Numerical Valuation of European Options Under Two-Asset Infinite-Activity Exponential Lévy Models

Abstract

We propose a numerical method for the valuation of European-style options under two-asset infinite-activity exponential Lévy models. Our method extends the effective approach developed by Wang et al. [Robust numerical valuation of European and American options under the CGMY process. Journal of Computational Finance, 2007;10(4):31–69] for the 1-dimensional case to the 2-dimensional setting and is applicable for general Lévy measures under mild assumptions. A tailored discretization of the non-local integral term is developed, which can be efficiently evaluated by means of the fast Fourier transform. For the temporal discretization, the semi-Lagrangian θ-method is employed in a convenient splitting fashion, where the diffusion term is treated implicitly and the integral term is handled explicitly by a fixed-point iteration. Numerical experiments for put-on-the-average options under Normal Tempered Stable dynamics reveal a favourable convergence behaviour of our method whenever the exponential Lévy process has finite-variation. In addition, a relevant theoretical convergence result for the discretization of the integral term is proved.