Pricing Foreign Exchange Options: Incorporating Purchasing Power Parity

Incorporating Purchasing Power Parity Second edition i«

David W.K.Yeung Michael Tow Cheung

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Incorporating Purchasing Power Parity

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tan ma Incorporating Purchasing Power Parity Second edition

David W.K.Yeung Michael Tow Cheung

# m *, * tt t& *t H O N G KONG U N I V E R S I T Y P R E S S

Hong Kong University 14/F Hing Wai Centre 7 Tin W a n P r a y a Road Aberdeen, Hong Kong

Press

© Hong Kong University Press 1992, 1998 First edition 1992 Second edition 1998

ISBN

962 209 454 6

All rights reserved. No portion of this publication may be reproduced or t r a n s m i t t e d in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage or retrieval system, without permission in writing from the publisher.

Printed in Hong Kong by Caritas Printing Training Centre.

F O R E W O R D B Y STEVEN N.S. C H E U N G Reading this book reminds me of that glad morning thirty years ago, when a coterie of young lions under Armen Alchian and Jack Hirshleifer was hard at work, laying the foundations of finance. Bill Sharpe invented CAPM (for which he was awarded the Nobel Prize the year before last), while I can still remember the thrill of discovering the first application of decision making under uncertainty to contracts, to solve the classic problem of sharecropping. I have always believed finance is a branch of economics. Now that our new School of Economics is looking to develop teaching and research in finance, I am delighted Michael Cheung and David Yeung have come up with a new and interesting piece of work in the field. Their book originated in a problem Milton Friedman and I have been mulling over for years, of how "virtual" transactions in foreign exchange futures affect spot prices. Cheung and Yeung propose to synthesise international monetary theory with the Samuelson-Black-Scholes insight that asset prices follow diffusion processes, and obtain a system of stochastic differential equations to model exchange rate dynamics under the influence of purchasing power parity. This analysis suggests a way in which my problem may be approached: contracts bought and sold in the futures market affect individual estimates of a currency's purchasing power parity. The authors present an exact formula to price options on a currency, which incorporates the influence of its purchasing power parity. Because they obtain a closed form expression for the exchange rate's transition density function, this result is operational. Ronald Coase and I have spent a goodly part of our time teaching that good economics must be operational economics, so I await with interest practical applications of Cheung and Yeung's model in the financial markets.

ACKNOWLEDGEMENTS The ideas underlying Chapters 7 and 8 originated in remarks made by Merton H. Miller. We are much indebted to Steven N.S. Cheung for writing the Foreword, to P.R. Chandrasekaran, Michael C.H. Chan, Raymond S Y . Cheung, the Rt. Hon. Lord Griffiths, Ziqi Liao, Kah Hwa Ng and Lars Werin for encouragement and comments, and to Irene M.L. Cheung for expert typing of the final text. Mistakes are, of course, the authors' responsibility. Financial support for the first edition of this book was generously supplied by the Centre of Urban Planning and Environmental Management (University of Hong Kong) and Cypress International Investment Advisors Ltd.

TABLE OF C O N T E N T S Page Chapter 1

Preamble

References Chapter 2 2.1 2.2 2.3 2.4

1 4

Definitions and Terminology

Calls and Puts Options Trading Hedging and Speculating with Options Foreign Currency Options

References

6 6 7 7 8 9

Chapter 3 Technical Glossary 3.1 Introduction 3.2 Stochastic Processes 3.3 Martingales 3.4 Markov Stochastic Processes 3.5 Random Walks 3.6 Brownian Motion 3.7 Geometric Brownian Motion 3.8 Formulae from Stochastic Calculus References

10 10 10 12 12 13 15 16 17 18

Chapter 4 References

19 23

Stochastic Assumptions and Option Pricing

Chapter 5 The Black-Scholes Options Theory 5.1 Introduction

24 24

5.2 5.3 5.4

24 25

The Geometric Brownian Motion Assumption The Black-Scholes Option Pricing Formula Kolmogorov's Backward Equation and the Transition Density Function of the Stock Price 5.5 Transition Density for Geometric Brownian Motion . . . . 5.6 Transition Density and Option Pricing 5.7 Static and Dynamic Assumptions in Option Pricing . . . .

28 29 32 35

Page 5.8 Conclusions References Appendix

37 37 40

Chapter 6

Geometric Brownian Motion, "Almost Certain Ruin", and Asset Markets Equalibrium in Options Pricing 6.1 Introduction 6.2 GBM Sample Path and Moments Behavior 6.3 Interpreting Asset Markets Equilibrium 6.4 Conclusions References Figures N o n Random Walk Effects and a New Stochastic Specification 7.1 Introduction 7.2 A New Stochastic Specification 7.3 Dynamics of Stock Price and Premium Rate of Return 7.4 An Exact Option Pricing Formula 7.5 Conclusion References Appendix

43 43 44 46 50 50 52

Chapter 7

Chapter 8

. . .

57 57 58 59 64 65 66 68

Pricing Foreign Exchange Options Incorporating Purchasing Power Parity 71 8.1 Introduction 71 8.2 Stochastic Dynamics of the Exchange Rate 73 8.3 An Exact Formula to Price Forex Options 78 8.4 How to Use the Exact Formula 79 8.5 Conclusion 80 References 80 Appendix 84

Page Chapter 9 Index

Conclusions

87

90

CHAPTER 1 PREAMBLE This book is a revised and re-written version of part of a study, sponsored (from 1992 onward) by Cypress International Investment Advisors Ltd. Its purpose is to describe a new approach to the valuation of options on foreign exchange. Though the core of the original text remains, comments, especially from professional practitioners, have led the authors to re-orientate the exposition. In particular, the present edition has been recast with applications very much in mind. The reason underlying this change in exposition would be clear if one remembers a bit of methodology. According to Friedman's well known view (1953), in positive economics assumptions do not matter. When one's purpose is to test a theory, what is important is that its predictions are not empirically falsified. However, in applied economics the situation is different, for in this case assumptions matter very much indeed. When one is applying a theory to study economic growth in Hong Kong and produce policy recommendations, it would not do if it assumes a closed economy or an infinitely elastic supply of land. The same may be said about option pricing, which is essentially an application of capital theory. In this case, as Cox & Ross (1976) have shown, what one assumes about the stochastic specifications governing the price of the underlying asset is of fundamental importance. The use of stochastic processes1 to model the price of assets was pioneered by Bachelier (1900). The idea was that in a continuous competitive market, the asset price would be subject to so many independent influences that we can imagine it to fluctuate randomly along a continuous path. As a result, Bachelier assumed the price of the representative asset to follow (what is now called) a Brownian motion. Brownian motion, however, allows asset prices to go negative. Since this would violate the condition of limited liability, the assumption cannot be used when applying the analysis to an equity. (It would also be difficult to 1

Technical terms are explained in Chapter 3 below.

1

Pricing Foreign Exchange Options apply to any good which has a positive price and therefore a market). This result led Sarnuelson (1965) to introduce the idea of geometric Brownian motion, in which the asset price is restricted to take positive values with nonzero probabilities. Since that time, geometric Brownian motion has become a paradigm for financial research. In the particular subject of options, the seminal work of Black & Scholes (1973) is based on the assumption that the representative stock price follows geometric Brownian motion. Recent research (to which the authors contributed 2 ) has uncovered a number of problems, which suggests that the scope of application of geometric Brownian motion is not as wide as first envisaged. Institutionally, it is clear that the assumption cannot be applied to the bond market, and to value fixed income options and the options embedded in callable bonds. Since geometric Brownian motion allows asset prices to go infinite with non-zero probability but every bond has a finite maximum price, as long as interest rates are non-negative it would be inappropriate to model the representative bond price in such terms (Dyer & Jacob 1996). In addition, there are serious problems in theory. For example, if an asset price follows geometric Brownian motion, it is possible for its sample path to drop to 0 with probability 1, and yet all the time the expected price of the asset would be increasing without limit. An individual who holds such an asset according to standard portfolio (mean-variance) critieria would be "almost certainly ruined", with a zero price the market would disappear, and options on the asset would yield distorted values. In this book we propose an alternative assumption to geometric Brownian motion, and show how it can be applied to perhaps the largest financial market in the world, that for foreign exchange. This new stochastic specification is free from the theoretical problems noted above. "Almost certain ruin" and the disappearance of markets are excluded for a representative foreign currency. In addition, the non-random effects of standard economic theory (in particular changes in purchasing power parity) can be incorporated, both in the description of the stochastic process for the spot price of the currency and in a new formula for pricing foreign exchange options. 2

See Cheung & Yeung (1994a) and (1994b). 2

Preamble Chapter 2 introduces definitions and terminology. To save on time spent looking up textbooks, a technical glossary is supplied in Chapter 3. Chapter 4 attempts to impress upon the reader the importance of assumptions in option theory, by presenting an example in which an option price is obtained without making stochastic assumptions at all, and inviting the reader to compare it with the classic Black-Scholes formula. Black and Scholes contribution (1973), which is fundamental to all modern work in options, is discussed in Chapters 5 and 6. Since (as noted above) the underlying asset price is assumed to follow geometric Brownian motion, two serious problems are seen to arise. First, the technique commonly used to solve Black-Scholes differential equations does not exclude "almost certain ruin", so that it is difficult to maintain the required general equilibrium interpretation of the resulting option prices. Secondly, under geometric Brownian motion the asset price displays the characteristics of a random walk, in the sense that its value at any future point of time depends solely on what it is at present. This property is beginning to be called into question by recent research. (See e.g. McQueen & Thorley 1991, Samuelson 1991, Kaehler & Kugler eds. 1994, Haugen 1995, Malkiel 1996, Campbell, Lo & Mackinlay 1997). For example, it is suggested that returns to U.S. common stocks in the post-war period show statistically significant nonrandom walk behavior, especially that runs of high and low returns have been found to follow one another. To meet the problems which arise from assuming geometric Brownian motion, we propose (in Chapter 7) an alternative stochastic process to model the dynamic behavior of asset prices. The solution of the resulting stochastic differential equation is characterised completely, in the form of a closed form expression for the asset price's transition density function. It is shown that "almost certain ruin" is excluded, and non-random walk effects from standard economic theory — for example, the changes in the representative firm's equilibrium balance sheet which underlie the ModiglianiMiller Theorem — can be taken into account. An option pricing formula is also obtained, by taking mathematical expectation in terms of the asset price transition density. Chapter 8 shows how the stochastic specification of Chapter 7 can be applied to model the spot price of a representative foreign 3

Pricing Foreign Exchange Options currency, and to price options on it. In contrast to the random walk restriction imposed by geometric Brownian motion, we are able to incorporate a fundamental theorem of international finance, that in the long run the exchange rate converges to purchasing power parity. "Almost certain ruin", the disappearance of markets and their consequences are excluded, and finally a computable closed-form formula to price foreign exchange options is obtained. References Bachelier, L. (1900). Theorie de la Speculation, Annales de I'Ecole Normale Superieure 17, 21-86. Translated as The Theory of Speculation in P.H. Cootner ed., The Random Character of Stock Market Prices, Cambridge MA: MIT Press 1967. Black, F. k M. Scholes (1973). The Pricing of Options and Corporate Liabilities, Journal of Political Economy 81, 637-654. Campbell, J.Y., A.W. Lo & A.C. Mackinlay (1997). The Econometrics of Financial Markets, Princeton NJ: Princeton University Press. Cheung, M.T. & D.W.K. Yeung (1994a). A Non-Random Walk Theory of Exchange Rate Dynamics with Applications to Option Pricing, Stochastic Analysis and Applications 12, 141-159. Cheung, M.T. & D.W.K. Yeung (1994b). Divergence Between Sample Path and Moments Behavior: an Issue in the Application of Geometric Brownian Motion to Finance, Stochastic Analysis and Applications 12, 277-291. Cox, J.C. & S.A. Ross (1976). The Valuation of Options for Alternative Stochastic Processes, Journal of Financial Economics 3, 145-166. Dyer, L.J. & D.P. Jacob (1996). An Overview of Fixed Income Option Models. In F.J. Fabozzi ed., Handbook of Fixed Income Options. Chicago: Irwin Professional Publishing. Friedman, M. (1953). The Methodology of Positive Economics. In Essays in Positive Economics. Chicago: University of Chicago Press. Haugen, R.A. (1995). The New Finance: the Case Against Efficient Mar4

Preamble kets, Englewood Cliffs NJ: Prentice Hall Kaehler, J. & P. Kugler eds. (1994). Econometric Analysis of Financial Markets, Heidelberg: Physica-Verlag. Malkiel, B.G. (1996). A Random Walk Down Wall Street 6th ed., New York: Norton & Co. McQueen, G. & S. Thorley (1991). Are Stock Returns Predictable? A Test Using Markov Chains, Journal of Finance 46, 239-263. Samuelson, P.A. (1965). The Rational Theory of Warrant Pricing (with Appendix by H.P. McKean), Industrial Management Review 6, 13-31. Reprinted in R.C. Merton ed., The Collected Scientific Papers of P.A. Samuelson Vol 3, Cambridge MA: MIT Press 1972. Samuelson, P.A. (1991). Long Run Risk Tolerance when Equity Returns are Mean Regressing: Psuedoparadoxes and Vindication of "Businessmans Risk". In W.C. Brainard, W.D. Nordhaus k H.W. Watts eds., Money, Macroeconomics and Economic Policy: Essays in Honor of James Tobin, Cambridge MA: MIT Press 1991.

5

CHAPTER 2 DEFINITIONS A N D TERMINOLOGY CaUs and P u t s 1

2.1

Very briefly, let us summarise the definitions and terminology that will be used in the following chapters. There are two basic types of options: call options (or simply calls), and put options (or puts). A call option on an asset gives the owner the right to buy the asset on or before a certain date at a certain price. If the right is exercisable only on that date, the option is "European": otherwise it is an "American" option. A put option gives the owner the right to sell an asset on or before a certain date. Puts can also be European or American in nature. The price specified in the option contract is known as the exercise price or strike price. The date specified in the contract is the exercise date, expiry (expiration) date, or the maturity date of the option. There are four types of participants in options markets: buyers of calls; sellers of calls; buyers of puts; sellers of puts. Buyers are referred to as having long positions and sellers short positions. Selling an option is also known as writing it. An option confers upon the holder the right but not the obligation to buy or sell an asset at a certain price on or before a certain date. This fact distinguishes options from futures. For example, the holder of a long futures contract commits himself to buying an asset at a certain price at a certain date from now. In contrast, the holder of a European call option has a right to buy an asset at a certain price at a certain date in the future, but he is not contractually bound to exercise this right. 1

A classic exposition of the properties, structure and trading procedures of options and option markets is Cox & Rubinstein (1985). An excellent recent survey can be found in Hull (1997). 6

Definitions and Terminology 2.2

Options Trading

To illustrate the above concepts, suppose an investor instructs his broker to buy one European call option on Hongkong Bank stock with an exercise price of $100 and an exercise date of 31 July for $10, and the firm's trader is able to find another trader on the floor willing to sell a 31 July call option on Hongkong Bank stock with an exercise price of $100, for a price of $10. The most important piece of information in the contract is the price the buyer is willing to pay for the call, and the price at which the seller is willing to write the call. Only if the two are the same would the deal be possible. (On the other hand, the price of the stock at the time of the deal does not have to be equal to the exercise price: Hongkong Bank shares may be trading at $102, greater than the exercise price of $100.) The sixty-four dollar question is then: How much should an individual pay for the option? How much should another individual sell it for? Or, what is the value of an option with a given exercise price and maturity? 2.3

Hedging and Speculating with Options

If the above question can be answered, then myriad possibilties become open to investors. For example, options can be used for hedging. Consider an individual who owns (it is now April) 500 Hongkong Bank shares, the current price of which is $102. He is worried that the share price may decline sharply in the next two months. To hedge against this possibility, the individual can buy 500 30 June American put options, to sell Hong Kong Bank shares for an exercise price of $100 (per share). If the price of the option is $10, the contract would cost 500 x $10 = $5000. The hedging strategy costs $5000, but in return it guarentees that the individual would be able to sell Hongkong Bank shares at $100 during the life of the option (from now to the end of June). If the price of Hongkong Bank shares does fall to below $100, the options are exercised and 500 x$100 = $50000 realised on the portfolio. Taking the cost of the hedging option into account, the sum realised becomes $45000. If the price of Hongkong Bank shares stays above $100 the individual would not exercise the options, and allow them to expire worthless. However, in this case his portfolio would be worth more than $50000 (or $45000 when the cost of hedging is taken into account). 7

Pricing Foreign Exchange Options Next, consider how a speculator can make use of options. An individual has $9000 and expects Hongkong Bank shares to rise at the end of July, from its current price of $90. Suppose 31 July European call options for one Hongkong Bank share with exercise price $95 costs $3, so that the speculator is able to buy 3000 of them. If his hunch proves to be correct and the price of Hong Kong Bank shares rises to $100 by the end of July, then exercising the options would allow an asset worth $100 to be bought for $95. Net of cost, this would yield a profit of (3000 x $5 =)$15000 - $9000 = $6000. If the individual had bought 100 shares outright with $9000, his gains would have been $1000: it is seen that the options alternative is six times more profitable (if things turn out as expected). 2.4

Foreign Currency Options

The Philadelphia Exchange began trading in foreign currency options in 1982. The size of the market has grown very rapidly, so that by 1990 European and American options are available for the Australian dollar, British pound, Canadian dollar, German mark, Japanese yen, French franc, Swiss franc, and the European Currency Unit. A large volume of trading in foreign currency options takes place outside the organised exchanges. Banks and financial institutions are often prepared to buy or sell foreign currency options with exercise prices and exercise dates tailored to meet the needs of clients (especially multi-national corporations). For an individual or corporation wishing to hedge against a foreign exchange exposure, currency options provide a more flexible alternative to forward contracts. For example, suppose a company is due to receive £1000 at a given date in the future. It can hedge the risk by buying European put options on sterling, to mature on that date. This guarentees that the value of each £ in the sum (in terms of U.S. dollars) will not be less than the exercise price, and also allows the company to benefit from favourable movements in the exchange rate. Similarly, a company due to pay £1000 at a given time in the future can hedge the risk by buying European call options on sterling, to mature on that date. This guarentees that the cost of each £ in the sum (in terms of U.S. dollars) will not exceed a certain amount, while allowing the company to benefit from favourable movements in the exchange rate. 8

Definitions and Terminology Compared to forward contract, which locks in the exchange rate for a future transaction, a currency option represents a flexible form of insurance. The price of the option can be looked upon as the cost of this insurance. References Cox, J.C. & M. Rubinstein (1985). Options Markets, Englewood Cliffs, NJ: Prentice Hall. Hull, J.C. (1997). Options, Futures, and Other Derivatives 3rd. ed., Englewood Cliffs NJ: Prentice Hall.

CHAPTER 3 TECHNICAL GLOSSARY Introduction 1

3.1

The technical terms and results which will be used in the exposition are summarised in this Chapter. For details and proofs, the reader is referred to any good text on stochastic processes, e.g., Karlin & Taylor (1975, 1981). 3.2

Stochastic Processes

Let (Q, A, P) be a probability space, and T an arbitrary set of numbers. Suppose we define the function: X{tyuj),

t GT, w E f i .

(3.1)

A stochastic process is a family {X(t,w)} of such functions. For any given t G T, X{ty •) denotes a random variable (or a random vector) on the probability space (fl,.4,P). For any fixed u G fi, X(-yu) is a real valued function (vector valued function) defined on T, called a sample path or realisation of the stochastic process. The standard notation suppresses the variable u>, so that a stochastic process is written {X(t)}. The theory of stochastic processes is concerned with the structure and properties of {X(t,u)} under different assumptions. The main elements which distinguish stochastic processes are the state space 5, which is the set of values the random variable X(t, •) may take, the index set X1, and the dependence relationships among the random variables X(t, •). If the state space 5 = {0,1,2, • • • } , the stochastic process {X(t)} is described as integer valued. If S is the real line (—00,00), {X(t)} is a real valued stochastic process. If S is a fc dimensional Euclidean space, {X(t)} is a k-vector stochastic process. If the index set T = {0,1,2, • • •}, {X(t)} is a discrete stochastic process. If T = (—00,00), the stochastic process {X(t)} is continuous. Often, the variable t is interpreted to be time. Then, if T = (—00,00), {X(t)} would be a continuous time stochastic process. 1

The notation is special to this Chapter.

10

Technical Glossary Different dependence relationships among the X(t) give rise to different stochastic processes. If we wish to characterise the stochastic process {X(t)}} this requires the knowledge of (countably or uncountably many, depending on the nature of T) joint distributions of the random variables (or random vectors) X(t). The set of all such joint distributions: Fxiuixfa),"

)x(tn)(xi,x2i'"

,Xn)

= [ P r X f a ) < * i , X(t2) < x2y • • • , X(tn)

< xn],

for all ti,t2, • * * >*n £ T} ti ^ tj, i / j , constitutes the probability law of the stochastic process. Generally, the random variables X(t) are interdependent. If, for all choices of t\, • • • , tn, t{ G T such that *i < t2 < • • • < tn the random variables X(t2) - X(h),

X(t3) - X(t2), • • • , X(t„) - X ( t „ - i )

are independent, then {X(t)} is a stochastic process with independent increments. If the index set T contains a smallest element to, it is assumed that the random variables X(to)y X(t\) — X(£o), • • • , X(tn) — X(tn^i) are independent. If the index set T = {0,1,2,•••}, then a stochastic process with independent increments reduces to a sequences of independent random variables Z(0) = X(0), Z(i) = X(i)-X(i-l), 2 = 1,2,.-. ,n. If (for any t) the distribution of the random variables X(t + h) — X(t) depends only on the length h of the interval and not on t, the stochastic process {X(t)} is said to possess stationary increments. Given a stochastic process with stationary increments, the distribution of X(t\ + h) — X(t\) is the same as the distribution of X(t2 -f h) — X(t2), no matter what the values of
{X(t1),X(ti),--',X(tn)}

Pricing Foreign Exchange Options are the same for all h > 0 and all choices of {ti,t2j • • • ,tn} from T. This condition says that the stochastic process is in probabilistic equilibrium, so that the specific instances at which we examine the process are irrelevant. In particular, the distribution of X(t) is the same for each t ET. If the stochastic process {X(t)} possesses finite second moments and if cov[X(t), X(t + h)] depends only on h for all t G T, it is said to be wide sense stationary. A stationary stochastic process with finite second moments is wide sense stationary, but there are wide sense stationary stochastic processes which are not stationary. In economics, stationary stochastic processes are frequently used in rational expectations models, to characterise stochastic equilibrium (in the macroeconomic sense). 3.3

Martingales

Let {X(t)} be a real valued stochastic process with a discrete parameter set T. Then it is a martingale if:

(a) #[|x(t)|]
tf[X(tn+i)|X(ti) = au X(t2) = a 2 , .. , X(tn) U
= a„] = a n> for any

More generally, if {X(t)} and {Y(t)} are stochastic processes with the discrete parameter set T, X{(t)} is a martingale with respect to {Y(t)}, if: (a) £[|X(t)|] < 00, V t G T , (b) £7[X(t n + i)|y(*i) = bly Y(t2) = 6 2 ,..- , Y(tn) h
<Wi.

= bn] = 6 nj for any

UeT.

Martingales are considered to be appropriate models for fair games, in which the random variable X(t) represents the amount of money a player possesses at time t. The martingale property states that the average amount the player would have at time tfn+i, given that he has an at time tn, is equal to an regardless of what his past fortune has been. 3.4

Markov Stochastic Processes A Markov stochastic process has the property that, given the value of 12

Technical Glossary X(s)y the values of X(t)y t > sy do not depend on the values of X(u), u < s. That is, the probability of any particular future behavior of the process, when its present state is known exactly, is not changed by knowledge about its past behavior. More formally, if for any h < t2 < •..

Pi[a < X(t) < b\X(h) = xu X(t2) = * 2 l • • • , X(tn) = Pr[a < X(t)
= xn]

= xn]y

is a Markov stochastic process.

Suppose T — (—00,00), and A = (a, 6] is an interval of the real line. The function P{xys\tyA)

= Pr[X(*) G A\X(s) = x]y

t>sy

is called the transition probability function of the Markov stochastic process {X(t)}. In particular, it can be proved that the probability distribution of the set of random variables {X(ti)yX(t2)y • • • yX(tn)} (the probability law of the stochastic process) can be found in terms of the transition probability function of the process and the initial distribution function of X(t). A Markov stochastic process {X(t)} with a finite or count ably infinite state space S = {0,1,2, • • • , n} or S = {0,1,2, • • • } is called a Markov chain. A Markov stochastic process {X(t)} for which all sample functions {X(tyw)} t G T = (—oo,0]} are continuous in t is called a diffusion process. Under certain conditions, the transition probability function P(xys;ty •) has a transition density function p(xys]ty-). A Markov process is said to possess stationary transition probabilities if the transition probability function P(xys]ty-) (and the transition density function p(xys]t,-)y if it exists) is a function only of (t — s). Notice that a stochastic process with stationary transition probabilities is not necessarily stationary. 3.5

R a n d o m Walks

A discrete time Markov chain is a Markov stochastic process with a finite or countably infinite state space S = {0,1,2, • • • }, and index set T = 13

Pricing Foreign Exchange Options { 0 , 1 , 2 , . . - } . It is common to write a Markov chain as {X(n)} instead of {X(t)}y and to say that Xn is in state i if Xn = i.

or

{Xn}

The probability of Xn+i being in state j , given that Xn is in state i, called a one-step transition probability, is denoted by: J # n + 1 = P r ( X B + 1 = j | X B = i). The notation emphasises that in general, the transition probabilities of the Markov chain depend on the initial state i and final state j t and on the time interval over which the transition occurs (n,n + 1). If one-step transition probabilities are independent of the time of transition (n), then (as we have seen) the Markov chain possess stationary transition probabilities. In this case, pn,n+l

p

Uj

r

— ij •

Since P{j is a probability, oo

Pij>0> i,j = 0 , 1 , 2 , . . . ,

£ i ^ = l, 2 = 0,1,2,..i=o (The summation condition expresses the fact that some transition occurs in each step., or each trial, of the process.) It can be shown that the Markov chain is completely determined once P{j is known for all i and jy and the probability distribution of XQ is specified. A random walk is a Markov chain in which Xn, if it is in state iy can in a single transition either remain in state i, or move to one of the adjacent states i -f 1 or * — 1. In this case:

Pi(Xn+1=i+l\Xn

= i)=pi,

Pl(Xn+l=i-l\Xn=i)=qiy Pr(X n + i = i\Xn = i) = riy where pi > 0, qi > 0, rt- > 0, pi + qi + n = 1, i = 1,2, • • • , p0 > 0, r 0 > 0, po + ^o = 1- K Pi =

0 and rt- = r > 0, the random walk is symmetric. 14

Technical Glossary The fortune of an individual in a game is often depicted by a random walk stochastic process. Suppose the player has fortune k and plays a game against an (infinitely rich) adversary, with the probability pk of winning one dollar and probability q^ = 1 — Pk of losing one dollar in each trial, and ro = 1. If the random variable Xn represents the individual's fortune after n trials, the stochastic process {Xn} is a random walk, known as gambler's ruin. (Once state 0 is reached, the process will remain in it.)

3.6

Brownian Motion

The study of Brownian motion began with the observation by the Scottish botanist R. Brown in 1827, that small particles like pollen grains immersed in a liquid exhibit ceaseless irregular motions. In 1905, Einstein explained this phenomenon by a theory in which the particles under observation are subject to perpetual collisions with the molecules of the surrounding medium. Einstein's results were later extended by various physicists and mathematicians, for example N. Wiener and S. Chandrasekhar. (Brownian motion is also known as a Wiener stochastic process.) At time t G T = (—oo,0], let X(t) denote the displacement (from a starting point along a fixed axis) of a Brownian particle. The displacement X(t) — X(s) over the time interval (syt) can be regarded as the sum of a large number of small displacements. The central limit theorem is then applicable, so we can assert that the random variable X(t)—X(s) is normally distributed. It is intuitively clear that the displacement X(t)—X(s) depends only on (t — s) and not on the time we begin the observation. Moreover, it is reasonable to assume that the Brownian motion is in stochastic equilibrium, in the sense that the distribution of X(t -f h) — X(s -f h) is the same as the distribution of X(t) - X(s)y for all h > 0. Given these observations, the Brownian motion stochastic process {X(t)y t > 0} possesses the following characteristics: (a) Given to < t\ • • • < tn, the increments X(t\) — X(tQ)y X(tn_i) are (mutually) independent random variables; (b) the probability distribution function of X(t) — X(s)y 15

• • • , X(tn) —

t > sy depends

Pricing Foreign Exchange Options only on (t — s) and not on f or 5; (c) ?i[X(t)-X(s) <x]= l/ s and cr is a positive constant. Assume that for each sample path of the process, X(0) = 0. It can then be proved that, conditional upon X(0) = 0, £[*(*)] = 0,

var[X(*)] = (T2ty

and that for 0 < ti < t2 < • • • < tn < ty the conditional probability distribution of X(t) given X(ti)y X(t2)y • - • , X(tfn) is Pr[X(t) < *|X(*i) = * i , X(t 2 ) = x2y • • • , X(tn) = xn]

^pTr^-^^y.oo

eXP

l

2
U

'

A discrete approximation to Brownian motion is provided by a symmetric random walk. 3.7

G e o m e t r i c Brownian Motion

Let {X{i)y t G [0, 00)} be a Brownian motion stochastic process. Brownian motion with drift is a stochastic process {U(t), t G [0,00)}, where U(t) = X(t) + iit, and the drift parameter fj, is a constant. Alternatively, we can define a Brownian motion with drift to be a stochastic process {U(t)y t G [0,oo)} with the properties: (a) the increments X(t -f s) — X(s) are normally distributed with mean /i and variance a2ty where fi and cr > 0 are constants; (b) for every tx < t2 < • • • < tny the increments X(t2)-X(ti)y X(tn-\)

• •• yX(tn) —

are independent random variables with distributions given in

(a); (c) for every sample path, X(0) = 0, and X(tyu>) is continuous at t = 0. 16

Technical Glossary Let {U(t)y t G [0,oo)} be a Brownian motion stochastic process with drift coefficient \i. The stochastic process defined by: Y(t) = exp[U(t)]y

*>0,

is called geometric Brownian motion. (The state space of the process is (0,oo).) It can be shown that the random variable Y(t) is lognormally distributed with mean and variance: E[Y(t)\Y(0)

1 2 = YQ] = Y0exp fit + -
and

vaz[Y(t)\Y(®) = YQ] = Y02 [exp(2/i* + a2t)] [exp(
3.8

Formulae from Stochastic Calculus

Consider a probability space (Q, Ay P) and a stochastic process {X(ty CJ)}, t G [0,T]. Let tr(t,cj) be a non-anticipating function and f(tyw) be another function (both defined by properties which we can take for granted here). Then the stochastic process {X(tyu)} or {X(t)} has a stochastic differential denoted by: dX(t) = f(t)dt + a(t)dz(t)y where (see §3.6 above) {dz(t)} is a Wiener (Brownian motion) stochastic process with E[dz(t)] = 0, va,i[dz(t)] = dt. Let u[tyX) be a continuous non-random function with continuous partial derivatives. Then ltd's lemma states: if the stochastic process {Y(ty u)} or {y(*)} is such that we have Y(t) = u(tyX(t))y it also possesses a stochastic differential given by: dY(t) =

du . du „,,x . 1 ,, x2 d 2 u 1

9f

+ 9^/(*)+^w I'W 2 WSI2 17

dt +

^a(t)dz(t).

Pricing Foreign Exchange Options As we will see below, Ito's lemma is fundamental to Black and ScholesJ theory of option pricing. References Karlin, S. & H.M. Taylor (1975). A First Course in Stochastic Processes, 2nd ed., New York: Academic Press. Karlin, S. & H.M. Taylor (1980). A Second Course in Stochastic Processes, New York: Academic Press.

18

CHAPTER 4 STOCHASTIC A S S U M P T I O N S A N D OPTION P R I C I N G

In Chapter 1 we suggested that, since option pricing is applied economics, the choice of assumptions is a matter of primary importance. In particular, we referred to the view of Cox h Ross (1976), that if a different assumption is introduced regarding the stochastic behavior of an asset price, generally a different formula to price options on the asset would follow. The objective of present chapter is to convince the reader of the truth of this observation. We present an example, in which a call option on an equity is priced without any stochastic assumptions about the behavior of the equity price. The reader is then invited to compare (in the next chapter) this expression with the classic Black- Scholes formula for pricing the same option, in which the equity price is assumed to follow a Geometric Brownian Motion. Following Cox, Ross & Rubinstein (1979), suppose that over a single period of time (say, one "day") the price of a representative equity can change in one of two ways. From the current level 5, it can either go up to hSy or go down to kS. No probabilities are introduced, and no restrictions apart from the fact that h > 1 and 0 < k < 1 (and later on, that h > 1 + r, where r is the one period risk-free rate of interest). Consider a (European) call option on the equity in the current period, which will mature in n days. What would be its equilibrium price C(Sy n)? Imagine a portfolio which is short one call option of this type, and long N units of the underlying stock. It is called a "hedge portfolio", for reasons which will be clear later. At current market prices, the hedge portfolio is worth: NS-C(Syn)

(1)

In one day's time, the stock price becomes either hS or kSy and with n — 1 days remaining to maturity the price of the call option would be C(hSy n— 1) 19

Pricing Foreign Exchange Options or C(kS, n — 1). The value of the hedge portfolio at this time is then: NhS-C(hS,n-l)

(2.1)

or (2.2)

NkS - C(kS, n-1)

Since N is left open by construction, let us choose it such that these two values are equal. We then have: NhS - C(hS, n - 1) = NkS - C(kS,

n-1)

which gives: N =

C(hS,n-1)

C(kS,n-1) (h - k)S

(3)

The value of the hedge portfolio one period hence is then: NhS-C(hS,n-l)

=

'C(hS,n-l)-C(kS,n-l) (h - k)S

hS-C(hS,n-l)

kC(hS,n - 1) - hC(kS,n - 1) h-k

(4)

No matter what the stock price is, hS or kSy the value of the portfolio would be the same. Though there is no explicit uncertainty in our example, the portfolio plays the part of a hedge against changes in the stock price. Since (4) represents a certain sum of money, to prevent arbitrage profits the current value of the hedge portfolio must be equal to its value one period hence, discounted at the (given) one period risk-free rate of interest r. Therefore: NS - C(5, n) =

C(hS1n-l)-C(kSyn-l) (h - k)S

-(£)[

S - C(5, n)

kC(hS,n - 1) - hC(kS,n - 1) h-k

(5)

This yields:

c(s n)

> = (rb) H^c^

n

-^+"-i^k1^n ~ i

(6)

20

Stochastic Assumptions and Option Pricing (6) is then the current price of the call option, when it has n days remaining to maturity, in terms of its price tomorrow, when it has n—l days remaining to maturity. Since there are no arbitrage opportunities, it is an equilibrium price. As we have seen in Chapter 2, on the day the call option expires it would be worth C(Sy 0) = max(S — K, 0), where the stock price for the current period (maturity day) is S and K is the exercise price. Using (6) and remembering that the stock price for the current period is denoted by 5 , we obtain the price of the option one day before maturity as: C(S,1)

1 + r

[l + rj

k

h—k

'

C(hS,0)+hu\rC(kS,0) h—k

[eC(hSy 0) + (1 - 0)C(kS, 0)].

(7)

where (since at maturity one day hence the stock price would either go up from S to hS or go down from S to kS): C(AS,0) =

max(hS-K,Q)t

C(kSy0)=

jmx(kS-K,0)t 1+ r —k 0= — —, 0 < 0 < 1. h—k

and

Similarly, the price of the option two days before maturity is:

C(5}2) =

( r b ) [9C{hS>l)

+ (1

" 9)C{kS>1)]'

C(hSy 1) = (^^j

[0C(h2SyO) + (1 - 0)C(hkSy 0)] ,

C(kSy 1) = (jl-A

[0C(hkSy 0) + (1 - 0)C(k2Sy 0)]

(8)

where if S is today's stock price, the price tomorrow would be either hS or kSy and the price the day after tomorrow (when the option expires) would be h2Sy hkS or k2S. 21

Pricing Foreign Exchange Options We can re-write (8) as: C(Sy 2) = (

J-^-J;)

[02C(h2Sy 0) + 20(1 - 0)C(hkSy 0) +(l-0)2C(k2S,O)]

(9)

where:

C(/i 2 S, 0) = max(h2S - KyQ)y C(/ifc5,0) = max(hkS - Ky 0), C(ib2S, 0) = max(Jb2S - Ky 0) Proceeding recursively, we therefore obtain:

C n)

^ = (ihT [Zl^^-^i^-'S-K)}

(10)

where I is the set of indices i for which max(hikn — z'S — K, 0) > 0. (10) is then a formula for pricing the call option on the stock in the current period, when there are n days remaining to maturity. It is clear that in derving (10), we have not introduced any stochastic assumptions about the behavior of the stock price over time. In particular, neither hy k nor 6 is a probability. However, since (10) resembles the expected value of a binomial random variable with "probability of success" 0y one can interpret it as the expected value of a random stock price. However, as Cox, Ross & Rubinstein (op. cit.) pointed out, in this case 0 is merely an "artificial" probability, and it does not represent the true probability that the stock price would change from S today to hS tomorrow. In the next chapter we will see how Black and Scholes (1973), applying the same reasoning (with a hedge portfolio and the requirement of no arbitrage opportunities) but assuming that the stock price follows a Geometric Brownian Motion, obtained a completely different formula to price the same call option. As noted above, if one introduces a different stochastic assumption regarding the behavior of the underlying asset price, a different option pricing formula emerges. 22

Stochastic Assumptions and Option Pricing References Black, F. & M. Scholes (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy 81, 637-54. Cox, J.C. & S.A. Ross (1976). The Valuation of Options for Alternative Stochastic Processes. Journal of Financial Economics 3, 145-66. Cox, J . C , S.A. Ross k, M. Rubinstein (1979). Option Pricing: a Simplified Approach. Journal of Financial Economics 7, 383-402.

CHAPTER 5 T H E BLACK-SCHOLES OPTIONS THEORY 5.1

Introduction

As we have seen in Chapter 4, the question is: what is the equilibrium price of an option, given its nature (whether it is American or European), exercise price, exercise date, the current price of the underlying asset, the discount rate, and given an assumption regarding the behavior of the asset price over time? A major breakthrough was achieved when Black &; Scholes (1973) obtained a closed-form expression to price an European option on a stock which does not pay dividends, assuming that its price follows a Geometric Brownian Motion 1 . 5.2

The Geometric Brownian Motion Assumption

In their seminal work (1973), Black and Scholes adopted the assumption, that the price of the representative stock follows a Geometric Brownian Motion. As we have seen in §3.6 above, the standard discrete approximation to Brownian Motion is the symmetrical random walk. In the case of Geometric Brownian Motion is the symmetric random walk (see §3.6 above). In the case of geometric Brownian motion, this means that over a small length of time, proportional changes in the price of a stock are normally distributed. It then follows that at any future point of time, the stock price is lognormally distributed (see §3.7 above). Writing S(t)y t > 0 for the price of the stock (a random variable) and t = 0 for current time, we can write its probability density function as: (SQ)tyS) = - — 7 = = = e x p

{logS- [log Sp + (jt -
-

(5-1)

where fi and a > 0 are constants.

1

Though the analysis has been extended to take dividends into account and a number of other assumptions have been relaxed, for ease of exposition we will centrate on the classic simplicity of the original formulation. 24

The Black-Scholes Options Theory 5.3

The Black-Scholes Option Price Formula

In addition to the assumption that stock prices follow geometric Brownian motion, Black Sz Scholes assume that: (a) trading in asset markets is continuous; (b) transaction costs and taxes are zero; (c) assets are perfectly divisible; (d) dividends are not paid on the stock; (e) riskless arbitrate opportunities are absent; (f) the risk free rate of interest r is given and constant. (As we have noted above, many of these assumptions have been relaxed by later researchers.) Black and Scholes demonstrate that it is possible to create a riskless hedge by forming a portfolio which contains units of the stock and European call options on it. At any point of time, since the quantities of assets in the portfolio are fixed, the sources of change in its value must be the prices. If the call price is a function of the stock price and its time to maturity, then changes in the call price can be expressed as a function of changes in the stock price and changes in the time to maturity. Black and Scholes then observe that at any point of time, the portfolio can be made into a riskless hedge by choosing an appropriate mixture of the stock and European calls: e.g., if the hedge portfolio is established with a long position in the stock and a short position on the option and if the stock price rises, the increase in value from the profit gained by the long position would be offset by the loss on the short position. If the quantities of the stock and the option in the hedge portfolio are continuously adjusted in the appropriate manner as asset prices change over time, then it becomes riskless and hence would earn the risk free rate of interest. Since the prices of stocks and options are random variables, the value of the hedge portfolio is a random variable, which we may write as: VH(t)=S(t)Q8+C(t)Qc, 25

(5.2)

Pricing Foreign Exchange Options where S(t) is the price of the stock, C(t) the price of the option, Qs the quantity of stocks and Qc the quantity of European calls in the hedge portfolio. The change in the value of the hedge portfolio is then: dVH(t) = QsdS(t) + QsdC(t).

(5.3)

By assumption the stock price S(t) follows geometric Brownian motion, so that its dynamic behavior can be described by the stochastic differential equation (§3.8 above): dS(t) = fiS(t)dt + aS(t)dz(t)y where {dz(t)}

(5.4)

is a Wiener process with E[dz(t)] = 0 and var[dz(2)] =

dty fiS(t) the (multiplicative) drift coefficient, and
applying Ito's lemma

03.8): Substituting (5.5) into (5.4), we have: dVH = Q,dS + Qc

dCJa

ld2C

dCJ±

-_-.•

(5.6)

For arbitrary quantities of the stock and the option, dVn is stochastic, but if the quantities Qs and Qc are chosen so that Q.dS

+

Q™dS

=0

or

|

= - g ,

(5.7)

the hedge portfolio would become risk free. Setting Q5 = 1 and Qc = — l/{dC/dS) in (5.6), we then have:

dv

»-(licks) (w+1S°'s')«-

<5'8>

By the Modigliani-Miller theorem (1958), in equilibrium perfect substitutes must earn the same rate of return. Therefore, since the hedge portfolio is riskless,

If = rdt26

(59)

The Black-Scholes Options Theory Substituting (5.8) into (5.9), we obtain dC

_

adC

ld2C

2a2

,c,n.

subject to the boundary condition that at the exercise date T, the option would either be worthless or it would have a value equal to the difference between the stock price at that date and the exercise price K: C(T) = max[0, S(T) - K\.

(5.11)

The differential equation (5.10) and (5.11) must be solved to give the equilibrium price of an European option on the stock, given the exercise price K and exercise date T. Black k, Scholes (1973) obtain a solution by transforming (5.10) into a heat transfer equation. However, as Merton (1989) noted: "Apart from some notable exceptions, [Black-Scholes] type differential equations cannot be solved analytically for a closed form solution". An alternative approach is offered in a remarkable paper by Cox Sz Ross (1975)? To solve (5.10) subject to (5.11), they make two observations. First, whatever is the solution to (5.10), it would be a function only of the variables {r, 5, T, a2 and K}. Second, in constructing the Black-Scholes hedge portfolio, only one assumption is made concerning the preferences of individuals in the market, that in equilibrium, perfect substitutes must earn the same rate of return. In particular, no assumption involving individuals' attitudes towards risk is involved. This suggests that if a solution is found on the basis of one particular assumption about preferences, it must be the solution to (5.10) for any preference structure which permits asset markets equilibrium. It follows that (5.10) may be solved by adopting the most tractable assumption about preferences. To apply this technique, Cox and Ross choose the assumption that all individuals in the market are risk neutral. In this case, the equilibrium rate of return of all assets would be equal to the risk free rate of interest. In particular, the price of an European option with exercise price K and 2

See also Smith (1976), and IngersoE (1989). 27

Pricing Foreign Exchange Options exercise date T must be such that its current value is equal to its discounted expected value, or: C(i) = exp(-rT)E[C(T)]. (5.12) The solution then depends on /•CO

E[C(T)] = /

[5(T) - K]

f(S(T))dS(T)

JK

where f(S(T)) S(T). 5.4

is the probability density function of the terminal stock price

Kolmogorov's Backward Equation and the Transition Density Function of the Stock Price

Formula (5.12) can be applied to find the equilibrium price of an European option with exercise price K and exercise date T (remember the current date is t = 0), if the probability density function of the stock price at the exercise date T, 5(T), is known. The problem is then: given that the dynamics of S(t) is described by (5.4), what is the (transition) probability density function of the stock price stochastic process? In particular, is it the case that given any t, this density would have the lognormal form specified by (5.1)? The most comprehensive way to characterise a continuous time stochastic process is to derive its transition density function. As Kac wrote: "We may interpret the variation of the [transition] density function over time as a macroscopic description of many microscopic particles undergoing motions governed by a [continuous time] stochastic process. The FokkerPlanck equation describes these variations, [and] the solution to this differential equation determines the process" (see Iranpour &; Chacon 1988, §7.5). Up to the present time, it has been possible to explicitly solve the Fokker-Planck equation, or its dual the Kolmogorov backward equation, to derive transition density functions in only a few instances. 3 A closed form solution has yet to be obtained for the case of geometric Brownian 3 See Uhlenbeck & Omstein (1930), Chandrasekhar (1943), Wang & Uhlenbeck (1945), Bharucha-Reid (1960, and Gihman & Skorohod (1972). For an explicit solution to a n-dimensional system in steady state, see Yeung (1988).

28

The Black-Scholes Options Theory motion, and authors have studied its properties by means of a number of other approaches, in particular, by sample paths or moments methods. 4 As we have seen in §5.3, though Black &, Scholes (1973) assumed that the stock price underlying an option follows the GBM dynamics (5.4), they chose to transform (5.10) into a heat equation. This allowed a solution to be found (though as McKean (1965) has shown, there is a problem with the boundary condition (5.11)), without requiring one to know the stock price's transition density function. However, as Goldenberg noted in a recent contribution, "solving the partial differential equation arising from the [Black and Scholes] approach is as difficult as solving the Kolmogorov backward equation" (1991, p.6). The remainder of this Chapter derives from Cheung & Yeung (1992) and presents, we believe for the first time, a closed form solution of Kolmogorov's backward equation for geometric Brownian motion. Since "solving the backward equation solves the European option pricing problem" (Goldenberg 1991, p.6), the transition density function for the stock price is used to write the Black-Scholes option valuation equation directly in terms of expected values. This derivation is even more straightforward than the binomial random walk "simplified approach" offered by Cox, Ross & Rubinstein (1979). In addition, we show how it is possible to reconcile option valuation formulae based on GBM stock price dynamics, with those which follow early writings like Sprenkle (1964) and Boness (1964), in adopting the static assumption that the stock price is lognormally distributed. 5.5

Transition Density for Geometric Brownian Motion

Suppose the continuous time stochastic process {X(t)y t > 0} describes a geometric Brownian motion. The dynamic behavior of the random variable X(t)y t > 0, then follows the stochastic differential equation: dX(t) = aX(t)dt +
(5.13)

where {dz(t)} is a Wiener process with E[dz(t)] = 0, vai[dz(t)] = dty aX(t) the (multiplicative) drift coefficient, and a2X(t)2 the diffusion coefficient. 4

See e.g. Kozin (1972), Arnold (1974), and Gard (1988). 29

Pricing Foreign Exchange Options Given the initial condition X(0) = ^Oj let the stochastic (diffusion) process {X*(t)} denote the solution to (5.13). For t > 0, the transition density function of this stochastic process, (Xoy ty X) then satisfies the Kolmogorov backward equation (see e.g. Gihman & Skorohod 1972):

^{Xo,t,X) •£(X 0,t,X)

X00^(X = aaX ^:(X ,t,X) 0>0t,X)

++

^ X 0i,t,X), ^ -a'Xi^(X

t > 0. (5.14)

Up to the present time, a general closed form solution algorithm for (5.14) does not seem to be available. Formulae for the transition density functions for diffusion processes of the form dX(t) = ct{X{i))dt -f c{X{i))dz{t) may be found in Gihman and Skorohod's classic text (1972, p.91-113). In particular, one is required to solve explicitly for the function:

where g(X) is such that the integral equation JQ \/(r{u)du = X holds. Corresponding to the GBM process (5.13), we then have:

-I

gW

1 , logti
9(X) =

log g{X)-

logO

Since logO is not defined there is no solution for g(X) in (5.16), so that Gihman and Skorohod's technique is not applicable to geometric Brownian motion. To obtain a closed form solution to (5.14), let us introduce the transformation x(t) = \ogX(t). By Ito's lemma, (5.13) may then be written: dx(t) = (a - y J dt + crdz(t).

(5.17)

Given the initial condition x(0) = XQ = logXo, let the stochastic process {x*(t)} denote the solution to (5.17). For t > 0, its transition density function ip(xo,tyx) would then satisfy the Kolmogorov backward equation: dip, /
30

7

(5.18)

The Black-Scholes Options Theory Lemma. The transition density function which satisfies (5.18) has the form: ip(x0ytyx)

=

1 ayThd

{x-[x0

exp

+ (<x-
(5.19)

lo-H

Proof. Consider the standard Brownian motion {F(<)} with zero drift and unit variance. Its dynamical equation may be written: dY(t) = dz(t).

(5.20)

In this case, as e.g. Arnold (1974, Ch.2) has shown, the transition density function for the stochastic process {Y(t)}, 0(yo,t,y), t > 0, satisfies the Kolmogorov backward equation: 89 ( j / 0 2 / ) = id2e. & ''' 2^(2/0,i,y)-

(5.21)

Moreover, 0(yo,t, y) is unique and has the form of the Gauss kernel:

••'•»>=TM-T}

e{m

<«2>

As may be verified by straightforward substitution, it is possible to express the function ip(x0ytyx) in (5.19) in the form of a Gauss kernel, by replacing the arguments (yo,tyy) of the function 0(yoytyy) in (5.22) by (x0 + (or - cr2/2)ty
It must then be shown that (5.23) (and therefore (5.19)) satisfies (5.18). Differentiating (5.23) with respect to XQ and t: dip dxQ

(5.24)

where subscripts indicate partial differentiation. We then have:

/



i 2a2v>

/

31


i ,„

(5.25)

Pricing Foreign Exchange Options From (5.21), 02 = 0 n / 2 . Substituting into (5.25):

K)£^g=(-£)'-'*- <"•> As may be seen from (5.24), the RHS of (5.26) is dip/dt. Therefore, dip

(


1 2d2ip

/rnrt.

so that (5.23) or (5.19) indeed satisfies the Kolmogorov backward equation (5.18). The Lemma tells us the transition density function of the stochastic process {x*(t)} is given by (5.19). Since for any ty x(t) = log X{i) is a 1:1 mapping of X{i) G (0,oo) to x(t) E (—00,00), a standard variate transform allows us to write the transition density function for the solution stochastic process {X*(t)} to (5.13) as: |21

^'x'*o) = w^ e x p

{ l o g X - [ l o g X o + (a -
(5.28) The transition density function (5.28) is then the closed form solution to the Kolmogorov backward equation (5.14), for geometric Brownian motion. One may readily observe that given any ty the random variable X(t) is lognormally distributed with (compare §3.7): mean variance 5.6

E[X(t)] = {exp(logXo -\- at)},

and

var[X(t)] = {exp[2(logX0 + arf)][exp(<72*) - 1]}.

Transition Density and Option Pricing

Suppose the price of a (common) stock S(t) follows a geometric Brownian motion with the dynamics: dS(t) = aS(t)dt +
(5.29)

As we have seen, following the approach of Cox & Ross (1975) and Smith (1976), the transition density function (5.28) may be used directly, to derive Black and Scholes' valuation equation for an European option on the stock. 32

The Black-Scholes Options Theory If we look forward from the present time t = 0, the expected value of such an option at the expiry date T > 0 is: poo rOO

C(T)=

/

(S-K)
(5.30)

JK

where S 6 $}+) are realisations of the random variable S(T), So is the current price of the stock, and K the exercise price of the option. Substituting for the transition density <j>(So,T,S) from (5.28), (5.30) may be written: ,00

C(T)= V

'

JK

/

{ l o g g - [ l o g S o + (a-
J

.

exp

JK
exp

{log5-[log5o + (a-
dS

dS. (5.31)

Define k = log if, s = logS. Transforming the variable in (5.31) from S to s = log 5, we have:

C(T) =

,00

j

I ^

exp

f°° 1 exp V2irT Jk ~o^/2

{s-[S0

+

|2"

(a-a2/2)T]Y 2 2
sxp(s)ds

{ s - [ S o + (a-
(5.32)

Denote the constant [so + (a — a2/2)T] by \i. Then: 2
exp(s)ds (s-tf 2
ds.

Expanding (5.33), s 2 - 2/is + fi2
2a2Ts

( s - M ) 2 1 ds. 2
33

(5.33)

Pricing Foreign Exchange Options Upon re-arrangement, the first term on the RHS of this expression becomes: 1\ (s - fi -
exp

T;vb [B)

:



ds.

Since
/

f°°

1

\(

( . - / • - I T ' T2 T) \-2 "

1

cm = -»("+— J/, ^ w K - i (s - //) exp

-^r^ [(-0

r2T

C?S

21

2

(7 T

(5.34)

ds.

Transforming variables from s to ( = [s — (/i + a2T)]/(aVT)y £ = (s — fi)/(cry/T) in (5.34), we may write:

and

c m -«p(, + ^jT-£.«p(-i?)«

where a = [k - (/* + <7-2T)]/(<7Vr), b = (k- /i)/(aVT).

Substituting back

2

// = [so + (a -
#

- * f vfe exp B c2 ) dc -

(5 36)

-

With a standard normal distribution function N(u), l—N(u) = N(—u). Therefore, from (5.36): C(T) = So exp(aT) £

-KN

-j== exp (- ^

flog(S0/K)

+ (a -

rVT 34

^

a2l2)T )

(5.37)

The Black-Scholes Options Theory Discounting C(T) back T periods with the given rate of interest r, the current value of the (European) option is then: C(T,S0,K)

= S0eM-rT)eM«T)N

( * * * ' « > ^

+

T

^

)

^lK)+£-
-eM-rT)KN

(5.38) Introducing the risk neutral equilibrium condition a = r into (5.38), we obtain Black and Scholes' option price equation:

C(r,s„,W)

2

= S „ * ( ! ? « t ^' /2)T\ - e x p ( - r T ) * ^ V

5.7

°vT

J

(539)

Static and Dynamic A s s u m p t i o n s in Option Pricing

Explicit formulae to evaluate options (warrants) were first published in Sprenkle (1964) and Boness (1964). Instead of assuming the price of the underlying stock to follow the stochastic differential equation (5.24), these early papers begin with the assumption that at any point of time, it is a lognormally distributed random variable. The relationship between models based on static lognormal and dynamic GBM specifications has been explored in the famous survey article by Smith (1976, §3). Here, we apply the results of the previous Section to study one further question of this nature: viz., under what conditions is it possible to reconcile option valuation formulae based on the two different assumptions? 5 In Sprenkle's model, a warrant expires on date ty at which time the underlying stock price P is assumed to be lognormally distributed. If the conversion price is H, the expected value of the warrant is then: fOO

W = /

(P-H)f(v)dvy

(5.40)

JH 5

Using 'static' and 'dynamic' in the classic Samuelsonian sense that in a dynamic model, "time enters in an essential way" via difference and/or differential equations (1983). 35

Pricing Foreign Exchange Options where 1 / ( p ) =

^ ^

F e

x

p

. 2o l

11

- ^2 T3-(lo8p-A«p)

F>0,

is the lognormal density of the stock price P. On the assumption that investors anticipate the mean stock price at date t would be K times the current price PQ, we have: KP0 = i£(P) = exp f / i p + - ^ J ,

( - * ) •

such that

2

/i p = log VQ + log * - °^-.

(5.41)

As shown in the Appendix, a warrant valuation equation may then be obtained from (5.41):

w KP

= °L ^

exp

B"2)dr] - H L vfeexp B*2)dx

flogPo -logH

_

+ log/c +
AogPo^og*

+

log*-^/2\

^

where iV(-) is the standard normal distribution function. Due to the fact that Sprenkle's model is static, the expiry date t does not enter into (5.42). As Smith (1976) has shown, to tie up the static lognormal option pricing model with the dynamic GBM model, the first step is to find the present value of W:

_ eM.rt)HN

(togft-fagg + fagic-XM a \ " J

where (as in §5.3) r is the given rate of discount. 36

(5.43)

The Black-Scholes Options Theory In addition, consider the conditions: K

= exp (at),

2

a = (T2t.

(5.44)

(5.44) implies that the price of the stock is expected to grow at a constant rate a, and its variance (conditional upon its current price) changes over time. If (5.44) holds, it would be possible to introduce dynamic concepts into Sprenkle's equation. Writing P = 5, P0 = S 0 , E -Ky T = t in (5.43) so that the warrant becomes an European option, and substituting the risk neutral equilibrium condition a = r, the Black and Scholes valuation equation (5.39) obtains 5.8

Conclusions

This Chapter summarises the stochastic specification underlying the classic Black and Scholes option pricing model. We also solve the Kolmogorov backward equation for geometric Brownian motion, to find explicitly the transition density function for the price of the stock which underlies the representative option. The technique of Cox and Ross is then applied, to obtain the Black-Scholes option valuation equation, by directly taking expected values with the transition density function of the underlying stock price. In addition, we show how it is possible to reconcile option pricing formulae based on geometric Brownian motion dynamics, with those which derive from the static assumption that the stock price has a lognormal distribution. References Arnold, L. (1974). Stochastic Differential Equations, New York: Wiley & Sons. Bachelier, L. (1900). Theorie de la Speculation, Annales de VEcole Normale Superieure, 17, 21-86. Bharucha-Reid, A.T. (1960). Elements of the Theory of Markov Processes and Their Applications, New York: McGraw-Hill. 37

Pricing Foreign Exchange Options Black, F. &; M. Scholes (1973). The Pricing of Options and Corporate Liabilities, Journal of Political Economy, 8 1 , 637-654. Boness, A.J. (1964). Elements of a Theory of Stock-Option Value, Journal of Political Economy, 72, 163-175. Chandrasekhar, S. (1983). Stochastic Problems in Physics and Astronomy, Review of Modem Physics, 15, 1-89. Cheung, M.T. & D, Yeung (1992). Problems in the Application of Geometric Brownian Motion to Finance, forthcoming in the Proceedings of the 17th Symposium on Operations Research, Berlin: Physica Verlag. Cox, J.C. & S.A. Ross (1976). Valuation of Options for Alternative Stochastic Processes, Journal of Financial Economics, 3, 145-166. Cox, J . C , S.A. Ross & M. Rubinstein (1979). Option Pricing: A Simplified Approach, Journal of Financial Economics, 7, 229-263. Gihman, I. & A.V. Skorohod (1972). Stochastic Differential Equations, New York: Springer Verlag. Goldenberg, D.H. (1991). A Unified Method for Pricing Options on Diffusion Processes, Journal of Financial Economics, 29, 1-34. Ingersoll, J.E. (1989). Option Pricing Theory, in J. Eatwell, M. Milgate & P.K. Newman eds., The New Palgrave Finance, New York: Norton, 1989. Iranpour, R. & P. Chacon (1988). Basic Stochastic Processes: the Mark Kac Lectures, New York: Macmillan. Merton, R.C. (1989). Options, in J. Eatwell, M. Milgate and P.K. Newman eds., The New Palgrave Finance, New York: Norton, 1989. McKean, H.P. (1965). A Free Boundary Problem for the Heat Equation Arising from a Problem of Mathematical Economics, Appendix to P.A. Samuelson, Rational Theory of Warrant Pricing, Industrial Management Review, 6, 1965, 13-31. Reprinted in R.C. Merton ed., Collected Scientific Papers of P.A. Samuelson Vol.3, Cambridge MA: MIT Press 1972.

38

The Black-Scholes Options Theory Modigliani, F. & M.H. Miller (1958). The Cost of Capital, Corporation Finance, and the Theory of Investment, American Economic Review, 48, 261-297. Samuelson, P.A. (1965). Rational Theory of Warrant Pricing, Industrial Management Review, 6, 13-31. Reprinted in R . C Merton ed., Collected Scientific Papers of P.A. Samuelson Vol. 3, Cambridge MA: MIT Press, 1972. Smith, C.W. (1976). Option Pricing: a Review, Journal of Financial Economics, 3, 3-51. Sprenkle, C M . (1964). Warrant Prices as Indicators of Expectations and Preferences, in P. Cootner ed., The Random Character of Stock Prices, Cambridge MA: MIT Press, 1964. Wang, M . C & G.E. Uhlenbeck (1945). On the Theory of Brownian Motion II, Review of Modern Physics, 17, 323-342. Uhlenbeck, G.E. k L.S. Ornstein (1930). On the Theory of Brownian Motion, Physical Review, 36, 823-841. Yeung, D.W.K. (1988). Exact Solution for Steady State Probability Distribution of a Simple Stochastic Lotka-Volterra Food Chain, Stochastic Analysis and Applications, 6, 103-116.

39

Pricing Foreign Exchange

Options

Appendix Define Q = P/Po, and q = logQ. Since l o g P is normally distributed with mean fip and variance <j2py q would be normally distributed with mean Hq = fj,p — logPoj

an<

l variance a2 = a2. We may then write the density

function of q as:

By assumption, KP0 = E(P) (5.45), E(Q) = exp(tiq + a2/2).

(\ogq-fiq)2 ,

-—

/(«) = qa ^/27r exp q

= P0E(P/PQ)

q > 0.

= P0E(Q),

(5.45)

and from

Therefore:

Hq=logK-

-j-.

(5.46)

Substituting (5.46) into (5.45) and then into (5.40), we have:

W := f ° JH/PO

+
l(\ogQ-\ogK

(PoQ-^—^exp Qo-,V2ir

dQ


2

(5.47) Transforming the variable in (5.47) from Q to q = l o g Q :

W

Po =

r

l ( g - l o g « +
, v

7= / exp(?)exp aqV2n J\og(H/P0) 1 (g - log K +
*_ A

'2

/log(ff/P„)

o-i

dg

dg. (5.48)

Since / i g = logK — 0"?/2:

W =

T =

/

exp

aQy/27T J\logWPo) c

lq2-2tiqq "2

exp 40

+

ii2q-2qa2

<7? 1 (g ~ A*g)

"2

*?


21

c?g.

(5.49)

The Black-Scholes

Options

Theory

Upon re-arrangement,

W =

l(q-

Z

/ exp (T J\c qy/2TT J\og(H/P 7^V27T 0)


\iq-


dq

T=

H -=

1 (q -

f°° I

exp

2

fiq)

21

dg

0-2


1 (<7 " ^ "

exp

21

tf/2)

dq

\og(H/P0)

y= I

exp /i9 + -f] H

21

l (
/log(if/P 0 )

r

Gq\J2i J\c Write 7} = (q — fiq — (T2)l^qy

x = (<j — fiq)/aq.

dq.

(5.50)

Transforming g to these

variables in (5.50) and substituting log/c = fiq + 0 ^ / 2 , we have: W = exp(log K ) P 0 /

-j=

exp f - - ^ 2 J dr)

-H£w,^Hx2)dx-

(551)

where: A =

B =

logP0-logH

+ logK +
logP0-logH

+

logK-
(5.52)

Invoking the property that with the standard normal distribution, 1 — N(u) = N(-u)

in (5.52):

41

Pricing Foreign Exchange Options

w=

=

*p°L«,7s°**{-\'i')d'>

KP0N

logPo-logH

- HN

+ logK + <x2/2

/log Po - log H + log K - 0-2/2 \

-5-J!

5

which is equation (5.42).

42

5

«L_

f

(5.53)

CHAPTER 6 GEOMETRIC B R O W N I A N MOTION, "ALMOST CERTAIN RUIN", A N D ASSET M A R K E T S EQUILIBRIUM IN OPTIONS P R I C I N G

6.1

Introduction

At the same time that he introduced geometric Brownian motion into finance, Samuelson (1965) pointed out that if asset prices are modelled in this way, a bias in the (Brownian) random walk must be taken into consideration. In this Chapter, we first explore some implications of Samuelson's observations for the interpretation of asset markets equilibrium, in particular with respect to the assets which underlie options. Attention is drawn to the fact that given the assumption of geometric Brownian motion, the sample path behavior of a stock price (stochastic) process may differ from the behavior of its moments. Most importantly, a O-equilibrium stable process has a divergent mean: the expected price of the stock is (in the limit) infinite, and yet the individual who holds it long enough would be "almost certainly ruined" (Samuelson 1965, p.795), in the sense that typical sample values of the stock price would drop to 0 with probability 1. The existence of this divergence between the moments and sample path behavior of asset price processes poses a serious problem, if we wish to interpret an equilibrium for continuous asset markets. In particular, the widely used "risk neutrality" interpretation in options theory (see §4.6 above), that in equilibrium the expected rate of return on each asset is equal to the risk free rate of interest, does not exclude "almost certain ruin" assets. To resolve this problem, we propose an interpretation of asset markets equilibrium, in which risk aversion is compensated by a premium. If adjusted expected rates of return are equalised in such an equilibrium, assets would have typical sample paths which exclude the possibility of "almost certain ruin". With this interpretation of equilibrium, the standard technique for solving Black-Scholes type options differential equations (see §4.6 above), which derives from a risk neutral equilibrium, is no longer applicable. An alternative stochastic specification is then presented in the next Chapter. 43

Pricing Foreign Exchange Options 6.2

G B M S a m p l e P a t h and M o m e n t s Behavior

As we have seen in Chapter 2, as a result of Samuelson's seminal work (1965), the price of a stock traded in a continuous market is generally assumed to follow the geometric Brownian motion dynamics: dS(t) = aS(t)dt +
(6.1)

where S(t) is the stock price at time ty a a drift term, a a scaling parameter for the variance of S(t), and {dz(t)} a Weiner process. Given the initial condition 5(0) = SQ, the solution to (6.1) is given by: S(t) = So exp

(6.2)

" - y )< + «(<)

As we have seen in Chapter 5, S(t) has the transition density function: <j>(tySo,S) =

1 exp 5CTV2TT7

{logS-[logSo + (a-<7 2 /2)t]K 2
.

(6.3)

From (6.3), it may be seen that S(t) is lognormally distributed with mean exp(logSo+af) and varianceexp[2(logSo+at)][exp(o-2t) — l]. Furthermore, the nth moment of S(t) is: 1 E[Sn(t)] = S%exp

/

a2\ j nt+

{«--2)

a2n2t

—

(6.4)

In particular, the mean stock price is: E[S(t)] = exp(log5 0 + at) = S0 exp(crf).

(6.5)

Let us re-write (6.1) as: ^

= adt + *dz(t).

(6.6)

If cr = 0 in (6.6), it is clear the stock price would grow at the deterministic proportional rate a. Generalising, we may therefore interpret a as the 1

See e.g. Kozin (1972), and Arnold (1974), §11.3.

44

Geometric Brownian Motion expected proportional rate of growth of the GBM stock price S(t)y or the expected rate of return on the stock.2 Even if a > 0, it is possible for the GBM stock price process (6.1) to bring about the "virtual certainty of ruin". This may be seen if we first define s(t) = log 5(2), and apply Ito's lemma to (6.1) to obtain: ds(t) = (a-?—)dt

+
(6.7)

Given the initial condition s(0) = so = log So, the solution to (6.7) may be written: s(t) = «

0

+ f a - y ) < +
(6.8)

The solution process has transition density function: ^(tys0,s)=

—7^exP aV2irt

{s-[s0

(a-cr2/2)t]}2

+ 2a2t

(6.9)

From (6.9), we can see that s(t) is normally distributed with mean so + (a — a2/2)ty and variance a2t. Typical sample paths for the solution process (6.8), and the corresponding sample paths for the process (6.2) are shown in Figs. 6.1a and 6.1b. As Kozin (1972) has shown, in the case 0 < a < a2 /2, the mean of S(t) diverges to infinity but the equilibrium (in the mathematical sense) solution 5 = 0 is asymptotically stable with probability 1. An individual who holds such a stock long enough would be "almost certainly ruined", a positive expected rate of return and in the limit infinite expected price notwithstanding. Kozin's result can be briefly explained as follows.3 The solution (stochastic) process for the GBM stock price S(t) is obtained as: T 2>

5(0) exp f a - y j * +
(6.10)

where 5(0) is the given initial price. Since the sample function of the process grows no faster than ^(tlnlnt) with probability one, the behavior of S(t) 2 3

See also Smith (1976), §3. For more details, see Cheung & Yeung (1994). 45

Pricing Foreign Exchange Options depends only on the sign of a — er2/2. If 0 < a <
Interpreting Asset Markets Equilibrium

As we have seen in §5.6 above, one of the most convenient and widely used interpretations of asset markets equilibrium is that with risk neutral investors, the expected rate of return on each asset would be equal to the risk free rate of interest. In particular, this interpretation underlies the standard technique to solve Black-Scholes type differential equations for option prices. An equation of the form: a = ry

or

E[S(t)] = S0 exp(orf) = 5 0 exp(rt)

(6.11)

then holds for each stock. In particular, whatever the diffusion parameter cr for a stock, in equilibrium it is only required that its a is equal to the risk free rate of interest r. The "almost certain ruin" results obtained by Samuelson and Kozin suggest a serious problem would arise in the financial analysis which employs geometric Brownian motion to characterise the behavior of asset prices. In a continuous stock market, this specification fails to exclude stocks which give rise to the prospect of "almost certain ruin". In particular, it is possible for two stocks to command the same initial price 5(0), and yet one would lead to "almost certain ruin" for the investor, while typical sample values of the other stock price would fluctuate around a rising growth path over time. Moreover, looking forward from the initial time t = 0, (as we have seen above) at any future time t > 0 the expected price E[S(t)] of the stocks are the same. The co-existence of these two stocks is at odds with efficient asset markets behavior, for why should individuals with rational expectations be 46

Geometric Brownian Motion willing to pay the same price for them? Such observations suggest that the risk neutral interpretation of equilibrium, which is widely used in option theory, is not completely free from difficulties. Generally, stocks traded in the asset markets may have different cr's. If (6.11) alone is assumed to hold, over time typical sample points for the prices of these stocks would become dispersed around increasing divergent growth paths. An individual holding stocks with different cr's is therefore faced with the possibility of different degrees of downward drag on prices over time. Geometrically this is clear from Figs. 6.1a and 6.1b: the following example, though based on a simpler stochastic process, may serve to illustrate the point in terms of arithmetic. Suppose the risk free rate of interest is 5%. There are three stocks, with initial price S\o = S20 = S30 = $100 per share. In each of the subsequent periods, their rates of return fluctuate around 5% by ± 1 % , ±4% and ±40%, each with probability | . Over n periods, the expected price of stock j (j = 1,2,3) is then: 3 3

3

E{Sjn) = 100 E E - E f ^ f X ^

Ajin

where: Au

= 1.04,

Al2 = 1.05,

A13 = 1.06,

A2i = 1.01,

A22 = 1.05,

A23 = 1.09,

A31 = 0.65,

A32 = 1.05,

A33 = 1.45,

Expanding this expression, we have: E(Sjn)

= 100

£¥

= 100(1.05)n.

li=l

This gives an expected rate of return of 5% per period on each stock. Consider stock 1, and suppose that each of the outcomes {4%, 5%, 6%} occurs once over the next three periods. Calculating the corresponding sample points, the price of the stock would evolve to: 513 = 100(1.04)(1.05)(1.06)= 115.752= 100(1.04997)3, 47

Pricing Foreign Exchange Options so that an individual holding it over the 3 periods would earn a yield of 4.997% per period. Since 5 i j 3 n = 100(1.04) n (1.05) n (1.06) n = 100(1.04997)3n, this result generalises to any integer n. In the same way, we obtain for stocks 2 and 3: 5 2j3 n = 100(1.01) n (1.05) n (1.09) n = 100(1.0494918)3n, 53>3n = 100(0.65) n (1.05) n (1.45) n = 100(0.989625)3n, = (1 - 0.010375)3n. If, for equilibrium, we only require the expected rate of return on an asset to be equal to the risk free rate of interest in each period, all three stocks would exist in the market. For sufficiently large n, by the law of large numbers the price of stock j would approach Sj)3n. An individual holding any or all of these stocks would be faced with the possibility of different degrees of downward drag on prices over time. In particular, if the third stock is held long enough, it would lead to "almost certain ruin". The above observations suggest that, given the availability of a risk free asset with a deterministic rate of return equal to the expected rate of return on each stock, the risk neutral investor would have no reason to prefer holding the first stock, and a fortiori to prefer holding the other two stocks. Contrary to what the equilibrium condition tells us, there is no market for the three stocks. In sum, if for consistency with the Samuelsonian price dynamics (6.1) we assume stock markets exist continuously and take the divergence between sample path and moments behavior of GBM stock price processes into account, it would be difficult to maintain a general equilibrium interpretation of the condition a = r. To resolve this difficulty, it may be convenient to assume that investors are risk averse. As long as a stock's a is not 0, a premium on the risk it carries would appear. 4 In other words, an individual would require: a = r + risk premium 4

(6.12)

We follow Miller (1991) in adopting the simplest measure of volatility. For a n excellent discussion of other measures and issues, see Shiller (1989). 48

Geometric Brownian Motion to hold the stock. The risk premium increases as the stock becomes more volatile, as reflected in a higher cr2. If high volatility stocks are to co-exist in the market with low volatility stocks, the former must yield higher expected rates of return. A simple way to capture this property is to write (6.12) in the form: a = r + 7<72, 7 > 0 . (6.13) The expected rate of return on a stock a is adjusted by a risk premium ycr2. let us denote the value of 7 produced by equilibrium in the asset markets by 7*. The price of the stock would then develop according to the dynamical equation: dS(t) = aS(t)dt + aS(t)dz(t)y

(6.14)

where the risk-adjusted equilibrium condition: a = r + 7V2

(6.15)

is satisfied at each point of time. Substituting (6.15) into (6.14): dS(t) = (r + y*cr2)S(t)dt + aS(t)dz(t).

(6.16)

Transforming to s(t) = log S(t) as before, we have: ds(t) = Ir-

(^-7Aa2\dt

+ crdz(t).

(6.17)

Since rational investors would behave to exclude the possibility of "almost certain ruin", in (6.17) the risk coefficient 7 must have a market equilibrium consistent value 7* > 1/2 — r/cr2. In this case, we may readily observe: (i) If 7* > 1/2, the prices (typical sample points) of a stock with a higher cr are dispersed around a higher growth curve. Moreover, each of these curves lies above the growth curve of the risk free asset. (See Fig. 6.2a.) (ii) If 7* = 1/2, the growth curves of all stocks would be the same as that of the risk free asset. (Fig. 6.2b.) (iii) If 1/2 — r/cr2 < 7* < 1/2, the prices (typical sample points) of a stock with a higher a are dispersed around a lower growth curve, each of 49

Pricing Foreign Exchange Options which lies below the growth curve of the risk free asset. However, none of these sample paths would decay. (Fig. 6.2c.) 6.4

Conclusions

A serious problem arises when geometric Brownian motion is applied to financial modelling. Because of the divergence between the moments and sample path behavior of the stochastic process of the stock price, a stock with an exponentially increasing expected value and a positive expected rate of return may have a typical sample path which approaches zero asymptotically with probability one. An individual who holds such an asset for a sufficient length of time will be "almost certainly ruined" in Samuelson's sense. There is then a difficulty in the interpretation of asset markets equilibrium: in particular, the widely used interpretation in options theory, that with risk neutral individuals the expected rate of return on each asset is equal to the risk free rate of interest, does not exclude "almost certain ruin" assets. We suggest an interpretation of asset markets equilibrium which is free from this problem. References Chandrasekhar, S. (1943). Stochastic Problems in Physics and Astronomy, Review of Modern Physics, 15, 1-89. Cheung, M.T. & D. Yeung (1994). Divergence Between Sample Path and Moments Behavior: an Issue in the Application of Geometric Brownian Motion to Finance, Stochastic Analysis and Applications 12, 277-291. Cox, J.C. & S.A. Ross (1976). The Valuation of Options for Alternative Stochastic Processes, Journal of Financial Economics, 3, 145-166. Gard, T.C. (1988). Introduction to Stochastic Differential Equations, New York: Marcel Dekker. Goldenberg, D. (1991). A Unified Method for Pricing Options on Diffusion Processes, Journal of Financial Economics, 29, 1-34. Iranpour, R. & P. Chacon (1988). Basic Stochastic Processes: the Mark Kac Lectures, Now York: Macmillan. 50

Geometric Brownian Motion Kozin, F. (1972). Stability of Linear Stochastic Systems, in R. Curtain ed., Stability of Stochastic Dynamic Systems, New York: Springer Verlag. McQueen, G. & S. Thorley (1991). Are Stock Returns Predictable? A Test Using Markov Chains, Journal of Finance, 46, 239-263. Miller, M.H. (1991). Financial Innovations and Market Volatility, Oxford: Black well. Samuelson, P.A. (1965). Rational Theory of Warrant Pricing, Industrial Management Review, 6, 13-31. Reprinted in R.C. Merton ed., Collected Scientific Papers of P.A. Samuelson Vol.3, Cambridge Mass.: MIT Press, 1972. Samuelson, P.A. (1991). Long Run Risk Tolerance when Equity Returns are Mean Regressing: Pseudoparadoxes and Vindication of "Businessman's Risk", in W.C. Brainard, W.D. Nordhaus k H.W. Watts eds., Money, Macroeconomics and Economic Policy: Essays in Honor of James Tobin, Cambridge, Mass.:MIT Press, 1991. Schiller, R.J. (1989). Market Volatility, Cambridge, Mass.: MIT Press. Smith, C.W. (1976). Options Pricing: A Review, Journal of Financial Economics, 3, 3-51. Soong, T.T. (1973). Random Differential Equations in Science and Engineering, New York: Academic Press.

51

Pricing Foreign Exchange Options

s0+(a-<^/2)t

I

1 Fig. 6.1a

Typical sample paths of the stochastic process s(t). Here cr3 > a2 > 0, a -
52

Geometric Brownian Motion

Fig. 6.1b Typical sample paths of the stochastic process S(t)y corresponding to s(t) in Fig. 6.1a.

53

Pricing Foreign Exchange Options

s(t)

S0ap{[r+(f-l/2)a{lt} Sflexp{[r+(f. 1/2)0^} S0exp(rt)

Fig. 6.2a Typical sample paths for two stocks with a\ > a2, given the condition a = r + j*a2, and 7* > 1/2.

54

Geometric Brownian Motion

S(t)

:S0e^rt) t

:

s.

0 Fig 6.2b Typical sample paths for two stocks given the condition a = r + 7*cr2, with 7* = 1/2. The dotted path is associated with a higher cr2.

55

Pricing Foreign Exchange Options

^oq>(rt)

So»p {[r+(y*. V2)(ffc)

S0aq){[r+(f-l/2)agt}

Fig 6.2c Typical sample paths for two stocks with cr2 > cr2, given the conditions a = r + 7 V 2 , and 1/2 - r/cr2 < 7* < 1/2.

56

CHAPTER 7 N O N R A N D O M WALK EFFECTS A N D A N E W STOCHASTIC SPECIFICATION

7.1

Intro duction

As we have seen in Chapter 5, a serious problem arises when geometric Brownian motion is used to model asset prices. In addition, recent research is beginning to "question the random walk dogma" associated with geometric Brownian motion (Samuelson 1991). One factor which may account for "runs" in the prices of assets like stocks is provided by standard economic theory, which tells us that in long period equihbrium, the value of the firm's balance sheet is determined by exogenous variables like technology and tastes. If this value changes in response to changes in any one exogenous variable, the 'intrinsic value' of a firm's shares will change in a systematic way. However, by virtue of the assumption of geometric Brownian motion in the Black-Scholes theory, it is not possible to take such standard results into account. The above observations suggest it is in order to consider an alternative system of stochastic asset price dynamics. Following the approach of Cox k Ross (1976), we propose a candidate for addition to the list of "plausible alternative [to geometric Brownian motion] forms of stochastic processes", which are useful in financial analysis. The remainder of this Chapter draws on Cheung & Yeung (1994), and introduces a two dimensional stochastic process to model the dynamic behavior of the representative equity. To characterise the solution of the resulting system of stochastic differential equations in the most complete form known, we obtain a closed form expression for its transition density function. It is shown that the solution (stochastic) process of the stock price excludes the possibility of almost certain ruin. Since "the option valuation problem is equivalent to the problem of determining the distribution of the stock price" (Cox k Ross 1976, p. 154), we proceed to take mathematical expectations directly in terms of the transition density function. An exact formula to price options on the stock is then obtained. 57

Pricing Foreign Exchange Options One important feature of this option pricing formula is that non random walk effects can be incorporated. In particular, it is shown that a stock's intrinsic value enters in an essential way into the valuation of options on it. 7.2

A New Stochastic Specification

As we have seen, the concept of geometric Brownian motion was introduced into financial modelling by Samuelson (1965). The market price of a stock S(t) is assumed to behave according to the stochastic differential equation dS(t) = aS(t) + aS(t)dz(t)y

(7.1)

where a and a are constant parameters and {dz(t)} is a Wiener process. In their classic work on option pricing, Black and Scholes (1973) also adopted the assumption that stock prices follow geometric Brownian motion. A fundamental result following from this assumption is that the stock price would display the characteristics of a random walk. In particular, the price of the stock at any time in the future depends solely on its present price. This property is beginning to be called into question by recent research (see e.g. McQueen k Thorley 1991, Samuelson 1991, Kaehler k Kugler eds. 1994, Haugen 1995, Malkiel 1996, Campbell, Lo k Mackinlay 1997). For example, it is suggested that postwar returns to U.S. common stocks exhibit statistically significant non-random walk behavior, especially in the sense that runs of high and low returns have been found to follow one another. These findings suggest that an alternative stochastic assumption is required, to model the behavior of the prices of the assets (in particular equities) which underlie options. In addition, recent events in financial markets direct our attention to two new questions: (a) Is it possible for the Black k Scholes-GBM model to accomodate insiders' information, which seems to a not uncommon phenomenon in Hong Kong and some less sophisticated markets like Taiwan, and in newly emerging markets like Shenzhen? (b) Is it possible for the model to accomodate world-wide stock market 58

A New Stochastic Specification crashes like 1987 or sustained bear markets like Tokyo 1992-7, or sustained bull markets like New York 1992-97? A two dimensional stochastic process is introduced in this and the following sections, to model the dynamic behavior of the market price of a stock and its premium rate of return. Non random walk effects are incorporated into the model, by way of standard results of economic theory. In particular, we introduce the concept of 'intrinsic value' of a stock, which reflects (as standard economic theory tells us) the value of the firm's balance sheet in long period equilibrium. Over time, fluctuations in the price of a stock are then anchored, so to speak, by its intrinsic value. To characterise the solution of the resulting system of stochastic differential equations in the most complete form known, we obtain a closed form expression for the transition density function. It is also shown that the stock price process excludes the possibility of "almost certain ruin". Since "the option valuation problem is equivalent to the problem of determining the distribution of the stock price" (Cox and Ross 1976, p.154), we proceed to take (mathematical) expectations directly in terms of the transition density function of the stock price. An exact formula to price options on the stock is obtained, conditional upon its current price and current premium rate of return. 7.3

Dynamics of Stock Price and Premium Rate of Return

Suppose the price of a stock traded in a continuously existing market develops over time according to the process S(t). The dividend policy of the stock offers the owner an instantaneous payout, at a rate equal to the market rate of interest p(t). If a total rate of return b(t) is realised on the stock at time t, a premium rate of return R(t) = b(t) — p(t) obtains. Net of dividend, the growth equation of the stock price is then: dS(t) = R(t)S(t)dty

(7.2)

where the initial price 5(0) = So is given. To characterise the behavior of the premium rate of return, we follow Samuelson's general dynamic method (1983), and identify three factors 59

Pricing Foreign Exchange Options which affect the way R(t) changes over time. If the current value of R is positive (negative), with competition it would tend to decrease (increase). The second factor is the potential wealth of the corporation, as reflected in the value of its balance sheet in long period equilibrium, or in the intrinsic value of its shares. If the current price of the stock is above (below) its intrinsic value sy the premium rate of return would decrease (increase). Finally there is a random factor, which following standard practice we assume to be represented by a Wiener process. Given these considerations, the premium rate of return is then governed by the dynamics: dR(t) = {-aR(t)

- /?pn5(*) - lns]}dt + adz(t)y

(7.3)

where {dz(t)} is a Wiener process with E[dz(t)] = 0, var[cb(£)] = dty a and a are serial correlation and variance scaling parameters for R(t)} j3 > 0 is an adjustment coefficient, and the initial value R(0) = RQ is given. The dynamic behavior of the stock price and the premium rate of return is described by the system of stochastic differential equations (7.2) and (7.3). The most comprehensive way to characterise a solution (a continuous time stochastic process) to such an equation is to derive its transition density function. As Kac pointed out: "We may interpret the variation of the [transition] density function over time as a macroscopic description of many microscopic particles undergoing motions governed by a [continuous time] stochastic process. The Fokker-Planck equation describes these variations, [and] the solution to this differential equation determines the process" (see Iranpour and Chacon 1988, §7.5). In the present case, the joint transition density function of S(t) and R(t), x(5,R y t]So,Ro), must satisfy the two dimensional Fokker-Planck equation: •fiX(S,R,t;S0,Ro)

=

-^SRX(S,R,t;S0,R0) +

-^{[-aR-S0(lnS-lns)]x(S,R,t;S0,Ro)}

"^
(? 4)

-

A New Stochastic Specification However, as Goldenberg (1991) observed, "solving Fokker-Planck equations is difficult" (p.6). To date, explicit solutions have been found for only a small number of equations, which do not include (7.4). To obtain a closed form expression for the transition density function in the present case, let us introduce the transformation V(i) = In S(t). (7.2) and (7.3) become: dV(t) = R(t)dt} dR(t) = [-aR(t)

(7.5) - f3(V(t) - V)\dt + adz(t)y

(7.6)

where V = In s. Performing a second transformation W(t) = V(t) — V, we have: dW(t) = R(t)dt,

(7.7)

dR(t) = - [aR(t) + /3W(t)]dt +
(7.8)

(7.7) and (7.8) then yield the second order Langevin equation:


(7.9)

where A(t) is a Gaussian white noise with a zero mean, covariance
«W,t,W0,Ro) = ^2^F(<) eXp {-2^W) — Wo exp I —— ) (cosh-j— H sinh-1— ) \ 2J \ 2 <*i 2 ) 2R0 ( cd\ . .a^]2)

-^T exp V~Tj sm/l ^-J ) 61

(710)

Pricing Foreign Exchange Options where F(t) = -a/3 — expf — — aaj/? 2a 2 sm/n —- J -f- aaisinh(ait)

-f a\\

and W(0) = Wo is the initial value for W(t). For a\ imaginary, the terms cosh(ait/2), [sinh(ctit/2)]/ai and [sinh(ait)]/ai in (5.25) should be replaced by cos jity (sin jit)/2ji, and (sin 2jit)/2ji respectively, where 71 = \J(f3 — a2/4. If a i = 0, the replacement terms become 1, t/2y and t respectively. (See Chandrasekhar 1943, Ch.2, §3.) As Chandrasekhar (1943) noted, physically (5.25) characterises a harmonic oscillator. Given Wo, W(i) has the expected value c*i*

a

2

<*i

E[W(t)] = W0exp(-?p) cosh—-— 2Ro

(

at\

.

. ait sinn—2

ait

From (7.11), limJ5[W(<)] = 0 as t -+ 00 (Chandrasekhar 1943, p.30). The time path of ^[PV^)] depends on which of the three possibilities for c*i = \Jcx2 — 4/? obtains. If a\ is real and positive, or equal to zero, £'[W(t)] would be damped over time towards zero. If a\ is imaginary, E^W^)] would fluctuate towards zero over time. 1 Properties of harmonic oscillation are well documented in standard texts on stochastic processes (see e.g. Soong 1973, §7.3 and Gard 1988, §4.2). In particular, the behavior of W(t) over time is that of the harmonically bound particle, which can be approximated by a white noise excitation around zero as t —• 00. In the present case, where W(t) = In 5(£)—In 5, the behavior of ^ [ W ^ ) ] reflects the nature of intrinsic value. As information becomes complete so that in the limit probability densities collapse on their means, the market would value the corporation's shares at exactly what they would earn in long period equilibrium. Moreover, since W(t) excites around zero, the stock price S(t) would describe a typical sample path under attraction around its 1

Details are omitted, as Chandrasekhar (1943) contains a long discussion. 62

A New Stochastic Specification intrinsic value s, even in the limit. Individuals holding such a stock do not face the prospect of becoming "almost certainly ruined", because unlike the case of geometric Brownian motion prices, there is no possibility that its price would suffer a persistent downward drag towards zero over time. 2 Since W(t) = V(t) — V is a 1:1 mapping, applying a standard variate transform to (7.11), we obtain the transition density function of V(t) as:

(

ai\ (

. ait

a . . a\t\

where V(0) = VQ = W0 + V is the initial value of V(t). Similarly, V(t) = In 5(2) is a 1:1 mapping, therefore applying a variate transform to (7.12), we obtain the transition density function of S(t) as: In 5

e(Syt;So,Ro)= ( | ) y ^ c ^ f, ~

/

(*t\ (

. ait

a . . aiA

— < In s -f- cexpl ——- ) l cosh—-—I I \ 2 / \ 2 ai

sinh-— J 2 )

where c = (ln5o — Ins), and 5(0) — So — exp(Vb) is the initial value of S(t). Because of the relationships between (7.10), (7.11) and (7.12), the stock price has a transition density function which depends on its intrinsic value s and the initial premium rate of return RQ , as well as on its initial 2 It should be noted that if the stock price is assumed to follow GBM, then given a dividend policy similar to the above (together with the standard condition expected rate of return = market rate of interest), "almost certain ruin" must arise. This is because the diffusion parameter b must be more than twice the instantaneous post-dividend expected rate of return (i.e., a — r = 0).

63

Pricing Foreign Exchange Options value So- From inspection of (7.13), it is clear that 0(5,£;5o,i?o) is the marginal density function of S(t)y which is obtained from the joint density function x(5, R,t\S0iRo) of S(t) and R(t) by setting R(t) = RQ. The transition density function of a stock price is essential to many financial decisions. In particular, as Smith (1976) has shown, it allows one to adopt a direct approach when evaluating options, by taking the expected value of the difference between stock price and exercise price. This is the subject of the next Section. 7.4

A n Exact Option Pricing Formula

Following Smith (1976), if we look forward from the current period (2 = 0), at the expiry date T > 0 an European call option on the stock with exercise price K would have the value:3

J'

/*oo

(S-K)0(SyT-ySo,ro)dS.

(7.14)

K

Since in (7.13), s, 5o, Ro, a and ai are given parameters, for any t the expression: In s + c exp

(*)[•

. ait cosh——| 2

a . ait smh——ai 2

+

2R0

ai

f at\ . , ait exp P I — — J smh——-

V

27

2

is a constant. Denoting it by fi(t)y we may re-write (7.14) as: 0(Syt-ySoyRo)

=

1 Say/2irF(t)

exp

[In S-/*(<)] * 2a2F(t)

(7.15)

It is seen that for any t, the stock price S(t) is lognormally distributed with mean exp[fi(t) + cr2F(t)/2]y and variance exp[2/i(t) +
}

xp

^W>-* wbr 3

[lnS-„(t)]21 2
dS.

(7.16)

In the derivation, we take the case of ot\ real and positive. Formulae for the remaining cases of a\ = 0 and a\ imaginary may be obtained by making the appropriate substitutions in (5.25). 64

A New Stochastic Specification It follows that (see Appendix): C(T) = h(T)S0N

_

RN

^S

0

+

lnh(T)-^+*2F(T)/2\

A n So + In h(T) - In K -

{


,y/m

) (7.17)

where h(T) = E[S(T)]/So is the number of times the stock is expected to rise in price, at the date the option expires. Substituting for E[S(T)]y h(T) = {exp[/i(T) + a2F(T)/2]}/S0, so that h(T) depends on the stock's initial price 5o, its intrinsic value s, and the initial premium rate of return Ro (via fi(T) and F(T)). Following Smith (1976), discounting (7.17) back to t = 0 using the market rate of interest p(t)y we obtain the exact option valuation formula: C*(TyK,syS0,Ro) exp

_ KN ^ S o + h t m - ^ - ^ D / i ) | .

(7 . 18)

In addition to the stock's current price, the presence of the easily computable terms F(T) and h(T) in (7.18) captures the influence of its intrinsic value s and the premium rate of return Ro on the value of the option. In particular, it is possible for non random walk effects to be introduced via the variable

7.5

Conclusion

This Chapter proposes a modelling alternative to the "work horse" (Cox and Ross 1976, p. 165) of modern financial analysis — geometric Brownian motion. By its means, we are able to introduce non random walk effects into the analysis, and avoid the Samuelson-Kozin "almost certain ruin" problem, which are present when the price of a stock is assumed to follow geometric Brownian motion. Our results, which include a closed form expression 65

Pricing Foreign Exchange Options for the transition density function of the stock price and an exact formula for pricing options, offer a new theoretical framework for further work in financial analysis. Developments, extensions, and in particular empirical investigation of parameter values are very much awaiting. References Campbell, J.Y., A.W. Lo k A.C. Mackinlay (1997). The Econometrics of Financial Markets, Princeton NJ: Princeton University Press. Chandrasekhar, S. (1943). Stochastic Problems in Physics and Astronomy, Review of Modern Physics, 15, 1-89. Cheung, M.T. k D.W.K. Yeung (1994). A Non-Random Walk Theory of Exchange Rate Dynamics with Applications to Option Pricing, Stochastic Analysis and Applications 12, 141-159. Cox, J.C. k S.A. Ross (1976). The Valuation of Options for Alternative Stochastic Processes, Journal of Financial Economics, 3, 145-166. Gard, T.C. (1988). Introduction to Stochastic Differential Equations, New York: Dekker. Goldenberg, D. (1991). A Unified Method for Pricing Options on Diffusion Processes, Journal of Financial Economics, 29, 1-34. Haugen, R A . (1995). The New Finance: the Case Against Efficient Marketsy Englewood Cliffs NJ: Prentice Hall. Iranpour, R. k P. Chacon (1988). Basic Stochastic Processes: the Mark Kac Lectures, Now York: Macmillan. Kaehler, J. k P. Kugler eds. (1994). Econometric Analysis of Financial Markets, Heidelberg: Physica-Verlag. Kozin, F. (1972). Stability of Linear Stochastic Systems, in R. Curtain ed., Stability of Stochastic Dynamic Systems, New York: Springer Verlag. Malkiel, B.G. (1996). A Random Walk Down Wall Street 6th ed., New York: Norton k Co. McQueen, G. k S. Thorley (1991). Are Stock Returns Predictable? A Test 66

A New Stochastic Specification Using Markov Chains, Journal of Finance, 46, 239-263. Miller, M.H. (1991). Financial Innovations and Market Volatility, Oxford: Black well. Samuelson, P.A. (1965). Rational Theory of Warrant Pricing, Industrial Management Review, 6, 13-31. Reprinted in R.C. Merton ed., Collected Scientific Papers of P.A. Samuelson Vol.3, Cambridge Mass.: MIT Press, 1972. Samuelson, P.A. (1991). Long Run Risk Tolerance when Equity Returns are Mean Regressing: Pseudoparadoxes and Vindication of "Businessman's Risk", in W.C. Brainard, W.D. Nordhaus k H.W. Watts eds., Money, Macroeconomics and Economic Policy: Essays in Honor of James Tobin, Cambridge, Mass.:MIT Press, 1991. Schiller, R.J. (1989). Market Volatility, Cambridge, Mass.: MIT Press. Smith, C.W. (1976). Options Pricing: A Review, Journal of Financial Economics, 3, 3-51. Soong, T.T. (1973). Random Differential Equations in Science and Engineering, New York: Academic Press.

67

Pricing Foreign Exchange Options Appendix From (7.16): ,00

[lnS-^t)]21

1

2
K

JK

S(r^/2irF(t)

exp

dS

[In S - / ! ( < ) ] 2 1 c/S 2
(7.19)

Denning h(t) = E[S(t)]/S0 and substituting exj>[((i(t)+cr2F(t)/2] for £[S(*)], we may write: H{t) = In S0+In

h(t)-

c2F{i)

(7.20)

Let us introduce the random variable G(t) = S(t)/So- g = logG(t) is then normally distributed with mean fig = n(t) — logSo> and variance cr,2 _
f(G) =

G
exp

[In G - In h(t) + <x2F{t)/2\ 2a2F(t)

(7.22)

Recall that S0G(t) = S(t). (7.22) then becomes: /•OO

C(t) = / IK/SO

[]nG-]nh(t) +
S0g-K exp G
dG. (7.23)

Transforming the variable in (7.23) to g = In G:

C

®

=

/o wrt /JA
ex

[g- In h(t) + cr2F(t)/2]2 dg 2c 2 f , (<)

ex

P(») P

[g-lnh(t)

K f° exp
-

68

+
A New Stochastic Specification where A = (In K — In SQ). Re-arranging the first term on the RHS of (7.24), and writing F for F(t)y h for h(t) to make the following lengthy expressions less cumbersome, we obtain: So exp
g2 - 2(ln h - <x2F/2)g + (in h -
dg

So
/;

[g - (In h -
exp

Since -

dg

is independent of g, (7.24)

may be written: C(t) = exp(ln h - a2F + a2F)

—r-l

exp

(Ty/2irF JA hSp f exp fc


exp

(g-lnh- 2a2F

dg

( ( / - l n / i + (T 2 F/2) 21 dg 2
[g-(\nh

~
K

50 exp aV2nF /J A;

+ cr2F/2)]: 2a2F

dg

[g-(lnh-a2F/2)} 2a2F

21

dg

(7.25)

where once again F(t) and h(t) are shortened to F and h. Let us transform the normal distribution functions on the RHS of (7.25) to standard normal form by means of the substitutions: _ y=

{g-[lnh(t)-j-a2F(t)/2]}

__

{g-[\nh(t)-
cry/Flfi


69

Pricing Foreign Exchange Options where _ [In Q - In So - In h(t) -

B=

e2F(t)/2]

*VW)

'

_ [In Q - In So - In h(t) +
D=

^ W

'

Since with the standard normal distribution, 1 — N(£) = N(—£), we have: C(t) = k(t)S0N{-B) - QN(-D) (lnSo + lnh(t)-lnQ + a2F(t)/2\

=mSoN _

QN

{

o-jm

J

(\nSp+lnh(t) -InQ-
\

°Vm

Putting t = T in (7.27), (7.17) obtains.

70

J

(7.27)

CHAPTER 8 P R I C I N G FOREIGN E X C H A N G E OPTIONS INCORPORATING PURCHASING POWER PARITY Introduction 1

8.1

One of the most useful and widely used applications of the theory of options is to price options on foreign exchange.2 As we have seen in Chapter 4, according to a fundamental assumption in Black and Scholes' theory, the stochastic process governing the (spot) price of the underlying foreign currency follows geometric Brownian motion. 3 As a result, the exchange rate displays the characteristics of a random walk.4 On the other hand, a fundamental theorem in the international trade predicts that over time, the spot (relative) price of a currency would converge to its purchasing power parity. 5 Fluctuations in the exchange rate are anchored, so to speak, by its long period equilibrium value. To synthesise the theory of foreign currency options and the theory of international trade, the present Chapter introduces the element of non random walk behavior represented by purchasing power parity into the stochastic specification of the exchange rate. An alternative to geometric Brownian motion, in the form of a two dimensional stochastic process, is introduced to model the dynamics of foreign currency prices. A closed form expression for the transition density function is obtained, to characterise the solution of the resulting system of stochastic differential equations in the most complete form. Since "option valuation is equivalent to the problem of determining the distribution of the [underlying] asset price", 6 we proceed to take mathematical expectation in terms of the transition density function. An exact formula to price options on the currency is obtained, which in particular is conditional upon its purchasing power parity. This result is line line with recent research (see e.g. McQueen k Thorley 1991, Samuelson 1991, Kaehler k Kugler eds. 1994, Haugen 1

This chapter is an expanded version of Cheung & Yeung (1994a). See e.g. Biger & Hull (1983), Garman & Kohlhagen (1983), Grabbe (1983), Bodurtha & Courtadon (1987), and Hull (1991). 3 See e.g. Bodurtha & Courtadon (1987). 4 See e.g. Samuelson (1965) and McKean's Appendix. 5 See. e.g. Frenkel & Mussa (1984), Levich (1984), and Lothian and Taylor (1996). 6 See Cox & Ross (1976), p.154, and Goldenberg (1991), p.6. 2

71

Pricing Foreign Exchange Options 1995, Malkiel 1996, Campbell, Lo k Mackinlay 1997), which is beginning to "question the random walk dogma" (Samuelson 1991) so much a part of the geometric Brownian motion assumption. A second problem arises when exchange rates (and other asset prices) are assumed to follow geometric Brownian motion. In a classic paper, Samuelson (1965) pointed out that when modelling asset prices in this way, one must take into account a bias in the (Brownian) random walk. In particular, a theorem on "the virtual certainty of relative ruin" (Samuelson 1965, p.795) was obtained, such that if the diffusion parameter is more than twice the instantaneous expected rate of return of the asset price, its typical sample values would drop to zero with probability one over time, a positive expected rate of return notwithstanding. A detailed mathematical verification of the existence of this phenomenon was provided by Kozin (1972), in the context of a general demonstration that the behavior of geometric Brownian motion moments diverges from the behavior of its sample paths. Under the GBM assumption, asset price dynamics is described by the stochastic differential equation dP(t) = aP(t)dt-\-bP(t)dz(t)y where a is the asset's instantaneous rate of return, b is the diffusion parameter, and {^(t)} is a Wiener process with E[dz(t)] = 0, va.r[dz(t)] = dt. Given the initial price P(0)> the solution stochastic process of the asset price can then be obtained as P(t) = P(0) exp [(a — b2/2)t -f bz(t)]. Since the sample function of the process {P(t)} grows on faster than y/(t\oglogt) with probability 1, the sample behavior of the asset price depends solely on the relative magnitudes of the parameters a and 6. If a — 6 2 /2 < 0, the typical sample path of P(t) approaches 0 with probability 1, as t —• oo. On the other hand, the nth moment of P(t) is found to be E[P(t)n] = P(0) n exp [(a - b2/2)nt + (b2n2/2)t). The expected growth of the asset price over time is then E[P(t)] = P(0)exp(at)y despite the fact that if a — b2/2 < 0, in the limit its typical sample values drop to 0 with probability 1. Recently, "almost certain ruin" phenomena associated with geometric Brownian motion have also been discovered in biology (Gard 1988, §6.2).7 7

Further implications are explored in Cheung & Yeung (1994b).

72

Pricing Foreign Exchange Options Samuelson's and Kozin's results suggest that a serious problem would arise, in the financial analysis which employs geometric Brownian motion to characterise the behavior of asset prices. In particular, this specification does not exclude currencies which give rise to almost certain ruin from the characterisation of equilibrium. It is logically possible for two currencies to command the same initial spot price P(0), and yet because of differences in the relative magnitudes of a and 6, one currency would lead to almost certain ruin for the investor, while typical sample values of the other exchange rate would fluctuate around a rising growth path over time. 8 Moreover, looking forward from the initial time t = 0, (as we have seen above) at any future time t > 0 the expected spot price E[P(t)] of the two currencies are the same. The co-existence of such currencies is at odds with efficient markets condition of equilibrium, for why should individuals with rational expectations be willing to pay the same initial spot price for them? The above observations suggest it is in order to consider an alternative system of stochastic exchange rate dynamics. Following the approach of Cox and Ross (1976), we propose the model of §8.2 below as an addition to the list of "plausible alternative [to geometric Brownian motion] stochastic processes", which are useful in financial analysis. 8.2

Stochastic Dynamics of T h e Exchange R a t e

By definition, a foreign currency's exchange rate or spot price relative to the home currency can be written as B(t)/H(i)y where B(t) is the accounting price of the foreign currency, and H(t) the accounting price of the home currency.9 Following the literature, let us regard a currency as "analogous to a stock paying a continuous dividend equal to the constant interest rate in that country" . 10 We then have [d(B/H)/dt]/(B/H) = rf - r, where rf and r are foreign and domestic rates of interest respectively. An individual holding the foreign currency then earns a net dividend or rj — r. If a rate of 8 This apart from the problem of whether the sample path of each spot price would converge to the respective purchasing power parities. 9 Hull (1991), C h . l l . Following standard practice, we assume markets are perfect and continuous, and transactions cost is zero. See e.g. Bodurtha and Courtadon (1987), and Hull, Ch.10. 10 Accounting prices can be measured in terms of gold.

73

Pricing Foreign Exchange Options change Q(t) actually obtains in the spot price of the foreign currency, the individual would realise a premium rate of return R(t) = Q(t) — (rf — r). With net dividend paid, the growth equation of the currency's exchange rate is then: dP(t) = R(t)P(t)dty (8.1) where the initial spot price P(0) = Po is given. To characterise the behavior of the premium rate of return R(t)y we identify the following factors. The first factor is the intrinsic value of the currency, as reflected in its purchasing power parity. If the current spot price of the currency is above (below) its purchasing power parity p, its premium rate of return would decrease (increase). Second, if the current value of the premium rate of return is positive (negative), with competition it would tend to fall (rise). Finally there is a random factor, which following standard practice we assume to be represented by a Wiener process. Given these considerations, the dynamics of R(t) is then specified as: dR(t) = {-aR(t)

- /3\inP(t) - \np]}dt + adz(t)y

(8.2)

where {dz(t)} is a Wiener process with E[dz(t)] = 0, var[cfer(tf)] = dt, a and a > 0 are serial correlation and variance scaling parameters, /? > 0 is an adjustment coefficient, and the initial value R(Q) = i^o is given.11 The dynamic behavior of the exchange rate and its rate of change is described by the system of stochastic differential equations (8.1) and (8.2). The most comprehensive way to characterise a solution process to such an equation is to derive its transition density function. As Kac pointed out: "We may interpret the variation of the [transition] density function over time as a macroscopic description of many microscopic particles undergoing motions governed by a stochastic process. The Fokker-Planck equation describes these variations, [and] the solution to this differential equation determines the process" (see Iranpour and Chacon 1988, §8.5). In the present case, the joint transition density function of P(t) and R(t)y

X(^OJ^OJ^-P)^))

must satisfy the two dimensional Fokker-Planck

11

In addition to the second factor in the text, a enters with a negative sign in (6.2), to exclude the possibility of un-directional runs in the exchange rate brought about by constant positive correlation.

74

Pricing Foreign Exchange Options equation:

-^dtxx(.)^±PR ( '- ) -x{.)^{[R-Po{\nP-\np)]x{-)}-\^-yX{-). XK dP a dR

2 8R2 (8.3)

However, as Goldenberg observed, "solving Fokker-Planck equations is difficult" (1991, p.6). To date, explicit solutions have been found for only a small number of equations, which do not include (8.3). 12 To obtain a closed form expression for the transition density function in the present case, let us introduce the transformation W(t) = In P(t)— In p. 1 3 (8.1) and (8.2) become: dW(t) = R(t)dt,

(8.4)

dR(t) = - [aR(t) + pW(t)} dt +
(8.5)

(8.4) and (8.5) then yield the second order Langevin equation: d2W(t) dW(t) + a dt2 dt

+ (3W(t) = aA(t),

(8.6)

where A(t) is a Gaussian white noise with zero mean, covariance a26(t — s), and S(-) is Dirac's delta function. Transforming the two dimensional first order system (8.1) and (8.2) to the one dimensional second order equation (8.6), we can then invoke the classic results of Chandrasekhar (1943). For a\ = \/OL2 — 4/3 real and positive, W(i) has the transition density function: (WytyW0lRo) =

2wa2F(t)

(

L

W 2 * \1 2
•exp

-WQ

/ at\ ( , a\t a . , a>it\ '• e x P — TT ) cosh——- -\ stnn—— 1 V 2J\ 2 <*i 2 ) 2Ro

•exp

/

at\

. ,aitl2

(8.7)

12 See Uhlenbeck & Ornstein (1930), Chandrasekhar (1943), Wang & Uhlenbeck (1945), Bharucha-Reid (1960), Gihman & Skorohod (1972), and Cheung & Yeung (1992). 13 As will be shown below, this allows us to find the transition density function of the exchange rate, without having to solve the Fokker-Planck equation (6.3).

75

Pricing Foreign Exchange Options where F(t) = -a/3 — exp ( — — 1 aa\/3 \2a2sinhl

—- j -f aaisinh(ait)

+ a2

and W(0) = Wo is the initial value for W(t). For ai imaginary, the terms cosh(ait/2)y [sinh(ait/2)]/ai and [sinh(ait)]/ai in (8.7) should be replaced by cos7it, (sin7it)/27i, and (sin27i^)/27i respectively, where 71 = y/(3 — a214. If a i = 0, the replacement terms become 1, t/2, and t respectively. (See Chandrasekhar 1943, Ch.2, §3.) As Chandrasekhar (1943) noted, physically (8.6) characterises a harmonic oscillator. Given Wo, W(t) has the expected value ct\t E[W(t)] = Wo exp (—} Icosh^- + —sinh \ 2J I 2 ax ~2 2Ro f at\ . , axt H exp —— smh-—. ai \ 2J 2

(8.8)

From (8.8), X\mE[W(t)} = 0 as t -+ oo (Chandrasekhar 1943, p.30). The time path of J£[W(tf)] depends on which of the three possibilities for a\ = \JOL2 — 4/3 obtains. If a\ is real and positive, or equal to zero, ^ [ W ^ ) ] would be damped over time towards zero. If ot\ is imaginary, E[W(t)] would fluctuate towards zero over time. 14 Properties of harmonic oscillation are well documented in standard texts on stochastic processes (see e.g. Soong 1973, §8.3 and Gard 1988, §4.2). In particular, the behavior of W(t) over time is that of the harmonically bound particle, which can be approximated by a white noise excitation around zero as t —> oo. In the present case, where W(t) = \nP(t)—lnp, the behavior of E'fW^)] reflects the nature of intrinsic value of the currency or its purchasing power parity. In long period equilibrium, as information becomes complete so that in the limit probability densities collapse on their means, the market would value the currency at exactly what it would be worth in terms of relative purchasing power. Moreover, since W(t) excites around zero, the exchange rate P(t) would describe a typical sample path under attraction around its purchasing power parity py even in the limit. Individuals holding such a 14

Details are omitted, as Chandrasekhar (1943) contains a long discussion. 76

Pricing Foreign Exchange Options currency do not face the prospect of becoming "almost certainly ruined", because unlike the case of GBM, there is no possibility that its spot price would suffer a persistent downward drag towards zero over time. Since W(t) = V(t) - V, where V(t) = lnP(t), V = lnp, is a 1:1 mapping, applying a standard variate transform to (8.7), we obtain the transition density function of V(t) as:

mt,V0,Ro) = X_J_«xp{__L_V[v

+

+

(Vo - ^ ) e x p ( - f )

(coshf+^-sinhtf)

^-(-fh^}f}

™

where V(0) = V0 = W0 + V is the initial value of V(t). Similarly, V(t) = In P(t) is a 1:1 mapping, so applying a variate transform to (8.9), we obtain the transition density function of the exchange rate P(t) as:

e(p,t,p0,Ro)= PV2Tto- ^ / o _2F{ty~*\ _ 2 ^ e x p { - 22a^2F{t)[\nPfi ( <xt\ ( , ait a < lnp + cexpl — — I I cosh-—- -\

L

\

2

/ V

2

a!

. , «it\ sinh—— I

2 )

where c = (lnP 0 — lnp) 3 a n ( i -P(O) = PQ = exp(Vb) is the initial value of P(t). Because of the relationships between (8.7), (8.8) and (8.9), the exchange rate P(t) has a transition density function which depends on its purchasing power parity or intrinsic value p and the initial premium rate of return R0y as well as on its initial value PQ. From inspection of (8.10), it is clear that 6(PytyPoyRo) is the marginal density function of P(i)y which is obtained from the joint density function x(P, Ryt, Po,Ro) of P(t) and R(t) by setting R(t) = RQ. 77

Pricing Foreign Exchange Options Knowledge of an exchange rate's transition density function is essential to many financial decisions. In particular, as Smith (1976) has shown, it allows one to adopt a direct approach when evaluating foreign exchange options, by taking the mathematical expection of the difference between the currency's spot price and the exercise price. This is the subject of the next Section. 8.3

A n E x a c t Formula t o Price Forex Options

Following Smith (1976), if we look forward from the current period t = 0, at the expiry date T > 0 an European call option on the stock with exercise price Q has the value: /•OO

C(T)=

(P-Q)9(PyTyPo,Ro)dP.

(8.11)

JQ

Since in (8.10), py Po, RQ, a. and a\ are given parameters, 15 for any t the expression: . cx\t a . . ait 2R0 ( ott\ . , ait cosh—-—| smh—— + lnp + cexp exp — — smh—— 2 c*i 2 ai *V 2 7 2 is a constant. Denoting it by fi(t)y we may re-write (8.10) as:

(?)

0(P,tyPo,Ro)

=

1 Pay/2irF(t)

[lnP-MQ]

exp

21

(8.12)

2
(8.12) shows that for any ty the exchange rate P(t) is lognormally distributed with mean exp[/i(t) + a2F(t)/2]y and variance equal to exp[2fi(t) + <7 2 F(*)]{exp[<7 2 F(*)]-l}. Substituting (8.12) into (8.11), we have for t = T: /•OO

C(T)=

/ JQ

[lnP-/*(T)]: 2(r 2 F(T)

1 (P-Q)exp P(Ty/2irF(T)

dP.

(8.13)

'lnP0 + In h(T) - In Q - o-2F(T)/2 '

(8.14)

It follows that (see Appendix): C(T) =

h(T)P0N

'lnP0 + Inft(T) - InQ +

<x2F(T)/2\


^\/^) 15 For details, see Chandrasekhar (1943), Ch.2, §3.14. In the derivation, we take the case of a\ real and positive. Formulae for the remaining cases of a\ = 0 and ot\ imaginary may be obtained by making the appropriate substitutions in (8.7).

78

Pricing Foreign Exchange Options where h(T) = E[P(T)]/PQ

is the number of times the currency is expected

to rise in (spot) price, up to the date the option expires. for E[P(T)l

Substituting

we have h(T) = {exp|/i(T) +
depends on the currency's initial spot price P 0 , and (through fi(T) and F(T)) its intrinsic value py and the initial premium rate of return RQ. Following Smith (1976), discounting (8.14) back to t = 0 using an appropriate (given) rate of interest py we obtain the exact option valuation formula for the currency: 16 C*(T,Q,p,p,Po,i2o)

= exp(-pT) „

ATfhi

QN

{

W

^

Pp + In h(T) - In Q -


*2F(T)/2

(8.15)

The value of currency option depends on the expiry date T, exercise price Q, the discount rate py and the initial values of the exchange rate P 0 and premium rate of return Po. Moreover, formula (8.15) captures the influence of the currency's purchasing power parity or intrinsic value p on the value of the option. 8.4

How t o Use t h e Exact Formula Let us briefly discuss the computation of currency option prices using

formula (8.15). To use the formula, apart from information on T, Q, p, Po and Po, we must have estimates of the parameters a, /?, a and p for a currency. 17 The purchasing power parity p can be estimated using standard techniques (see e.g. Levich (1984)).18 Once it is known, using data on the exchange rate P(t) we can construct a time series for W(t) =• In P(t) — In p. Values for PQ and Po can also be obtained from the time series on P(t). Since 16

For a variable discount rate, the exponential term in (8.15) becomes exp[/J t p(t)dt]. In computational procedure, (8.15) is therefore similar to the classic Black-Scholes formula (see Cox and Rubinstein (1985), Ch.6). 18 "Special market knowledge" (Cox and Rubinstein 1985,p.55) can be incorporated into estimates of purchasing power parity. 17

79

Pricing Foreign Exchange Options in (8.6) A(t) is a Gaussian white noise, the second order Langevin equation can be estimated to yield unbiased and consistent estimators for a, /? and a. Given estimates for p, ay (3 and
Conclusion

In modelling exchange rate dynamics, this Chapter proposes an alternative stochastic specification to the "work horse" of modern finance — geometric Brownian motion. A two dimensional stochastic process is specified, to synthesise the theory of foreign exchange options and the theory of international trade, by incorporating the influence of a currency's intrinsic value or purchasing power parity. Moreover, we are able to avoid the Samuelson-Kozin "almost certain ruin" problem, which is present when asset prices are assumed to follow geometric Brownian motion. Our results, which include a closed form expression for the transition density function of the exchange rate and an exact formula to value currency options which incorporates the non random walk effects of its purchasing power parity, offer a theoretical framework for further study of foreign exchange markets, and a partial framework for computation of the prices of foreign currency options. In both directions developments, extensions, and in particular empirical investigation of parameter values are very much awaiting. References Biger, N. k J. Hull (1983). The valuation of Currency Options, Financial Management, 12, 24-28. Bharucha-Reid, A.T. (1960). Elements of the Theory of Markov Processes and Their Applications. New York: McGraw Hill. Black, F. k M. Scholes (1973). The Pricing of Options and Corporate Liabilities, Journal of Political Economy, 8 1 , 637-654. Bodurtha J.N. k G.R. Courtadon (1987). Tests of an American Option Pricing Model on the Foreign Currency Options Market, Journal of Fi80

Pricing Foreign Exchange Options nancial and Quantitative Analysis, 22, 153-167. Campbell, J.Y., A.W. Lo k A.c. Mackinlay (1997). The Econometrics of Financial Marketsy Princeton NJ: Princeton University Press. Chandrasekhar, S. (1943). Stochastic Problems in Physics and Astronomy, Review of Modern Physics, 15, 1-89. Cheung, M.T. k D.W.K. Yeung (1992). Problems in the AppUcation of Geometric Brownian Motion to Finance, Proceedings of the Ilth Symposium on Operations Research, Berlin: Physica Verlag. Cheung, M.T. k D.W.K. Yeung (1994a). A Non-Random Walk Theory of Exchange Rate Dynamics with Applications to Option Pricing, Stochastic Analysis and Applications 12, 141-159. Cheung M.T. k D.W.K. Yeung (1994b). Divergence Between Sample Path and Moments Behavior: an Issue in the Application of Geometric Brownian Motion to Finance, Stochastic Analysis and Applications 12, 277-291. Cox, J.C. k S.A. Ross (1976). The Valuation of Options for Alternative Stochastic Processes, Journal of Financial Economics, 3, 145-166. Cox, J.C. k M. Rubinstein (1985). Options Markets, New Jersey Prentice Hall. Frenkel, J.A. k M.L. Mussa (1984). Asset Markets, Exchange Rates, and the Balance of Payments: The Reformulation of Doctrine, in R.W. Jones and P.B. Kenen eds., Handbook of International Trade, Amsterdam: North Holland. Gard, T.C. (1988). Introduction to Stochastic Differential Equations, New York: Dekker. Garman, M.B. k S.W. Kohlhagen (1983). Foreign Currency Option Values, Journal of International Money and Finance, 2, 231-253. Gihman, I. k A.V. Skorohod (1972). Stochastic Differential Equations, New York: Springer Verlag. ' Goldenberg, D.H. (1991). A Unified Method for Pricing Options on Diffu-

81

Pricing Foreign Exchange Options sion Processes, Journal of Financial Economics, 29, 1-34. Haugen, R.A. (1995). The New Finance: the Case Against Efficient Markets, Englewood Cliffs NJ: Prentice Hall. Hull, J. (1991). Introduction to Futures and Options Markets, New Jersey: Prentice Hall. Iranpour, R. k P. Chacon (1988). Basic Stochastic Processes: the Mark Kac Lectures, New York: Macmillan. Kaehler, J. k P. Kugler (1994). Econometric Analysis of Financial Markets, Heidelberg: Physica-Verlag. Kozin, F. (1972). Stability of Linear Stochastic Systems, in R. Curtain ed., Stability of Stochastic Dynamic Systems, New York: Springer Verlag. Levich, R.M. (1984). Empirical Studies of Exchange Rates: Price Behavior, Rate Determination, and Market Efficiency, in R.W. Jones k P.B. Kenen eds., Handbook of International Trade, Amsterdam: North Holland. Lothian, J.R. k M.P. Taylor (1996). Real Exchange Rate Baheviour: The Recent Float from the Perspective of the two Past Two Centuries, Journal of Political Economy, 104, 488-509. Malkiel, B.G. (1996). A Random Walk Down Wall Street 6th ed., New York: Norton k Co. McQueen, G. k S. Thorley (1991). Are Stock Returns Predictable? A Test Using Markov Chains, Journal of Finance, 46, 239-263. Samuelson, P.A. (1965). Rational Theory of Warrant Pricing (with Appendix by H.P. McKean), Industrial Management Review, 6, 13-31. Reprinted in Merton, R.C. ed., The Collected Scientific Papers of P.A. Samuelson, Vol.2. Cambridge: MIT Press. Samuelson, P.A. (1991). Long Run Risk Tolerance when Equity Returns are Mean Regressing: Pseudoparadoxes and Vindication of "Businessman's Risk", in W.C. Brainard, W.D. Nordhaus k H.W. Watts eds., Money, Macroeconomics and Economic Policy: Essays in Honor of James Tobin, Cambridge, Mass.:MIT Press, 1991. 82

Pricing Foreign Exchange Options Smith, C.W. (1976). Options Pricing: A Review, Journal of Financial Economics, 3, 3-51. Soong, T.T. (1973). Random Differential Equations in Science and Engineering, New York: Academic Press. Uhlenbeck, G.E. k L.S. Ornstein (1930). On the Theory of Brownian Motion, Physical Review, 36, 823-841. Wang, M.C. k G.E. Uhlenbeck (1945). On the Theory of Brownian Motion II, Review of Modern Physics, 17, 323-342.

83

Pricing Foreign Exchange Options Appendix From (8.13):

[kp-tit)]:

f°° 1 exp C(t) = / Q — —
r

2
Q

Q P
exp

[in

dP

p-m)21

2a2F(t)

dP.

and substituting exp[(fi(t)+(r2F(t)/2]

Defining h(t) = E[P{t)]/P0 we may write:

fi(t) = In PQ + In h(t)

a2F(t)

(8.16) for E[P(t)],

(8.17)

Let us introduce the random variable G(t) = P(t)/Po. g = In G(t) is then normally distributed with mean \i3 = fi(t)—In P0, and variance cr2 =
f(G) =

G
exp

[ l n G - l n / i ( t ) + o- 2 F(0/2] : 2a2F{t)

(8.19)

Recall that P0G(t) = P(i). (8.16) then becomes:

C(t) = f

IQ/P0 JQ,

[inG -lnh(t) + cx2F(t)/2]' 2a2F(t)

PoG-Q :exp G
dG. (8.20)

Transforming the variable in (8.20) to g = In G: C(t) =

Po as/2itF{t)

/•OO

/ JA

exp(flf) exp

[g-\nh(t)

+ a2F(t)/2]2 dg 2
[g- Inh(t)+ a2F(t)/2]'J 2
L

Q
dg (8.21)

Pricing Foreign Exchange Options where A= (In Q — In PQ). Re-arranging the first term on the RHS of (8.21), and writing F for F(t)y h for h(t) to make the following lengthy expressions less cumbersome, we obtain: Po exp aV2irF J A

J:

g2 - 2(\nh-

a2F/2)g -f (in/i - <j2F/2) - 2a2Fg 2<72P

dg

Po
f

exp

[g - (Inft - <72P/2) -
JA

Since -a4F(t)2

a2FI2)a2F

dg

- 2 [in h(t) -
may be written:

f

Po C(t) = exp(ln h - a F + a F) exp ay/2irF J A 2

2

(g-lnh-
21

dg

JA

(g-\nh

Q 0-V27TF J A

exp

hp0 r exp aV2irF J A Q n-KF

i

exp

+
[g-(\nh + c2F/2)X 2
dg

21

dg,

(8.22)

JA

where once again F(t) and h(t) are shortened to F and h. Let us transform the normal distribution functions on the RHS of (8.22) to standard normal form by means of the substitutions:

_ {g - [Inh(t) +
{g-[lnh(t)-*2F(t)/2]}

This gives:

"W-^JM-SK&JM-T)* 85

*«

Pricing Foreign Exchange Options where _ [In Q - In PQ - In h(t) -
B=

WW)

_ [In Q D=

'

In P 0 - In h(t) + <x2F(t)/2]

*vm

'

Since with the standard normal distribution, 1 — N(£) — N(—£), we have: C(t) = h(t)P0N(-B) = h(t)PoN

_ QN

hnPo

-

QN(-D) +

lnhW-pQ +
f\nP0+\nh(t)-lnQff

V

V^W

Putting t = T in (8.24), (8.15) obtains.

86


/

(8.24)

CHAPTER 9 CONCLUSIONS

The appropriate choice of assumptions is a matter of primary importance in applied economics. In option pricing, it is generally assumed that the underlying asset has a price which fluctuates over time as a geometric Brownian motion. Since a number of problems then arise, the present volume proposes an approach to the valuation of foreign exchange options based on an alternative assumption. Definitions and terminology are introduced in Chapter 2. A technical glossary is given in Chapter 3. The remainder of the book is devoted to a new stochastic specification to model the spot price of the representative currency underlying foreign exchange options, and to showing that it is free from the problems which arise under the standard geometric Brownian motion assumption. An example is presented in Chapter 4 to convince the reader of the importance of stochastic assumptions in option pricing, a n d t o demonstrate how a different specification leads in general to a different valuation formula. Chapter 5 surveys the Black-Scholes (1973) theory, which is the basis of modern work on option pricing. It is shown in Chapters 6 and 7 that two serious problems arise in this theory. First, since it is assumed that the price of the underlying asset follows geometric Brownian motion, it is possible for assets to exist which lead the investor to "almost certain ruin" (Samuelson 1965), in the sense that over time their prices would drop to zero with probability one, positive and increasing expected rates of return notwithstanding. The technique commonly used to solve Black-Scholes differential equations for option prices does not exclude such assets, which makes it difficult to maintain an (general) equilibrium interpretation of the resulting option price. Second, given the geometric Brownian motion assumption, the price of an asset displays the characteristics of a random walk. Its price at any point of time in the future depends solely on the present price. This property has been called into question by recent research. For example, it is shown that returns to U.S. common stocks in the postwar period show significant non-random walk behavior, in the sense that runs of high and low returns tend to follow one another. 87

Pricing Foreign Exchange Options To overcome the difficulties arising from the assumption of geometric Brownian motion, Chapter 7 proposes an alternative stochastic specification to model the dynamic behavior of asset prices, in particular the price of the asset which underlies the representative option. We characterise the solution of the resulting system of stochastic differential equations in the most complete form known, by obtaining a closed form expression for the transition density function of the asset price. It is shown that non random walk effects can be incorporated into the analysis, and that the stochastic process of the asset price excludes the possibility of almost certain ruin. Since "the option valuation problem is equivalent to the problem of determining the distribution of the asset price", we proceed to take mathematical expectations directly in terms of the transition density function, to illustrate how an exact formula can be found to evaluate options on the asset. Chapter 8 shows in detail how the new stochastic specification can be used to price options on foreign exchange. In contrast to the random walk-restriction imposed by the standard theory, we are able to incorporate a fundamental result of the theory of international trade, that over time the spot price of a currency (its exchange rate) converges to its purchasing power parity. It is also shown that the exchange rate process excludes the possibility of almost certain ruin. Computational aspects of the formula are also discussed, in particular, the availability of data, and the methods by which the parameters in the option price formula can be estimated.

88

A Note on Software A software package to compute the prices of foreign exchange options using formula (7.15) is being developed by the authors, under sponsorship from Cypress International Investment Advisors Ltd.

89

Index Almost certain ruin, 36, 38, 39, 41, 50, 52, 55, 65, 70. Black, F., 17, 18, 20, 28, 39, 50, 51. Call options, 4. Chandrasekhar, S., 54, 55, 68, 69, 71. Cheung, M.T., 21, 38, 59. Foreign exchange (rate), 64, 66, 69, 70, 71. Foreign exchange options, 6, 71, 72. Geometric Brownian motion, 14, 17, 18, 19, 23, 24, 25, 37, 45, 46, 55, 64, 65. Harmonic oscillator, 55, 69. Hedge portfolio, 18, 19. Kolmogorov backward equation (Fokker Planck equation), 21, 23, 24, 25, 53, 67, 68. Kozin, F., 37, 38, 39, 65. Option pricing equation, 18, 28, 62, 63, 76, 77. Purchasing power parity, 69, 70, 72, 74, 75, 77. Premium rate of return, 57, 58, 71, 72. Put options, 4. Random walk, 11, 50, 64. Risk adjusted equihbrium, 41, 42. Risk neutral equilibrium, 21, 28, 36, 39, 41. Samuelson, P.A., 17, 29, 37, 50, 51, 65. Scholes, M., 18, 20, 28, 39, 50, 51. Stochastic process, 8, 50, 51, 64, 67. 90

Pricing Foreign Exchange Options Stochastic differential equations, 15, 19, 23, 37, 51, 54, 65, 68. Transition density function, 21, 23, 24, 25, 26, 37, 38, 53, 56. Yeung, D.W.K., 21, 38, 59.