Mathematical methods for foreign exchange a financial engineers approach

The exposition is restricted mainly to problems occurring in the foreign exchange forex context not only due to the fact that it is the author’s current area of responsibility but also b

Mathematical Methotis

for Foreign Exchange

A Financial Engineer’s Approach

Alexander Lipton

Mathematical Methods for Foreign Exchange

Mathematical Methods for Foreign Exchange

A Financial Engineer's Approach

Published by

World Scientific Publishing Co Pte Ltd

5 Toh Tuck Link, Singapore 596224

USA office: Suite 202, 1060 Main Street, River Edge, NJ 07661

UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data

A catalogue record for this book is available from the British Library

First published 2001

Reprinted 2003

MATHEMATICAL METHODS FOR FOREIGN EXCHANGE: A FINANCIAL

ENGINEER’S APPROACH

Copyright © 2001 by World Scientific Publishing Co Pte Ltd

All rights reserved This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA In this case permission to photocopy is not required from the publisher

ISBN 981-02-4615-3

ISBN 981-02-4823-7 (pbk)

Printed in Singapore by World Scientific Printers

This Book is Dedicated

To Marsha

The importance of money essentially flows from it

being a link between the present and the future

John Maynard Keynes

Money and currency are very strange things

They keep on going up and down and no one knows why;

If you want to win, you lose, however hard you try

Gilles li Muisis

Preface

This book is devoted to some mathematical problems encountered by the au- thor in his capacity as a mathematician turned financial engineer at Bankers Trust and Deutsche Bank The exposition is restricted mainly to problems occurring in the foreign exchange (forex) context not only due to the fact that

it is the author’s current area of responsibility but also because mathematical methods of financial engineering can be described more vividly when the expo- sition is centered on a single topic Studying forex is interesting and important because it is the grease on the wheels of the world economy Besides, while the meaning of some financial instruments is difficult to comprehend without prior experience, everyone who has ever travelled abroad has had to exchange currencies thus acquiring direct experience of such concepts as spot forex rate, bid-ask spread, transaction costs (in the form of commissions), etc At the same time, the reader who acquires working knowledge of the material pre- sented in this book should be able to handle efficiently most of the problems occurring in equity markets and some of the problems relevant for fixed income markets

If one were to choose just one word in order to characterize financial mar- kets, that word would be uncertainty since it is their dominant feature Some investors consider uncertainty a blessing, while others think it a curse, yet both groups participate in the intricate inner workings of the markets The fact that foreign exchange rates (relative prices of different currencies), as well

as prices of bonds (government or corporate obligations to repay debts) and stocks (claims on future cash flows generated by companies) are random and financial investments are risky was realized long ago and has been a source

of fascination for economists, mathematicians, speculators, philosophers, and moralists, not to mention laymen

Due to the random nature of financial markets, trying to predict future prices of individual financial instruments makes little sense However, one can introduce so-called derivative instruments (the name indicates that their

ix

This book is devoted to various problems which financial engineers face

in the market place and gives a detailed account of mathematical methods necessary for their solution Even though the exposition is presented from

a financial engineer’s prospective, the author tried his best to expose all the necessary details At the same time, mathematical rigor as such was not high

on the author’s priority list In particular, most of the results are not formu- lated as theorems and lemmas since this traditional format is not adequate for the purposes of the present exposition We start with a brief survey of relevant mathematical concepts After that we present an in-depth discussion

of discrete-time models of forex We distinguish between single-period and multi-period models In both cases the corresponding models are too stylized

to be of practical importance but they do allow the reader to understand some

of the issues which occur in more complicated situations For this reason, and because of their aesthetic appeal, these models deserve a careful study We analyze conditions which guarantee that a particular model is financially rea- sonable and show how to price derivatives and solve the optimal investment problem for such models Once discrete-times models are mastered, we switch our attention to more practically useful continuous-time models We describe

in detail a variety of models, starting with the standard Black-Scholes mode! and ending with rather involved stochastic volatility models with a special em- phasis on practical aspects We then show how to price derivatives and solve the optimal investment problem in the continuous setting

Recently, several very good (and some not so good) books dealing with various aspects of financial engineering were published The author hopes that the present book can complement the existing literature on the subject and will be useful to the reader in more than one way In fact, when deciding whether to write this book, he followed the advice of Franz Kafka who once said “Such books as make us happy, we could, if need be, write ourselves”

In the process of writing this book the author enjoyed help, advice, and support of various individuals First and foremost, he is grateful to his wife Marsha, father Yefim, mother Eugenia, and daughter Rachel Next, he is also deeply grateful and much indebted to his fellow quants, especially to Christo-

Preface xi

pher Berry, Stewart Inglis, and William McGhee, as well as to Peter Carr, Brian Davidson, Vladimir Finkelstein, Ken Garbade, Arvind Hariharan, Tom Hyer, Andrew Jacobs, Bin Li, Dmitry Pugachevsky, Eric Reiner, and Paul Romanelli Last but not least, he greatly benefited from the interactions with a group of outstanding managers and traders including Hal Herron, Dan Almeida, Jim Turley, Kevin Rodgers, Matt Desselberger, Perry Parker, and Andrew Baxter Reasonable efforts were made to publish reliable information However, in

a book like this one typing and other errors are unavoidable The author and the publisher do not assume any responsibility or liability for the validity of the information presented in this book and for the consequences of its use or misuse The book represents only the personal views of the author and does not necessarily reflect the views of Deutsche Bank, its subsidiaries or affiliates Finally, a few words about the epigraphs J M Keynes needs no introduc- tion The Abbot Gilles li Muisis of Tournai lived in the fourteenth century His wonderful verse is quoted by P § Lewis in “Later Medieval France” and

by B W Tuchman in “A Distant Mirror”

Alexander Lipton New York and London March 2001

2.5 References and further reading

Discrete-time stochastic engines

xiv Contents

3.56 References and further reading 57

4 Continuous-time stochastic engines 59 4.1 Introduction 2 0 ee 59 4.2 Stochastic processes .0 0 0000 0000004 61 4.3 Markov processes .0 0000022000 vee 63 4.4 Diffusions 2 ee v2 65 4.5 Wiener processes 20.0.0 0 000000 e epee eae 76 4.6 Poisson processes 2 2 ee 81 4.7 SDE and Mappings 0 0.000 0000.4 84 4.8 Linear SDEs 0 0 00000000004 91 4.9 SDEs for jump-difusions 96

4.10 Analytical soluion of PDES 98

4.10.1 Introduction .0 0 0000.2 20 000 98 4.10.2 The reductlon method 98

4.10.3 The Laplace transform method 103

4.10.4 The eigenfunction expansion method 104

4.11 Numerical solution of PDEs 106

4.11.1 Introduction .0.02 0.0.00 000 2 eee 106 4.11.2 Explicit, implicit, and Crank—Nicolson schemes for solving one-dimensional problems 107

4.11.3 ADI scheme for solving two-dimensional problems 109

4.12 Numerical solution of SDEs .- 4 112 4.12.1 Introduction .-.2 2.2-20205 112 4.12.2 Formulation of the problem 113

4.12.3 The Eule-Maruyama scheme 115

4.124 The Misten scheame - 114

4.13 References and frtherreading 116

III Discrete-time models 119 5 Single-period markets 121 5.1 Introduction 0.0 2.2- 20000 ee ee eee 121 5.2 Binomial markets with nonrisky investments 123

5.3 Binomial markets without nonrisky investments 140

5.4 General sinpleperlod markets 145

5.5 Economic constraints - - 00522.0-02 0004 147 5.6 Pricng of contingent clams 154

5.7 Elementary portfolio theory . .- 162

Contents

5.8

5.9 The optimal investment problem Elements of equilibrium theory

5.10 References and furtherreading

6 Multi-period markets 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 7.1 Tntroduclion Q Q Q HQ HQ và Stationary binomial markets

Non-stationary binomial markets

6.3.1 Introduction .0 220058

6.3.2 The nonrecombining case .0.4 6.3.3 The recombiingcase

General multi-period markels

Contingent claims and their valuation and hedging

Portfolio theory 2 2 ee ee The optimal investment problem

References and further reading

IV Continuous-time models 7 Stochastic dynamics of forex Introduction 2 ee Two-country markets with deterministic investments

7.2 7.3 7.4 7.5 7.6 7.7 7.8 Two-country markets without deterministic investments

Multi-country markets 2 0.-0.0.0000002007 | The nonhnear difusion model

The jump difusion model

The stochastic volatility model

The general forex evolution model

7.9 References and further reading .- -.-

8 European options: the group-theoretical approach 8.1 Introduclilon Ặ QẶ LH SH ha 8.2 The two-country homogeneous problem,Ï

8.3 8.2.1 Formulation of the problemn

8.2.2 Reductions of the pricing problem

8.2.3 Continuous hedging and the Greeks

Forwards, calls and puts QC 8.3.1 Definitions .2 2 22.0 20000 8.3.2 Pricing via the Feynman—Kac formula

XV

166

168

169

171

171

172

194

194

195

197

202

207

208

210 212

xvi Contents

8.3.3 A naive pricing attempt .0 000 256

8.3.4 Pricing via the Fourier transform method 257

8.3.5 Pricing via the Laplace transform method 259

8.3.6 The limiting behavior of calls and puts 261

8.4 Contingent claims with arbitrary payofs 263

8.41 Introduction .02.000 263 8.4.2 The decomposition formula 263

8.4.3 Calland put bets 00 264

8.4.4 Log contracts and modified log contracts 265

8.5 Dynamic asset allocation .0 00.000% 266 8.6 The two-country homogeneous problem, lÏ 275

8.7 The multi-country homogeneous problem 277

8.71 Introduction .00000 277 8.7.2 The homogeneous pricing problem 278

8.7.3 Reductions .0 0 000000000004 279 8.7.4 Probabilistic pricing and hedging 280

8.8 Some representative multi-factor options 281

8.8.1 Introduction 0.02.2 0000.5 281 8.8.2 Outperformance options 282

8.8.3 Options on the maximum or minimum of several FXRs 284 8.8.4 Basket options .0 00000 286 8.8.5 Index options 0200022 ee eee 290 8.8.6 The multifactor decomposition formula 291

8.9 References and further reading 292

European options, the classical approach 293 9.1 Introduction .-2.02.22 0200000002 293 9.2 The classical two-country pricing problem,I 294

921 The projectionrmethod 294

9.2.2 The classical method 296

9.23 The impact of the actual driÍt 297

9.3 Solution of the classical pricing problem 298

9.3.1 Nondimensionalization - 04 298 9.3.2 Reductions .20 0000 eee eee ee 298 9.3.3 The pricing and hedging formulas for forwards, calls and puts 2 ee 299 9.3.4 European options with exotic payofis 306

9.4 The classical two-country pricing problem, I 310

9.5 The multi-country classical pricing problem 315

9.5.1 Introduction .0.0.0 0.000 eee ee ee 315

Contents xv1

9.5.2 Derivation 2 0.0.0.0 2 ee ee 315 9.5.8 Reductions .2 0000002 ee 315 9.5.4 Pricing and hedging of multi-factor options j17

9.6 References and further reading .-. 2 0-.2 317

10 Deviations from the Black-Scholes paradigm I: nonconstant volatility 319 10.1 Introduction 2 2 ee ee 319 10.2 Volatility term structures and smiles 321

10.2.1 Introduction .2 2.2.-0- 321

10.2.2 The implied volailty 321

10.23 The local volatlty 323

10.2.4 The inverse problem 325

10.2.5 How to deal with the smile 329

10.3 Pricing via implied t.p.df’s 2 2.0.00 00.2 0.00 329 10.3.1 Implied t.p.d-f.’s and entropy maximization 329

10.3.2 Possible functional forms oft.p.dÊ'§S 332

10.3.3 The chi-square pricing formula,I 335

10.3.4 The Edgeworth-type pricing formulas 338

10.4 The sticky-strike and the sticky-deltamodels 341

10.5 The general local volatility model 344

10.5.1 Introducliion Q Q Q Q Q HQ V2 344 10.5.2 Possible functional forms of local volatility 345

10.5.3 The hyperbolic volatility model 348

10.5.4 The displaced difusion model 350

10.6 Asymptotic treatment of the local volatility model 353

10.7 The CEV model .0.0.2.02.0.020200.0 040 359 10.7.1 Introduction .02 00 359 10.7.2 Reductions of the pricing problem 360

10.7.3 Evaluation of thet.p.df 362

10.7.4 Derivative pricing .0.0 364 10.7.5 ATMF approximation 368

10.8 The Jump difusion model 371

10.8.1 Introduction .0 0.0.0.0 000.0 371 10.8.2 The pricing problem 371

10.8.3 Evaluation of the t.p.df 372

10.8.4 Risk-neutral pricing .2 002022 373

10.9 The stochastic volatillty model 375 10.9.1 Introduction .20 0 00.0 375

Xxvii Contents

10.9.3 Evaluation of thet.pd.F 379

10.9.4 The pricing formula 384

10.9.5 The case of zero correlation 386

10.10 5mail volatility of volatiity 387

10.10.1 Introduction .0 0 0 000004 387 10.102 Basicequations 388

10.10.3 The martingale formulaton 388

10.10.4 Perturbative expansion 389

10.10.5 SummaryofODESs 393

10.10.6 Solution of the leading order pricing problem 394

10.10.7 The square-root model 394

10.10.8 Computation of the implied volatility 397

10.11 Multi-factor problems 398

10.11.1 Introduction 1 0.00.2 ee 398 10.11.2 The chi-square pricing formula, II 399

10.12 References and further reading 404

11 American Options 405 11.1 Introduction Q Q Q SH HQ nh V 405 11.2 General considerations - TQ 407 11.2.1 The early exercise constraint 407

11.2.2 The early exercise Dremiun 408

11.2.3 Some representative examples 410

11.24 Rationalbounds 411

11.2.5 Parity and symmetry -.- -2 20 414

11.3 The risk-neutral valuation - 415

11.4 Alternative formulations of the valuation problem 416

11.4.1 Introduction .- .2 2 -.2 416

11.4.2 The inhomogeneous Black-Scholes problem formulation 416 11.4.3 The linear complementarity formulation 418

11.4.4 The linear program formulation - 420

11.5 Duality bebtween puts and cals 421

11.6 Application of Duhamel’s princple 422

11.6.1 The value of the early exercise premium 422

11.6.2 The location of the early exercise boundary 423

11.7 Asymptotic analysis of the pricing problem 425

11.7.1 Short-dated options .- 425

11.7.2 Long-dated and perpetual options 430

11.8 Approximate solution of the valuation problem 434

11.8.1 Introduction .002 2 0.022200 434

Contents xix

11.8.2 Bermudan approximation and extrapolation to the limit 434

11.8.3 Quadratic approximation 440

11.9 Numerical solution of the pricing problem 442

11.9.1 Bermudan approximation 442

11.9.2 Linear complementarity 443

11.9.3 Integral equation So 444 11.9.4 Monte-Carlo valuation 444

11.10 American options in a non-Black-Scholes ramework 445

11.11 Multi-factor American options .- 0005 445 11.11.1 Formulation .0.0 0.002.202 eee eee 445 11.11.2 Two representative examples 446

11.12 References and further reading 449

12 Path-dependent options I: barrier options 451 12.1 Introduction © ee 451 12.2 Single-factor, single-barrier options 452

12.2.1 Introduction 2.00.02 00088 452 12.2.2 Pricing of single-barrier options via the method of images 453 12.2.3 Pricing of single-barrier options via the method of heat poteniials Q Q Q Q HQ HQ HQ v2 462 12.3 5tatichedging Q Q Q Q QC 469 12.4 5ingle-factor, double-barrieroplions 472

12.4.1 Introduction .0.2 020000004 472 12.4.2 Formulation 2.02 00004 473 12.43 The pricing problem withoutrebates 474

12.4.4 Pricing of no-rebate calls and puts and double-no-touch options 2 2 ee ee A77 12.4.5 Pricing of calls and puts with rebate 482

12.5 Deviations from the Black-Scholes paradigm 484

12.5.1 lmroduclon Q Q Q Q HQ vi 484 12.5.2 Barrier options in the presence of the term structure of volatility 2 ee 484 12.5.3 Barrier options in the presence of constant elasticity of variance 2 ee ee 486 12.5.4 Barrier options in the presence of stochastic volatility 492 12.6 Multi-factor barrier optiions 498

12.7 Options on one currency with barriers on the other currency 499 12.7.1 Introduction .020000 499 12.7.2 Formulation 2.0.0.0 .0.0 00000000 499

13 Path-dependent options II: lookback, Asian and other options527 13.1

13.6.1 Description .0.0.0 2.000.000.0000 553 13.6.2 Geometric averaging .200.2 553 13.6.3 Arithmetic averaging 556 13.6.4 Exact solution via similarity reductions 558 13.6.5 Pricing via the Laplace transform 561 13.6.6 Approximate pricing of Asian calls revisited 562 13.6.7 Discretely sampled Asian options 565 Timer, fader and Parisian options 566 13.7.1 Introduction .0.- 200022000000 566 13.7.2 Timer options 2.000220 0000 0G 566 13.7.3 Fader options .-.0.2004 573 13.7.4 Parisian options .- 02+ 000.% 973 Standard passport options Lo Q 578 13.8.1 Description 2 22.0022 22 0000.0 2 ee 578 13.8.2 Similarity reductions and splitting 579

14 Deviations from the Black-Scholes paradigm II: market fric-

14.2 Imperfect hedging .0.2.02.00000.4 618 142.1 P&Ldistrbutons 618 14.2.2 Stop-loss start-gain hedging and local times 621 14.2.3 Parameter misspeciicaion 622 14.3 The uncertain volatility model Ð 627 14.4 Transaction €COSES Q Q Q HQ HQ x2 630

14.6 Defaultrisk Q Q Q Q0 Q0 HQ es 635 14.6.1 Introduction .0 0.02 00.4 635 146.2 The pricingmodel 635 14.6.3 Pricing of defaultable Puropean calls 637 14.6.4 Pricing of defaultable forward contracts 640 14.7 References and further reading 643

Part I

Introduction

$1.5 trillion Changes in foreign exchange rates (FXRs) (i.e., relative prices of different currencies) are caused by both deep structural shifts in the respective economies and a variety of less fundamental factors These changes have a profound impact on the world economy at large An adequate formalism for studying the dynamics of FXRs has been developed by a number of researchers over the last thirty years In addition to explaining the qualitative behavior

of FXRs, this formalism can be used in order to develop consistent pricing of various derivative instruments, such as forwards, calls, puts, etc., whose value depends on the value of the underlying FXRs By necessity this formalism

is probabilistic in nature and requires a solid grasp of several mathematical disciplines for its efficient usage In the present book we use these to solve a number of fundamental problems of financial engineering, such as derivative pricing, asset management, etc

This Chapter is organized as follows In Section 2 we give the a brief overview of the historic development of financial engineering In Section 3 we discuss properties of forex as an asset class In Section 4 we describe properties

of spot FXRs In Section 5 we introduce and discuss derivatives in the forex

3

4 Foreign exchange markets

context In Section 6 we give relevant references to the literature

1.2 Historical background

In different disguises, problems of financial engineering have been discussed since classical times in a variety of sources stretching from Plato’s Dialogues to the Old and New Testaments Many illustrious noble houses in Medieval Italy made their fortunes by astute dealings in money lending and foreign exchange

An intimate relation between money lending (fixed income in modern language) and money changing (foreign exchange) was used by Italian bankers collectively known as the Lombards, in order to circumvent the church prohibition on charging interest on loans They lent money in one currency and received it back in another one with the FXR artificially lowered to accommodate the interest The blossoming of the Netherlands in the seventeenth century is, at least partly, due the introduction of new financial instruments such as forward contracts, calls and puts, etc The rise of the British Empire resulted in further advances in finance, particularly, on the fixed income side The economic growth in the United States was facilitated by the unprecedented growth of the financial system, especially, by the introduction of limited liability companies The modern development of financial engineering starts with the work of the French mathematician Louis Bachelier who, in the year 1900, published the now famous memoir entitled “Theorie de la spéculation” Bachelier’s achieve- ments are remarkable in several respects To mention just one, he developed (before Einstein and others) the first theory of Brownian motion which he used

in order to quantify the evolution of stock prices In modern terms, Bachelier assumed that the stock price follows an arithmetic Brownian motion and, con- sequently, is distributed normally at any given time He derived the pricing formulas for call and put options on such stocks However, since Bachelier’s theory predicted that stock prices can become negative and because of the sheer complexity of its mathematical apparatus, the theory was neglected by the mainstream economists for more than fifty years

Fundamental contributions to modern financial engineering were made in the 1950s by several authors Arrow (1953) and Debreu (1959) extended the existing economic models by incorporating uncertainty and showed how to solve the corresponding asset allocation problem Modigliani and Miller (1958) proved that the financial structure of the firm, i.e., the firm’s choice between equity and debt financing, does not affect its value The method of financial arbitrage they used turned out to be even more useful than the theorem itself and became the method of choice for generations of financial engineers Finally,

Historical background 5

Markowitz (1959) developed the mean-variance portfolio selection theory

In the 1960s financial engineering continued to grow rapidly In particu- lar, Sprenkle (1961), Boness (1964), and Samuelson (1965) proposed a more adequate description of the stock price evolution by assuming that this price follows the geometrical Brownian motion and, consequently, is distributed log- normally which guarantees its positivity Although they obtained a closed form formula for pricing options on lognormally distributed stocks, this formula was difficult to use in practice because it contained too many free parameters, namely, the volatility of the stock, and the growth rates of the stock and option In the meantime, Sharpe (1964), Lintner (1965), and Mossin (1966) extended the Markowitz theory and created the so-called capital asset pricing model (CAPM)

The major breakthrough was achieved the early 1970s when Black and Sc- holes (1973) and Merton (1973) discovered a consistent pricing formula for stock options depending on the volatility of the underlying stock and the risk- less interest rate at which the money can be borrowed overnight (rather than the growth rates of the underlying stock and option) The Black-Scholes- Merton pricing methodology is based on the idea of dynamic hedging of deriva- tives which allows the seller of an option to become indifferent to the changes

in the underlying stock price In order to hedge himself, the seller of an op- tion has to maintain positions in both stock and bond and to adjust them dynamically when the stock price changes For their discovery Scholes and Merton were awarded the Nobel Prize for Economics in 1997 (Black died in 1995) The Black-Scholes formula instantly became extremely popular among practitioners and academics alike, and within a few years helped to create a multi-trillion dollar market in financial derivatives, currently estimated at $65 trillion (The impact of Black-Scholes discovery on financial markets is a great example of the influence of mathematics on society at large.)

Many important contributions to financial engineering were made after pub- lication of the seminal papers by Black and Scholes and Merton For instance, Merton himself and later Rubinstein and Reiner (1991), used the well-known method of images in order to price the so-called barrier options which dis- appear when the price of the underlying hits a predetermined barrier Black (1976) derived the formula for the valuation of options on futures Ross (1976) developed the arbitrage-pricing theory (APT) as an alternative to CAPM Margrabe (1978) valued the right to exchange one risky asset for another (via

an elegant application of the principle of homogeneity) Harrison and Kreps (1979) and Harrison and Pliska (1981) developed an approach to pricing and hedging of derivatives which complements the one developed by Black and Sc- holes and Merton They showed that the price of an option (provided that

6 Foreign exchange markets

it is hedged appropriately), can be written as the expectation (with respect

to the so-called risk-neutral probability measure) of its discounted payoff at maturity In probabilistic terms, they showed that the discounted value of an option is a martingale so that options can be priced via probabilistic meth- ods (which complement partial differential equations methods originally used

by Black-Scholes) Mathematical aspects of this approach were elucidated by Delbaen and Schachermayer (1994) Garman and Kohlhagen (1983) extended the Black-Scholes valuation formula in order to incorporate options on forex Although the original approach to problems of financial engineering was predominantly analytical, numerical methods necessary to solve more com- plicated problems were developed as well The earliest were the so-called binomial, explicit finite difference, and Monte Carlo methods introduced by

Cox, Ross and Rubinstein (1979), Schwartz (1977) and Brennan and Schwartz

(1978), and Boyle (1977), respectively Although these methods were very in- tuitive, they were not sufficiently refined and could not price certain derivatives accurately In a due course they were complemented by the implicit finite dif- ference method which became the method of choice for solving more advanced problems,

In spite of its many successes, the Black-Scholes formula is too idealized and does not capture certain features of the market The most important

of those features are the non-lognormal distribution of the underlying stock prices, transaction costs, liquidity, and discontinuous nature of trading When this formula is used in practice, different volatilities are used to price options with different strikes which gives rise to the so-called volatility skew (in equity markets) and smile (in forex markets) Appropriate modifications of the Black- Scholes paradigm which would account for skews and smiles observed in actual markets is a major challenge which is only partly met at present

It proved to be more difficult to model fixed income derivatives, such as bond options, caps, floors, etc., than forex and equity derivatives This dispar- ity is due to the fact that the dynamics of the underlying short term interest rate (or the bond price) is much more complicated than the forex and stock dynamics and cannot be approximated by the geometrical Brownian motion (Indeed, empirical observations suggest that interest rates are mean-reverting, while, by definition, bond prices exhibit the so-called pull-to-par property and approach bond’s notional value at maturity) In addition, one has to deal with bonds of all maturities (the so-called yield curve) at once Vasicek (1977) was the first to propose an analytically tractable model taking into account the mean reversion property of the instantaneous interest rate and the pull-to-par property He assumed that the interest rate follows the Ornstein-Uhlenbeck process and obtained analytical expressions for prices of bonds and bond op-

Forex as an asset class 7

tions However, Vasicek model is not without some limitations; namely, it cannot be fitted sufficiently accurately to the market data, and it allows in- terest rates to become negative, i.e., it suffers from the same difficulty as the original Bachelier model for equities While the first drawback is easy to rectify,

as was done by Hull and White (1990), the second one is an inherited feature of the model and cannot be helped In order to guarantee the positivity of inter- est rates one needs to use other stochastic processes with mean reversion, such

as the Feller square-root process used by Cox, Ingersoll and Ross (1985), or the log-Ornstein-Uhlenbeck process used Black and Karasinski (1991) An al- ternative idea of dealing with fixed income derivatives is based on studying the yield curve in its entirety, as described by Heath, Jarrow and Morton (1992) and Brace, Gatarek and Musiela (1997) in continuous and discrete settings, respectively In spite of the progress made by the above mentioned authors and many others, an adequate versatile model for pricing and hedging of fixed income derivatives is still missing

1.3 Forex as an asset class

As was mentioned earlier, the daily turnover of forex markets is approximately

$1.5 trillion The majority of transactions with foreign exchange are executed from London and New York (about 1/3 and 1/5 of the total, respectively), as well as Tokyo, Singapore, Frankfurt, Zurich, etc Market participants include governments, banks, international corporations, mutual and hedge funds, and individual investors With increasing globalization of the world financial sys- tem, the role of forex as an important asset class in its own right on a par with more traditional equity and fixed income instruments becomes more and more apparent Indeed, it is necessary to have an exposure to forex in order to be able to invest in the global markets and create a well-balanced portfolio When investors based in a particular country put their money in domestic bonds and equities, they are not (directly) affected by the forex rate fluc- tuations and all the uncertainties they have to face are domestic in nature However, if they decide to invest their money in foreign bonds and equities, they first have to convert the domestic currency (say US dollars) into the for- eign currency (say euros) at the deterministic spot rate in order to purchase foreign securities, and then, at some time in the future, they have to convert the foreign currency generated by these securities into the domestic currency (at the unknown rate prevailing at that time) Thus, investors have to deal with uncertainties both at home and overseas, and, in addition, with changes

in forex rate Nonetheless, foreign investments can generate returns which are

8 Foreign exchange markets

so attractive that additional risks are worth taking

For simplicity we consider only investments in domestic and foreign fixed income instruments In order to quantify uncertainties associated with these investments, we need to know the prices of domestic and foreign zero coupon bonds, i.e., the price in dollars (euros) at time ¢ = 0 of the obligation to pay one dollar (euro) at time T in the future We denote the price of the domestic and foreign bonds maturing at time T by Bj and Bj,p, respectively Assuming that the bond prices are deterministic (this is a strong assumption which is made in order to simplify the exposition), we can represent the bond prices at time ¢ as

(a Bor 1 _ Bor 1,0 0°? t,t — 1°

In addition, we need to know the forex rate S;, ie., the number of dollars one needs to pay at time t in exchange for one euro The dimension of S; is dollar/euro In contrast to bond prices which are deterministic in our simplified framework, the forex rate is random, so that its value at some future time T is uncertain We use the domestic bond as a benchmark for measuring the rate

of return on different investments It is clear that the relative rate of return on investment in domestic bonds is zero The relative rate of return on investment

0 —rÈ(T~t 1 _ „—rl(T-~t

Bop =e" | ) Bip =e" ( ),

where r° and r! are constant domestic and foreign interest rates, respectively

Derivatives: forwards, futures, calls, puts, and all that 9

so-called bid-ask spread, i.e., the difference between the lower rate at which the foreign currency can be sold to market makers and the higher rate at which it can be bought from them (This concept will be familiar to anyone who has exchanged currency abroad.) Depending on the particular currency pair, the rate is agreed upon today while the actual transfer takes place either

in two business days (US dollar/yen, euro/US dollar, pound/US dollar) or in

one business day (US dollar/Canadian dollar) Instantaneous FXRs (as well as stock and bond prices) are determined by market forces (through the “invisible hand” envisioned by Adam Smith) and reflect their intrinsic values, as well as considerations of supply and demand FXRs constantly fluctuate around their moving equilibrium values Their most important characteristics are rates of return on investments in foreign currency (yearly, monthly, daily, hourly, etc.) Rates of return are random and have to be treated via statistical methods for studying time series To give the reader an idea of what can be expected, we just mention that the distribution of daily returns for the foreign exchange rate USD/DEM over a period of ten years from 1986 to 1996 has volatility 0.11, skewness -0.1, kurtosis 5, and no daily deviations exceeded five standard devi- ations, while for the S&P 500 Index over the same period the distribution of returns has volatility 0.16, skewness -5, kurtosis 111, and that one (five) daily deviation (deviations) exceeded ten (five) standard deviations The distribu- tion of daily returns for the FXR is reasonably close to Gaussian, while for the S&P 500 Index the corresponding distribution is strongly non-Gaussian

In most cases it is not necessary to explain the observed FXRs and rates of return on investments in foreign currencies in fundamental terms; the main objective is to develop a model for pricing derivative instruments in terms of the underlying ones and to solve the asset management problem Even though,

in general, the distribution of the daily returns for underlying instruments is non-Gaussian, it is frequently assumed to be Gaussian for practical purposes Surprisingly, more often than not, this approach produces satisfactory results Typical behavior of FXRs is illustrated in Figure 1.1

1.5 Derivatives: forwards, futures, calls, puts,

and all that

Exchange of currencies at spot rates serves only the most obvious and the most immediate needs of market participants Their more sophisticated needs are met by derivative instruments The basic types of forex and equity deriva- tives are forward and futures contracts, and calls and puts; their fixed income counterparts are known as forward rate agreements, Euros, caps, and floors

10 Foreign exchange markets

40.9 104.00 J

+ 0.85 102.00 4

Options on a financial instrument are different from forward and futures

Derivatives: forwards, futures, calls, puts, and all that 11

contracts because they give the buyer of an option the right to buy (call) or

to sell (put) the underlying instrument and impose on the seller of an option the obligation to sell (call) or to buy (put) this instrument at a predetermined strike price at maturity (European options) or at any moment between the inception of the option and its maturity (American options) Since the buyer

of an option receives a right and its seller accepts an obligation, options have

to be purchased for an upfront premium with the subsequent settlement at (or before) maturity The buyer of an option cannot lose more than the premium paid, while the losses of the seller can, in certain cases, be unlimited The

holder of a call (put) option will benefit from the rise (fall) of the price of

the underlying instrument Knowing prices of forward and futures contracts, and calls and puts with different strikes and maturities, we can obtain detailed probabilistic information about the future spot price of the underlying Pric- ing and risk-managing derivatives is one of the most important objectives of financial engineering which we discuss in detail below We emphasize that it

is necessary for market makers to know fair prices of derivatives because they need to quote these prices (with bid-ask spread) before being told if they are going to be a seller or a buyer of the corresponding derivative This situation is not dissimilar to the one discussed in the old mathematical problem: how two persons should divide a cake in such a way that each one receives a piece which

is perceived to be greater or equal to half the cake The answer is that the first person (the market maker) divides the cake in two parts, while the second person (the investor) chooses the part which he thinks is bigger ' The way the market maker earns a living is via the bid-ask spread Under normal cir- cumstances the fair price is sandwiched between the bid and ask prices There are at least three reasons for an investor to buy a derivative instrument: (A) for protection against unfavorable forex changes; (B) for leveraging his market

views; (C) for speculation

We consider forward contracts and European calls and puts, which are called plain vanilla instruments, and denote the prices of these claims at time t for the spot FXR equal S by FO(t,S,T, K), C(t, $,T, K), P(t, S,T, K), where the second pair of arguments emphasizes their dependence on the strike K and maturity T When t = T we have

12 Foreign exchange markets

in exchange for For units of domestic currency To avoid the exposure to random fluctuation of the FXR, we have to buy one foreign zero coupon bond maturing at T Accordingly, we have to borrow eT TS units of domestic currency which we need to repay in the amount of e~”'7 So / eT at time T

At maturity our bond produces one unit of foreign currency which we deliver

to the buyer and receive Fo 7 units of domestic currency which we have to pay

Derivatives: forwards, futures, calls, puts, and all that 13

back to the lender of domestic currency, so that

Pop = el )T go (1.5)

This important relation, known as the interest rate parity theorem, establishes the fair forward FXR, because of the perfect reversibility of the entire procedure which we used to establish it (This reversibility is similar, in more than one respect, to the reversibility in thermodynamics.) We emphasize that the forward FXR depends only on the spot FXR, and domestic and foreign bond prices A foreign currency is called a discount currency if Bor < Bor: and

a premium currency otherwise For discount currencies the forward FXR is lower than the spot one, while for premium currencies the opposite is true The reader should think about this seemingly simple statement at some length because it is not obvious For an arbitrary (nonequilibrium) strike, we can write

FOC, S, T, K) = en (Tt) ¢ _ ent (T-t) Ke Comparison of expressions (1.1), (1.5) shows that if a domestic investor uses the forward FXR rather than the spot FXR for repatriating his profits at time T’, the rate of return on foreign investment becomes deterministic and vanishes (No pain, no gain.)

We use the forward rate Fo,r in order to classify calls and puts We say

that calls and puts with K = Fo 7 are at the money forward (ATMP), calls (puts) with K < For (K > Fo,r) are in the money forward (ITMF), while calls (puts) with K > Fo,r (K < Fo,r) are out of the money forward (OTMF)

To find the bounds for the price of calls and puts we can use the comparison principle for two portfolios which says that the portfolio which pays more at maturity should be more expensive at any time t before maturity It is clear that the portfolio consisting of a European call with strike K and of K domestic bonds pays at maturity no less than a portfolio consisting of a foreign bond This is true at any time t before maturity At the same time the portfolio consisting of a call pays at maturity no more than the portfolio consisting of one unit of foreign currency Accordingly, we can conclude that

(z-r'Œ-9s _ ert) x) < Cứ,S,T,K) < 8 + (1.6)

Similarly,

(er K — eg) < P(t, S, T, K) <K (1.7)

14 Foreign exchange markets

It is interesting to note that in the special case when e~" (T-*) = 1 (so that the foreign interest rates is equal to zero) we have

(S- K), <CŒ,8,T,K) < 5,

1.e., the call price never falls below its intrinsic value Similarly, when e~"’ (7-4) —

1 (so that the domestic interest rates is equal to zero) the put price does not fall below its intrinsic value,

(K —S), < P(t,S,T,K) <K

It is easy to establish relations between European calls and puts It follows from contract definition that at maturity we have

The pricing problem for European options is linear, i.e., any linear combination

of its solutions is a solution, too Accordingly, equation (1.8) is valid for any

t, so that

C(t, S,T, K) — P(t,S,T,K) =e" TF -9g —e PT -OK,

This relation is known as put-caill parity

In addition to put-call parity there exists one more important relation be- tween calls and puts which we call put-call symmetry The buyer of a call has the right to buy one euro (the notional amount) at the rate of K dollars/euro Alternatively, he has the right to sell K dollars at the rate of 1/K euros/dollar The right to buy euros is worth C° (0,5, 7, K) dollars (Here and below we use superscripts to show the currency in which the corresponding option is valued.) The right to sell dollars is worth K P} (0,1/5,T,1/K) euros Since buying eu- ros is equivalent to selling dollars the corresponding amounts have to coincide once they are expressed in the same currency (domestic, say) Accordingly,

which is probably more elegant but definitely less useful To switch from P® to

P? one needs to interchange the domestic and foreign interest rates, r° — r1, 1_, ,0

rear’,

Derivatives: forwards, futures, calls, puts, and all that 15

American calls and puts are determined by the same terminal payoffs as their European counterparts but can be executed at any time between the inception and maturity of the option; their prices are denoted by C’(t, S,T, K), P'(t,.S,T, K) Since American options can be executed at any time t, 0 <t<

T, their prices cannot fall below their intrinsic values, additionally, they are always more expensive that the corresponding European options Accordingly,

max{(S — K), ,C(t,S,T, K)} < C’'(,S,T, K) < 5, (1.9)

max{(K —S),,P(t,S,T,K)} < P'(t,S,T,K) < K (1.10) When e~" (T-*) = 1, the prices of American and European calls coincide,

C(t, S,T, K) = C(t, S,T, K), when e~" (T~#) = 1, the prices of American and European puts coincide,

It is difficult to say more about option prices without constructing an ade- quate model for the behavior of S We discuss different possibilities throughout the book

Certain linear combinations of European calls and puts with the same ma- turity are particularly popular in the market place These are strangles, strad- dles, and risk reversals which are discussed in Chapter 9

16 Foreign exchange markets

In addition to options mentioned above, several others are important in the forex context The reason is that European and American calls and puts can be rather expensive, so that their cheaper versions can be attractive to investors

One popular way of making them cheaper, is to take a view on the distri- bution of the future FXR and to finance the purchase of one option by selling another one For instance, an investor who thinks that the foreign currency will be traded above a certain level Ky but below some other level K2, where

ky < Ko, can buy a call with strike Ky and sell a call with strike Ko, thus creating the so-called call spreads with payoffs of the form

0, S< ky payoff = S— ki, Ki, <S< Ke

Ko-ki, Kaạ<®

Another approach is to add various barrier features to ordinary calls and puts For instance, a down-and-out call (up-and-out put) disappears if the FXR hits a certain level below (above) the strike level Even though the corresponding option provides less protection than its European or American analogue, it can be attractive to investors having specific market views, or those who can adjust to low (high) FXRs Other options with mild barrier features which are frequently traded in practice are the so-called timers and faders (also known as time trades) with payoffs depending on the amount of time the FXR spends above (or below) a certain barrier

In addition, there are many investors whose needs are better served by the so-called lookback calls (puts) which give the buyer the right to buy (sell) foreign currency at the best FXR observed between the inception and maturity

of the corresponding option, and Asian options which give the right to buy or sell currency at the average FXR We also study the so-called passport options

In contrast to all other options considered in this book, passport options are written on the trading account rather than on the underlying FXR The buyer

of such an options has the right to keep all the profit and is compensated for any loss generated by buying and selling a specific amount of foreign currency between the inception of the option and its maturity Passport options require active participation of the buyer in the trading process, so that their value depends not only on the evolution of the FXR but also on the buyer’s strategy Barrier, lookback, Asian, and passport options are examples of the so-called path-dependent options whose payoff depends on the entire trajectory of the FXR between the inception and maturity of an option rather than on the terminal value of the FXR Since it is more difficult to price and hedge such