As option pricing become increasingly sophisticated in a world-wide market, so do the methods of evaluating and calculating them. Achdou (U. Denis Diderot, Paris) and Pironneau (U. Pierre et Marie Curie, Paris) describe modern numerical techniques useful in financial simulations, focusing on finite difference and finite element methods for partial differential equations to ascertain their reliability, to control and improve their accuracy, and to implement them efficiently. They describe the mechanics of option pricing, including the Monte Carlo methods generally used, but work mainly from the Black-Scholes equation, describing adaptive mesh refinements, the special considerations of American options, sensitivities and calibration, including that of local volatility with European options, and close by describing the calibration of local volatility with American options.