# A contour integral method for the Black-Scholes and Heston equations

Article

A contour integral method recently proposed by W eideman [IMA J. Numer. Anal., 30 (2010), pp. 334-350] for integrating semidiscrete advection-diffusion PDEs is improved and 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. An improvement on the existing method is the use of Krylov methods for the shifted linear systems, the solution of which represents the major computational cost of the algorithm. A parallel implementation is also considered. Test examples presented are the Black-Scholes PDE in one space dimension and the Heston PDE in two dimensions, for both vanilla and barrier options. In the Heston case efficiency is compared to ADI splitting schemes, and experiments show that the contour integral method is superior for the range of medium to high accuracy requirements. © 2011 Society for Industrial and Applied Mathematics.