Springer-Verlag Berlin Heidelberg GmbH

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1 U niversitext

2 Springer-Verlag Berlin Heidelberg GmbH

3 Fred Espen Benth Option Theory with Stochastic Analysis An Introduction to Mathematical Finance i Springer

4 Fred Espen Benth Centre of Mathematics for Applications University of 0510 Department of Mathematics P.O. Box 1053 Blindem Norway Tide of the o riginal Norwegian edition: Matematisk Os I0,2002 This book has bccn fundcd by NORLA-Norwegian Literature Abroad, Fiction and Non-fiction Cataloging-in Publication Data appliw. for A catalog record for Ihis book i5 available from the library ofcongres5. Bibliographie information published by Die!Xutsche Bibliothek Die Deutsche Bibliothek lists tbis publieation in th e Deutsche Nationalbibliografie; detailed bibliographie data i5 available in the Internet at hup,lidnb.ddb.de Mathematics Subj«t Classification (2000): 91828, 60H30, 65C05, 60G35 ISBN ISBN ( ebook) DOI / This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights oflranslation, reprinting, reuse of illustrations, recitation, broadcasting, reproduclion on microfilm or in any otber war, and storage in data bank. DlIplicalion ofthis publication or parts th ert ofis permitted only under thc provisions of the German Copyright Law of September 9, '965, in its CUTTe nt version, and permission fo r lise must always be obtained from Springer-Verlag. Violations are Hable for prosecution under the German Copyright Law. springeronline.oom o Springer-Verlag Berlin Heidelberg 2004 OriginaHy published by Springer-Verlag Berlin Heidclbcrg Ncw York in 2004 The lise of general descriptive names, registere<! names, trademarks, elc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and tberefore free for general lise. Cover design: design 0- producrion GmbH, Heidelberg Type5ct by the 3uthor using a Springer NE?< macro paclulge Printe<! on acid-free paper db- S 431\ 0

5 To my wife Jurate

6 Preface Since 1972 and the appearance of the famous Black & Scholes option pricing formula, derivatives have become an integrated part of everyday life in the financial industry. Options and derivatives are tools to control risk exposure, and used in the strategies of investors speculating in markets like fixed-income, stocks, currencies, commodities and energy. A combination of mathematical and economical reasoning is used to find the price of a derivatives contract. This book gives an introduction to the theory of mathematical finance, which is the modern approach to analyse options and derivatives. Roughly speaking, we can divide mathematical finance into three main directions. In stochastic finance the purpose is to use economic theory with stochastic analysis to derive fair prices for options and derivatives. The results are based on stochastic modelling of financial assets, which is the field of empirical finance. Numerical approaches for finding prices of options are studied in computational finance. All three directions are presented in this book. Algorithms and code for Visual Basic functions are included in the numerical chapter to inspire the reader to test out the theory in practice. The objective of the book is not to give a complete account of option theory, but rather relax the mathematical rigour to focus on the ideas and techniques. Instead of going deep into stochastic analysis, we present the intuition behind basic concepts like the Ito formula and stochastic integration, enabling the reader to use these in the context of option theory. To comprehend the theory, a background in mathematics and statistics at bachelor level (that means, calculus, linear algebra and probability theory) is recommended. This book is a revision of the Norwegian edition which appeared in It is used in a course for students at the University of Oslo preparing for a master in finance and insurance mathematics. The manuscript for the Norwegian edition grew from lecture notes prepared for an introductory course in modern finance for the industry. Several people have contributed in the writing of this book. I am grateful to Jurate Saltyte-Benth for carefully reading through the manuscript and significantly improving the presentation, and Neil Shephard for providing me with Ox software to do statistical analysis of financial time series. Furthermore, the advice given and corrections made by Jeffrey Boys, Daniela Brandt,

7 VIn Preface Catriona Byrne and Susanne Denskus at Springer are acknowledged. All remaining errors are of course the responsibility of the author. Oslo, August 2003 Fred Espen Benth

8 Table of Contents 1 Introduction An Introduction to Options in Finance Empirical Finance Stochastic Finance Computational Finance Some Useful Material from Probability Theory Statistical Analysis of Data from the Stock Market The Black & Scholes Model Logarithmic Returns from Stocks Scaling Towards Normality Heavy- Tailed and Skewed Logreturns Logreturns and the Normal Inverse Gaussian Distribution An Alternative to the Black & Scholes Model Logreturns and Autocorrelation Conclusions Regarding the Choice of Stock Price Model An Introduction to Stochastic Analysis The Ito Integral The Ito Formula Geometric Brownian Motion as the Solution of a Stochastic Differential Equation Conditional Expectation and Martingales Pricing and Hedging of Contingent Claims Motivation from One-Period Markets The Black & Scholes Market and Arbitrage Pricing and Hedging of Contingent Claims X = f(s(t)) Derivation of the Black & Scholes Partial Differential Equation Solution of the Black & Scholes Partial Differential Equation The Black & Scholes Formula for Call Options Hedging of Call Options Hedging of General Options 70

9 X Table of Contents Implied Volatility The Girsanov Theorem and Equivalent Martingale Measures Pricing and Hedging of General Contingent Claims An Example: a Chooser Option The Markov Property and Pricing of General Contingent Claims Contingent Claims on Many Underlying Stocks Completeness, Arbitrage and Equivalent Martingale Measures Extensions to Incomplete Markets Energy Markets and Incompleteness Numerical Pricing and Hedging of Contingent Claims Pricing and Hedging with Monte Carlo Methods Pricing and Hedging of Contingent Claims with Payoff of the Form f(st)' The Accuracy of Monte Carlo Methods Pricing of Contingent Claims on Many Underlying Stocks Pricing of Path-Dependent Claims Pricing and Hedging with the Finite Difference Method 112 A Solutions to Selected Exercises 121 References 157 Index 161