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**Introduction to stochastic calculus applied to finance. Transl. from the French by Nicolas Rabeau a. François Mantion.**
*(English)*
Zbl 0847.60001

London: Chapman & Hall. xi, 185 p. (1995).

This book is an excellent introduction to the modern theory of financial mathematics and in particular to the pricing of options. It is written very clearly and manages to be both rigorous and intuitive at the same time. The important ideas are explained in simple situations, and proofs are either complete or sketched with precise references for those details which have been left out. A large number of guided exercises serve to deepen the understanding of the material presented in the text and cover quite an amount of additional material.

The first two chapters are elementary in the sense that they restrict themselves to models for asset prices with a finite time index set over a finite probability space. This has the great advantage that the presentation is not obscured by any technical considerations. Chapter 1 defines trading strategies and arbitrage opportunities and proves by elementary arguments a version of the fundamental theorem of asset pricing which states that the absence of arbitrage is equivalent to the existence of an equivalent martingale measure for discounted asset prices. It also shows in this finite case that completeness of the model is equivalent to uniqueness of the equivalent martingale measure and explains how completeness can be used to price arbitrary contingent claims by arbitrage arguments. The well-known binomial model of Cox, Ross and Rubinstein is included as a particular example. Chapter 2 addresses the pricing and hedging of American options and explains the link to the theory of optimal stopping.

Chapter 3 gives an introduction to stochastic calculus based on Brownian motion. It presents those mathematical concepts which are subsequently used in financial models, namely Brownian motion, the construction and properties of stochastic integrals, the Itô formula, stochastic differential equations and the Markov property of their solutions. Further topics like the Girsanov transformation and Itô’s representation theorem are discussed in Chapter 4 where they are used to explain the pricing and hedging of contingent claims in the well-known Black-Scholes model of geometric Brownian motion. This chapter also includes a section on the pricing of American options and in particular the American put.

The Black-Scholes formula derived in Chapter 4 gives a closed-form expression for the price of a European call option. But this is an exceptional result since it relies heavily on the particular model of geometric Brownian motion and especially on the restrictive assumption that the model’s coefficients are constant (or at most time-dependent functions). Chapter 5 therefore discusses the relation between diffusion processes and partial differential equations which allows one to study more general diffusion models for asset prices. As a consequence of the Feynman-Kac formula, option prices in such a model typically satisfy a fixed partial differential equation (associated with the generator of the diffusion used to describe asset prices) with variable boundary conditions (depending on the option under consideration). To link this to more practical aspects, it is shown how to attack such a partial differential equation by numerical methods. In the case of American options, this approach leads to a parabolic system of partial differential inequalities which can also be approximately solved by numerical methods.

Chapter 6 is an introduction to the modelling of the term structure of interest rates. After explaining some general modelling principles and their consequences under the assumption of a Brownian filtration, the authors present and discuss the well-known models of Vasicek and Cox, Ingersoll and Ross for the short rate. The approach of Heath, Jarrow and Morton to model the entire yield curve is also briefly explained. While this chapter does contain some of the main ideas, it cannot of course do justice to the whole spectrum of ideas and approaches in the term structure literature that have appeared in the last years.

Chapter 7 considers asset models with jumps and more precisely an extension of the Black-Scholes model to a situation where the driving processes are finitely many Brownian motions and Poisson processes with jumps of random size. Since this model is typically incomplete, there is no unique price determined by arbitrage arguments alone, and so the authors resort to a concept of risk-minimization to define option values and hedging strategies. For the case of constant coefficients, this leads again to explicit expressions in particular cases. Finally, Chapter 8 describes some methods which can be used to simulate financial models and compute prices numerically.

The present book is an English translation of the French version which has been on the market since 1991. It is an ideal textbook for an advanced first course on modern financial mathematics and can be highly recommended both to students and instructors in this field. The new version contains quite a number of new exercises as well as updated references in comparison to the French edition. My only regret is that the difference between normal and small print explicitly mentioned in Chapter 3 has inadvertently been omitted during the printing process.

The first two chapters are elementary in the sense that they restrict themselves to models for asset prices with a finite time index set over a finite probability space. This has the great advantage that the presentation is not obscured by any technical considerations. Chapter 1 defines trading strategies and arbitrage opportunities and proves by elementary arguments a version of the fundamental theorem of asset pricing which states that the absence of arbitrage is equivalent to the existence of an equivalent martingale measure for discounted asset prices. It also shows in this finite case that completeness of the model is equivalent to uniqueness of the equivalent martingale measure and explains how completeness can be used to price arbitrary contingent claims by arbitrage arguments. The well-known binomial model of Cox, Ross and Rubinstein is included as a particular example. Chapter 2 addresses the pricing and hedging of American options and explains the link to the theory of optimal stopping.

Chapter 3 gives an introduction to stochastic calculus based on Brownian motion. It presents those mathematical concepts which are subsequently used in financial models, namely Brownian motion, the construction and properties of stochastic integrals, the Itô formula, stochastic differential equations and the Markov property of their solutions. Further topics like the Girsanov transformation and Itô’s representation theorem are discussed in Chapter 4 where they are used to explain the pricing and hedging of contingent claims in the well-known Black-Scholes model of geometric Brownian motion. This chapter also includes a section on the pricing of American options and in particular the American put.

The Black-Scholes formula derived in Chapter 4 gives a closed-form expression for the price of a European call option. But this is an exceptional result since it relies heavily on the particular model of geometric Brownian motion and especially on the restrictive assumption that the model’s coefficients are constant (or at most time-dependent functions). Chapter 5 therefore discusses the relation between diffusion processes and partial differential equations which allows one to study more general diffusion models for asset prices. As a consequence of the Feynman-Kac formula, option prices in such a model typically satisfy a fixed partial differential equation (associated with the generator of the diffusion used to describe asset prices) with variable boundary conditions (depending on the option under consideration). To link this to more practical aspects, it is shown how to attack such a partial differential equation by numerical methods. In the case of American options, this approach leads to a parabolic system of partial differential inequalities which can also be approximately solved by numerical methods.

Chapter 6 is an introduction to the modelling of the term structure of interest rates. After explaining some general modelling principles and their consequences under the assumption of a Brownian filtration, the authors present and discuss the well-known models of Vasicek and Cox, Ingersoll and Ross for the short rate. The approach of Heath, Jarrow and Morton to model the entire yield curve is also briefly explained. While this chapter does contain some of the main ideas, it cannot of course do justice to the whole spectrum of ideas and approaches in the term structure literature that have appeared in the last years.

Chapter 7 considers asset models with jumps and more precisely an extension of the Black-Scholes model to a situation where the driving processes are finitely many Brownian motions and Poisson processes with jumps of random size. Since this model is typically incomplete, there is no unique price determined by arbitrage arguments alone, and so the authors resort to a concept of risk-minimization to define option values and hedging strategies. For the case of constant coefficients, this leads again to explicit expressions in particular cases. Finally, Chapter 8 describes some methods which can be used to simulate financial models and compute prices numerically.

The present book is an English translation of the French version which has been on the market since 1991. It is an ideal textbook for an advanced first course on modern financial mathematics and can be highly recommended both to students and instructors in this field. The new version contains quite a number of new exercises as well as updated references in comparison to the French edition. My only regret is that the difference between normal and small print explicitly mentioned in Chapter 3 has inadvertently been omitted during the printing process.

Reviewer: M.Schweizer (Berlin)

### MSC:

60-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to probability theory |

90-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operations research and mathematical programming |

91B28 | Finance etc. (MSC2000) |