DISCRETIZATION OF THE BLACK-SCHOLES OPERATOR WITH A NATURAL LEFT-HAND SIDE BOUNDARY CONDITION

Lutz Angermann (Germany)

Received September 11, 2007

Abstract

In this paper, we propose a fitted finite volume discretization of the Black-Scholes differential operator, where the left-hand side boundary condition is treated as a natural boundary condition. For the case of European options, we describe a fully discrete method based on an implicit time stepping technique. The analysis is performed within the framework of the vertical method of lines, where the spatial discretization is formulated as a Petrov-Galerkin finite element method with each basis function of the trial space being determined by a set of two-point boundary value problems. We establish the stability and an error bound for the solutions of the fully discretized system.

Keywords and phrases: Black-Scholes equation, option valuation, fitted finite volume method.