Appraisal of a contour integral method for the Black-Scholes and Heston equations

TL;DR

This paper extends a contour integral method to solve key equations in mathematical finance, demonstrating its efficiency and accuracy for the Black-Scholes and Heston PDEs, especially at medium to high precision levels.

Contribution

The paper adapts and analyzes a contour integral approach for financial PDEs, deriving optimal parameters and comparing its performance to existing methods.

Findings

Superior accuracy for medium to high precision requirements

Effective application to Black-Scholes and Heston equations

Efficiency gains over ADI splitting schemes in two-dimensional cases

Abstract

A contour integral method recently proposed by Weideman [IMA J. Numer. Anal., to appear] for integrating semi-discrete advection-diffusion PDEs, is extended for application to some of the important equations of mathematical finance. Using estimates for the numerical range of the spatial operator, optimal contour parameters are derived theoretically and tested numerically. Test examples presented are the Black-Scholes PDE in one space dimension and the Heston PDE in two dimensions. In the latter case efficiency is compared to ADI splitting schemes for solving this problem. In the examples it is found that the contour integral method is superior for the range of medium to high accuracy requirements. Further improvements to the current implementation of the contour integral method are suggested.