Tri-diagonal preconditioner for pricing options

Authors:
Hong-Kui Pang;Ying-Ying Zhang;Xiao-Qing Jin
Affiliations:
School of Mathematical Sciences, Jiangsu Normal University, Xuzhou, China;College of Mathematics and Statistics, Chongqing University, Chongqing, China;Department of Mathematics, University of Macau, Macao, China
Venue:
Journal of Computational and Applied Mathematics
Year:
2012

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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 T"nx=b in general is a nonsymmetric Toeplitz system. We then solve this system by the normalized preconditioned conjugate gradient method. A tri-diagonal preconditioner L"n is considered. We prove that under certain conditions all the eigenvalues of the normalized preconditioned matrix (L"n^-^1T"n)^*(L"n^-^1T"n) are clustered around one, which implies a superlinear convergence rate. Numerical results exemplify our theoretical analysis.