A semilinear Black and Scholes partial differential equation for valuing American options: approximate solutions and convergence
In [Finance Stoch. 7, No. 3, 277--298 (2003; Zbl 1064.60080)] the authors 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, the authors propose and analyze an upwind finite difference scheme of ``predicator-corrector type'' for the semilinear Black and Scholes equation. The authors prove that the approximate solutions generated by the predicator-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.
upwind finite difference scheme of predicator-corrector type
scientific article; zbMATH DE number 2148304