Second order finite volume IMEX Runge-Kutta schemes for two dimensional parabolic PDEs in finance

TL;DR

The paper develops a second-order finite-volume method combined with IMEX Runge-Kutta time stepping to solve two-dimensional parabolic PDEs with mixed derivatives in financial contexts. By treating diffusion implicitly and convection explicitly, the approach achieves stability with larger time steps while preserving non-oscillatory behavior and accurate Greeks, even for non-smooth payoffs. Validation on basket options and the Heston model demonstrates robust second-order convergence and substantial speedups over fully explicit schemes, highlighting practical applicability to multi-asset and stochastic-volatility pricing problems. The framework is extendable to higher-order schemes and broader financial models.

Abstract

We present a novel and general methodology for building second-order finite volume implicit-explicit Runge-Kutta numerical schemes for solving two-dimensional financial parabolic PDEs with mixed derivatives. The methods achieve second-order convergence even in the presence of non-regular initial conditions. The IMEX time integrator allows to overcome the tiny time-step induced by the diffusive term in the explicit schemes, also providing accurate and non-oscillatory approximations of the Greeks.

Figures (2)

• Figure 1: Cuts of price surfaces. Numerical solution, continuous; COS solution with squares.
• Figure 2: Cuts of prices surfaces. Numerical solution, continuous; COS solution, squares.