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Demystifying the Mathematics of Derivatives

The Mathematics of Financial Derivatives: A Student Introduction

By Paul Wilmott, Sam Howison, Jeff Dewynne
Cambridge University Press, 1995, 1996 (paperback)

Reviewed by Viva Hammer

The year 1973 was a turning point in the theory and practice of finance, with the publication by Fischer Black and Myron Scholes of "The pricing of options and corporate liabilities," their seminal paper on the pricing of options. With this paper it could be said that the science of finance was born. Also in 1973, Robert Merton published a paper-"Theory of rational option pricing"-that extended the results reached by Black and Scholes. Since then there has been, in Merton's own words, an explosion of theoretical, applied and empirical research on option pricing. The impact of that explosion has been felt beyond the borders of economics.

The Mathematics of Financial Derivatives presents an excellent overview of the directions in which theoretical finance has expanded. In this regard, the title is a little misleading, because the book has a much wider scope than merely presenting the mathematics of finance, as I hope the reader will glean from this review.

The authors of The Mathematics of Financial Derivatives had very specific aims in writing this book: to present a unified approach to the modeling of financial derivatives using partial differential equations (any equation that has derivatives in it is a differential equation, and a differential equation that has more than one independent variable is a partial differential equation, with a minimum of unrealistic simplifying assumptions); to explain why this model is most appropriate for the particular problem under discussion; and to provide a numerical solution to this more realistic model rather than a closed, explicit solution to a simpler and less realistic model.

The authors address these objectives systematically, in the four parts of the book.

Part 1 introduces the reader to option theory. In Chapter 1, the authors define fundamental concepts, such as shares and dividends, then options and futures, and explain interest rates and the calculation of present values. There is even a guide to reading the financial press. Building on these concepts, Chapter 2 describes the movement of asset prices as a random walk. The mathematical representation of asset price generation is modeled as a stochastic differential equation. The methods of stochastic calculus are introduced via Ito's lemma of options. The authors' heuristic approach to the lemma is based on the Taylor series of expansion.

The authors then lay out the derivation of the Black-Scholes partial differential equation and the corresponding boundary and initial or final conditions for the value of plain vanilla European options.

Almost all the partial differential equations used in modeling financial derivatives are second-order linear parabolic equations, all derived or generalized from the original Black-Scholes equation. Using linear parabolic equations as the vehicle for developing these mathematical representations of financial derivatives is particularly advantageous because there is almost two centuries worth of experience using these equations in the physical sciences, for instance in heat (or diffusion) equations.

In order to provide readers with a good background to the model upon which the book is based, the authors deal in detail with the mathematical properties of the second-order linear parabolic partial differential equation. They explain through examples why certain boundary and initial conditions are relevant on finite or infinite intervals. The examples are taken from physics and then applied to the Black-Scholes model.

The Black-Scholes analysis is extended to be able to model futures and forward pricing as well as the effect of dividends on financial instrument pricing. The authors also discuss the free boundary problems that arise because of the special features of American options-that they can be exercised at any time between the date of entering into the option contract and the exercise date. This requires the introduction of time-dependent parameters in the boundary conditions of the Black-Scholes model.

The authors then present numerical methods for solutions of the boundary and initial value problems derived earlier in the book. Two methods are discussed at length-the finite-difference method and the binomial tree method-and in both, the authors rely heavily on the experience from the physical sciences. In this section we are presented with many pseudo computer programs for the evaluation of the algorithms. This part of the book will be of great assistance to financial practitioners. Although the authors do not recommend any specific software package, readers could design their own tailor-made packages from the pseudo programs detailed in the book.

In the last part of the text, a second variable enters into the analysis, namely interest rates-the authors assume that interest rates are unpredictable. This entails the introduction of a stochastic model for the short-term interest rate. In Chapter 17, one finds an analysis of interest rate derivative products, and in the last chapter convertible bonds with random interest rates are discussed. Some of the problems discussed in these sections take the reader to the perimeters of research in finance.

The book is a stimulating read. It turns out to be much more than an introductory text, yet less than a compendium. It is clear, accessible and concise. It will serve teachers, practitioners, undergraduates in finance or applied mathematics-indeed anyone seeking an introduction to the rich literature on theoretical mathematical finance. As a prerequisite, the reader should have some knowledge of advanced calculus, elements of stochastic differential equations, a course in linear algebra and some experience in computer programming.