# A Course in Financial Calculus

## Book Preface

Financial mathematics provides a striking example of successful collaboration between academia and industry. Advanced mathematical techniques, developed in both universities and banks, have transformed the derivatives business into a multi-trillion-dollar market. This has led to demand for highly trained students and with that demand comes a need for textbooks.

This volume provides a first course in financial mathematics. The influence of Financial Calculus by Martin Baxter and Andrew Rennie will be obvious. I am extremely grateful to Martin and Andrew for their guidance and for allowing me to use some of the material from their book.

The structure of the text largely follows Financial Calculus, but the mathematics, especially the discussion of stochastic calculus, has been expanded to a level appropriate to a university mathematics course and the text is supplemented by a large number of exercises. In order to keep the course to a reasonable length, some sacrifices have been made. Most notable is that there was not space to discuss interest rate models, although many of the most popular ones do appear as examples in the exercises. As partial compensation, the necessary mathematical background for a rigorous study of interest rate models is included in Chapter 7, where we briefly discuss some of the topics that one might hope to include in a second course in financial mathematics. The exercises should be regarded as an integral part of the course. Solutions to these are available to bona fide teachers from [email protected]

The emphasis is on stochastic techniques, but not to the exclusion of all other approaches. In common with practically every other book in the area, we use binomial trees to introduce the ideas of arbitrage pricing. Following Financial Calculus, we also present discrete versions of key definitions and results on martingales and stochastic calculus in this simple framework, where the important ideas are not obscured by analytic technicalities. This paves the way for the more technical results of later chapters. The connection with the partial differential equation approach to arbitrage pricing is made through both delta-hedging arguments and the Feynman– Kac Stochastic Representation Theorem. Whatever approach one adopts, the key point that we wish to emphasise is that since the theory rests on the assumption of absence of arbitrage, hedging is vital. Our pricing formulae only make sense if there is a ‘replicating portfolio’.

An early version of this course was originally delivered to final year undergraduate and first year graduate mathematics students in Oxford in 1997/8. Although we assumed some familiarity with probability theory, this was not regarded as a prerequisite and students on those courses had little difficulty picking up the necessary concepts as we met them. Some suggestions for suitable background reading are made in the bibliography. Since a first course can do little more than scratch the surface of the subject, we also make suggestions for supplementary and more advanced reading from the bewildering array of available books. This project was supported by an EPSRC Advanced Fellowship. It is a pleasure and a privilege to work in Magdalen College and my thanks go to the President, Fellows, staff and students for making it such an exceptional environment. Many people have made helpful suggestions or read early drafts of this volume. I should especially like to thank Ben Hambly, Alex Jackson and Saurav Sen. Thanks also to David Tranah at CUP who played a vital rˆole in shaping the project. His input has been invaluable. Most of all, I should like to thank Lionel Mason for his constant support and encouragement.

Alison Etheridge, June 2001

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