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**Option pricing: mathematical models and computation.**
*(English)*
Zbl 0844.90011

Oxford: Financial Press. xii, 457 p. (1995).

This book is entirely devoted to the theory of option pricing in continuous time and to the numerical methods for calculating the options’ fair price. Readers should not expect too much in terms of applications of option pricing to portfolio management, efficient market theory, derivative asset design, ... Although the book is self-contained, it will be hard reading without some background in mathematics and finance. The pre-requisites in mathematics would be some exposure to partial differential equations (PDE) and some familiarity with the elementary concepts of stochastic processes although, unlike other texts in the area, the authors de-emphasize the stochastic calculus treatment of the topic, which may suit the prejudices of some instructors. On the finance side, the book does not provide an in-depth coverage of the concepts and institutional aspects of the financial markets. The novice reader would do well to heed the author’s advice to supplement their reading with a more general textbook on investments. Generally speaking, the mathematical presentation is clear and at the lowest level consistent with the subject matter, consciously avoiding the notions of filtration, equivalent martingale measure, ... The authors make an effort to introduce and motivate their main results by using heuristic finance arguments, although more could have been done to provide the reader with a feeling for purely financial intuition.

The first three chapters introduce the vocabulary and the basic concepts of option markets, Wiener processes, Ito’s lemma, the put-call parity theorem, and the PDE approach to the Black-Scholes framework including the complications generated by American options and the distribution of dividends. The theory of PDE used in finance is presented in the next four chapters, diffusion equation for European options, free boundary problems and its variation inequality treatment for American options, jump condition for the integration of dividend payments. Exotic (barrier, Asian, and lookback) options and their pricing theory are presented in chapters nine to twelve. Chapters thirteen to fifteen deal with miscellaneous issues: transaction costs, interest rate derivatives, and convertible bonds. The last seven chapters are devoted to the finite-difference methods and their application to the numerical calculation of the fair option prices. Lattice and finite-element methods are described in the appendices. General considerations concerning numerical methods (discretization, stability, convergence, and efficiency) are introduced in chapter sixteen. Chapter seventeen introduces forward, backward and central differences with their associated methods (explicit, implicit, Crank-Nicolson, and theta) being presented in chapters eighteen and nineteen. Extensions to methods for American (free-boundary) and exotic options are presented in chapters twenty to twenty-two.

The first three chapters introduce the vocabulary and the basic concepts of option markets, Wiener processes, Ito’s lemma, the put-call parity theorem, and the PDE approach to the Black-Scholes framework including the complications generated by American options and the distribution of dividends. The theory of PDE used in finance is presented in the next four chapters, diffusion equation for European options, free boundary problems and its variation inequality treatment for American options, jump condition for the integration of dividend payments. Exotic (barrier, Asian, and lookback) options and their pricing theory are presented in chapters nine to twelve. Chapters thirteen to fifteen deal with miscellaneous issues: transaction costs, interest rate derivatives, and convertible bonds. The last seven chapters are devoted to the finite-difference methods and their application to the numerical calculation of the fair option prices. Lattice and finite-element methods are described in the appendices. General considerations concerning numerical methods (discretization, stability, convergence, and efficiency) are introduced in chapter sixteen. Chapter seventeen introduces forward, backward and central differences with their associated methods (explicit, implicit, Crank-Nicolson, and theta) being presented in chapters eighteen and nineteen. Extensions to methods for American (free-boundary) and exotic options are presented in chapters twenty to twenty-two.

Reviewer: G.Talmain (Vancouver)

### MSC:

91-02 | Research exposition (monographs, survey articles) pertaining to game theory, economics, and finance |

91G20 | Derivative securities (option pricing, hedging, etc.) |

91B24 | Microeconomic theory (price theory and economic markets) |

91G60 | Numerical methods (including Monte Carlo methods) |