Tri-diagonal preconditioner for pricing options

The value of a contingent claim under a jump-diffusion process satisfies a partial integro-differential equation (PIDE). We localize and discretize this PIDE in space by the central difference formula and in time by the second order backward differentiation formula. The resulting system Tnx=b in general is a nonsymmetric Toeplitz system. We then solve this system by the normalized preconditioned conjugate gradient method. A tri-diagonal preconditioner LnLn is considered. We prove that under certain conditions all the eigenvalues of the normalized preconditioned matrix (Ln−1Tn)∗(Ln−1Tn) are clustered around one, which implies a superlinear convergence rate. Numerical results exemplify our theoretical analysis.