Here is a book for anyone who would like to become better acquainted with the modern tools of numerical analysis for several significant computational problems arising in finance. The authors review some important aspects of finance modeling involving partial differential equations and focus on numerical algorithms for the fast and accurate pricing of financial derivatives and for the calibration of parameters.

Option pricing has become a technical topic that requires sophisticated numerical methods for robust and fast numerical solutions. This book explores the best numerical algorithms and discusses them in depth, from their mathematical analysis up to their implementation in C++ with efficient numerical libraries. Much of this information is not available elsewhere. In particular, this is one of the few books that gives detailed coverage of 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.

This is a book for postgraduate students, professional scientists in the field of finance, researchers, numerical code developers, and those well versed in numerical analysis desiring to learn about numerical and mathematical finance.