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A fast, stable and accurate numerical method for the Black-Scholes equation of American options

Ehrhardt, Matthias; Mickens, Ronald E.

Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin

In this work we improve the algorithm of Han and Wu (SIAM J. Numer. Anal. 41 (2003), 2081-2095) for American Options with respect to stability, accuracy and order of computational effort. We derive an exact discrete artificial boundary condition (ABC) for the Crank-Nicolson scheme for solving the Black-Scholes equation for the valuation of American options. To ensure stability and to avoid any numerical reflections we derive the ABC on a purely discrete level. Since the exact discrete ABC includes a convolution with respect to time with a weakly decaying kernel, its numerical evaluation becomes very costly for large-time simulations. As a remedy we construct approximate ABCs with a kernel having the form of a finite sum-of-exponentials, which can be evaluated in a very efficient recursion. We prove a simple stability criteria for the approximated artificial boundary conditions. Finally, we illustrate the efficiency and accuracy of the proposed method on several benchmark examples and compare it to previously obtained discretized ABCs of Mayfield and Han and Wu.