A Textbook for the Education of Tomorrow’s Quants

This is a textbook for a master’s degree finance course with a significant quantitative element. Mr. Cerny is a lecturer in finance at The Business School, Imperial College, London. He received his Ph.D. in economics from the University of Warwick. He is highly regarded, and the textbook is likely to make its mark.

It is notable for its subtitle and for the emphasis that it implies. A “complete market” (the kind assumed by the Black-Scholes-Merton model) is one in which any derivative product can be dynamically replicated by means of cash and the underlying asset. An incomplete market, then, is one is which the world of derivatives and their underlyings do not match each other in the point-by-point replicable manner implied by that definition of completeness. This failure to match makes for a necessary imperfection in hedging. That, of course, is the real world, where traders practice, as Messrs. Scholes and Merton famously discovered.

A variety of illustrations of this practical emphasis might be adduced. In the preface, for example, Mr. Cerny tells us frankly that in his experience “is it hard to understand the Itô calculus, but it is possible to get used to it and to apply it quickly and consistently….”

The road to Itô, though, is long and Sharpe. Mr. Cerny gives a lot of attention to the Sharpe ratio and its weakness as a reward-for-risk measure. The Sharpe ratio, first suggested by William Sharpe in 1966, is the ratio of the mean excess return of a risky security to the standard deviation of that return. Standard deviation is supposed to serve as a surrogate for risk. But does it?

Mr. Cerny asks us to consider two assets A and B, such that asset B performs no worse that asset A in all states. One would expect from a reward-for-risk measure, then, that it will always rate B as equal to or better than A. But the Sharpe ratio fails this test. There is a “bliss point” in excess returns beyond which the Sharpe ratio punishes success, rating A above B. Mr. Cerny solves this problem on Mr. Sharpe’s behalf, first “intuitively” and then in mathematical formalization. He offers what he calls an arbitrage-adjusted Sharpe ratio, defined as “the maximum Sharpe ratio when part of the return can be set aside into an arbitrage fund.”

The book includes a variety of exercises and two appendices, reviewing foundational materials in calculus and probability theory, respectively, as well as a valuable bibliography.

Reviewed by Christopher Faille, Reporter, January 19, 2004
© Copyright Hedge World 2004, 2005

This book is intended as a text for students in masters programs in quantitative finance. Several features distinguish it from others in this market:

1. Concepts are introduced through detailed numerical examples, then developed more generally.

2. There is a web site with extensive supplementary materials and exercises.

3. Many of the computational procedures are implemented and demonstrated using Gauss programs. Some of these are displayed in the text and others are on the web site.

4. Each chapter ends with a summary of the main points, in enough detail as to actually be useful.

The able student could get through most of this book with little prior mathematical knowledge. Elementary concepts from matrix algebra, calculus, and probability are introduced through examples and reinforced through applications. Appendices give systematic overviews of calculus and probability, accompanied by exercises. On the other hand, some talent for abstract, formal reasoning is certainly required, as the applications take students into topics as advanced as stochastic calculus, Fourier transforms and dynamic programming.

Reviewed by Thomas W. Epps
© Copyright American Mathematical Society 2004, 2005