Elementary Calculus of Financial Mathematics (Monographs on Mathematical Modeling & Computation) (Monographs on Mathematical Modeling and Computation)

140 Pages·2008·1.72 MB·English

Elementary
Calculus
of Financial
Mathematics
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
Elementary
Calculus
of Financial
Mathematics
A. J. Roberts
University of Adelaide
Adelaide, South Australia, Australia
Society for Industrial and Applied Mathematics
Philadelphia
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: [phone],
To Barbara, Sam, Ben, and Nicky
for their support over the years
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
✐ ✐
✐ ✐
emfm
✐ ✐
2008/10/22
page viii
✐ ✐
viii Contents
B.2 Kolmogorov backward equation . . . . . . . . . . . . . . . . . . . . . 121
Bibliography 125
Index 127
✐ ✐
✐ ✐
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
✐ ✐
✐ ✐
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
✐ ✐
✐ ✐