Numerical valuation of European options under two-asset infinite-activity exponential L\'evy models

TL;DR

This paper introduces a numerical method for valuing European options under two-asset exponential Lévy models, extending previous 1D approaches to 2D with efficient FFT-based integral evaluation.

Contribution

It extends a 1D numerical approach to 2D for two-asset models and develops a tailored discretization for the non-local integral term.

Findings

The method demonstrates second-order convergence for finite-variation Lévy processes.

Numerical experiments show the method's efficiency and accuracy for put-on-the-average options.

The approach is applicable under mild assumptions for general Lévy measures.

Abstract

We propose a numerical method for the valuation of European-style options under two-asset infinite-activity exponential L\'evy models. Our method extends the effective approach developed by Wang, Wan & Forsyth (2007) for the 1-dimensional case to the 2-dimensional setting and is applicable for general L\'evy 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 theta-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 favourable second-order convergence of our method whenever the exponential L\'evy process has finite-variation.