×

Tri-diagonal preconditioner for pricing options. (English) Zbl 1244.91094

Summary: 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 \(T_n\mathbf {x}= \mathbf {b}\) 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 \((L_n^{-1}T_n)^{\ast}(L_n^{-1}T_n)\) are clustered around one, which implies a superlinear convergence rate. Numerical results exemplify our theoretical analysis.

MSC:

91G20 Derivative securities (option pricing, hedging, etc.)
65F08 Preconditioners for iterative methods
91G60 Numerical methods (including Monte Carlo methods)
65F10 Iterative numerical methods for linear systems
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
15B05 Toeplitz, Cauchy, and related matrices
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Samuelson, P., Rational theory of warrant pricing, Industrial Management Review, 6, 13-32 (1965)
[2] Black, F.; Scholes, M., The pricing of options and corporate liabilities, Journal of Political Economy, 81, 637-654 (1973) · Zbl 1092.91524
[3] Hull, J.; White, A., The pricing of options on assets with stochastic volatilities, Journal of Finance, 42, 281-300 (1987)
[4] Chan, T., Pricing contingent claims on stocks driven by Lévy processes, The Annals of Applied Probability, 9, 504-528 (1999) · Zbl 1054.91033
[5] Kou, S., A jump-diffusion model for option pricing, Management Science, 48, 1086-1101 (2002) · Zbl 1216.91039
[6] Merton, R., Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3, 125-144 (1976) · Zbl 1131.91344
[7] S. Raible, Lévy Processes in Finance: Theory, Numerics, and Empirical Facts, Ph.D. Thesis, Albert-Ludwigs-Universität Freiburg i. Br., 2000.; S. Raible, Lévy Processes in Finance: Theory, Numerics, and Empirical Facts, Ph.D. Thesis, Albert-Ludwigs-Universität Freiburg i. Br., 2000. · Zbl 0966.60044
[8] Almendral, A.; Oosterlee, C., Numerical valuation of options with jumps in the underlying, Applied Numerical Mathematics, 53, 1-18 (2005) · Zbl 1117.91028
[9] Cont, R.; Voltchkova, E., A finite-difference scheme for option pricing in jump diffusion and exponential Lévy models, SIAM Journal on Numerical Analysis, 43, 1596-1626 (2005) · Zbl 1101.47059
[10] Matache, A.; Schwab, C.; Wihler, T., Fast numerical solution of parabolic integro-differential equations with applications in finance, SIAM Journal on Scientific Computing, 27, 369-393 (2005) · Zbl 1098.65123
[11] Sachs, E.; Strauss, A., Efficient solution of a partial integro-differential equation in finance, Applied Numerical Mathematics, 58, 1687-1703 (2008) · Zbl 1155.65109
[12] Y. Zhang, Preconditioning Techniques for a Family of Toeplitz-Like Systems with Financial Applications, Doctoral Dissertation, University of Macau, Macao, 2010. Available at http://netfiles.uiuc.edu/fenglm/www/pdf/Zhang_2010_Thesis.pdf; Y. Zhang, Preconditioning Techniques for a Family of Toeplitz-Like Systems with Financial Applications, Doctoral Dissertation, University of Macau, Macao, 2010. Available at http://netfiles.uiuc.edu/fenglm/www/pdf/Zhang_2010_Thesis.pdf
[13] Chan, R.; Ching, W., Toeplitz-circulant preconditioners for Toeplitz systems and their applications to queueing network with batch arrivals, SIAM Journal on Scientific Computing, 17, 762-772 (1996) · Zbl 0859.65030
[14] Serra, S., Toeplitz preconditioners constructed from linear approximation processes, SIAM Journal on Matrix Analysis and Applications, 20, 446-465 (1998) · Zbl 0919.65031
[15] Chan, R.; Jin, X., An Introduction to Iterative Toeplitz Solvers (2007), SIAM: SIAM Philadelphia · Zbl 1146.65028
[16] Serra, S., On the extreme eigenvalues of Hermitian (block) Toeplitz matrices, Linear Algebra and its Applications, 270, 109-129 (1998) · Zbl 0892.15014
[17] Horn, R.; Johnson, C., Matrix Analysis (1985), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0576.15001
[18] Feng, L.; Linetsky, V., Pricing options in jump diffusion models: an extrapolation approach, Operations Research, 56, 304-325 (2008) · Zbl 1167.91367
[19] Chan, R.; Ng, M., Conjugate gradient methods for Toeplitz systems, SIAM Review, 38, 427-482 (1996) · Zbl 0863.65013
[20] Chan, T., An optimal circulant preconditioner for Toeplitz systems, SIAM Journal on Scientific Computing, 9, 766-771 (1998) · Zbl 0646.65042
[21] Jin, X., Developments and Applications of Block Toeplitz Iterative Solvers (2002), Science Press, Kluwer Academic Publishers: Science Press, Kluwer Academic Publishers Beijing, Dordrecht
[22] Ng, M., Iterative Methods for Toeplitz Systems (2004), Oxford University Press: Oxford University Press Oxford · Zbl 1059.65031
[23] Strang, G., A proposal for Toeplitz matrix calculations, Studies in Applied Mathematics, 74, 171-176 (1986) · Zbl 0621.65025
[24] Tyrtyshnikov, E., Optimal and super-optimal circulant preconditioners, SIAM Journal on Matrix Analysis and Applications, 13, 459-473 (1992) · Zbl 0774.65024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.