JournalsifbVol. 6 , No. 4DOI 10.4171/ifb/106

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

  • F. E. Benth

    University of Oslo, Norway
  • Kenneth Hvistendahl Karlsen

    University of Oslo, Norway
  • K. Reikvam

    University of Oslo, Norway
A semilinear Black and Scholes partial differential equation for valuing American options: approximate solutions and convergence cover

Abstract

In \cite{BKR

}, 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 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 as well.