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Elementary Calculus of Financial Mathematics

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Mathematical Modeling Editor-in-Chief and Computation Richard Haberman Southern Methodist About the Series University The SIAM series on Mathematical Modeling and Computation draws attention to the wide range of important problems in the physical and life sciences and Editorial Board engineering that are addressed by mathematical modeling and computation; promotes the interdisciplinary culture required to meet these large-scale challenges; Alejandro Aceves and encourages the education of the next generation of applied and computational mathematicians, physical and life scientists, and engineers. Southern Methodist University The books cover analytical and computational techniques, describe significant mathematical developments, and introduce modern scientific and engineering Andrea Bertozzi applications. The series will publish lecture notes and texts for advanced University of California, undergraduate- or graduate-level courses in physical applied mathematics, biomathematics, and mathematical modeling, and volumes of interest to a wide Los Angeles segment of the community of applied mathematicians, computational scientists, and engineers. Bard Ermentrout University of Pittsburgh Appropriate subject areas for future books in the series include fluids, dynamical systems and chaos, mathematical biology, neuroscience, mathematical physiology, epidemiology, morphogenesis, biomedical engineering, reaction-diffusion in Thomas Erneux chemistry, nonlinear science, interfacial problems, solidification, combustion, Université Libre de transport theory, solid mechanics, nonlinear vibrations, electromagnetic theory, Brussels nonlinear optics, wave propagation, coherent structures, scattering theory, earth science, solid-state physics, and plasma physics. Bernie Matkowsky A. J. Roberts, Elementary Calculus of Financial Mathematics Northwestern University James D. Meiss, Differential Dynamical Systems Robert M. Miura E. van Groesen and Jaap Molenaar, Continuum Modeling in the Physical Sciences New Jersey Institute Gerda de Vries, Thomas Hillen, Mark Lewis, Johannes Müller, and Birgitt of Technology Schönfisch, A Course in Mathematical Biology: Quantitative Modeling with Mathematical and Computational Methods Ivan Markovsky, Jan C. Willems, Sabine Van Huffel, and Bart De Moor, Exact and Michael Tabor Approximate Modeling of Linear Systems: A Behavioral Approach University of Arizona R. M. M. Mattheij, S. W. Rienstra, and J. H. M. ten Thije Boonkkamp, Partial Differential Equations: Modeling, Analysis, Computation Johnny T. Ottesen, Mette S. Olufsen, and Jesper K. Larsen, Applied Mathematical Models in Human Physiology Ingemar Kaj, Stochastic Modeling in Broadband Communications Systems Peter Salamon, Paolo Sibani, and Richard Frost, Facts, Conjectures, and Improvements for Simulated Annealing Lyn C. Thomas, David B. Edelman, and Jonathan N. Crook, Credit Scoring and Its Applications Frank Natterer and Frank Wübbeling, Mathematical Methods in Image Reconstruction Per Christian Hansen, Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion Michael Griebel, Thomas Dornseifer, and Tilman Neunhoeffer, Numerical Simulation in Fluid Dynamics: A Practical Introduction Khosrow Chadan, David Colton, Lassi Päivärinta, and William Rundell, An Introduction to Inverse Scattering and Inverse Spectral Problems Charles K. Chui, Wavelets: A Mathematical Tool for Signal Analysis

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Elementary Calculus of Financial Mathematics A. J. Roberts University of Adelaide Adelaide, South Australia, Australia Society for Industrial and Applied Mathematics Philadelphia

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Copyright © 2009 by the Society for Industrial and Applied Mathematics. 10 9 8 7 6 5 4 3 2 1 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 Market Street, 6th Floor, Philadelphia, PA 19104-2688 USA. Trademarked names may be used in this book without the inclusion of a trademark symbol. These names are used in an editorial context only; no infringement of trademark is intended. Maple is a registered trademark of Waterloo Maple, Inc. MATLAB is a registered trademark of The MathWorks, Inc. For MATLAB product information, please contact The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098 USA, 508-647-7000, Fax: 508-647-7001, [email protected], www.mathworks.com. Library of Congress Cataloging-in-Publication Data Roberts, A. J. Elementary calculus of financial mathematics / A. J. Roberts. p. cm. -- (Mathematical modeling and computation ; 15) Includes bibliographical references and index. ISBN 978-0-898716-67-2 1. Finance--Mathematical models. 2. Stochastic processes. 3. Investments-- Mathematics. 4. Calculus. I. Title. HG106.R63 2009 332.01'51923--dc22 2008042349 is a registered trademark.

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To Barbara, Sam, Ben, and Nicky for their support over the years

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emfm ✐ ✐ 2008/10/22 page vii ✐ ✐ Contents Preface ix List of Algorithms xi 1 Financial Indices Appear to Be Stochastic Processes 1 1.1 Brownian motion is also called a Wiener process . . . . . . . . . . . . 3 1.2 Stochastic drift and volatility are unique . . . . . . . . . . . . . . . . 9 1.3 Basic numerics simulate an SDE . . . . . . . . . . . . . . . . . . . . 14 1.4 A binomial lattice prices call option . . . . . . . . . . . . . . . . . . . 20 1.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2 Ito’s Stochastic Calculus Introduced 39 2.1 Multiplicative noise reduces exponential growth . . . . . . . . . . . . 39 2.2 Ito’s formula solves some SDEs . . . . . . . . . . . . . . . . . . . . . 43 2.3 The Black–Scholes equation prices options accurately . . . . . . . . . 48 2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3 The Fokker–Planck Equation Describes the Probability Distribution 61 3.1 The probability distribution evolves forward in time . . . . . . . . . . 65 3.2 Stochastically solve deterministic differential equations . . . . . . . . 76 3.3 The Kolmogorov backward equation completes the picture . . . . . . 84 3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4 Stochastic Integration Proves Ito’s Formula 93 ∫b 4.1 The Ito integral afdW . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.2 The Ito formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Appendix A Extra MATLAB/SCILAB Code 115 Appendix B Two Alternate Proofs 119 B.1 Fokker–Planck equation . . . . . . . . . . . . . . . . . . . . . . . . . 119 vii ✐ ✐ ✐ ✐

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emfm ✐ ✐ 2008/10/22 page viii ✐ ✐ viii Contents B.2 Kolmogorov backward equation . . . . . . . . . . . . . . . . . . . . . 121 Bibliography 125 Index 127 ✐ ✐ ✐ ✐

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emfm ✐ ✐ 2008/10/22 page ix ✐ ✐ Preface Welcome! This book leads you on an introduction into the fascinating realm of ﬁ- nancial mathematics and its calculus. Modern ﬁnancial mathematics relies on a deep and sophisticated theory of random processes in time. Such randomness reﬂects the erratic ﬂuctuations in ﬁnancial markets. I take on the challenge of introducing you to the crucial concepts needed to understand and value ﬁnancial options among such ﬂuctuations. This book supports your learning with the bare minimum of necessary prerequisite mathematics. To deliver understanding with a minimum of analysis, the book starts with a graph- ical/numerical introduction to how to adapt random walks to describe the typical erratic ﬂuctuations of ﬁnancial markets. Then simple numerical simulations both demonstrate the approach and suggest the symbology of stochastic calculus. The ﬁnite steps of the numeri- cal approach underlie the introduction of the binomial lattice model for evaluating ﬁnancial options. Fluctuations in a ﬁnancial environment may bankrupt businesses that otherwise would grow. Discrete analysis of this problem leads to the surprisingly simple extension of classic calculus needed to perform stochastic calculus. The key is to replace squared noise by a mean drift: in effect, dW2= dt. This simple but powerful rule enables us to differentiate, integrate, solve stochastic differential equations, and to triumphantly derive and use the Black–Scholes equation to accurately value ﬁnancial options. The ﬁrst two chapters deal with individual realizations and simulations. However, some applications require exploring the distribution of possibilities. The Fokker–Planck and Kolmogorov equations link evolving probability distributions to stochastic differential equations (SDEs). Such transformations empower us not only to value ﬁnancial options but also to model the natural ﬂuctuations in biology models and to approximately solve differential equations using stochastic simulation. Lastly, the formal rules used previously are justiﬁed more rigorously by an introduc- tion to a sound deﬁnition of stochastic integration. Integration in turn leads to a sound interpretation of Ito’s formula that we ﬁnd so useful in ﬁnancial applications. Prerequisites Basic algebra, calculus, data analysis, probability and Markov chains are prerequisites for this course. There will be many times throughout this book when you will need the concepts and techniques of such courses. Be sure you are familiar with those, and have appropriate references on hand. ix ✐ ✐ ✐ ✐

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emfm ✐ ✐ 2008/10/22 page x ✐ ✐ x Preface Computer simulations Incorporated into this book are MATLAB/SCILAB scripts to enhance your ability to probe the problems and concepts presented and thus to improve learning. You can purchase MATLAB from the Mathworks company, http://www.mathworks.com. SCILAB is available for free via http://www.scilab.org. A. J. Roberts ✐ ✐ ✐ ✐

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