“Mathematical Techniques in Finance: Tools for Incomplete Markets,” by Ales Cerny?, Princeton, N.J.: Princeton University Press, 2004, 378 pp., US$80 (cloth).
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 It? calculus, but it is possible to get used to it and to apply it quickly and consistently….”
The road to It?, 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.