Tri-Diagonal Preconditioner for Pricing Options

Abstract

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 Ln is considered. We prove that under certain conditions all the eigenvalues of the normalized preconditioned matrix (Ln(-1)Tn) (Ln(-1)Tn) are clustered around one, which implies a superlinear convergence rate. Numerical results exemplify our theoretical analysis.

Keywords: European call option, partial integro-differential equation, nonsymmetric Toeplitz system, normalized preconditioned system, tri-diagonal preconditioner, family of generating functions