Valuation of American options using direct, linear complementarity-based methods

Abstract

The Linear Complementarity Formulation of the American Option Valuation problem, which is based on the Black-Scholes partial differential operator, is often used to assist implicit solution methods. Taking advantage of numerical methods, we can obtain approximate solutions, where a "moving index" determines the approximate position of the moving boundary which corresponds to the optimal exercise boundary of the option. Both Direct Inverse Multiplication (DIM) and Stable DIM (SDIM) methods, which are presented in this paper, use the inverse of the coefficient matrix to locate the moving index and then solve a fixed boundary problem. DIM proves to be more than 100 times faster than very popular iterative competitors like Projected Successive OverRelaxation (PSOR). Due to stability problems from which DIM suffers in some cases, a stable version of it, SDIM, has been proposed. Although SDIM is slightly slower than DIM, it is still much faster compared to PSOR. © Springer-Verlag 2003.