An introduction to derivative pricing

THES Book review

The following is an unedited review of Financial Calculus which appeared in the Times Higher Educational Supplement of 24 January 1997.

## Get to know your tool kit

## reviewed by Rudi Bogni

The last two years have seen an upsurge in books meant to bring financial calculus to the attention of a wider public and within that public's range of understanding. This book fits neatly into that category. Andrew Rennie, one of the authors, studied mathematics at Cambridge and is a quantitative analyst at the Union Bank of Switzerland. His co-author Martin Baxter is a professional academic at Cambridge and a Fellow of the Cambridge Philosophical Society.I suspect that Cambridge University Press will be rewarded for its patience in tolerating the deadlines missed by the authors, to which they confess. The book seems to sell itself. Indeed, when I happened to leave it on top of my desk at Imperial College, there was soon a queue of research associates and postgraduate students begging to read it before it hit the bookshelves.

But when the authors say in their preface that "a reader is not expected to have any particular prior body of knowledge, except for some (classical) differential calculus and experience with symbolic notation", I must demur. Baxter and Rennie have done a formidable job in making difficult and/or counter-intuitive concepts accessible and user friendly, but their text still requires a good basis of mathematical general culture, without which the classicist-turned-investment-banker will be stopped in his or her tracks by concepts such as measure and filtration or forced to make several attempts before digesting the subject.

At the outset, the authors state that "the goal of this book is to explore the limits of arbitrage". It open with the "parable of bookmaker", a succinct method, borrowed from the world of bookmaking on horse races, to explain the theory of how a market-maker of a stock, bond, currency or commodity goes about quoting a price, with a view to making a profit through his/her intermediation rather than by taking a directional view. Some may find the parable patronising, I do not: I have met many bankers who have never grasped its point in a professional lifetime.

The first chapter describes how randomness does not necessarily mean lack of structure and explains the concepts of time value for money and arbitrage. It introduces expectations and construction strategies. In the next chapter we learn about the binomial branch and tree models and the binomial representation theorem. We meet the idea of backwards induction and the authors show that the tree structure is not entirely arbitrary, as it "embodies a one-to-one relationship between a node and the history of the stock's path up to and including that node". It points out that whereas construction portfolios are random, like the stock, "there is a vital structural difference - [construction portfolios] are known just in time to be useful, unlike the stock value they are known one step in advance." Filtration, conditional expectations, martingales and processes under a given measure, noise and shift are also explained.

Chapter three covers continuous processes, introduces random walks and Brownian motion, with the warning that "shaping Brownian motion with functions may be powerful, but it brings a dangerous complexity". It introduces stochastic processes and the uniqueness of volatility and drift. Though the presentation of Ito's lemma is traditional, it is done better than most.

Measure reappears with the continuous Radon-Nikodym derivative. The importance of the Cameron-Martin-Girsanov theorem is highlighted by the statement: "The C-M-G theorem applies to Brownian motion, but all our processes are disguised Brownian motions at heart. Now we can see the rewards of our Brownian calculus instantly - C-M-G becomes a powerful tool for controlling the drift of any process." Soon after, we back into Black-Scholes on the basis that "we need a model to cut our teeth on".

Chapter four deals with the pricing of market securities, starting from currencies, moving on to equities with dividends and to bonds. It does so by converting the foreign exchange process into a tradeable discount bond process, and recombining dividends and, separately, coupons. It then covers the general market price of risk and, finally, quanto models.

Interest rate models are the focus of the fifth chapter. All the usual suspects are duly covered: single-factor HJM, Ho and Lee, Vasicek, Cox-Ingersoll-Ross, multi-factor HJM and even the Black-Karasinski and Brace-Gatarek-Musiela models.

At last, in chapter six, we reach the bigger models. In the authors' words: "The Black-Scholes stock model assumes that the stock drift and stock volatility are constant. It assumes that there is only a single stock in the market. And it assumes that the cash bond is deterministic with zero volatility. None of these assumptions is necessary. The subsequent sections tackle these restrictions one by one and show how a more general model can still price and hedge derivatives... if a model is driven by Brownian motions, and has no transaction costs, it is analysable in this framework."

Throughout the text, there are exercises, with answers in appendix three, which allow a book to be used both as a teaching and a self-teaching aid. The layout is well conceived and lends itself to efficient and continuous review of previous material. The glossary in appendix four is concise, but effective; particularly useful is the summary of notation used, listed in appendix two.

Full marks for academic generosity to the authors for listing books for further reading that are in partial competition with their own book. But I am less persuaded by some of their other recommendations. I would have thought that A First Course in Probability, for instance, would be useful prereading to Financial Calculus, rather than the other way around. However the acquisition of knowledge never follows a straight line.

One should not look to a text of this type for the latest theoretical leap forward. That said, the authors have succeeded in both keeping the material up-to-date and in providing some novel insights into well-walked territory. I imagine that all their work and effort was intended to bridge the gap they perceive between the world of practitioners and the academic world. They comment early on: "Such `seat of the pants' practices are more suited to the pioneering days of an industry, rather than the mature $15 trillion market which the derivatives industry has become. On the academic side, effort is too often expended on finding precise answers to the wrong question." Although I cannot subscribe to this diagnosis, as I know personally many practitioners with a solid academic background and many academics well steeped in the problems of the market, I belive nevertheless that Baxter and Rennie have produced a good product, which will make practitioners more rigorous in their thinking. But they have not really pushed forward the current frontier in the attack on the unsolved empirical problems of finance. In this they are just as constrained by the "tool-kit" available as the rest of the academic community.

To the prospective reader interested in the existing state of thinking about quantitative finance, I would say buy the book and read it from cover to cover, perhaps twice, if you do not know why nonmartingales are nontradables. If you know all about that and like to take Harrison-Pliska apart in your leisure time, buy the book regardless for ready reference. And, by the way, I do not know the authors personally - despite the Swiss connection - so my enthusiasm for Financial Calculus is unbiased.

Rudi Bogni, formerly chief executive, Swiss Bank Corporation, London, is on sabbatical at Imperial College, London, studying mathematics.© Times Higher Educational Supplement, Friday 24 January 1997.

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