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Financial calculus. An introduction to derivative pricing. (English) Zbl 0858.62094

Cambridge: Cambridge Univ. Press. ix, 233 p. (1996).
This book gives an introduction to the pricing and hedging of financial derivatives in a market with asset prices given by Itô processes. It is aimed mainly at practitioners who want to gain a better understanding of the mathematical ideas and concepts underlying the modern theory of derivative markets. With this goal in mind, the authors have written a very clear account which combines clean mathematical statements with understandable formulations of their meanings and implications.
The book splits naturally into two main parts of roughly the same length. After a brief introduction emphasizing that it is arbitrage and not expectation that matters for derivative pricing, Chapters 2 and 3 explain the required mathematical background. Chapter 2 does this in discrete time by considering in detail the binomial tree known as the Cox-Ross-Rubinstein model. In this elementary setting, the authors introduce concepts like filtrations, martingales, previsible processes and the martingale representation theorem. This is then applied to construct for any given derivative a replicating self-financing strategy which leads to a price for the derivative by absence of arbitrage. Chapter 3 repeats all this in continuous time, but almost without giving proofs. The authors work with analogies and the intuition developed in the discrete-time setting to introduce and explain Brownian motion, the basics of Itô calculus, changes of measure via the Girsanov theorem and the martingale representation theorem in a Brownian filtration. The Black-Scholes model of geometric Brownian motion is then used as an example to set up as in Chapter 2 a three-step procedure for dealing with derivatives: 1) Find an equivalent martingale measure \(Q\) for the discounted stock price process, 2) take the process \(V\) of conditional expectations of the discounted derivative under \(Q\), and 3) represent this martingale as an integral of the discounted stock price to get a replication strategy and therefore \(V\) as discounted price for the derivative.
Having explained the theory, the book then turns to more practical considerations. Chapter 4 explains by well-chosen examples how to adapt the basic approach in order to deal with market securities. These examples are foreign exchange options, stocks with dividends, a simple model for bonds, and quantos. Chapter 5 is a thorough introduction to the interest rate market. After introducing the usual terminology for zero coupon bonds as the basic instruments, the authors consider a simple example of a model to explain how absence of arbitrage restricts the choice of drift in the possible model specification. The Heath-Jarrow-Morton framework is presented and several popular models (Ho-Lee, Vasicek/Hull-White, Cox-Ingersoll-Ross, Black-Karasinski) are discussed in this connection. The chapter also contains a multi-factor version of the HJM approach as well as an overview of some important interest rate products. The final Chapter 6 briefly mentions possible extensions of the basic model to random coefficients, multidimensional models and stochastic calculus in several dimensions. It also explains the invariance of prices under numeraire changes and ends with a brief survey of the relations between absence of arbitrage, completeness and their connections to equivalent martingale measures.
For the intended audience, this book should certainly turn out to be very useful and will most probably justify its price. For teaching a course on financial mathematics, one would prefer more details on proofs and definitions or at least more precise references, but the book could very well be used as an applied complement to a more theoretical text. For a practitioner-oriented text, the comments on the limitations of the basic model are rather too short: problems like transaction costs, incomplete markets, the presence of jumps or constraints on strategies are simply assumed away, and no mention is made of recent research and advances in these areas. I also did not like the terminology “stochastic process” for what is in fact an Itô process. But on the whole, I very much enjoyed reading this book and warmly recommend it to anyone who wants to learn the essential ideas behind pricing by absence of arbitrage.

MSC:

62P05 Applications of statistics to actuarial sciences and financial mathematics
91B28 Finance etc. (MSC2000)
60G35 Signal detection and filtering (aspects of stochastic processes)
91-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to game theory, economics, and finance
62-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics
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