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Tri-Diagonal Preconditioner for Toeplitz Systems from Finance

Published online by Cambridge University Press:  28 May 2015

Hong-Kui Pang ,
Ying-Ying Zhang  and
Xiao-Qing Jin
Hong-Kui Pang*
Affiliation:
Department of Mathematics, University of Macau, Macao
Ying-Ying Zhang
Affiliation:
Department of Mathematics, University of Macau, Macao
Xiao-Qing Jin
Affiliation:
Department of Mathematics, University of Macau, Macao
*
Corresponding author. Email: [email protected]
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Abstract

We consider a nonsymmetric Toeplitz system which arises in the discretization of a partial integro-differential equation in option pricing problems. The preconditioned conjugate gradient method with a tri-diagonal preconditioner is used to solve this system. Theoretical analysis shows that under certain conditions the tri-diagonal preconditioner leads to a superlinear convergence rate. Numerical results exemplify our theoretical analysis.

Keywords

65F1065M0691B7047B35European call optionpartial integro-differential equationnonsymmetric Toeplitz systemnormalized preconditioned system (matrix)tri-diagonal preconditioner
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 1 , Issue 1 , February 2011 , pp. 82 - 88
Copyright
Copyright © Global-Science Press 2011

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References

[1]Almendral, A. and Oosterlee, C., Numerical valuation of options with jumps in the underlying, Appl. Numer. Math., Vol. 53 (2005), pp. 1–18.Google Scholar
[2]Chan, R. and Jin, X., An Introduction to Iterative Toeplitz Solvers, SIAM, Philadelphia, 2007.Google Scholar
[3]Cont, R. and Voltchkova, E., A finite-difference scheme for option pricing in jump diffusion and exponential Lévy models, SIAM J. Numer. Anal., Vol. 43 (2005), pp. 1596–1626.Google Scholar
[4]Golub, G. and Van Loan, C., Matrix Computations, 3rd edition, Johns Hopkins University Press, Baltimore, 1996.Google Scholar
[5]Jin, X., Developments and Applications of Block Toeplitz Iterative Solvers, Science Press, Beijing; and Kluwer Academic Publishers, Dordrecht, 2002.Google Scholar
[6]Jin, X., Preconditioning Techniques for Toeplitz Systems, Higher Education Press, Beijing, 2010.Google Scholar
[7]Merton, R., Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, Vol. 3 (1976), pp. 125–144.CrossRefGoogle Scholar
[8]Raible, S., Lévy Processes in Finance: Theory, Numerics, and Empirical Facts, PhD thesis, Albert-Ludwigs-Universität Freiburg i. Br., 2000.Google Scholar
[9]Saad, Y., Iterative Methods for Sparse Linear Systems, PWS Publishing Company, Boston, 1996.Google Scholar
[10]Sachs, E. and Strauss, A., Efficient solution of a partial integro-differential equation in finance, Appl. Numer. Math., Vol. 58 (2008), pp. 1687–1703.Google Scholar
[11]Zhang, Y., Preconditioning Techniques for a Family of Toeplitz-Like Systems with Financial Applications, Doctoral Dissertation, University of Macau, Macao, 2010.Google Scholar