Computational methods for quantitative finance. Finite element methods for derivative pricing.

*(English)*Zbl 1275.91005
Springer Finance. Berlin: Springer (ISBN 978-3-642-35400-7/hbk; 978-3-642-35401-4/ebook). xiii, 299 p. (2013).

This book is a comprehensive treatise on finite difference and finite element methods for financial derivative pricing. I appreciate that the introduction about mathematical finance, on which there are plenty of books, is very short; the authors basically assume that the stage for derivative pricing is set. After introductory chapters on the numerical schemes for PDEs, they are applied to European, American, and exotic options in the Black-Scholes model. The following chapters cover numerical schemes for stochastic volatility models, Lévy models, and combinations thereof, also in several dimensions. American options lead to boundary value problems, and Lévy models to partial integro-differential equations (PIDEs). Many results about discretization error estimates are included. The focus of the book is on pricing, to some extent also on computing Greeks; it seems that the authors do not comment on whether they also propose to use these methods for calibration. Each chapter begins with a succinct, but informative paragraph on the topics that are to be expected, and ends with suggestions for further reading. At least the methods for one-dimensional models should be accessible to practitioners, whereas the later chapters use more advanced concepts and tools (e.g., wavelet discretization, tensor product finite element spaces, Feller processes).

Reviewer: Stefan Gerhold (Wien)

##### MSC:

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

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

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

65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |

65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |

65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |