A second order finite volume IMEX Runge-Kutta scheme for two dimensional PDEs in finance

TL;DR

This paper introduces a second order finite volume IMEX Runge-Kutta scheme for two-dimensional PDEs in finance, effectively handling mixed derivatives and improving accuracy and stability in option pricing models.

Contribution

The paper develops a novel second order IMEX finite volume scheme for 2D financial PDEs, addressing challenges with mixed derivatives and non-regular initial conditions.

Findings

Achieves true second order convergence in complex financial models

Overcomes tiny time-step restrictions of explicit schemes

Provides highly accurate, non-oscillatory Greeks approximations

Abstract

In this article we present a novel and general methodology for building second order finite volume implicit-explicit (IMEX) numerical schemes for solving two dimensional financial parabolic PDEs with mixed derivatives. In particular, applications to basket and Heston models are presented. The obtained numerical schemes have excellent properties and are able to overcome the well-documented difficulties related with numerical approximations in the financial literature. The methods achieve true second order convergence with non-regular initial conditions. Besides, the IMEX time integrator allows to overcome the tiny time-step induced by the diffusive term in the explicit schemes, also providing very accurate and non-oscillatory approximations of the Greeks. Finally, in order to assess all the aforementioned good properties of the developed numerical schemes, we compute extremely accurate semi-analytic solutions using multi-dimensional Fourier cosine expansions. A novel technique to truncate the Fourier series for basket options is presented and it is efficiently implemented using multi-GPUs.