MPS-RR 2001-30
September 2001

On a Semilinear Black and Scholes Partial Differential Equation for Valuing American Options.

Part II: Approximate Solutions and Convergence

Fred E. Benth, Kenneth H. Karlsen, and Kristin Reikvam

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 aslo presented. Besides the predictor-corrector schemes, other methods for constructing approximate solution sequences are discussed and analyzed as well.

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