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 options: a change of numeraire approach, J Comput Fin 2, 5-30 ARROW, K AND DEBREU , G ( 954) Existence of equilibrium for a competitive economy, Econometrica 2 , 265-290 BACHELlER, L ( 900) Theorie de la speculation, Ann Sci Ecole Norm Sup , 21-86 BALDUZZI , P , DAS , S , FORESI, S , AND SUNDARAM, R ( 996) A simple approach to three-factor term structure models, J Fixed Income 6, 43-53 BAXTER, M W AND RENNIE, A ( 996) Financial Calculus: An Introduction to Derivative Pricing, Cambridge University Press, Cambridge, UK BENSOUSSAN , A ( 984) On the theory of option pricing, Acta Appl Math , 139- 158 10 BILLINGSLEY, P ( 986) Probability and Measure, 2nd ed , Wiley, New York 1 BJORK , T ( 998) Arbitrage Theory in Continuous Time, Oxford University Press, Oxford, UK 12 BJORK , T , KABANOV, Y , AND RUNGGALDIER, W ( 997) Bond market struc ture in the presence of marked point processes, Math Fin 7, 21 1-239 13 BLACK , F ( 1976) The pricing of commodity contracts, J Fin Econ , 1671 79 14 BLACK , F ( 986) Noise, J Fin 41, 529-543 15 BLACK , F , DERMAN, E , AND TOY , W ( 990) A one-factor model of interest rates and its applications to treasury bond options, Fin A nal J 46( ) , 33-39 16 BLACK , F AND KARASINSKI , P ( 99 ) Bond and option pricing when short rates are lognormal, Fin Anal J 47(4), 52-59 BLACK, F AND SCHOLES , M ( 973) The pricing of options and corporate liabilities, J Polit Econ 81, 637-659 18 BORODIN, A AND SALMINEN , P ( 996) Handbook of Brownian Motion - Facts and Formulae, Birkhii.user, Boston 538 References 19 BRACE, A , G4TAREK , D , AND MUSIELA, M ( 997) The market model of interest rate dynamics, Math Fin 4, 27-155 20 BRACE, A AND MUSIELA , M ( 994) A multifactor Gauss-Markov implemen tation of Heath, Jarrow and Morton, Math Fin 2, 259 283 BRIGO, D AND MERCURIO, F (200 ) Interest Rate Models: Theory and Prac tice, Springer-Verlag, Berlin 22 BROADIE, M , GLASSERMAN, P , AND Kou , S ( 999) Connecting discrete and continuous path-dependent options, Fin Stochastics 3, 55-82 23 BROCKHAUS, , FARKAS, M , FERRARIS, A , LONG, D , AND 0VERHAUS , M (2000) Equity Derivatives and Market Risk Models, Risk Books, London 24 BRU, B (2000) Un hiver en campagne, C R Ser I 3 , 1037-1058 25 BRU , B AND YOR, M (2002) Comments on the life and mathematical legacy of Wolfgang Doeblin, Fin Stochastics 6, 3-47 26 CAMERON , R H AND MARTIN , W T ( 944) Transformation of Wiener inte grals under translations, Ann Math 45, 386-396 27 CARR, P , JARROW, R , AND MYNENI, R ( 992) Alternative characterizations of American put options, Math Fin 2(2) , 87-106 28 CHAN , K , KAROLY, A , LONGSTAFF, F , AND SANDERS, A ( 992) An empir ical comparison of alternative models of the short-term interest rate, J Fin 47, 209-1 227 29 CHEN , L ( 996) Stochastic Mean and Stochastic Volatility - A Three-Factor Model of the Term Structure of Interest Rates and Its Application to the Pncing of Interest Rate Derivatives, Blackwell, Oxford and Cambridge 30 CHEN , R AND SCOTT, L ( 992) Pricing interest rate options in a two-factor Cox-Ingersoll-Ross model of the term structure, Rev Fin Stud 5, 613-636 CHEN , R AND ScoTT, L ( 993) Maximum likelihood estimation for a multi factor equilibrium model of the term structure of interest rates, J Fixed Income 3, 14-31 32 CHEN , R AND SCOTT, L ( 995) Interest rate options in multifactor Cox Ingersoll-Ross models of the term structure, J Derivatives 3, 53-72 33 CHERIDITO, P (2003) Arbitrage in fractional Brownian motion models, Fin Stochastics 7, 533-553 34 CHEYETTE, ( 996) Markov representation of the Heath-Jarrow-Morton model, BARRA, Berkeley, CA 35 CHUNG, K L ( 968) A Course in Probability Theory, Academic Press, Or lando 36 CHUNG, K L AND WILLIAMS , R J ( 983) Introduction to Stochastic Inte gration, Birkhiiuser, Boston 37 CLEMENT, E , LAMBERTON, D , AND PROTTER, P (2002) An analysis of a least squares regression algorithm for American option pricing, Fin Stochastics 6, 449 471 38 COLLIN-DUFRESNE, P AND GOLDSTEIN, R (2002) Pricing swaptions in the affine framework, J Derivatives 10, 39 COLLIN-DUFRESNE, P AND GOLDSTEIN, R (200 ) Generalizing the affine framework to HJM and random fields, Graduate School of Industrial Admin istration, Carnegie Mellon University 40 Cox, J C , INGERSOLL, J E , AND Ross , S ( 98 ) The relation between forward prices and futures prices, J Fin Econ 9, 32 1-346 Cox, J C , INGERSOLL , J E , AND Ross, S ( 985) A theory of the term structure of interest rates, Econometrica 53, 373-384 References 539 42 Cox, J C , Ross, S A , AND RUBINSTEIN, M ( 979) Option pricing: a sim plified approach, J Fin Econ 7, 229-263 43 Cox, J C AND RUBINSTEIN, M ( 985) Options Markets, Prentice-Hall, En glewood Cliffs, NJ 44 DAI , Q AND SINGLETON , K (2000) Specification analysis of affine term struc ture models, J Fin 55, 1943-1 978 45 DALANG, R C., MORTON , A., AND WILLINGER, W ( 990) Equivalent martin gale measures and no-arbitrage in stochastic security market models, Stochas tics , 185-201 46 DAs , S ( 999) A discrete-time approach to Poisson-Gaussian bond option pricing in the Heath-Jarrow-Morton model, J Econ Dynam Control 23, 333369 47 DAS, S AND FORESI, S ( 996) Exact solutions for bond and option prices with systematic jump risk, Rev Derivatives Res 1, 7-24 48 DEGROOT, M ( 986) Probability and Statistics, 2nd ed , Addison-Wesley, Reading, MA 49 DELBAEN , F AND SCHACHERMAYER, W ( 997) Non-arbitrage and the funda mental theorem of asset pricing: summary of main results, Proceedings of Sym posia in Applied Mathematics, American Mathematical Society, Providence, Rl 50 DERMAN, E AND KANI, I ( 994) Riding on a smile, Risk (2) , 98-10 DERMAN , E , KANI, , A N D CHRISS, N ( 996) Implied binomial trees o f the volatility smile, J Derivatives 3, 7-22 52 DOEBLIN, W ( 940) Sur l'equation de Kolmogoroff, C R Ser I 33 , 10591 102 53 DooB, J ( 1942) Stochastic Processes, Wiley, New York 54 DOTHAN , M U ( 990) Prices in Financial Markets, Oxford University Press, New York 55 DUFFEE, G (2002) Term premia and interest rate forecasts in affine models, J Fin 57, 405-444 56 DUFFIE, D ( 992) Dynamic A sset Pricing Theory, Princeton University Press, Princeton, NJ 57 DUFFIE, D AND KAN , R ( 994) Multi-factor term structure models, Philos Trans R Soc London, Ser A 34 , 577-586 58 DUFFIE, D AND KAN , R ( 994) A yield-factor model of interest rates, Math Fin 6, 379-406 59 DUFFIE, D , PAN , J , AND SINGLETON , K (2000) Transform analysis and option pricing for affine jump-diffusions, Econometnca 68, 1343-1376 60 DUFFIE, D AND PROTTER, P ( 992) From discrete- to continuous-time fi nance; weak convergence of the financial gain process, Math Fin , 1-15 61 DUPIRE, ( 994) Pricing with a smile, Risk (3) , 8-20 62 EINSTEIN, A ( 905) On the movement of small particles suspended in a sta tionary liquid demanded by the molecular-kinetic theory of heat, Ann Phys 17 63 ELLIOTT, R AND KOPP , P ( 990) Option pricing and hedge portfolios for Poisson processes, Stochastic Anal Appl 9, 429-444 64 FAMA , E ( 1965) The behavior of stock-market prices, J Business 38, 34-104 65 FEYNMAN , R ( 948) Space-time approach to nonrelativistic quantum mechan ics, Rev Mod Phys 20, 367-387 540 References 66 FILIPOVIC, D (200 ) Consistency Problems for Heath-Jarrow-Morton Interest Rate Models, Springer, Berlin 67 Fu, M , MADAN , D , AND WANG, T ( 998/ 1999) Pricing continuous Asian op tions: a comparison of Monte Carlo and Laplace transform inversion methods, J Comput Fin , 2(2) , 49-74 68 GARMAN , M AND KOHLHAGEN , S ( 983) Foreign currency option values, J Int Money Fin 2, 231-237 69 GEMAN , H ( 989) The importance of the forward neutral probability in a stochastic approach of interest rates, Working paper, ESSEC 70 GEMAN , H , EL KAROUI , N , AND ROCHET, J ( 995) Changes of numeraire, change of probability measure and option pricing, J Appl Prob 32, 443-458 GEMAN , H AND YOR, M ( 993) Bessel processes, Asian options, and perpe tuities, Math Fin , 349-375 72 GIRSANOV, I V ( 960) On transforming a certain class of stochastic processes by absolutely continuous substitution of measures, Theory Prob Appl 5, 285301 73 GLASSERMAN , P AND Kou, S (2003) The term structure of simple forward rates with jump risk, Math Fin 13, 383-4 10 74 GLASSERMAN , P AND MERENER, N (2003) Numerical solution of jump diffu sion LIBOR market models, Fin Stochastics 7, -27 75 GLASSERMAN , P AND Yu, B Number of paths versus number of basis func tions in American option pricing, Fin Stochastics to appear 76 HAKALA, J AND WYSTUP, U (2002) Foreign Exchange Risk, Risk Books, London 77 HARRISON , J M AND KREPS, D M ( 1979) Martingales and arbitrage in multiperiod security markets, J Econ Theory 20, 381-408 78 HARRISON , J M AND PLISKA, S R ( 98 ) Martingales and stochastic in tegrals in the theory of continuous trading, Stochastic Processes Appl 1 , 5-260 79 HARRISON , J M AND PLISKA, S R ( 983) A stochastic calculus model of continuous trading: complete markets, Stochastic Processes Appl 15 , 313-316 80 HAUG, E ( 998) The Complete Guide to Option Pricing Formulas, McGraw Hill, New York HEATH , D , JARROW, R , AND MORTON, A ( 990) Bond pricing and the term structure of interest rates: A discrete time approximation, J Fin Quant Anal , 419-440 82 HEATH , D , JARROW, R , AND MORTON , A (1 990) Contingent claim valuation with a random evolution of interest rates, Rev Futures Markets 9, 54-76 83 HEATH , D , JARROW, R , AND MORTON, A ( 992) Bond pricing and the term structure of interest rates: a new methodology, Econometrica 60, 77- 84 HESTON , S ( 993) A closed-form solution for options with stochastic volatility and applications to bond and currency options, Rev Fin Stud 6, 327-343 85 Ho, T AND LEE, S ( 986) Term structure movements and pricing interest rate contingent claims, J Fin , 101 1-1029 86 HUANG, C - F AND LITZENBERGER, R ( 988) Foundations for Financial Eco nomics, North Holland, Amsterdam 87 HULL, J (2002) Options, Futures, and other Derivative Secunties, 5th ed , Prentice-Hall, Englewood Cliffs, NJ 88 HULL, J AND WHITE, A ( 990) Pricing interest rate derivative securities, Rev Fin Stud 3, 573-592 References 54 89 HULL, J AND WHITE, A ( 994) Numerical procedures for implementing term structure models II: two-factor models, J Derivatives 2, 37-47 90 HUNT, P , KENNEDY, J , AND PELSSER, A (2000) Markov-functional interest rate models, Fin Stochastics 4, 391-408 INGERSOLL, J E ( 987) Theory of Financial Decision Making, Rowman and Littlefield, Savage, MD 92 IT