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