Computational methods for option pricing. (English) Zbl 1078.91008

Achdou, Yves; Pironneau, Olivier

Frontiers in Applied Mathematics 30. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (ISBN 0-89871-573-3/pbk; 978-0-89871-749-5/ebook). xviii, 297 p. (2005).

The book to be reviewed is concerned with option pricing algorithms and implementation in C++. Therefore the book is addressed to both practitioners and graduate students in the field of mathematical finance.
The first chapter outlines the general option pricing methods in the Black-Scholes model and briefly introduces numerical techniques like Monte Carlo simulation and binomial trees. Another two numerical techniques, i.e. the methods of finite differences and finite elements, are fully discussed in the following chapters and are used in the remaining chapters. The aim of the book is not to present financial models or their validity, but to present efficient algorithms for selected problems. These problems contain asset price dynamics driven by Lévy-processes, stochastic volatility and local volatility. Furthermore, algorithms with adaptive mesh refinement are studied in order to derive prices and error estimates for European and American options. A great emphasis is placed on calibration methods for the volatility from market data, e.g. prices for European and American options. In order to deal with possible unstability problems of the calibration methods, the authors come up with the Tikhonov regularization. In this context the authors introduce a tool called automatic differentiation, which is also used for calculating sensitivities of option prices.
To summarize, the book provides an interesting source to the implementation of selected problems. Much of this information is not available elsewhere. All computer programs are given partially in the book and fully on the authors’ web site. Thus, the book is highly recommendable for readers with interest in computational finance.

Reviewer: Klaus Schürger (Bonn)

MSC:

91-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to game theory, economics, and finance
91G60	Numerical methods (including Monte Carlo methods)
91-04	Software, source code, etc. for problems pertaining to game theory, economics, and finance

Keywords:

Black-Scholes model; Monte Carlo simulation; finite differences; finite elements; efficient algorithms; asset price dynamics; Lévy processes; stochastic volatility; local volatility; adaptive mesh refinitement; European options; American options; Tikhonov regularization; automatic differentiation; computational finance