Quadratic finite element and preconditioning methods for options pricing in the SVCJ model

We consider option pricing problems in the stochastic volatility jump diffusion model with correlated and contemporaneous jumps in both the return and the variance processes (SVCJ). The option value function solves a partial integro-differential equation (PIDE). We discretize this PIDE in space by the quadratic FE method and integrate the resulting ordinary differential equation in time by an implicit-explicit Euler based extrapolation scheme. The coefficient matrix of the resulting linear systems is block penta-diagonal with penta-diagonal blocks. The preconditioned bi-conjugate gradient stabilized (PBiCGSTAB) method is used to solve the linear systems. According to the structure of the coefficient matrix, several preconditioners are implemented and compared. The performance of preconditioning techniques for solving block-tridiagonal systems resulting from the linear FE discretization of the PIDE is also investigated. The combination of the quadratic FE for spatial discretization, the extrapolation scheme for time discretization, and the PBiCGSTAB method with an appropriate preconditioner is found to be very efficient for solving the option pricing problems in the SVCJ model. Compared to the standard second order linear finite element method combined with the popular successive over-relaxation (SOR) linear system solver, the proposed method reduces computational time by about twenty times at the accuracy level of 1 cent and more than fifty times at the accuracy level of 0.1 cent for the barrier option example tested in the paper.