A semilinear Black and Scholes partial differential equation for valuing American options: approximate solutions and convergence

Abstract

In The semilinear Black and Scholes equation constitutes a starting point for designing and analyzing a variety of "easy to implement" numerical schemes for computing the value of an American option. To demonstrate this feature, we propose and analyze an upwind finite difference scheme of "predictor-corrector type" for the semilinear Black and Scholes equation. We prove that the approximate solutions generated by the predictor-corrector scheme respect the early exercise constraint and that they converge uniformly to the American option value. A numerical example is also presented. Besides the predictor-corrector schemes, other methods for constructing approximate solution sequences are discussed and analyzed.