Convergence of Numerical Schemes for Degenerate Parabolic Equations Arising in Finance Theory

Abstract

Introduction The aim of this article is twofold: on one hand, we describe a general convergence result which applies to a wide range of numerical schemes (‘monotone schemes’) for nonlinear possibly degenerate elliptic (or parabolic) equation; this type of equation arises naturally in Finance Theory as we will show first. This convergence result was obtained in an article written in collaboration with P.E. Souganidis (1991). On the other hand, we present several simple numerical schemes for computing the price of different types of ‘simple’ options: American options, lookback options and Asian options. These schemes are all based on ‘splitting methods’ and we want to emphasize the fact that this allows also easy extensions for computing the price of more complex options with complicated contracts (cap, floor, … etc). These schemes also provide examples for which the convergence result of the first part applies. This second part reports on several works in collaboration with J. Burdeau, Ch. Daher & M. Romano (cf. references) which were done in connection with the Research and Development Department of the Caisse Autonome de Refmancement (CDC group). The article is organized as follows: since the convergence result for numerical schemes relies strongly on the notion of ‘viscosity solutions’, which is a notion of weak solutions for nonlinear elliptic and parabolic equations, we are first going to present this notion of solutions. In order to introduce it, as a motivation, we examine in the first section several examples of equations arising in Finance Theory, and more particularly in options pricing, and we describe the theoretical difficulties in studying them.