Yves Achdou, Olivier Pironneau “Computational Methods for Option Pricing"
Society for Industrial and Applied Mathematic | 2005-07-01 | ISBN: 0898715733 | 297 pages | PDF | 11,2 MB
Society for Industrial and Applied Mathematic | 2005-07-01 | ISBN: 0898715733 | 297 pages | PDF | 11,2 MB
This book is a must for becoming better acquainted with the modern tools of numerical analysis for several significant computational problems arising in finance. Important aspects of finance modeling are reviewed, involving partial differential equations and numerical algorithms for the fast and accurate pricing of financial derivatives and the calibration of parameters. The best numerical algorithms are fully explored and discussed, from their mathematical analysis up to their implementation in C++ with efficient numerical libraries.
This is one of the few books that thoroughly covers the following topics: mathematical results and efficient algorithms for pricing American options; modern algorithms with adaptive mesh refinement for European and American options; regularity and error estimates are derived and give strong support to the mesh adaptivity, an essential tool for speeding up the numerical implementations; calibration of volatility with European and American options; the use of automatic differentiation of computer codes for computing greeks.
Mathematical finance is an old science but has become a major topic for numerical analysts since Merton [97], Black-Scholes [16] modeled financial derivatives. An excellent book for the mathematical foundation of option pricing is Lamberton and Lapeyre's [85]. Since the Black-Scholes model relies on stochastic differential equations, option pricing rapidly became an attractive topic for specialists in the theory of probability, and stochastic methods were developed first for practical applications, along with analytical closed formulas. But soon, with the rapidly growing complexity of the financial products, other numerical solutions became attractive. Applying the Monte-Carlo method to option pricing is very natural and not difficult, at least for European options, but speeding up the method by variance reduction may become tricky. Similarly, tree methods are very intuitive and fast but also rapidly become difficult as the complexity of the financial product grows
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