
'r.).
s
=
c-rw/
-1
(22)
,
(z?)
.
Here the up and down probabilities are assumed Wstates'' and across and are given by'.
Puj
=
1 + (r );
Pd)
=
l
'r
Flsv, t,
:GVZ
sf-t
=
-
5)
-
1tr2
-
2
2c.
to be constant across n
,Z
'
#(u).
(24) (25)
That is, once the process reaches a point Si-, the next stage is either u#, 57, or down, with a probability equal to Puj or Pqdj.l With this choice of discretization parameters the discretized system con0, nat is, the drift and the difhlsion verges to the geomctric process as parameters would be the same and the Si would follow the same trajectory as dictated for St in thc SDE (19). Note that the up and down parameters u, d are constant and are given by: ,
q&/,
-->
z
5 A Simpllied Setting We continue studying stopplg times and the problcm of optimal stopping using the simplified setting of a binomial model for pricing a plain-vanilla and our Amcrican-style call option. Yet, although the setting Ls times, the related stopping to main purpose is thc understanding of tools frameworks within actual pridng of American-style options often proceeds uscful similar to the onc considered hcre. Thus, the discussion bclow is calculations numerical pricing as from the point of view of some simplc %Gsimplc''
wcll.
G
k=d
S/-A
#=c -<J
(28)
.
i'l-his is one possibie choice for discretizalion. Therc are others. For example, we, can let: 2j + u ) j. (( yX. jc S1 Si. (i()j ,:k
5.1 Thc Motlcl underlying The model is a binomial sctting for the pricc of an that behaves, in continuous time, ag a gcometrc Wiener process:
dSt
=
r -
=
(r
-
nlstdt+
a'stdk;,
t
i
((),x),
asset St
(19)
S,.d 7,4.s .
-->
=
for r.
ry
-
!
S,.-.Le
((r z)-
'r7 ... ,
pa-orvz'.i' .
0, the probabilities of up and down mtlvemelts become equal and 1 laj Pdj =
sI-hat is, it is the set of possible outcornes
rz
.-
=
.
r
C H A P T ER
496
22
*
Stopping Times and Arrterican-Type Securities
Also, we have, as usual, =
(29)
1.
That is, the tree is recombinng. ne likcly paths followcd by thc discretized St are shown in Figure 1. lt horontal movement is worthwhile to look at thc structure of thc tree. The process begins from the represents the path taken by Si ovcr six states. During cxpiration of the initial pojnt % and then ends up at one In fact, altotrajectorics. several 5 the Si can foliow 1, the times f number of where the n is gcther thcre are znpossible trajectories, trajcctories possible 32 this gives T. In this particular case, given by n that Si ean follow. For The call option's price will depend on the trajectory followed by that It carlier turns out chapters, European options, this was discussed in for the American-style options, there is a completely diffcrent way of looking at tlle same tree and seeing the dependence between the Si and G,,the price of the call option. rlnhe
Rtime.''
.
.
.
497
Tbe standard way of looking at binomial trees is as shown in Figure 1, which hozizontally mimicks the behavior of S; over For analyzing stopping times and understanding the complications introduced by interim decisions on it is worthwhilc to look at the same or tree from a diferent angle, literally. nis may be a bit ineonvenient at the beginning, but it greatly facilitates the understanding of some mathematical tools associated with stopping times. lnstead of looking at the tree over time as in Figure 1. consider now Figure 2, where on the horizontal axis we mark al1 possible values assumed by during the move from n 9 to zl 5, the expiration. During this period, can assume eleven possible values, Denoting this set by E and using thc condition ud 1 wc obtain: Rtime.''
ud
=
5 A Stmplifted Setting
,
Itstages''
=
''stopping''
''continuinp''
<j
=
=
vh,
=
.
E
=
fusvo,u4So, 13,o,
ulsov uso,
zs'a,
dsov J2c$'0,dYvo,
#4.S,d3soj.
(30)
Now, although binomial trees are normally visualized over time as in Figure 1, for stopping time problems, tme gains additional insights when the tree is visualized as a fundion of the value assumed by valucs arc represented tm the horizontal line shown in Figere 2. ne line is nonc othcr than the representation ol the set E. The hgure is the same as in Figurc 1, except we look at it from right to left. In case of American options, we have the riglt to exercise early. And early exercise will naturally depend tm the value of Si at the point that we find ourselves. Now consider the way the proccss behaves on tle set E as represented by the horizontal line in Figure 2. Initially, wc are at point So at the middle point. Next timc n 1 with will movc either to the left, to So, or with probability probability 'lnhese
uj.
I'sideways''
=
.5,
.5
,1
b'til1fl'' 0
l
2
3
4
5
cxpiration
iaitial poinl
F 1G U R E
FIGUR E
2
C H A PT E R
498
.
22
Stopping Times and American-Type Securities
will movc to the riglit, to uso. Once there, it can either come back to So, riglit or to the left. Hence, at each point, or move one step further to the states. The major excdptions are adjacent only to the process can move there, it must stop by necessity end the Once gets points. process the tw'o periods time get to those points and that is the exactly to 5 takes because it expiration of the option,s nis means that S; can also represent the position of a Markov C/fn at stage i. the experimcnt if we Now, remembcr that we can at any stage paycd: receive the this, do If desire to do so. we we
(31)
% - K.
in posIn contrast, if we decide othenvise, and continuea then we will be session of a security that is valucd at the pricc CSi). made. Lct us now conside.r the way an optimal decision to stop can be which decide to we We let the r represent the random time period at = 0 the r is exercise the option before expiration. At the initial point i
6 A Simple Example
According to this, we choose the so tbat when we stop, we stop optimally, in the sense that the value of te option now is maximized,g Thtts, we need to do two tlngs. First we Iteed to obtain the variable: 'r
maxl #(c(5'v ()1 ,
Second, we need to find a rule to dctermine the optimal stopping timc 'r* such that: E / (C(S , ()1s Cy. .r*
(32)
i.
rrhen, obviously this decision will be made by looking at the trajectory followed by S; until the stagc f. That is, wc will have the observations: ls'o Sk ,
(33)
xjl, .
.
.
,
tls, and the dvcision to stop will be a functiotl of this history, According to which of knowledge the depend on the decision to exercise early does not after i is still unknown, 'T'his i,s states will occ'ur ter stage i. ne what we mean when we gay v fs fi-measurable. able Now, if we can detcrmine a strates to chcose the r then we may be well. variable with a random associated te to obtain the probabilities will not be But if such a strategy is not defined, then the properties of r 'Ikus the random variable. well-delined known and the f'(.) will not be a done? be this is How to the choose r. srst task is to determine a strater to Consider tbe following criterion, where the conditional expectation operator is written in the expanded form: (tfuturen
'r
Flso)
EIT =
mjtx
l&q gct5-vl
EIT =
myx
(e-'?max (&
-
1
K, (11
.%l
called absorbent. 8In the termsnology of Markov chains the two end points are there. we get tizere, with probability one we stay
once
.
(34)
min
'r*
>
,
will be of
(k : Sk >
Bk,
(36)
thc form:
,,)1
,
where Bk, ) will be an optimal arercfac boundary that depends on the k and on the current (andpossibly past) values of Sk. nis boundary is to be determinedqlfl
6 A Simple Example We now discuss a simple, yet important examplc in order to undelstand some deeper issues associated with stopping time problems, as the one above. Recall the following problem faccd by the holder of an Americarlstyle call. At any instant during the life of the option, the option holder has the right to early exercisc, Hence, at a1l f G (0, F) a decision should be made concerning whether to exercise early or not. But this decisit'm is much more complicated than it looks, lt turns tlut that to make this choice. te invcstor has to calculate if he or shc is ikely to early exercise in the futureas well. This means that before reaching a decision today, the investor must evaluate the odds of making the same decision in the future.lt is only after analrzing possible h'ture gains from the option that a decision to continuc can be made, But a decision abottt early exercise possibilitics in the futurc depends on the same assessment of the morc distant future, and so on, At the end, the gAccording
to this setup, the F(.) is an objective
bounded,a1l tecimical conditions will be satisfied algorithmsassume this boundedness implicitly. l@ketkks see how we can intorpret cxpeuations
function. If such F(.) is assumed to be for the Iolltmring stcps. In practices pricing .!f#'i5
such as ). Heze one randorn variable is thc Hence possible values of the funclion Sv nccd to be multiplied by thc probability that k among othez th ings. nere are otlwr considerations, because evcn with m flxed S wll stll bc random. On the. other hand, in cxpectalions such as A-'lur) one wtluld multiply possible values of St by the probability tat OII assume these values. C
'r.
'r
Thal iss
=
ffar t
.%
.
=
),
When this is done the optimal strater
,
'r
(35)
.
T
random because whethcr we stop or not depends on the random trajectory followed by Suppose we considcr stopping at stage That is, let: uf
499
=
T
vk
500
C H A P T ER
.
Stopping Timts and American-Type Securties
22
Amcrican-optfon holdcr is left with a complex dccision that spans a1l time pcriods until expiration. Howshould such decisions bemadc? Are therc some mechanical rules tbat will help the decision maker to decidc on the early exercise or not? Finally how can we gain some insights into such interrelated complex decisions? 'rhe simple example below is expected to shed some lipt on these qucstions. The reader will notice that thc way the example is sct is similaz to the Nnomial tree model discussed in tbe previous section, ln fact, we usc thc same notation. Suppose we observe successive values of a random variable 5'j, i 5 and assume that there al'e eleven possible values 1, a. We 1et n that Si can take. These are given by the ordered set E1 =
.
.
.
=
,
E
.(Jl,
=
f73, tz4,
az,
a5, a,
Js.
as, t?9,
(J,7,
(38)
J1(;).-
is known to be a. The process is observed starng ne initial value with i 1, is Markovian, and behaves according to the following assumptions. 1. When the process asumes a particular value in E during stage i, in the next stage it can move either to the immediate left or to the immediate right. A11 other possibilities have zero probabiiity. nis wili 3 we have a(,, then means, for example, that if for i value the value a7. a or assume either the 2. Second, the statts tz1 and J:() are absorbent. lf the process reaches those states it will stay there with probability one. These states can be reached only at 'rhe next stage in describing these types of models is to state explicitly the relevant transition probabilities. According to the dcscription above, the transition probabilities are gjven by: zn
=
.5-3
=
=
,4
6 A Simple Example This implies that the process Si cnnnot jump across states and that it has to movc to an adjacent state. Finally, for the initial stage. we also have: (43)
PS,
m fu
ISo
=
tl)
PSi+3
=
41
=
&j+1
I'Si
P-b..
=
ai-,
Isi
lSi
=
t2:)
=
=
=
Psi-vq
Jj)
a!t
=
1
j',
=
-a,
tzlu
1
IS
=
(40) ,10)
=
1.
(41)
AIl other transitions carry zero probability: PSi+L
=
m
1'Si
=
J/)
=
0
kule! stop as you hi1 11:/ cnvclopc
(39)
=
prvtmbuity% =
Im I >
)- + 1.
(42)
l 2 -
.
(44)
This situation is shown in Figure 3. The holizontal axis represcnts the set E. The arrows indicate the possible moves and the corresponding probabilities. Note that if the process reaches the two end points it stays there. We need to introduce one more componcnt to discuss the optimal stopping decisions in tls context. When the Si visits a state, say, aj irl E, the decision maker is given an option to receive a payoff Faj). lf the decision maker aepts this payoff, then the game stops. If the payof is not accepted, thc gnrne continues and the Si moves to adjacent states. In Figure 3, tlte payoff associated with eaclt stato aj EE E is shown as the vertical line at the corresponding point. 'rhe problem faced by the decision maker is the following. Succcssive values of Si are observed and the corresponding Flajj are revealed. The dccison maker evaluates the payoff of stopplg l'mmediately, against the ewected payoff of continuing and ending up with a beuer Faj) in the future, How should this decision maker ad? We discuss the optimal decision using Figure 3.
Rexpiration.''
#(z%+1
=
B GURE
3
j'.
.
' .( :.' :jr
j
. L.''.
c H A PT E R
502
*
Scopping Times and Arnerican--l-pe Securities
22
Note that at eacb stage we know where we are. In obsen'e thc current vatue of k. But, we do not know the these cven thougb we do know the possibilities. Consider end points. reach the stage two at we Suppose some absorbent. We will no choice but to stop. 'T'he states are
other words, we
future outcomes, possibilities. Clearly wc have never visit other
statcs.
tzg, and Next consider thc state au. Should we stop once we have Si obviously, no (unless,of accept thc offered payoff Flagll ne ansavcr is, the that movc of clear It stop). is ncxt have and 5 to we course, n either of these Figure 3, will be to either ay t:r to tzg. As can be seen from guaranteed continuing, to wc are states has a payoff higher than Fas). By do better. We should not stop. Another obvious decision occus at statc a1. This state is associated with the hjghest payoff ever, and we should clearly stop as sooo as we reach it. Wc are not going to do any better by continuing. 'rhus far, the dccision to stop ws not complicated at all. Btlt nm' cxmsider the two states tzs and as. Here the dccisions will be more complicated. Anm Both of these states have the following property. ne payoff is a local and them, adjacent states reaclng we go to imum. If we continue aftez will off at maker be decision worse these states have lower payoffs, ne keeping the also stopping, is one least in the immediatc future. But by not possibili of reaching a payoff such as Faz) or F(a5) open, So which one is better? Should one accept the local mxima such as Fazj or F(B) and future stop, or should one continue at these points and expect to stop at a payoff? date when there is a higher The answer is not obvious at the outset, and requires careful evaluation of future possibilities, In fact, the tw'o states a, and as will givc differcnt answers. lt will be optimal to continue at as and stop at a5. a5. If we stop we Begin with state as. Suppose at some i we havc Si condnuc if much get we do expect to 8. How wc get Ffu5) It is easy to calculate the cxpected payoff of the immediate futurc: 1 f'sq i'F(J) 'jF(J4) (45) + E I)F(.%'+1)si =
usj
=
=
=
I
=
=
1
(46) rrhis is clearly worse than the 8 we can guarantee by stopping now. But and thcre is an additional point. Thc game does not end at the next stage payoff that the expectcd would ignore looking only at the immediatc futurc would result if we rcached the state a7. function dcWhat we would like to do is to obtain an optimal payoff namely, the pamffs; of two noted by, say. Pxcsl that rcpresents the greater
.;
6 A Simple Example
503
current payoff if we stop, or thc expected payoff if we decide to continue, assuming that l!tl continue in an optimalfashion. That is, we want: le'(tB) max (f gpayoff l Stopl E (Payoff I Continuell =
,
,
(47)
Here the expected payoff if we stop is lknown. It is the Fas). On the other hand, the expeced payof if wc continue is unknown. It should be calculated by using the same notion as P'(45) for future periods in an optimal fashion. In other words, we need to write: gF(tzs), jlyztujj I''(a5) ;z(u4) + (48) max
gj1
=
.
Note that this assumption assumes no discounting. Thus, before we can calculate P'(4s) we need to determille the P-(u4) and the #'(J). But, there is the same problem with these. ne F(.) that corresponds to future states seems to be unknown. Although the reasoning seems circular it really is not. The problem is set up in a way that there are some stages where calculation of the #'(Jj) is immediate. For example, as we already know that we will stop when we reach J1 or 47: Jz'(fz7)
=
FLa-;j =
8.
(49)
F'(Jj
=
F(cj)
=
0,
(50)
Also, we know that
)
Tiius, by substituting for 7(.) in (48),we can eventually get to #'(.)'s that ae known to us. nen, the P-(f?s) can be evaluated. So how can we take tbis into account in (48):? We do so by writing, I 1 E gF(5'j+:) 1Si a7, we stop optimally -LJ''(/z4)+ jyF(fJ) (51) 2 instead of acting as we did in (46)and taking into account on% the immediate fut-ure. In other words we want to substitute the F(tJ4) and F'(tJ) in place of #(tz4) and Faj in (46). nis is the case becauge the latter does not take into account the possibil of reaching the higher future payoffs and then stopping there, whereas the #'(c4) and P'(t?) do, In other words the F(tu) and F(c) irlcorporate the idea that the decision w111b made optimally in the f'umre. This way of reasoning is very similar to pricing derivatives by going in a binomial tree. When this is done it will be clear that we should continue at as but stop at as. Note that the expectation on the left-hand side of (51)is different f'rom (46)since it is now conditional on tlle fact tlmt we stop optimally. Hence, it is in fad F(Js). A final comment on the example. We should point out an interesting occurence in Figure 3. The pints at which we end up stopping are those 'sfuture''
1
=
=
504
C H A PT E R
.
22
Stopping Xmes and Anwrican-Type Securities
times when the payoff from the state is on a boundary that we denote as the envelope in Figure 3, Thus, if a market practitioner is given this envelope, then the rtde to pick optimal stopping times will be greatly sirliplilied. A11 one has to do is to see whether the payoff is below the envelope or on it. One sttps if the current payoff equals the value of the envelope at that point, and continues otherwise. Essentially, this is what is meant l7y Equation (37),which gives an optimal stopping rule.
10 Exercises
.50.5
Then we have E
' E/(B ) iB t)1
=
,.
8
7. 1 Muuingnlvu Suppose Mt rcpresents a continuous-tne probability #, with E EA6+u 1J'/1 Mt,
(52)
=
Would this martingale property be preserved if wc consider random;y selected timcs ag well? The answer is yes under some conditions. Let n and n bc two indepcndent stopping times measurable with resped to It and satisfjing: 1. #(c-1 < n) (53) =
Then, the martingale
property will still hold: (AVZ E l X Mh
)
=
(54)
.
rrhis property is clearly important in writing asset prices using equivafact tat the exercisc date of a derivative is lent martingale measures, random does not preclude the use of equivalent martingale measures. With random m, randomly stoppcd asset priccs will still be martingales under thc
?.
7
kin's Ftnvrkxlu
Let Bt be a process satisfying: dBt aBtjdt + G(#/)#I#7, =
Let
t
>
0.
(55)
Conclusions
rnw
10
Exercises
1. A player cortfronts thc following situation. A coi will be tossed at 1, 2, 3, F and the player will get a total reeward F;. every time f, t He or she can either decide to stop or to continue to play. If he or she continues, a new coin will be tossed at time t + 1, and so on. The question is, what is the best time to stop? We consider several cases. We begin witb the double-or-nothing gnrne. 'T'hc total reward received at time t T is givcn by: .
.
,
,
=
f (Bt) be a twfce-differentiable bounded function of this prccess. such that: Now consider a stopping time
'j'
'r
E z3 (zj < x.
(56)
.
There are three important topics in this chapter. First there is the issue of stopping times. nc early exercise is an optimal stopping problcm. We This tmeatment is classic but still very can recommend Dynkitl et al. (1999), intuitive. A rcader interested in learning more about classical stopping time problems can read the book by Shiryayev (1978)second major topic that we mcntioned is dynamic programming although this was a side issue for us. There are many excellent texts dcaling with dynamic progrnmming. Hnally there is the issue of numerical calculation of stopping times. Here the reader can go to references given in Broadie and Glasserman (1998).
=
.2
)dsI Bo
9 References
ne
probability
-4/'(A
The chapter also introduced the notion of stopping times. This concept was useful in prfcing American-style dervative products and in dynamic programming. We also illustrated the close relationship behveen binomial tree models and a certain class tf Markov chains.
with respect to a
martingalc
T
rrls expression is called Dynkin's formula. It gives a convenient representation for the expcdation of a hmction that depcnds on stopping time. a The operator A is as usual the generator.
7 Stopplng Times and Martingales We lnish this chapter by looking at the role played by stopping times in the theory of martingales. lt turns out that most of the results discussed in this book can be extended to stoppinj times. Below we simply give two such results withoet commenting exlenswely on them.
+ Ee
fBv)
lp'w
=
/=1
(z,
+ l ),
C HA P T ER
*
where the zt is a binonal
Stopping Times artd American--f'pe Securities
22
W
zt
1
ith probability
1 2
with probability 1.
Thes, acrding to this, the reward either doubles or becomcs zero at cvery stage. given (a) Can you calculate the expected reward at time F, A'(Wz-z.!, this information? (b) What is the best time to stop this game'? (c) Suppose now we sweeten the reward at every stage and we multiply the Fz by a number that increases and is greater than one. In fad supmse the reward is now given by:
2n frr (n+ jj =
r
Htz,+ 1),
.
,
.
dSt
gst
=
+ a'vd
. Volatility is 12% a year stock pays no dividends and the current stock price is 100. . 'rhe
Using these data you are asked to approximate the current value of an American call option on the stock. ne option has a strike price of 1* and maturity of 200 days, a (a) Detcrmining an appropriatc time inten'al , such that the binomial tree has four steps. What would be the implied U and D'? (b) What is the npiied probabili?
,.1
with T 1, 2, 3, Show that t-he expected reward if we stop at some timc Tk is given by: 2k k+ 1 (Here, Tk is a stopping time such that tne stops after the kth toss.) (d) What is the maximum valuc this reward can reach? (e) Is there an opfimal stopping rule? =
507
Suppose you arc given the following data: . Risk-free interest rate is 6% . The stock pzice follows:
random variable:
=
10 Exercises
.
(c) Determinc the tree for the stock price St. (d) Determine the tree for the call premium Ct. (e) Now the important question! would tis option ever bc exercised early?
'
2. Consider the problem above again. Suppose we tossed a coin T' times and the resulting zt were alI +1. The reward will be: T(2T'b1
H''w
=
(a)
)
(F + 1)
= u4, Show tbat the conditional expected reward as we just play one more time is; 2W+: F + J CiWT+I l W'F 1f)1 T+ 2 How does this compare with Fz? Should the jlayerthen But if the player never stops when he or she is m a winning streak, continue playirlg the jame? how long would the ylayer at some pomt? that probabihty zt What is the puzzle? explain this How do you =
=
(b) (c)
idstop''?
-1
(d) (e)
.
=
4. Suppose the stock discussed above pays dividends. Assume all parameters are the same. Consider these three forms of dividends paid by the Iirm. (a) The stock pays a continuous, known stream of dividends at a rate of 4% per tfme. stock pays 5% of the value of the stock at the third nodc. No (b) other dividends are paid. (c) The stock pays a $5 dividend at the third node. In each case determine if the option will be exercised early. 'rhe
5, Consider a polky maker who uses and instrument kf to control the path follfawed by some target variable 1$. ne policy maker has the followilg Objcctive function 4
U
=
kr-l)2 J7g2(k; I
=
+ 100(F;)2j.
1
The environment imposes the ftlllowing constraint on this pllicjr maker:
l'-r
.2
=
The initial Ff! is known to be 60-
t
+ .6Y)-1
.
;
'
L
508
C H A P T E 11
*
22
Stopping Times
and
American-Type Securities
t '
':
(a) What is thc best choice of kt for period t 4? (b) What is the best choice of kt for period f 37 (c) From these, can you iteratc and find the best choice of kt for t 1? (d) Determine the value function P'2that gives the optimal payoff for t 1, 2, 3, 4. (e) Plot the value function P'zand interpret it.
, .
=
E
t
=
=
'.:
:
=
' ,,,,m-
--
y.jj
.
.
.=
az
..
: .:
:
(:
.'
t
'
--
.
'$.. J': tj. '. ..? v
'
.r ' 7:
'
r. !.
i . (.
k'
(';
. L (.
('; ;
.C
)
i
E2.
,
J
.. k1 ml ''. y:' .. ,
'
'(
k!
..
gAl #' . q? Ef '
..
' ..',L'.
;.!' (.
: . t..
; i :
Bhattacbarya, S., and Constantinides, G. Theory of Valuation. Rowmand and Littlefield, 1989. Bjork, T. Arbitrage F/lwry in Continuoua Tzzlc. Oxford University Press, 1999. u'rhc Pricing of Options and Corporate Liabilitiesp'' J'ourBlack E, and Schdes, M. <1 of Political Economy, 81, 637-654, 1973. Model of Interest Rates and Its Blach F., Derman, E., and Troy, W $A One-ndor Application to Treasury Bond Optionsr'' Financial Analvsts Jsurlm/, 46, 3>39, 199 0. Bnace, A., Gatareh D. and Musida, M. ''The Market Model of Interest Rate I)ynamics,'' Mathematical Finance. 7, 1998. Bremaud, P, Bremaud (1979)p, 181. Point Processes J?'l'# Queues,Mcr/n,fzc Dy'in New namics, er-verlag, York 1981, , Pr g Brennan, M. J., and Sclxwarzy E. S. 'A Continous Time Approach to Pricing of Bondsp'' Journal W'Banking and Financc, 3, 135-155, 1979, Broadie, M.. and Glasserman, R American-style Securities Using Simulation, Journal of Economic flyrla/rsflo and Control, 21, 1323-1352, 1() py Brzeznialq Z., and Zasuwniakr T. Basic i/x&zylfc Proccsscs. Springer, UK, 1999, Cinlaq E. Stltastic Jrccswc-. Prentice-Hall, New York, 1978, Pricing and Its Applicationsr'' in Theor.v of U:/Cox! J. C.. and Huanp C. S, and Constantinides, G., eds,), Rowman and Littlefield, uation: (Bhattachaa, 1989. ctm J., and Rubinsjein, M. Options Markms. Prentice-Hall. New York, 1985, J. E.. and Ross, S. An Interternporal' Mset Pricing Model with cox,J. C.y Ingersoll, Econometrica Ratitmal Expectationsy'' 363-384, 1985. Das, S. Swap and Deriyurvc Financing, Revixd Edition. n'obus,1994. Dattatrezw R. Eo Venkatesh, R. S., and Venlia leshr M E. lnterest Smc and C'urrrncy swom.Probus, Clticago, 1994. Dellacheriw C., and Merep R Theorie fe. Mardngales. Hermannx Paris, 1980.
s
Rpricing
, j
.
.
,
.t
t
;' :
E:
.::
'
Soption
.$3s
; E
.:
.j rl
;
.t
509 '
v) ,
Bibliography
510
Due, D. Dynamic Asset Pricing, Second Edhion. lhinceton University Press, 1996, D.nkin, E. G., Yushkevicb, A. I!, Seitz, G. M., and Olzishchih A. L. Selected Pcllc?'sr #>'Ijynkin, ztpzcrfcza Mathematical Sociep, 1999. EI Khroui, No Jeanblancique, M. and Shrevw S. Robustness of Black and Scholcs Formula,'' Mathvmatical Fmznce. 8, 93-126, 1998. Equattbns. Springer-verlag, Gihman, I., and Skorohod, A. Stochastic .lfrprmk/ Berlins 1972.
Gihman, I., and Skorohod, A. The F/zcf?r..gof Stochastic Processes, Vol, I., SpringerVerlag, Berlin, 1974. Gihmany I., and Skorohod, A. The Theory of Stochtic Processcs, Vol. 11., SprimgerVerlag, Berlin, 197j, Wiley, New York Harrison, J. M. Brownitm Motion and Stvchastic Flow 'p/crn',
1985.
M., and Kreps, D. ttMartingales and Multiperiod Securities Markets.'' of Economic lheory, 1979. Harrison, M., and Pliska, S. RMartingales and Stochastic Integrals in neory of Continuous Trading,'' Stocllastic Processes and their Applicalions. 11, 313-316, Harrison,
Jurnal
1981, Prenticc Hall, 1993. J. Options, Futures cmf Other Derivative s'ecurrew, Prcntiee Hall, 20fX). Fourth Edition. Derieativcs, and Ofcr Futures, Options J. lnterest Rate Derivative Sectkritiesy'' Acvcw of FiJ., and Whitey A. Studies 3, 573-592, 1990. z'lzznclkz!
Hull, Hull, Hull,
Kpricing
Ingersoll, J. Theoty 0./ Finoncial Decision Making. Rosman and Liltlefield, 1987. JTiurzo Income Sccurities Jz'l# Interest Stztc Optms. Mcfiraw Jarrom R. J. Modelling Hill, Ncw York. 1996. South Western, Cincinnati, 1996. S. Dcrivative s'ccurried. Jarrow, R. J. und Valuation Ft/(?D.Wiley, Fixed-lncome Jegadeesh, N., and Tuckman, B. Advanced rfbrnbull,
20.
Kapne K., and Marshall, J. E The Swop Handbook. NYIE New York, 1992. Karatzas, I., and Shreve, S. E. Brownian Moon and Stochastic Calculus. SpringefVerlag, New York, 1991. Klein, R. A., and Lederman, J. Derivatives and Synthetics. Wtbus, Chiuago, 1994. Kloeden, E E., and Platen, E. Numerical Solution of Stochastic Dl///rezlliaf Equationa. Springcr-verlag, Berlin, 1992, Kloeden, R E., Platel, E., and Scburz. H. Numerical Solution OJMSDEThrough Computer Experiments. Springer-veylag, Berlin, 1994. Kushnep A. J. Numerical Methoda for Stochastic Control Problema frl Continuous FlWze.Springer-verlag,
New York, 1995.
1998, Kwoh Y-K. Mathematical Afotcls f Financial Derivatives. Springer-verlag. Gcneral 1: F/lzwr-p, Prtpcc-z'd Liptser, R., and Shiryaye: A. Stadstics of Random Springer-verlag, New York, 1977. Alwlicatiotu', Liptseri R., and Shiryayev, A. Statztics of Rtzltrprl Prxesses #; Springer-verlag, New York, 1978. 46, 1429-1445, Lucas, R. IAsset Pfices in arl Exchange Eeonomyy'' Econometrica
1978.
Bibliography
511
A. G. RIto Calculus in Hnancial Dedsion Maknp''
Malliaris,
481-496, 1983.
Merton,
R. GAn Intermporal Capital
SIAM Acvfew 25,
Asset Pricing Modelj'' Econometrica,
888, 1973.
41, 867-
Merton, R. Continuoua cme Finance. Blackwell, Cambridge, 1990. Medon, R. the Mathematics and Economic Assumptions of the C-ontinuous Time Models,'' in Financl Economics: Essays in Honot tp//t/l,l.l Cootner, (William Sharpe and Cathryn Cootner, eds.). Prentice-l-lali, 1982. Milne. E A General Equilibrium Asset Economy with Transacton Costs, Unpublished, 1999, Milstein, G. N. I'Pt?kApproxnation of Solutms ofsystcma of Stochastic Dt/.rwlkl/ Equations, Fetary of f'rtphfzNl/l.yand Applications, 1985, Musida, M., and Rutkowski, M. Martingak Mefhofa' n Financial Modelling. Springer, 1997. Nielsen, L. T. Pricing and Hedging of Dczyuffpr Securities. Oxford University Press, 'on
1999.
Oksendal, B. Stochastic Dl//rerl/ft?/ Equations, Third Edition. Springer-verlag, Berlin, 1992. Pliska, S. R. lntroduction to Mathematical Hnance: Dctete Fn: Modcls. Blackweli,
1997,
Prottep
R
1990.
Rebonato,
Jrlfcgrfzffon and Dlfcrcnfal
vtochastic
R. lnterest Rate Mode,
Equatiolw
Second Edition, Wilcy, 20.
ltehmatoy R. Volatilit.ycz?d Conrlation.
Springer-verlag,
Berlin.
Wiley, 1998,
Revuza D., and Yor, M. Continuous Martingales J?'IWBrownian Motion, Second Edition. Springer-verlag, Berlin, 1994, Rogers, C., and Williams, D. D@usions, Ikocesses dzrstf Marttgale, lto Cal1987. York Wiley, New tw1t/-, Ross, S. RA Simple Approach to Valuation of Risk'y Streamsy'' Journal of Buainess, 'ftzri)v'
1978.
Ross, S. lhnbability Models. Academic Press, San Diego, 1993. Ross, S. zllz Introduction to Mathemadcal Finance. Cxunbridge, 1999. Shiryayev: A. Optimal Stopping Rules, Springer-verlag, New York, 1978. Shiryayev, A. of Martingales,'' International Statistical Acvfo', 1983. Shiryayev, A. Probability Theory. Springer-verlag, New York, 1984, Smlth, G. D. Numerical Jt/zlffo?l of 'J?WJJ Dterential Equations Fiai/e D#erence Metho, Third Edition. Oxford University Press, 1985. Sussman, H. J. tbe Gap between Deterministir and Stochastic Ordinary Differential Equations,'' Arma r? Probability 1978. nomas, J. W Numerical fkr/i Differential Equations: Finite Dtcrence Methods. Springer-verlag, New York, 1995. Vasiceke 0. RAn Equilibrillm Characterization of the Term S'nlcture,'' Journal of Fl.fzrlcfz; Economics, 5, 177-188. 1977. Vasicek and Beyond. Risk publications, I-ondon, December 1996. Williams, D. Probability with Maningalkw Cambridge University Press, 1991. Wilmott, E Derivatives: Fe Thcory and Practice of Financial En,irlecrjrw. Wiley, Hneory
Gon
1998.
Fz
i !!
;; .
===
I
Absorbent states, 49%, 500 options, 7 American-style possibilitics, 490-491, 492 cxcrcise interest rate instruments and, 375 stopping time ands 371, 491-504 Arbitrage deMed, 13 i'Il
references
derivatives pricing, 77, 79, 88
arbitragc
Arbitrage-free prices, l4, 28, 29 Arbitrage opportunities, 13-14, 27-28,
concept
and, 13, 14
ctmverting into martimgales, 349-353
38
lto's lemma and. 230 ntrmalization and, 380 notation, 2, 15 unpredictable change in, 174 Asset pricing, see also Bond pricing', Derivatives pricing arbitrage theorem and, simple e'nvironment, 19-21, 381-383, complex cnvironment, 383-404
Arbitrage portfolios, 39 Arbitrage theorem in a complex environment,
on, 38
19-26, in a simple enronment, 381-383 states of the world, 36 synthetic probabilities and, 21-23 time index and, 36 Asia'n options, 303 Asset prices
383-403
liscountingand, 37 foreign currency investment and, 34-36 gcneralizaon of, 38-40 latlice models and, 29-32 mathematical tools required, 37-38 methtdology of, 37 no-arbitrage condition, 26 norm-l'lization fonard measure, 395-403, 404 with risk-neutral probabilities, 389-395 numerical ekample, 27-29 payouts and, 32-34
binomial modeling, 19*202 binomial process and, 9 1
Black-scholes 299-301,
equation 353-366
and,
diffcrential equaons and, 156-157 discountimg a'nd, 37 with dividend payouts, 32-34 equalization of ratcs of rdurn, 25
51
':
1
Subject Index
514 Asset pricing
ontinued)
equivalent martingale measures and, 77, 345-346. 354-366 whh foreign currency investment, 34-36 fomard measure and, 398-403, 't04
Girsanov theorem and, 312, 332-334, 345-346, 478-1.79 lattice models, 29-32
martingales and, 122-124, 133 methodologyfor, 17-29, 37-38 moment-generating functions and, 34*348 no-arbitrage assumption, 23, 26 normalation and, 389-404 partial differential equations and, 299-301 probability transformations and, 312, 315 rare and normal events modeling, 190-192, 19*202 references on, 1 1 risk-adjusted probabilities and, 21-23 stmtesof the world and, 14-15, 29, 36 stochastic differential equations and, 196, 253, 261-272 time index and, 36
Average options, 303 Averaging
operator,
98-99
Barrier options, 301, 302, 303 Basket options, 302 Benchmark prices, see Arbitragc-free
prices
Bermudan-style options, 492 BGM models. 437 Binomial distribution, 100-103, 139 Binomial modcls of accumulaled changes, 199-202 of normal events, 197-198 overviewof, 196-197 of rare events, 196-197, 198-199 and stopping times, 494-499
Binomial process, 91 limiting propcrties of, 101-102
moments, 102-103 overviewof, 100-101 patb of, 224-225 signihcamceof, 116
Binomial Binomial
Bt-md prices
continuous marlingales
ontinuedj
in pricing interest-sensitive 426-.430
securitics,
random variable, 46a, 100
468
with cmstant dividends, 300-301 exotic options and, 301-303 form of, 297n geometric description of, 29>299 mrtingale approach to, 354-358 cornparison to PDE approach, 358-366 oveniew of, 296-299. 451-452 pricing and, 299-301, 353-366 references on, 366 solulons of closcd-form,301-305 numerical, 306-309 underlying assumptions, 299-3, 368, 369
452,
HJM methodology, and, 441 market prico of equity risk, 456
to and,
't04
PDE analysis of, 454-465
Calculus, standard,
472M74
pridng cquation, 414-418, 422-4:23, 426, 428, 432-433 spot rates and, 375, 414.417, 418, 428, 432-433 stochastic differential equation for, 438-439
423
Bounded variatio'n, 50-52, 10 1 British Ba n'kdrs Association, 371n Brownian bridgea 115/1 Browlan motion, 106, 153, 175) scc atw Wiener p'rocess conditional expeelation of, 349
283
tzltp
Stochastic
derivatives, 53-57, 156, 238-232 chain rule, 53, 57-59, 158, 231 ordinary differential 72-73 paral, 66-73
equations.
deterministic and stochastic integration
compared,
209-213,
214 deterministic variables i'n, 86-87 functions,
coupon payments, 376 credit spread cutwes, 41 1.412 default zisks, 411-412 discount cun'e, 4 11, 427, 428, 429.430, 462 duration and convexity of, 70-72 forward rates and, 437.440 long, 385, 386, 387, 391-392, 408 options, 247-248, 37,1-375 in risk-free Nrtfolios, 453 risk prena, 456 short, 385, 386, 387, 391, 396-397, 399 three-period, 416-417 yield, 409, 418 (Jcc alao Yield cun'e)
301309, 451! 468 derivativespricing and, 80, 85-86, partial differential equations and, 279, 283-284, 308-309
see
calculus
Boundaries, optimal exercise, 499 Boundary conditions, 31 Black-scholes equation and,
Bond prices
arbitrage conditions, 43 arbitrage theorem andx 387 continuouslycompounded yield maturity, 422 fo-ard measure normalizatio'n 396-397, 399, 4, folward rates and, 409, 418-422, no-arbitrage condition, 436 notation, 408-409
and,
Bonds, 3', see zlstp Discount bonds; Zero-coupon bonds arbitrage theorem and, 3S7
methods
compared to classical pricing of interest-sensilivesecurities, 434-435 compared to PDE analysis tf nterest-senshivc secures, 459-460
arbitrage tbeorem on, 391-392 equivalent martingale measures 477-479 partial differcntial equalion for,
and, 126,
130-132
deflned, 177-178 Girsanov theorem and, 322
Bond pricing
trees. 30, 116, 183n,!495-499; Lattice models also see Black-scholes equation, 78, 79-80 conditions, 301309, 451, boundary
Black-scholes
515
Subject lndex
47-52
fundamental theorem of, 205 integrals, 5%.64 cc J/m Integrals) integration by parts, 65-66 Call options, 279., sce also American-style options; Options on bonds, 374-375
delined, 7
lookback, 30 1 mulli-asset, 302 pricing, 7, 80-84, 300-301, 353-366, 489-490 Capital gains, 278
Caplets oveniew of, 372-374 pricing, 398-399, 491-402 Cap rates, 372-374 Caps, 372-374 Cash-and-carry markcts, 3-4 Cash-llow, 470-472, 475 Cash market instruments, 2 Center of gravity, 94 Central limit theorems, 105, 17+, 200n, 202 Chain rule in standard calculus, 53, 57-59, 158, :131 in stochastic calculus, 231, 232-240 Circlcs, second-degree equation for. 290
Subject lndex
.516 Closed-form formulas, 7-9, 11, 78, 82,
447
3 Conmensated Poisson process, 124, Comrnodities,
127
fo-ard measure normalization and, 397-398 oveniew of, 9% 98-99 probability transformations and, 319-320, 321, 322
Conditional expectations, 370 Black-scholes equation and, 452 correspondence to PDF-S, 468-482 fomard measure normalization and.
397-398
of geometric processes, 348-349 martingales and, 121 overviewof, 97-99 probability measures and, 351 recursive property of, 134-135 stochastic difference equatms and, 169
Conditional probability 97-99 claim, 2 Continuous square integrable Contingent
martingales, 160a defmed, 126 trajecqoryproperties,
127-130 Wiener process as, 176-178 Continuous-timc martingales defuwd, 121-122, 12K126, 141
examples: 130-134 stopping times and, 504
Convergencc, see ct? Mean square
lirnit dened, 52 of random variables, 112-116 types of, 112-113 weak 104n, 105, 113-116, 17&
Convexiq', 70-72 Coupon bonds, 475
517
Cox-lmgersoll-Ross model, 464 CRB commodity index. 3 Credh spread culves, 411-412, Currencies, 2
Derivativcs pricing (continuedj probability transformations and.
Delta, 83 Delta hedging, 83, 2.31 Density ftmction, 93-94
Deterministic variables, 69, 86-.87 Differential equations, see Ordinary differential equations; Partial diferential equations; Stochastic differential equations Diferentials stochastic, 88, 170, 205
179a
Compensatory term, 179n Completing the square, 291 Composite functions, 58 Conditional density, 98 Conditional expectaton operator,
Subject Index
conditional probability and, 97-98 of random variables, 105, 313, 3l4
Derivatives
(Enandal), 2, 2769see zltp
individuall-vprboundary conditions, 283-284 de:ned. 2 expiratiomdate, 4-5 fonvards and futures. 5-7 index-linked,475 options, 7-9 referens on, 1, 11 swaps, 9-11 types of, 2-5
Detivatives (incalculus), 204 chai'n rtlle, 53, 57-59, 158, :231 oveniew of, 53-57, 156 partial, 6*73, 230. 231, 232 types of, 230-232
Derivatives
pricing, see also Asset
pricing',Bond pricing arbitrage concept and. 77. 79, 88
Black-scholes
equation
a'nd. 353-366
boundary conditions, 80, 85-86, 283 conditional expectations and, 467, 468-469 ctmtinuous-timestochastic processes and, 8*88 fomards, 78-80 general strategies in. 467-468 martingale representatons, 146-152 notation, 2 options, 7-9, 80-84, 390-301, 345-346, 353-366, 48*490 overdew of, 77-78, 88, 275 partial differential equations and, 77, 84-86, 282-289
341-342 reference.son, 88-89 risk-free pordblios, 276-284 synthctic probabilities and, 334-337
total, 67
Differentiation, 156, 157-161, 171 Diffusion coeflicient in PDES for intercst-sensitive sccurities, 45% 458 in spot ratc models, 430, 431 in stocbastic diffcrential equations, 157, 170, 253-254, 266, 267-268. 270, 271-272 Discontinuous squarc integrable
martingales, 179
Discount bonds discount cun'es and, 409, 41 1 forward rates and, 436 437 PDE mal'pis of prices, 454-465 price notatitm, 408-409
pricing equation, 41,1-418 simple martingales and, 136 yield, 409 scc alao Yield cunre)
Discount cunre, 411, 427, 428, 429-430,
462
Discount factors, 335
partial differential 469..472 random, 475
equation ftar,
risk-fTce, 319 Discountng, 37 in continuous time, 417 in discrete tfme, 416-417 normaiizatiomand, 390, 399, 404 Discount in riskless borrowing, 40 Distribution functions, 93. 94-97, 114
Dividends
irl asset pricing, 32-34
Black-choles equation and, 30(/-301 in risk-h'ee portfolios, 278 Doob-Meyer decomposition, 123, 124, 144/-143,145
Doob-Meyer theorem, 141 Down-and-out opeions, 301, 302, 303 Drift interest rate dmamix and, 376-377 partial differential equations and,
476-477 risk-adjusted, 402, 41 1, 476 of short rates, 370 of spot rates, 443
,177
Wiener process and, 332 Drih coefhcient, 183 arbitrage-free spot rate model, 433 for instantaneous fonvard rates, 441
in PDES for interest-sensitive securities, 457, 458 risk-free rates and, 353, 364 in spot rate models, 430, 431 in stochastic differentfal equations, 157, 170, 253-254, 266, 267-268, 270, 271-272
Dual strike call optionss 302 Duration, 70-72 Dynamic programming, 371, 505 stoppi'ngtimes and, 491-492, 495-499 'n's fonnula, 504-505 *
Econometrics, martingale probabilities and, 336-337 Ellipses, 290-291, 292 Emerging derivativcs, 302 Equilibrium pricing methodsa 13 Equivalent martingale measures, 123',
Probab measures see asset pricing and, 77. 345-346, 351-366 fz/xo
Black-scholes
equation and, 35K366
bond pricing and, 477-479 conditional expectation of geometric processes, 34<349 eonvenin.gasset pces, 349-353
Subject lndex
5l 8 Equivalent
martingale
Forecast,
measures
(connuedj derivativespricing and,
77 Girsanov theorem and, 329-334
method of, 77 moment-generating function, 346-348 oveniew of, 342 references on, 366 spzthetic probabilitics and, 334-337
Error terms, 193, :2.53 Euler schemc, 445 European options, 7, 302, 354-358 Events, 92, 93 Exchanges,
6
Exercise boundary, 499 Expectations, 263; see tztp
97, 472', see
(/1.75
Conditional
expectations
Foreign currency assets, 34-36 Folward loans, 371-372 Forward measure in market practice, 444
normalization, 385, 396.403, 404
Fonvard
rate agreemonls
arbitrage theorem and, 385, 387 twenricwof, 372, pricing, 392-394, 400-$01
Fomard
rates
bond prices and, 409, 418-422, continuouslycomNunded, 422
423
HJM pricing mcthods and, 436-441.
Conditional
expectations
Expiration datc, 2, 4-5 Exponential functions
derivative of, 54 i'n discounting, 37 overdcw of, 49
Taylor series approximatio'n,
442, 443 instantaneous. 420.421, 422, 437, 439-441 spot rates and, 442
Fomards.
70-72
2
compared to futures, 6-7 decomposing swap deals into, 9, 10, 11 defined, 5 cxpiration date, long position, 5-6 pricing, 78-80 short positiona 5-6 zl-.f
Fair price, sec Arbitrage-frcu prices Feynman-Kac theorem, 108n, 452, 482,
487
Filtration, 97, 120 Financial derivatives, sec Derivatives
(snancial)
Fixed-income
assets
basic comcepts in, 408-1.14 bond pricing equation, 414-418, 423 diversity of, 4O8 forward rate and bond price relationships 111,418-422, 423 Markov modeling ot 110-111
pricing classicalapproach, 427-435
435-144 referenccs on, 424 Fixed lookback options, 301 Floang lookback options, 301 HJM approach,
Floorlets, 374 Floors, 372, 374
FTLSEIOOindex, 3 Functions. 47-52 Fundamental heorem of calculus, 205 Fundamental Theorem of Finance, 467,
468 bond pricing cquatio'n a'nd, 415 in a complex environment, 383.403, 404 in a simple environment, 381-383
Futures
compared lo fomards, 6-7 expiration datc, 4-5 LiboT rates, 371-372 on marked to market, 475 payouts. 1% Gaussian modet 107 Generatorsa 452, 482, 483-485
Subject Index
519
Geometric processes, 201, 348-349 Instruments, see Cash market Girsanov theorem, 37, 38, 123-124, 150) instruments; Derivatives (fmancial) Intepals, see tzltp Riemarmtieltjes see am Equivalent martingale integral; Stochastic integrals measures Black-scholes equation and, 359, equations, 73, 205 integration by pan, 65-66 360-363, 364 discussionof, 331-334 oveniew of, 59-64 interest rate dynamics and. 376 pathwise, 224-226 Novikov condition, 365 random, 62 Riemann, 59-62, 209, 222, 224 overviewof, 322-323 Stieltjes, 62-63 pzicing and, 312, 332-334, 345-346, in stochastic calculus, 88 478-479 Intensity constant, 107 references on, 342-343 Interest rates, 3 statement of, 329-331 market price of interest rate risk
455-456
*
Heath-larrow-Morton (HJM) methods, 110,47.3, 427 advantages of, 444 abitragc-free dynamics 1, 437-440 forward rates and, 436-/37 interpretation of, 44+.441 in market practice, 444 rationale for, 435.436 references on, 447 spot rates in, 441-443 Heavy tailsa 96-97 Hedges, 147 HJM methods, see
Heath-larrow-Morton Hyperbolas, 292
metlmds
Indexes, 3
lndex-linked derivatives,
475 Indicator functions, 330 Infmitesimal generator, 485 Informatiom I'inside,''
252-253
stochastic dit-ferential equations and, :152-2,53,-:257 lnnovation terms, 157. 163, 169, 183, 184, 191, 195, 220-224, 276, 277, 279
RInsidg'' irifbrmation,
252-253
Markov processes and, 108a, 109-1 11
stochastic,375 swaps and, 10 volatility,464 yield c'urves, 247-248
Interest-sensitive secqlrities, seett/l-p Libor instruments analyticaltools and, 369-369, 379-380 arbitrage relations, 408 arbitrage theorem and, 383-403 complicatirtgcharacteristics of,
375-377 drift adjustment, 376-377 fonvard measure and, 398-403, 404 normalization and, 380, 389.403, 404 parlial differential equations for closed-fonu solutions, 460-465 derivation, 457-460 framework of, 454 market pricc of interest rate risky 455-156 mcthods of obtaining, 452-454 picing, 426-427 classical approach, 427.435 HJM approach,
435-444
referemceson, 378 simple martingales and, 134-136 term structure, 377 types of, 371-375 ln-the-money expiration, 8
Subject Index Ito diffusions, 452, 482-485 Ito integral,
46, 205
binomial process and, 102 correlation properties, 226-227 dehned, 213-214 evaluatingwith Ito's lemma, 242-244 cxistcnce of, 226 intepation by parts a'nd, 66 martingales and, 220-224, 228 mean
square convergence
and, 112,
114, 212, 213, 214-220, 228 pathwise integrals and, 224-226 properties of, 220-226 rcferenues on, 228 relevance of, 207-208 Riemann-stieltjes
methodolor
Black-scholes
equation
and,
and, 359,
360, 361, 364-365 defined, 240 derivation of, 232-240 generator of the Tto diffusion and, 484 intcgral form of, 244-245, 36.1-365 jumps and, 248-250 multivariate version, 245-248 twerviewof, 250 references on, 2,51 stocbastic differential equations and, 262, 263-265 uses of, 228, 241-242, 438-439 Jumps, 248-250 *
Knock-in
302 Knock-out options, 302. 303 Kolmogorov's backward equation, pptitms,
485.486
pricing, 392, 395 types of, 372-375
Libor rates, 385n, 388, 394 fonvard measure normalization
and,
396-397 origin of, 37 1/$
208, 209-211, 214 stochstic differential equations and, 170, 206-208, 228, 2,52 sthastic integration and, 211-213
Wiener process and, 481 Ito's lemma, 66 asset pricing and, 230
Ladder options, 301-302 Lattice models, 29-32 I-vy theorem, 176, 178 Libor instruments, 381, 385n,. see also Interest-sensitive securities
Limitss 52,1sce Tzlstp Convergence; Mean square limit Logarithmic fundions, 50 London Intcrbank Offered Rate, 371/;
see also Libor rate Long bonds, 385, 386, 387, 391-392, Long position, 5-6 Lo'ng rates, 433 Lookback options, 30l Mappings, 47 Market price of risk, 370 equity, 456 interest rates, 455-456, 459, 461 Markets, rare events in, 182-183 Mztrking to market, 16n lubitrage-free benchmarks and, 14
in futures, 6-7 interest rate derivatives and, 375-376
Markov chain, 498 Markov process HJM methods and, overviewof, 108-109 relevance of, 109-110 sbort rates and, 370-371 vector casc, 11(11 1 Markov property, 469, 476, 483 Martingalc difference, 124, 163, 221 Martingale probabilities, see Synthetic .144
probabilities 452,
408
Martingale
representations anda 146-152
derivativespicin:
Sobject lndex
521
Martingale representations (continuedj relevuce of, 112-1 13 Doob-Meye,r decomlyosition and, 123, stochastc integrais and, 212, 213 124, 148-143. 145 Method of equivalemt martingale examples ot 137-140 measures, 77., see also Equivalent overviewof, 123-124, 137, 152 martingale measures relevance of, 153 Moment-generating function, 34*348 stochastic integrals and, 143-145, 153 Moments Martingales, 38, 62 binomial process and, 102-103 in asset pricing, 122-124, 133 1to intcgrals and, 226-227 continuous square intepuble! 126, twerviewof, 94-97, 193
127-130, l6 continuous-time, 121-122, 124, 126, 13*134, 141, 504 converting asset prices into, 349-353 delined, 24, 120, 121-122 derivativespricing a'nd, 77 discontinuous squztre integrable, 179 examples of, 130-134 generating probabilities witb, 337-342
Girsanov theorem
and. 330, 331,
333-334
Ito integral and, 220-224, 228 lattice models and, 31 normalization and, Z5, 390, 398, 399 notation, 119, 120 oveniew of, 152-153
pathwise convergence and, 227 probability and, 24-2,5 references on, 153 relevance of, 153 right continuous, 124, 125, 134 simple, 134-136 stochastic differential equations 163 in stochastic modeling, 124-1 27 stoppi'ngtimes and, 504-505 submartingales, 120, 123, 127, 140-141, 142-143 supermartingales. 120, 122 trajectories of, 127-120
and,
Matrtx equations, 72 Mean square limit (convewgence) explicit calculation of, 216-219 Ito integrals and, 112, 114. 212, 213,
214-220, 228
rare and normal events, 193-195 Money market accounts, 384n Monte Carlo methods, 110a, 336, 479 options pricing and, 490 spot rates and, 445-446 Multi-asset options, 302 *
Normal
distzibution, 139, 357
continuous-time equivalents, heas'y tails and, 96-97 momenls and, 95 overviewof, 103-105 rare events and, 182, 196
Normal
106-107
events
binomial model of, 197-198, 202 charaderistics of, 174, 18K190 continuous paths, 188 moments and, 193-195 smoothness of sample paths, 188-189 stochastic differential equations and, 183-184
Wiener process amd, 176-178 370, 380, 388 fonvard measure, 395-403. 404 in martingalc representations,
No-alization,
149-152 martingales and, 24-25 with risk-neutral probabilities, 389-395 Notesx 3 Nutional assets, 3 Notional ptincipals, 10 Novikov condition, 329, 330, 365
Subject Index
522
derivation, 457-460 framework of, 454 market price of interest
*
Operatoo, 47., see also Conditonal expeuation operator averaging, 98-99 defmed, 97a unconditional expectation, 99 Optimal exercise boundary, 499 Options, 2) see tzz American-style options; Call options Bermudan-style, 492 Black-scholes equation and, 301-303
closed-formformulas,
7-9
7, 302, 35>358 expiration value, 303 nomlinearityof, 9 European,
path dependent, 489 pricing, 7-9, 80-84, 300-301, 345-346, 353-366, 489-490 probability transformations and, 327-328 stopping timrs and, 491-504 types of, 7, 301-303
Ordinary differential equatioms, 72-73, 156-157, 201-205, 259-260 Ornstein-uhlenbeck model, 271 Out-of-money expiratio'n. 8 Over-the-counter tradlg, 6 *
Parabolas, 292, 293 Partial derivatives, 66-73, 230, 231, 232 Partial diferential equations (PDEs). sec ato Black-scholes equation in asset pricing, 29*301 barrier derivatives and, 303 for bond pricing, 472-474 boundary conditions, 279, 283-284,
30+309 classificationof, 284-289, 292-293 correspondence to conditional expectations, 468.482 pricing. 77, 84-86, derivalives in 252-289 for djscount factorss 469-472 for interest-sensitive securities closed-form solutions, 460-465
455M56 methods of obtaining, oven'iew of, 293
Probability zrate
risk,
452-454
references on, 294, 310 risk-free portfolios and, 276-284 solutions of, 304-309
Pathwise integrzs, 22,1-226 Payofs, 1*17
diapams, 5-6 in forwards contrads, 5-6 of multi-asset options, 302 sirnple martingales and, 135-136 stopping times and. 501-504
Payouts, 16
asset pricing and, 32-34 for interest rate derivatives, 375-376 notation, 2
see Partial differential equations Piece-wise continuous functions, 115 Planes, 285-257 10*108 Poisson tlistribution, Poisson process, l07 compensated, 124, 179n PDES,
examples of, 180-182 overviewof, 175, 178-180 rare events and, 17>182, rcferences on, 202
196
Portfolio calls, 302 Portfolios, s'ee zlm Risk-free portfolios
defmed, 17, 39 delta neutral, 83 replicating, 147. 148-149, self-fmancinp 152, 279 weights, 280
152
523
ontinued)
basic concepts in, 91-94 bizmrnialprocess and, 100-103 conditional expectations and, 97-99 convergence and, 1 l 2-116 lattice models and, 30-31 Markov processes and, i()8-1 11
martingalcs and, 24. 25 moments and, 94-97 references on, 116
Probability distributiols,
123
normal, 103-105 Poisson,
transformations of,
314-315 Probability measures. see also Equivalent martingale measures conversion of asset prices into
martingales, 349-353 equival'ent, 328-329 Girsanov
changing means of, 3 1*322, 323-324 conditional expedations and, 97-99 convergence ot 112-i 16 density functions of, 15, 31 3, 314 moments and, 94-97 normal distribution and, 101-105 ovetviewof, 93-94 in stochastic differential equations, 162-161. stocbastic integrals as, 2 12 variance of, 159-160
Rare events
10*108
theorem and, 322, 323-329,
330 interest rate derivatives and, 380 oveniew of. 312-316 transforming, 320-322
Probability spacc, 92, 93, 137-139 Probability transformations by changing means, 316-322, 323-324 Girsanov theorcm and, 322-329
overviewof, 312-316 pricing and, 312, 315, 341-342 on probability mcasures, 320-322, 32.8-329
Radon-Nikodym
Positions, 5-6, 39 Premiums, 7 Price-discovery markets, 4 Pricing, see Asset pricing; Bmd pricing; Detivativcs pricing Probability see also Risk-adjustcd probabilities; Synthetic
probabilities addinp 139
Subject Index
derivative, 327-328
using martingalcs, 337-342 on vectors, 325-327
Put options, 7
characteristics of, l 73, 174, 184-187, 189-190 in derivative markets, 182-183 modeling, 190-192, 19*197, 198-199 moments and, 193, l95
Paisson process and, 178-182, 196 refercnces on, 202 relevanu'eof, 174-175 sample paths and, 190 stochastic differemtial equations and,
183-184
Wiener process and, 182, 196 Ratios, 380 Real-val' ued functions, 47 Rebates, 302 Replicating portfolios, 147, 148-149,
152 Representations. 19-29 Returns, 17, 25 Riemann integral, 59-62, 209, 222, 224 Riemanntieltjes intcgral, 63, 64, 180 binomial process and, 101, 1t)2 dehned, 209, 211 1to integral and, 208, 209-21 1 214 jump processes and, 227 Riemann sums and, 210-211 Riemann sums, 2161-213, 223 Right continuous martingales, 124, 125, ,
*
ladon-Nikodym derivative, 327-328, 329, 478 Random functions, 48-49 Random inteFals,62 Randomness, modeling of, 46-,17 landom variabks, see also Probability asymptotic negligibility and, 202 binomial pross and, 100-103
134
Right continuous submartingales, Risk-adjusted (risk-neutral) probabilities. 150, 151 arbitrage theorern and, 21-23, 381-383
141
Subject Index
524 Risk-adjusted probabilhies continued) equalzation of rates of return, ?.5 lattice models and, 30-31 martingales and, 24 normalizatjon and. 389-395 Risk-free discount factor, 319 Risk-free portfolios Black-choles equation and, 451 bonds ilt, 453
equivalent martingale measures and, 346 forming, 276-.284 market price of interest rate risk, 455.456
Risk-free rates, 133 arbitrage theorem and, 17, 18, 21-23,
381-383 probabilitytransformalions 318-320
and,
probability transformaAions
and, 318,
Risk-free returns. :25 Risk management, 14 Risk premiums, 133, 456
3 19-320, 327
*
Sample patlu, 138, 188-189. 190 Sltmple space, 138 Savings accounts
compared lo long-ter'm bonds, 408 d'risk-free,''384n, 385, 387, 388
Second-degree equations,
bivariate,
28*292
Self-:nancing portfolios, 152, 279 Self-hnancing tradlg strategies,
141-145
Short bonds, 385, 386, 387, 391, 39*397, 399 Short rates, 370-371, 388, 394, 416, 418 Sigma lields, 97 Smootbness, 188-189 S&P500 indtx, 3 Spot rates arbitrage tbeorem and, 383-384 in bond optjons, 375 bond pricing equation amd, 414-418
bond yields and, 418 constant, 414..415 defincd, 409 htting to initial term structures, 47 forward rates and, 422, 442
H.rM metimds and, 441-444 fntercst-sensitivesecuritics and in desmiptive PDES, 453, 454,
457-458, 459. 460-465 pricing with, 427-435 models of gcometric stochastic differential equation, 430.431 market practice, 444 i'n mean-reverting,431-432 using, 432..434
non-Markovial behavior, 443, 444 normalzation and, 399 stochastic,415-417 Spread calls, 302 Square integrable martingales, 160rI,
330 detined, 126 discontinuous, 179 trajectory properties, 127-130
Wiener process as. 176-178 Standard deviation, 94, 174 States of the world, 14-15, 29, 36,
92-93,
173
Stieltjes integral, 62-63: aee clta Ricrnamn-stieltjes integral Stochastic calculus, 37 chain rule in, 231, 232-240 delined, 58-59, 73 differentials in, 88, 170, 205 differentiation in, 157-161, 171 information flow and, 46 tz/5w integrals in, 88, 205-206 (see Stochastic integrals) intemation in, 66, 2*-213, 214
modeling random 46-47 Gnegligible''
bcavior
vadables
wilh,
in, 235-238
pal-tialderivalives ina 67 pricng methods and, 85
Subject Index
525
Stochastic calculus contintadj second-order terms in, 164 Taylor series approximaton and, 69 of, 45-47 uses Stochastic difference equation, 169-170,
208
Stochastic differential equations,
36, 38,
47 for bonds, 438.439 drift and diffusion ooef'ticien? in, 7.53-254,266, 267-268, 270, 271-272 error terms, 253 fmite dilerence approximatitm and, 161-164, 169, 183-184, 19 1 192, 20*207, 208 geometric paths implied by, 254 .
Girsanov theorem and, 332-334
infomalion and, 7.52-253, 257 for instamancous fonvard rates, 440.441
Ito integral and, 170, 206-208, 228,
252
Ito's lemma a'nd, 240, 245 Markov processes and! 109-11t) =d,
mean square convergence models of SDES, 265-271 geometric, 267-269, 43$1-431 Iinear constant coefficient, mean reverting, 270-271 Ortkstein-kThlenbeck 271
112
266-267
square root, 269-270 overviewof. 161-170, 171, 195-196, 271-272 probabilitymeasures and, 352-353 rare and nonnal events analyzed with, 183-1 84, 191-192, 196 references on, 171, 272 second-order terms in, 164-167 solutions of, 255-265 strtmg. 256, 2,57, 258-260, 272 verifying,39-260, 261-262 weak, 256-08, l11 for spot rates, 441-443 stocimsticintegrals andy 205 volatility and, 271-272
Sttxhastic
Sthastic
equivalence, 214 integrals, 66, 89.. Jee also Ito
integral jllmp processes arld, 227 ilt martingate representatioms, 143-145, 153 mean
Sqllare
and, 21 2,
Conv/rgence
213 overviewof, 205-206 pathwise integrals and, 224-226 as rudom variables, 212 simplc, 208 stfxhas&icdifference equation kmd, 208, 209
Sttchasdc
processes,
see J?
Binomal
procew, lkandom fundions derivativespricing and, 86-.88 expectations of, 263
Feynman-lac thcorem and, 482, 487 Ito diffusions and, 482-485 Kolmogorov's backavagd equation and. 485M86 martingales and, 120
references on. 116 stochastic differential 255-258
Stochstic
variables,
equations
and,
156
Sttxks, 2 Girsanov thetArem anda 332-334 Stopping times binomial model of, 494-499 defined. 492-493 example of, 499-504 martingales and, 50.1-505 references on, 505 significanceot 491-492 uses of, 493-494 Strike price, 7, 8-9, 279 Submartingales, 127 decomposing, 123, 140-141, 142-143
defmed, 24,
120
Supermartingales, 120, 122 Swap dealcrs, 11 Swap rates, 374 Swaps, 2
decomposing, 9, 10, 11 delined, 10
M Subjecr lndex
526 main groups of, 2-3 price-discovec markets!
ontinuedj interest-rate based, 374 interest rate example, 10-11 Swaptions, 9, 375 Synthetic probabilities, 21-23 Swaps
*
Valuation tbeory, 88-89 Value-at-rk measures, 175 Variance of probability functions, 314 of random variables, 159-160 Vasicek model, 461 Vecqors, probability transformatons
*
Taylor series approximation, 69-72, 159-161, 170, 207 Rylor series expansion closcd-formformulas and, 305,1 Ito's lemma and, 87-88, 233-240 ordinary differential cquations and,
and, 325-327
stochastic, 271-272 in stochastic diffcrential 165, 166
and,
and, 435-444
Time index, 36 Total derfvatives, 230-231, 232 Tota) differentials, 67 Trading gains, 144-145 Treasury bonds, 3-$, 416 Tree models. see tztp Lattice models
spot rates
and, 446
106, 108
adaptalion to informalion sets, 7.53 arbjtrage-free spot rate model and, 434 equation and, 359,
360-361, 362, 363 bond risk premia and, 456
Bruwnian
lztotion
and, 177-178
defincd, 177
density functftn and, 263 fonvard rates modeling and, generalized, 346n. 348
Trigger options, 302 Trinomial trees, 183a *
expiration date, 4-5
17&
Blacknscholes
-.447
emectation Unconditional Underlying securities cash-and-carrymarkets.
equations.
Wave cquation, 480, 482 Weak convcrgence, 1f)4a, 105, 113-1 16, Wealth, 248 Wiener pross,
opcrator, 3...4
99
Wiener process (continucdj ovezviewof, 175, 17*178 in PDES for interest-sensitive securities, 453, 457, 458, 459 Poisson press and, 181-182, 196 mobability measures and, 352, 353 rare events amd, 182, 196 rate of change and, 483 Riemann sums and, 223 stochastic diferential equations and, 195, 196, 73, 256-258, 263-264,
265 stochastic integrals and,
208, 219, 220
*
Volatility, 94 interest rate dynamics and, 377 spkt rates and, 430-43 1, 443 square-root specifkation, 464
T-bills, 3 Term structure modeling basic concepts in, 408-1. 14 bond pricing equation, 414-418, 423 fitting spot rates to, 444-447 forward rate and bond price relationships, 418-422, 423 HJM methods
E
Up-and-in options, 30M Up-and-out options, 3t)1, 30M
arbjlrage theozem and, 381-383 normalization, 395-399, 403 interest rate derivatives and, 380 pricing and, 334-337
20,1-205 overviewof, 68-72 stochastic difference equations 169-170 stochastic differentiation and, 158-159
4
Subject Index
402
Girsanov theorcm and, 330, 331, 334 interest rate dynarnfcs and, 376 Ito integrals and, 22*227, 481 Kolmogortw's backward equation anl, 486
Yield curve bond options and, 247-248
credit sprcad curves and, 411-412 defined, 410.411 fixed-incomeassets and, 1 10 movements on, 412-414 short rates and, 416 spot rates and, 462
*
Zero-coupon bonds arbitrage theorem and, 387 duration and convexity of, 70-72
folward measure normalization 39*397, 399, 4(k), 404 forward rates and, 436-437 pricing, 391-392
and,
