An introduction to derivative pricing

RISK Book review

The following is an unedited review of Financial Calculus which appeared in RISK Magazine March 1997.

## Financial Calculus

## reviewed by L.P. Hughston

This witty, elegant, compact book breaks fresh ground. I think it represents the first of what may come to be a new generation of industry-oriented introductory books on derivative pricing theory, based on the use of modern probabilistic methods.

In the past, books on derivative pricing have tended to be either distinctly academic or decidedly non-academic. The former include lengthy rigorous developments of the underlying theory in the theorem-proof style beloved of mathematicians. The latter tend to contain little more than a relatively soft introduction to the relevant finance theory, not going much further in mathematical content than, say, a simplified exposition of the rules of Itô calculus. For today's readers, this will no longer do, and a higher standard of direct exposition is required. There are several reasons for this.

First, in recent years, as a consequence of the advent of both new products and new sophisticated derivative pricing methodologies, there is now a widespread need for a framework of thought and discourse sufficiently robust to allow for a common understanding by traders, desk-quants and risk managers of the significance and limitations of the models they use. A simplistic familiarity with old-fashioned option pricing methods is not enough for this purpose.

Second, over the past few years we have seen increasing numbers of talented mathematicians, physicists, engineers and computer scientists moving from academia into the investment banking world. Many of these are ready to take on board the modern version of the theory, but they need an account that is both high level and reasonably succinct.

Third, there are many readers who may not be directly working in this area but who nevertheless want to know the real story behind derivative pricing, without hand-waving or superfluous simplification.

Baxter and Rennie have made an excellent step forward in satisfying these modern needs for a new generation of readers.

They key probabilistic concepts involved in a real understanding of derivative pricing are (1) martingale processes and (2) the change-of-measure technique. These are the pivotal ideas that are carefully explained in Financial Calculus, with a minimum of probabilistic prerequisites, and illustrated with a wide variety of pricing applications.

The book begins with a set of simple examples to demonstrate that a naïve expected payout argument for the price of a claim leads to an erroneous valuation, and that arbitrage-based considerations lead to the right value. From there we are led to the binomial model, and the relationship between the change-of-measure argument and the arbitrage pricing method.

In the third chapter, the reader finds a remarkably engaging introduction to the basics of stochastic calculus. Here we learn about stochastic integrals, martingales, Radon-Nikodym derivatives, changes of measure, filtrations, trading strategies and the Black-Scholes model.

As regards the latter, the more familiar approach via the Black-Scholes equation is bypassed. Instead, we are led directly to the value of the option as the expectation, under the risk-neutral measure, of the ratio of the terminal payout of the option to the value of a cash bond at that time. The risk-neutral measure is not given by the real-world system of probabilities, but rather is a changed measure characterised by the property that it makes the ratio of the underlying asset price process to the cash bond process a martingale.

Once these basic principles have been laid out, the book applies the theory in the remaining chapters, with numerous examples of specific structures from equity, foreign exchange and interest rate derivative markets.

The chapter on interest rate models is especially well developed, and is studded with gems of useful information. I liked, for example, the simple but crafty argument on page 150, which shows how to express an arbitrary short-rate diffusion model in Heath-Jarrow-Morton terms.

To help me form a more balanced opinion of the book, I took a poll of the views of some of my younger colleagues. One observed that to make a complex subject more approachable, the authors used the language of the mathematics tutorial, rather than the language of the mathematics textbook.

While the book was not for the mathematically untrained, he said, it was straight to the point, with a fine balance of mathematical formalism and intuitive arguments: the best kind of introductory book.

Another said that he thought it was quite good but that he was not the type who learned easily from purely theoretical argument. He would have preferred more worked numerical examples.

Another colleague thought that Financial Calculus was perhaps a bit slapdash in places and suffered slightly from having little market minutiae in it - but nevertheless was now the best of its kind on the market. And so it is.

L.P. Hughston, Merrill Lynch© RISK Magazine, Vol 10, No 3, March 1997

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