Skip to main content Accessibility help
×
Home
Hostname: page-component-684899dbb8-8hm5d Total loading time: 0.29 Render date: 2022-05-23T20:40:59.750Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true }

Tri-Diagonal Preconditioner for Toeplitz Systems from Finance

Published online by Cambridge University Press:  28 May 2015

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: ya87402@umac.mo
Get access

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.

Type
Research Article
Copyright
Copyright © Global-Science Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Almendral, A. and Oosterlee, C., Numerical valuation of options with jumps in the underlying, Appl. Numer. Math., Vol. 53 (2005), pp. 118.CrossRefGoogle Scholar
[2]Chan, R. and Jin, X., An Introduction to Iterative Toeplitz Solvers, SIAM, Philadelphia, 2007.CrossRefGoogle 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. 15961626.CrossRefGoogle 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. 125144.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. 16871703.CrossRefGoogle 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

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Tri-Diagonal Preconditioner for Toeplitz Systems from Finance
Available formats
×

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Tri-Diagonal Preconditioner for Toeplitz Systems from Finance
Available formats
×

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Tri-Diagonal Preconditioner for Toeplitz Systems from Finance
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *