The Bactra Review: Occasional and ecletic book reviews by Cosma Shalizi   127

Essentials of Stochastic Finance

Facts, Models, Thoery

by A. N. Shiryaev

translated by N. Kruzhilin

Advanced Series on Statistical Science and Applied Probability, volume 3
Singapore: World Scientific, 1999

Martingales for (normal) profit

[A modified version of this review appeared in Quantitative Finance, vol. 2, no. 3 (June 2002), p. 179.]

This huge book is intended as a comprehensive, self-standing introduction to the mathematical aspects of modern financial theory; this, in the author's mind, means a thorough course on the theory of real-valued stochastic processes in discrete and continuous time. Clearly, only a Russian probabilist could think such a thing, but this is not just an abstract course with superficial financial examples thrown in; it is sensible and surprisingly effective.

There are few people better suited to write such a book than A. N. Shiryaev. The pupil of A. N. Kolmogorov, one of the greatest probabilists of all time, he is an eminent authority on stochastic processes in his own right, and is co-author, with R. Lipster, of a fundamental two-volume monograph on time-series inference, The Statistics of Random Processes (2nd edition, Springer-Verlag, 2001). In the present book, as in previous works, he places considerable emphasis on the theory of martingales.

A martingale, as the learned readers of this journal will recall, is a stochastic process without a trend, one for which the best prediction we can make of the future is that it will be just like the present. More exactly, its expected value at time t + 1 is just its value at time t. Yet more exactly, a stochastic process {Xt} is a martingale if, and only if,

E[Xt + 1 | Xt, Xt - 1, ... X1] = Xt.
There is thus no trend to the process, and it is, in that sense, unpredictable (This uses a weak notion of prediction. Imagine the following random walk: take one step right with probability 0.99, and 99 steps left with probability 0.01. The walk is a martingale, but if you can just bet on going left or right, a winning strategy exists (bet right).). The origins of the name are lost in the mists of the history of English gambling.

Martingales are of interest mathematically largely because they lend themselves to proofs of convergence and the like; the machinery for dealing with them is almost as powerful and well-developed as that for dealing with sequences of independent, identically-distributed random variables (which are a genus of martingales). But of course not everything is a martingale; what to do then? Here Shiryaev brings out his main tool: the Doob (or Doob-Meyer) decomposition. This is the fact that every well-behaved, real-valued stochastic process can be represented as the sum of a martingale and a predictable process, and moreover the representation is unique.

The Doob decomposition lets Shiryaev treat reasonably arbitrary processes using tools of martingale theory. In particular, it lets him characterize a market without opportunities for arbitrage, or riskless profits: they are markets where the discounted values of risky assets are martingales, with no predictable components. It is easy to see that this is necessary --- a predictable component would present an opportunity for arbitrage --- but it is also sufficient. Given this fact, Shiryaev goes on to derive formulae for the pricing of different kinds of options, including, as a special case, the Black-Scholes formula.

Such is the essential line of argument. There is much more, of course, some of it quite standard (like the review of the usual classes of time-series models --- ARMA, ARIMA, ARCH, GARCH, etc), some of it much less standard (like the discussion of self-similar processes and the construction of martingale measures). To the extent that there is something missing here, it is, oddly enough, an adequate discussion of statistical inference for time-series. (``Martingale inference'' in the table of contents means using martingale methods to prove theorems.)

No background in finance is necessary to read this book --- the first chapter pretty comprehensively defines financial terms and goals --- but familiarity with stochastic processes, and comfort with measure-theoretic probability, are essential. While teachers who are willing to come up with all their own problems could use it as a textbook, or a very expensive supplementary text, the ideal audience consists of probabilists, or aspiring probabilists, who want to learn about finance, and financial engineers who want see what serious probability theory can teach them. My only hesitation about recommending the book whole-heartedly for self-study is that its binding isn't strong enough for the weight of pages; my copy cracked halfway through --- but I kept reading.

Disclaimer: I got a review copy of this book from Quantitative Finance, but I have no stake in the book's success.

xviii + 834 pp., figures, bibliography, index

Economics / Probability and Statistics

Currently in print as a hardback, ISBN 9-810-23605-0, US$113 [buy from Amazon]

Posted 5 April 2003