A novel fitted finite volume method for the Black-Scholes equation governing option pricing

In this paper we present a novel numerical method for a degeneratepartial differential equation, called the Black–Scholesequation, governing option pricing. The method is based on afitted finite volume spatial discretization and an implicittime stepping technique. To derive the error bounds for thespatial discretization of the method, we formulate it as a Petrov–Galerkinfinite element method with each basis function of the trialspace being determined by a set of two-point boundary valueproblems defined on element edges. Stability of the discretizationis proved and an error bound for the spatial discretizationis established. It is also shown that the system matrix of thediscretization is an M-matrix so that the discrete maximum principleis satisfied by the discretization. Numerical experiments areperformed to demonstrate the effectiveness of the method. Received 6 January 2003. Revised 15 January 2004.