This page intentionally left blank AN INTRODUCTION TO FINANCIAL OPTION VALUATION Mathematics, Stochastics and Computation This is a lively textbook providing a solid introduction to financial option valuation for undergraduate students armed with only a working knowledge of first year calculus. Written as a series of short chapters, this self-contained treatment gives equal weight to applied mathematics, stochastics and computational algorithms, with no prior background in probability, statistics or numerical analysis required. Detailed derivations of both the basic asset price model and the Black–Scholes equation are provided along with a presentation of appropriate computational tech- niques including binomial, finite differences and, in particular, variance reduction techniques for the Monte Carlo method. Each chapter comes complete with accompanying stand-alone MATLAB code listing to illustrate a key idea. The author has made heavy use of figures and ex- amples, and has included computations based on real stock market data. Solutions to exercises are made available at www.cambridge.org. D ES HIGHAM is a professor of mathematics at the University of Strathclyde. He has co-written two previous books, MATLAB Guide and Learning LaTeX.In2005 he was awarded the Germund Dahlquist Prize by the Society for Industrial and Applied Mathematics for his research contributions to a broad range of problems in numerical analysis. AN INTRODUCTION TO FINANCIAL OPTION VALUATION Mathematics, Stochastics and Computation DESMOND J. HIGHAM Department of Mathematics University of Strathclyde CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK First published in print format ISBN-13 978-0-521-83884-9 ISBN-13 978-0-521-54757-4 ISBN-13 978-0-511-33704-8 © Cambridge University Press 2004 2004 Information on this title: www.cambridge.org/9780521838849 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written p ermission of Cambrid g e University Press. ISBN-10 0-511-33704-3 ISBN-10 0-521-83884-3 ISBN-10 0-521-54757-1 Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not g uarantee that any content on such websites is, or will remain, accurate or a pp ro p riate. Published in the United States of America by Cambridge University Press, New York www.cambridge.org hardback paperback paperback eBook (EBL) eBook (EBL) hardback To my family, Catherine, Theo, Sophie and Lucas Contents List of illustrations page xiii Preface xvii 1 Options 1 1.1 What are options? 1 1.2 Why do we study options? 2 1.3 How are options traded? 4 1.4 Typical option prices 6 1.5 Other financial derivatives 7 1.6 Notes and references 7 1.7 Program of Chapter 1 and walkthrough 8 2 Option valuation preliminaries 11 2.1 Motivation 11 2.2 Interest rates 11 2.3 Short selling 12 2.4 Arbitrage 13 2.5 Put–call parity 13 2.6 Upper and lower bounds on option values 14 2.7 Notes and references 16 2.8 Program of Chapter 2 and walkthrough 17 3 Random variables 21 3.1 Motivation 21 3.2 Random variables, probability and mean 21 3.3 Independence 23 3.4 Variance 24 3.5 Normal distribution 25 3.6 Central Limit Theorem 27 3.7 Notes and references 28 3.8 Program of Chapter 3 and walkthrough 29 vii viii Contents 4 Computer simulation 33 4.1 Motivation 33 4.2 Pseudo-random numbers 33 4.3 Statistical tests 34 4.4 Notes and references 40 4.5 Program of Chapter 4 and walkthrough 41 5 Asset price movement 45 5.1 Motivation 45 5.2 Efficient market hypothesis 45 5.3 Asset price data 46 5.4 Assumptions 48 5.5 Notes and references 49 5.6 Program of Chapter 5 and walkthrough 50 6 Asset price model: Part I 53 6.1 Motivation 53 6.2 Discrete asset model 53 6.3 Continuous asset model 55 6.4 Lognormal distribution 56 6.5 Features of the asset model 57 6.6 Notes and references 59 6.7 Program of Chapter 6 and walkthrough 60 7 Asset price model: Part II 63 7.1 Computing asset paths 63 7.2 Timescale invariance 66 7.3 Sum-of-square returns 68 7.4 Notes and references 69 7.5 Program of Chapter 7 and walkthrough 71 8 Black–Scholes PDE and formulas 73 8.1 Motivation 73 8.2 Sum-of-square increments for asset price 74 8.3 Hedging 76 8.4 Black–Scholes PDE 78 8.5 Black–Scholes formulas 80 8.6 Notes and references 82 8.7 Program of Chapter 8 and walkthrough 83 [...]... method for an American put Optimal exercise boundary Monte Carlo for an American put Notes and references Program of Chapter 18 and walkthrough 174 176 177 180 182 183 19 Exotic options 19.1 Motivation 19.2 Barrier options 19.3 Lookback options 19.4 Asian options 19.5 Bermudan and shout options 19.6 Monte Carlo and binomial for exotics 19.7 Notes and references 19.8 Program of Chapter 19 and walkthrough... European put Payoff diagram for a bull spread Market values for IBM call and put options Another view of market values for IBM call and put options Program of Chapter 1: ch01.m Upper and lower bounds for European call option Program of Chapter 2: ch02.m Figure produced by ch02.m Density function for an N(0, 1) random variable Density functions for various N(µ, σ 2 ) random variables N(0, 1) density and. .. an early draft and made many helpful suggestions Vicky Henderson checked parts of the text and patiently answered a number of questions Petter Wiberg gave me access to his MATLAB files for processing stock market data Xuerong Mao, through animated discussions and research collaboration, has enriched my understanding of stochastics and its role in mathematical finance Additionally, five anonymous reviewers... method for an American put Error in binomial method for an American put Value P Am (S, T /4) for an American put, computed via the binomial method Exercise boundary for an American put Monte Carlo approximations to the discounted expected American put payoff with a simple exercise strategy Program of Chapter 18: ch18.m Two asset paths and a barrier Time-zero down -and- out call value Time-zero up -and- out... variates in option valuation 22.4 Notes and references 22.5 Program of Chapter 22 and walkthrough 229 229 229 231 232 234 23 Finite difference methods 23.1 Motivation 23.2 Finite difference operators 23.3 Heat equation 23.4 Discretization 23.5 FTCS and BTCS 23.6 Local accuracy 23.7 Von Neumann stability and convergence 23.8 Crank–Nicolson 23.9 Notes and references 23.10 Program of Chapter 23 and walkthrough... guide to MATLAB, see (Higham and Higham, 2000) I have not made use of any of the toolboxes that are available, at extra cost, to MATLAB users This is because (a) the emphasis in the book is on understanding the underlying models and algorithms, not on the use of black-box packages, and (b) only a small percentage of MATLAB users will have access to toolboxes However, those who wish to perform serious option. .. accuracy and clarity in mind, rather than efficiency or elegance I have made quite heavy use of MATLAB’s vectorization facilities, where possible working with arrays directly and eschewing unnecessary for loops This tends to make the codes shorter, snappier and less daunting than alternatives that operate on individual array components Meaningful comments have been inserted into the codes and a ‘walkthrough’... Cash-or-nothing options 17.3 Black–Scholes for cash-or-nothing options 17.4 Delta behaviour 17.5 Risk neutrality for cash-or-nothing options 17.6 Notes and references 17.7 Program of Chapter 17 and walkthrough 163 163 163 164 166 167 168 170 18 American options 18.1 Motivation 18.2 American call and put 173 173 173 Contents 18.3 18.4 18.5 18.6 18.7 18.8 xi Black–Scholes for American options Binomial method for an. .. and more attention paid to stochastic modelling and simulation Key features of this book are as follows (i) Detailed derivation and discussion of the basic lognormal asset price model (ii) Roughly equal weight given to binomial, finite difference and Monte Carlo methods In particular, variance reduction techniques for Monte Carlo are treated in some detail (iii) Heavy use of computational examples and. .. has two great selling points • From a student perspective, the topic is generally perceived as modern, sexy and likely to impress potential employers • From the perspective of a university teacher, the topic provides a focus for ideas from mathematical modelling, analysis, stochastics and numerical analysis There are many excellent books on option valuation However, in preparing notes for a lecture course, . Notes and references 11 1 11 .7 Program of Chapter 11 and walkthrough 11 1 12 Risk neutrality 11 5 12 .1 Motivation 11 5 12 .2 Expected payoff 11 5 12 .3 Risk neutrality 11 6 12 .4 Notes and references 11 8 12 .5. walkthrough 18 3 19 Exotic options 18 7 19 .1 Motivation 18 7 19 .2 Barrier options 18 7 19 .3 Lookback options 19 1 19 .4 Asian options 19 2 19 .5 Bermudan and shout options 19 3 19 .6 Monte Carlo and binomial. Chapter 10 and walkthrough 10 4 11 Mor e on the Black–Scholes formulas 10 5 11 .1 Motivation 10 5 11 .2 Where is µ? 10 5 11 .3 Time dependency 10 6 11 .4 The big picture 10 6 11 .5 Change of variables 10 8 11 .6