Springer Finance Editorial Board M Avellaneda G Barone-Adesi M Broadie M.H.A Davis E Derman C Kliippelberg E Kopp W Schachermayer Springer Finance Springer Finance is a programme of books aimed at students, academics, and practitioners working on increasingly technical approaches to the analysis of financial markets It aims to cover a variety of topics, not only mathematical finance but foreign exchanges, term structure, risk management, portfolio theory, equity derivatives, and financial economics M A mmann, Credit Risk Valuation: Methods, Models, and Applications (2001) E Barucci Financial Markets Theory: Equilibrium, Efficiency and Information (2003) N.H Bingham and R Kiesel Risk-Neutral Valuation: Pricing and Hedging of Financial Derivatives, 2nd Edition (2004) T.R Bielecki and M Rutkowski Credit Risk: Modeling, Valuation and Hedging (2001) D Brigo amd F Mercurio, Interest Rate Models: Theory and Pracbce (200 I) R Buff, Uncertain Volatility Models- Theory and Application (2002) R.-A Dana and M Jeanblanc, Financial Markets in Continuous Time (2003) G Deboeck and T Kohonen (Editors), Visual Explorations in Finance with Self­ Organizing Maps (1998) R.J Elliott and P.E Kopp Mathematics of Financial Markets (1999) H Gemon, D Madan, S.R Pliska and T Vorst (Editors), Mathematical Finance­ Bachelier Congress 2000 (2001) M Gundlach and F Lehrbass (Editors), CreditRlsk+ in the Banking Industry (2004) Y.-K Kwok, Mathematical Models of Financial Derivatives (1998) M Kii/pmonn, Irrational Exuberance Reconsidered: The Cross Section of Stock Returns, 2nd Edition (2004) A Pelsser Efficient Methods for Valuing Interest Ra te Derivatives (2000) J.-L Prigent, Weak Convergence of Financial Markets (2003) B Schmid Credit Risk Pricing Models: Theory and Practice, 2nd Edition (2004) S.E Shreve, Stochastic Calculus for Finance 1: The Binomial Asset Pricing Model (2004) S.E Shreve, Stochastic Calculus for Finance II: Continuous-Time Models (2004) M Yor, Exponential Funcbonals of Brownian Motion and Related Processes (2001) R Zagst,lnterest-Rate Management (2002) Y.-1 Zhu and 1-L Chern, Derivative Securities and Difference Methods (2004) A Ziegler, Incomplete lnfonnabon and Heterogeneous Beliefs in Continuous-Time Finance (2003) A Ziegler, A Game Theory Analysis of Options: Corporate Finance and Financial Intermediation in Conbnuous Time, 2nd Edition (2004) Steven E Shreve Stochastic Calcu I us for Finance II Continuous-Time Models With 28 Figures �Springer Steven E Shreve Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA 15213 USA shreve@cmu.edu Scan von der Deutschen Filiale der staatlichen Bauerschaft (KOLX03'a) Mathematics Subject Classification (2000): 60-01, 60HIO, 60165, 91B28 Library of Congress Cataloging-in-Publication Data Shreve, Steven E Stochastic calculus for finance I Steven E Shreve p em - (Springer finance series) Includes bibliographical references and index Contents v Continuous-time models ISBN 0-387-40101-6 (alk paper) I Finance-Mathematical models-Textbooks Textbooks I Title Stochastic analysis­ II Spnnger finance HGI06.S57 2003 2003063342 332'.01'51922-{lc22 ISBN 0-387-40101-6 Pnnted on acid-free paper © 2004 Spnnger Science+Business Media, Inc All nghts reserved This work may not be translated or copied in whole or in part without the wntten permission of the publisher (Springer Science+Business Media, Inc , 233 Spring Street, New York, NY 10013, USA), except for bnef excerpts in connection with revtews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use in this publication of trade names, trademarks, service marks and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to propnetary nghts Printed in the United States of America springeronline com To my students This page intentionally left blank Preface Origin of This Text This text has evolved from mathematics courses in the Master of Science in Computational Finance (MSCF) program at Carnegie Mellon University The content of this book has been used successfully with students whose math­ ematics background consists of calculus and calculus-based probability The text gives precise statements of results, plausibility arguments, and even some proofs, but more importantly, intuitive explanations developed and refined through classroom experience with this material are provided Exercises con­ clude every chapter Some of these extend the theory and others are drawn from practical problems in quantitative finance The first three chapters of Volume I have been used in a half-semester course in the MSCF program The full Volume I has been used in a full­ semester course in the Carnegie Mellon Bachelor's program in Computational Finance Volume II was developed to support three half-semester courses in the MSCF program Dedication Since its inception in 1994, the Carnegie Mellon Master's program in Compu­ tational Finance has graduated hundreds of students These people, who have come from a variety of educational and professional backgrounds, have been a joy to teach They have been eager to learn, asking questions that stimu­ lated thinking, working hard to understand the material both theoretically and practically, and often requesting the inclusion of additional topics Many came from the finance industry, and were gracious in sharing their knowledge in ways that enhanced the classroom experience for all This text and my own store of knowledge have benefited greatly from interactions with the MSCF students, and I continue to learn from the MSCF VIII Preface alumni I take this opportunity to express gratitude to these students and former students by dedicating this work to them Acknowledgments Conversations with several people, including my colleagues David Heath and Dmitry Kramkov, have influenced this text Lukasz Kruk read much of the manuscript and provided numerous comments and corrections Other students and faculty have pointed out errors in and suggested improvements of earlier drafts of this work Some of these are Jonathan Anderson, Nathaniel Carter, Bogdan Doytchinov, David German, Steven Gillispie, Karel Janecek, Sean Jones, Anatoli Karolik, David Korpi, Andrzej Krause, Rael Limbitco, Petr Luksan, Sergey Myagchilov, Nicki Rasmussen, Isaac Sonin, Massimo Tassan­ Solet, David Whitaker and Uwe Wystup In some cases, users of these earlier drafts have suggested exercises or examples, and their contributions are ac­ knowledged at appropriate points in the text To all those who aided in the development of this text, I am most grateful During the creation of this text, the author was partially supported by the National Science Foundation under grants DMS-9802464, DMS-0103814, and DMS-0139911 Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and not necessarily reflect the views of the National Science Foundation Pittsburgh, Pennsylvania, USA April 2004 Steven E Shreve Contents General Probability Theory Infinite Probability Spaces Random Variables and Distributions Expectations Convergence of Integrals Computation of Expectations Change of Measure Summary Notes Exercises 1 13 23 27 32 39 41 41 Information and u-algebras Independence General Conditional Expectations Summary Notes Exercises 49 49 53 66 75 77 77 Brownian Motion 3.1 Introduction : 3.2 Scaled Random Walks 3.2 Symmetric Random Walk 3.2.2 Increments of the Symmetric Random Walk 3.2.3 Martingale Property for the Symmetric Random Walk 3.2.4 Quadratic Variation of the Symmetric Random Walk 3.2.5 Scaled Symmetric Random Walk 3.2.6 Limiting Distribution of the Scaled Random Walk 83 83 83 83 84 1.1 1.2 1.3 1.4 1.5 1.6 1.8 Information and Conditioning 2.2 2.3 2.4 2.5 2.6 85 85 86 88 X Contents 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.2.7 Log-Normal Distribution as the Limit of the Binomial Model 91 Brownian Motion 93 3.3 Definition of Brownian Motion 93 3.3.2 Distribution of Brownian Motion 95 3.3.3 Filtration for Brownian Motion 97 3.3.4 Martingale Property for Brownian Motion 98 Quadratic Variation 98 3.4.1 First-Order Variation 99 3.4.2 Quadratic Variation 101 3.4.3 Volatility of Geometric Brownian Motion 106 Markov Property 107 First Passage Time Distribution 108 Reflection Principle 1 3.7 Reflection Equality 111 3.7.2 First Passage Time Distribution 112 3.7.3 Distribution of Brownian Motion and Its Maximum 113 Summary 115 Notes 116 Exercises 117 Stochastic Calculus 125 Introduction 125 4.2 Ito's Integral for Simple Integrands 125 4.2 Construction of the Integral 126 4.2.2 Properties of the Integral 128 4.3 Ito's Integral for General Integrands 132 4.4 ltO-Doeblin Formula 137 4.4.1 Formula for Brownian Motion 137 4.4.2 Formula for Ito Processes 143 4.4.3 Examples 147 4.5 Black-Scholes-Merton Equation 153 4.5 Evolution of Portfolio Value 154 4.5.2 Evolution of Option Value 155 4.5.3 Equating the Evolutions 156 4.5.4 Solution to the Black-Scholes-Merton Equation 158 4.5.5 The Greeks 159 4.5.6 Put-Call Parity 162 4.6 Multivariable Stochastic Calculus 164 4.6 Multiple Brownian Motions 164 4.6.2 ItO-Doeblin Formula for Multiple Processes 165 4.6.3 Recognizing a Brownian Motion 168 Brownian Bridge 172 Gaussian Processes 172 7.2 Brownian Bridge as a Gaussian Process 175 This page intentionally left blank References AIT-SAHALIA , Y ( 996) Testing continuous-time models of the spot interest rate, Rev Fin Stud 9, 385-426 AMIN , K AND JARROW, R ( 991) Pricing foreign currency options under stochastic interest rates, J Int Money Fin 10, 31Q-329 AMIN , K AND KHANNA , A ( 994) Convergence of American option values from discrete- to continuous-time financial models, Math Fin 4, 289-304 ANDREASEN , J ( 998) The pricing of discretely sampled Asian and lookback 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