KING’S COLLEGE LONDON DEPARTMENT OF MATHEMATICS Financial Mathematics An Introduction to Derivatives Pricing Lane P. Hughston Christopher J. Hunter Share Price $200 $150 $100 $50 $0 Date July Aug. Sept. Oct. Nov. Financial Mathematics An Introductory Guide Lane P. Hughston1 Department of Mathematics King’s College London The Strand, London WC2R 2LS, UK Christopher J. Hunter2 NatWest Group 135 Bishopsgate, London Postal Code, UK and Department of Mathematics King’s College London The Strand, London WC2R 2LS, UK copyright c 2000 L.P. Hughston and C.J. Hunter (cid:176) 1email: lane [email protected] 2email: [email protected] i Preface This book is intended as a guide to some elements of the mathematics of flnance. Had we been a bit bolder it would have been entitled ‘Mathematics for Money Makers’ since it deals with derivatives, one of the most notorious ways to make (or lose) a lot of money. Our main goal in the book is to develop the basics of the theory of derivative pricing, as derived from the so-called ‘no arbitrage condition’. In doing so, we also introduce a number of mathematical tools that are of interest in their own right. At the end of it all, while you may not be a millionaire, you should understand how to avoid ‘breaking the bank’ with a few bad trades. In order to motivate the study of derivatives, we begin the book with a discussion of the flnancial markets, the instruments that are traded on them and how arbitrage opportunities can occur if derivatives are mispriced. We then arrive at a problem that inevitably arises when dealing with physical systems such as the flnancial markets: how to deal with the ‘(cid:176)ow of time’. There are two primary means of parametrizing time|the discrete time pa- rameterization, where time advances in flnite steps; and the continuous time parameterization, where time varies smoothly. We initially choose the former method, and develop a simple discrete time model for the movements of asset prices and their associated derivatives. It is based on an idealised Casino, where betting on the random outcome of a coin toss replaces the buying and selling of an asset. Once we have seen the basic ideas in this context, we then expand the model and interpret it in a language that brings out the analogy with a stock market. This is the binomial model for a stock market, where time is discrete and stock prices move in a random fashion. In the second half of the notes, we make the transition from discrete to continuous time models, and derive the famous Black-Scholes formula for option pricing, as well as a number of interesting extensions of this result. Throughoutthebookweemphasisetheuseofmodernprobabilisticmeth- ods and stress the novel flnancial ideas that arise alongside the mathematical innovations. Some more advanced topics are covered in the flnal sections| stocks which pay dividends, multi-asset models and one of the great simpli- flcations of derivative pricing, the Girsanov transformation. This book ia based on a series of lectures given by L.P. Hughston at King’s College London in 1997. The material in appendix D was provided by Professor R.F. Streater, whom we thank for numerous helpful observations on the structure and layout of the material in these notes. ii For lack of any better, yet still grammatically correct alternative, we will use ‘he’ and ‘his’ in a gender non-speciflc way. In a similar fashion, we will use ‘dollar’ in a currency non-speciflc way. L.P. Hughston and C.J. Hunter January 1999 iii Contents 1 Introduction 1 1.1 Financial Markets . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Basic Assets . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Uses of Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Derivative Payofi Functions . . . . . . . . . . . . . . . . . . . 9 2 Arbitrage Pricing 13 2.1 Expectation Pricing . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Arbitrage Pricing . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3 Trading Strategies . . . . . . . . . . . . . . . . . . . . . . . . 17 2.4 Replication Strategy . . . . . . . . . . . . . . . . . . . . . . . 20 2.5 Currency Swap . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3 A Simple Casino 24 3.1 Rules of the Casino . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.3 No Arbitrage Argument . . . . . . . . . . . . . . . . . . . . . 26 4 Probability Systems 29 4.1 Sample Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.2 Event Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.3 Probability Measure . . . . . . . . . . . . . . . . . . . . . . . 31 4.4 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . 32 5 Back to the Casino 35 5.1 The Casino as a Probability System . . . . . . . . . . . . . . . 35 5.2 The Risk-Neutral Measure . . . . . . . . . . . . . . . . . . . . 35 5.3 A Non-Zero Interest Rate . . . . . . . . . . . . . . . . . . . . 37 6 The Binomial Model 41 6.1 Tree Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 6.2 Money Market Account . . . . . . . . . . . . . . . . . . . . . . 43 6.3 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 6.4 One-Period Replication Model . . . . . . . . . . . . . . . . . . 45 6.5 Risk-Neutral Probabilities . . . . . . . . . . . . . . . . . . . . 47 iv 7 Pricing in N-Period Tree Models 50 8 Martingales and Conditional Expectation 54 8.1 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . 54 8.2 Filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 8.3 Adapted Process . . . . . . . . . . . . . . . . . . . . . . . . . 56 8.4 Conditional Expectation . . . . . . . . . . . . . . . . . . . . . 56 8.5 Martingales . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 8.6 Financial Interpretation . . . . . . . . . . . . . . . . . . . . . 58 9 Binomial Lattice Model 60 10 Relation to Binomial Model 63 10.1 Limit of a Random Walk . . . . . . . . . . . . . . . . . . . . . 63 10.2 Martingales associated with Random Walks . . . . . . . . . . 64 11 Continuous Time Models 68 11.1 The Wiener Model . . . . . . . . . . . . . . . . . . . . . . . . 68 11.2 The Normal Distribution . . . . . . . . . . . . . . . . . . . . . 69 12 Stochastic Calculus 76 13 Arbitrage Argument 79 13.1 Derivation of the No-Arbitrage Condition . . . . . . . . . . . . 79 13.2 Derivation of the Black-Scholes Equation . . . . . . . . . . . . 83 14 Replication Portfolios 86 15 Solving the Black-Scholes Equation 89 15.1 Solution of the Heat Equation . . . . . . . . . . . . . . . . . . 90 15.2 Reduction of the Black-Scholes Equation to the Heat Equation 92 16 Call and Put Option Prices 97 16.1 Call Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 16.2 Put Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 17 More Topics in Option Pricing 104 17.1 Binary Options . . . . . . . . . . . . . . . . . . . . . . . . . . 104 17.2 ‘Greeks’ and Hedging . . . . . . . . . . . . . . . . . . . . . . . 105 v 17.3 Put-Call Parity . . . . . . . . . . . . . . . . . . . . . . . . . . 107 18 Continuous Dividend Model 109 18.1 Modifled Black-Scholes Equation . . . . . . . . . . . . . . . . 112 18.2 Call and Put Option Prices . . . . . . . . . . . . . . . . . . . 113 19 Risk Neutral Valuation 115 19.1 Single Asset Case . . . . . . . . . . . . . . . . . . . . . . . . . 115 20 Girsanov Transformation 121 20.1 Change of Drift . . . . . . . . . . . . . . . . . . . . . . . . . . 122 21 Multiple Asset Models 126 21.1 The Basic Model . . . . . . . . . . . . . . . . . . . . . . . . . 126 21.2 No Arbitrage and the Zero Volatility Portfolio . . . . . . . . . 129 21.3 Market Completeness . . . . . . . . . . . . . . . . . . . . . . . 131 22 Multiple Asset Models Continued 132 22.1 Dividends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 22.2 Martingales and the Risk-Neutral Measure . . . . . . . . . . . 133 22.3 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 A Glossary 137 B Some useful formulae and deflnitions 140 B.1 Deflnitions of a Normal Variable . . . . . . . . . . . . . . . . . 140 B.2 Moments of the Standard Normal Distribution . . . . . . . . . 140 B.3 Moments of a Normal Distribution . . . . . . . . . . . . . . . 141 B.4 Other Useful Integrals . . . . . . . . . . . . . . . . . . . . . . 141 B.5 Ito’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 B.6 Geometric Brownian Motion . . . . . . . . . . . . . . . . . . . 142 B.7 Black-Scholes Formulae . . . . . . . . . . . . . . . . . . . . . . 142 B.8 Bernoulli Distribution . . . . . . . . . . . . . . . . . . . . . . 143 B.9 Binomial Distribution . . . . . . . . . . . . . . . . . . . . . . . 143 B.10 Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . 143 C Solutions 145 vi D Some Reminders of Probability Theory 194 D.1 Events, random variables and distributions . . . . . . . . . . . 194 D.2 Expectation, moments and generating functions . . . . . . . . 195 D.3 Several random variables . . . . . . . . . . . . . . . . . . . . . 196 D.4 Conditional probability and expectation . . . . . . . . . . . . 199 D.5 Filtrations and martingales . . . . . . . . . . . . . . . . . . . . 204 E The Virtues and Vices of Options1 207 F KCL 1998 Exam 209 F.1 Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 F.1.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 210 F.2 Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 F.2.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 213 F.3 Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 F.3.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 216 F.4 Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 F.4.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 219 F.5 Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 F.5.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 222 F.6 Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 F.6.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 224 G Bibliography 225 vii 1 Introduction Thestudyofmostsciencescanbeusefullydividedintotwodistinctbutinter- relatedbranches, theoryandexperiment. Forexample, thebodyofknowledge thatweconventionallylabel‘physics’consistsoftheoretical physics, wherewe develop mathematical models and theories to describe how nature behaves, andexperimental physics, whereweactuallytestandprobenaturetoseehow it behaves. There is an important interplay between the two branches|for example, theory might develop a model which is then tested by experiment, or experiment might measure or discover a fact or feature of nature which must then be explained by theory. Finance is the science of the flnancial markets. Correspondingly, it has an important ‘theoretical’ side, called flnance theory or mathematical flnance, whichentailsboththedevelopmentoftheconceptualapparatusneededforan intellectually sound understanding of the behaviour of the flnancial markets, as well as the development of mathematical techniques and models useful in flnance; and an ‘experimental’ side, which we might call practical or applied flnance, that consists of the extensive range of trading techniques and risk management practices as they are actually carried out in the various flnan- cial markets, and applied by governments, corporations and individuals in their quest to improve their fortune and control their exposure to potentially adverse circumstances. In this book we ofier an introductory guide to mathematical flnance, with particular emphasis on a topic of great interest and the source of numerous applications: namely, the pricing of derivatives. The mathematics needed for a proper understanding of this signiflcant branch of theoretical and applied flnance is both fascinating and important in its own right. Before we can begin building up the necessary mathematical tools for analysing derivatives, however, we need to know what derivatives are and what they are used for. Butthisrequiressomeknowledgeoftheso-called‘underlyingassets’onwhich these derivatives are based. So we begin this book by discussing the flnancial markets and the various instruments that are traded on them. Our intention here is not, of course, to make a comprehensive survey of these markets, but to sketch lightly the relevant notions and introduce some useful terminology. Unless otherwise stated, all dates in this section are from the year 1999, and all prices are the relevant markets’ closing values. If no date is given for a price, then it can be assumed to be January 11, 1999. 1 1.1 Financial Markets The global flnancial markets collectively comprise a massive industry spread over the entire world, with substantial volumes of buying and selling occur- ring in one market or another at one place or another at virtually any time. The dealing is mediated by traders who carry out trades on behalf of both their clients (institutional and individual investors) and their employers (in- vestment banks and other flnancial institutions). This world-wide menagerie of traders, in the end, determines the prices of the available flnancial prod- ucts, and is sometimes collectively referred to as the ‘market’. The most ‘elementary’ flnancial instruments bought and sold in flnancial markets can be described as basic assets. There are several common types. 1.1.1 Basic Assets A stock or share represents a part ownership of a company, typically on a limited liability basis (that is, if the company fails, then the shareholder’s loss is usually limited to his original investment). When the company is profltable, the owner of the stock beneflts from time to time by receiving a dividend, whichistypicallyacashpayment. Theshareholdermayalsorealize a proflt or capital gain if the value of the stock increases. Ultimately, the sharepriceisdeterminedbythemarketaccordingtothelevelofconfldenceof investors that the flrm will be profltable, and hence pay further and perhaps higher dividends in the future. For example, the value of a Rolls-Royce share atthecloseoftheLondonStockExchangeonJanuary11was248.5p(pence), which was down 0.5p from the prior day’s closing value. In the previous 52 weeks the highest closing value was 309p, while the lowest was 176.5p. The company has declared a dividend of 6.15p per share for 1998 compared with 5.9p and 5.3p per share paid in the two years previous to that. Abondis, inefiect, aloanmadetoacompanyorgovernmentbythebond- holder, usually for a flxed period of time, for which the bond-holder receives a fee, known as interest. The interest rate charged is typically flxed at the time that the loan is made, but might be allowed to vary in time according to market levels and certain prescribed rules. The interest payments, which are typically made on an annual, semi-annual or quarterly basis, are called ‘coupon’ payments. If a 10-year bond with a ‘face-value’ of $1000 has a 6% annual coupon, that means that an interest rate payment of $60 is made every year for ten years, and then at the end of the ten year period the $1000 2