IMEX-RK finite volume methods for nonlinear 1d parabolic PDEs. Application to option pricing

TL;DR

This paper develops second-order IMEX-RK finite volume schemes for efficiently solving nonlinear 1D parabolic PDEs in option pricing, combining explicit advection and implicit diffusion discretizations for accuracy and computational efficiency.

Contribution

It introduces a novel second-order IMEX-RK finite volume method tailored for nonlinear 1D parabolic PDEs in option pricing, enhancing efficiency and accuracy.

Findings

Achieves second-order accuracy for nonlinear problems.

Overcomes strict time step restrictions with implicit diffusion.

Efficiently handles non-regular initial conditions.

Abstract

The goal of this paper is to develop 2nd order Implicit-Explicit Runge-Kutta (IMEX-RK) finite volume (FV) schemes for solving 1d parabolic PDEs for option pricing, with possible nonlinearities in the source and advection terms. The spatial semi-discretization of the advection is carried out by combining finite volume methods with 2nd order state reconstructions; while the diffusive terms are discretized using second-order finite differences. The time integration is performed by means of IMEX-RK time integrators: the advection is treated explicitly, and the diffusion, implicitly. The obtained numerical schemes have several advantages: they are computationally very efficient, thanks to the implicit discretization of the diffusion in the IMEX-RK time integrators, which allows to overcome the strict time step restriction; they yield second-order accuracy for even nonlinear problems and with non-regular initial conditions; and they can be extended to higher order.