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

ArXiv ID: 2410.02925 “View on arXiv”

Authors: Unknown

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.

Keywords: Finite Volume Methods, Financial Parabolic PDE, Basket Options, Heston Model, Fourier Cosine Expansion, Equities

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 7.0/10
  • Quadrant: Holy Grail
  • Why: The paper presents a sophisticated novel numerical scheme (IMEX Runge-Kutta for 2D PDEs) requiring advanced calculus and numerical analysis, but also includes extensive validation via semi-analytic solutions and GPU implementation, indicating strong empirical grounding.
  flowchart TD
    A["Research Goal: Develop a robust<br>2nd order FV IMEX Runge-Kutta scheme<br>for 2D financial PDEs"] --> B["Data/Inputs:<br>Basket Option & Heston Model PDEs<br>with mixed derivatives"]
    B --> C["Methodology: Second Order<br>Finite Volume IMEX Scheme<br>+ Fourier Cosine Expansion"]
    C --> D["Computational Process:<br>1. Spatial FV Discretization<br>2. IMEX Time Integration (2nd order)<br>3. GPU-Accelerated<br>Multi-dimensional Fourier Truncation"]
    D --> E["Key Findings/Outcomes:<br>• True 2nd order convergence<br>• Handles non-regular initial conditions<br>• Accurate Greeks (no oscillations)<br>• Overcomes tiny diffusive time-steps"]