An efficient sparse grid Galerkin approach for the numerical valuation of basket options under Kou's jump-diffusion model

Abstract

We use a sparse grid approach to discretize a multi-dimensional partial integro-differential equation (PIDE) for the deterministic valuation of European put options on Kou's jump-diffusion processes. We employ a generalized generating system to discretize the respective PIDE by the Galerkin approach and iteratively solve the resulting linear system. Here, we exploit a newly developed recurrence formula, which, together with an implementation of the unidirectional principle for non-local operators, allows us to evaluate the operator application in linear time. Furthermore, we exploit that the condition of the linear system is bounded independently of the number of unknowns. This is due to the use of the Galerkin generating system and the computation of L 2-orthogonal complements. Altogether, we thus obtain a method that is only linear in the number of unknowns of the respective generalized sparse grid discretization. We report on numerical experiments for option pricing with the Kou model in one, two and three dimensions, which demonstrate the optimal complexity of our approach.