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

Abstract

In [7], we proved that the American (call/put) option valuation problem can be stated in terms of one single semilinear Black and Scholes partial differential equation set in a fixed domain. 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.