Pricing Foreign Exchange Options: Incorporating Purchasing Power Parity

Incorporating Purchasing Power Parity Second edition i«

David W.K.Yeung Michael Tow Cheung

DI

I

1HHI

Incorporating Purchasing Power Parity

II

pi

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 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/ 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 - 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

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 + 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 0, its transition density function ip(xo,tyx) would then satisfy the Kolmogorov backward equation: dip, / =TM-T}

e{m



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

/

31

^

+

T

^

)

^lK)+£- 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- a2 > 0, a - 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)}

"^