Numerical inverse Laplace transform for convection-diffusion equations

TL;DR

This paper introduces a new contour integral method using elliptic arcs for numerically inverting the Laplace transform to solve linear convection-diffusion equations, demonstrating competitive performance in financial models.

Contribution

The paper presents a novel contour integral approach with an elliptic arc for Laplace inversion, adaptable to any linear convection-diffusion equation without prior pseudospectral information.

Findings

Method is effective for Black-Scholes and Heston models.

Shows fast convergence with trapezoidal rule.

Competitive with existing contour integral methods.

Abstract

In this paper a novel contour integral method is proposed for linear convection-diffusion equations. The method is based on the inversion of the Laplace transform and makes use of a contour given by an elliptic arc joined symmetrically to two half-lines. The trapezoidal rule is the chosen integration method for the numerical inversion of the Laplace transform, due to its well-known fast convergence properties when applied to analytic functions. Error estimates are provided as well as careful indications about the choice of several involved parameters. The method selects the elliptic arc in the integration contour by an algorithmic strategy based on the computation of pseudospectral level sets of the discretized differential operator. In this sense the method is general and can be applied to any linear convection-diffusion equation without knowing any a priori information about its pseudospectral geometry. Numerical experiments performed on the Black-Scholes ($1D$) and Heston ($2D$) equations show that the method is competitive with other contour integral methods available in the literature.