IDEAS home Printed from https://ideas.repec.org/a/eee/spapps/v120y2010i11p2258-2285.html
   My bibliography  Save this article

Jump-adapted discretization schemes for Lévy-driven SDEs

Author

Listed:
  • Kohatsu-Higa, Arturo
  • Tankov, Peter

Abstract

We present new algorithms for weak approximation of stochastic differential equations driven by pure jump Lévy processes. The method uses adaptive non-uniform discretization based on the times of large jumps of the driving process. To approximate the solution between these times we replace the small jumps with a Brownian motion. Our technique avoids the simulation of the increments of the Lévy process, and in many cases achieves better convergence rates than the traditional Euler scheme with equal time steps. To illustrate the method, we discuss an application to option pricing in the Libor market model with jumps.

Suggested Citation

  • Kohatsu-Higa, Arturo & Tankov, Peter, 2010. "Jump-adapted discretization schemes for Lévy-driven SDEs," Stochastic Processes and their Applications, Elsevier, vol. 120(11), pages 2258-2285, November.
  • Handle: RePEc:eee:spapps:v:120:y:2010:i:11:p:2258-2285
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0304-4149(10)00171-7
    Download Restriction: Full text for ScienceDirect subscribers only
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Jérémy Poirot & Peter Tankov, 2006. "Monte Carlo Option Pricing for Tempered Stable (CGMY) Processes," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 13(4), pages 327-344, December.
    2. NicolaBruti-Liberati & Eckhard Platen, 2007. "Strong approximations of stochastic differential equations with jumps," Published Paper Series 2007-7, Finance Discipline Group, UTS Business School, University of Technology, Sydney.
    3. Ernst Eberlein & Fehmi Özkan, 2005. "The Lévy LIBOR model," Finance and Stochastics, Springer, vol. 9(3), pages 327-348, July.
    4. Rubenthaler, Sylvain, 2003. "Numerical simulation of the solution of a stochastic differential equation driven by a Lévy process," Stochastic Processes and their Applications, Elsevier, vol. 103(2), pages 311-349, February.
    5. Peter Carr & Helyette Geman, 2002. "The Fine Structure of Asset Returns: An Empirical Investigation," The Journal of Business, University of Chicago Press, vol. 75(2), pages 305-332, April.
    6. Paul Glasserman & S. G. Kou, 2003. "The Term Structure of Simple Forward Rates with Jump Risk," Mathematical Finance, Wiley Blackwell, vol. 13(3), pages 383-410, July.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Arturo Kohatsu-Higa & Salvador Ortiz-Latorre & Peter Tankov, 2012. "Optimal simulation schemes for L\'evy driven stochastic differential equations," Papers 1204.4877, arXiv.org.
    2. Mikulevicius, R., 2012. "On the rate of convergence of simple and jump-adapted weak Euler schemes for Lévy driven SDEs," Stochastic Processes and their Applications, Elsevier, vol. 122(7), pages 2730-2757.
    3. Alaya, Mohamed Ben & Hajji, Kaouther & Kebaier, Ahmed, 2016. "Importance sampling and statistical Romberg method for Lévy processes," Stochastic Processes and their Applications, Elsevier, vol. 126(7), pages 1901-1931.
    4. Antonis Papapantoleon & John Schoenmakers & David Skovmand, 2011. "Efficient and accurate log-L\'evy approximations to L\'evy driven LIBOR models," Papers 1106.0866, arXiv.org, revised Jan 2012.
    5. Rey, Clément, 2019. "Approximation of Markov semigroups in total variation distance under an irregular setting: An application to the CIR process," Stochastic Processes and their Applications, Elsevier, vol. 129(2), pages 539-571.
    6. Rosenbaum, Mathieu & Tankov, Peter, 2011. "Asymptotic results for time-changed Lévy processes sampled at hitting times," Stochastic Processes and their Applications, Elsevier, vol. 121(7), pages 1607-1632, July.
    7. Rey Clément, 2017. "Convergence in total variation distance of a third order scheme for one-dimensional diffusion processes," Monte Carlo Methods and Applications, De Gruyter, vol. 23(1), pages 1-12, March.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Antonis Papapantoleon & John Schoenmakers & David Skovmand, 2011. "Efficient and accurate log-Lévi approximations to Lévi driven LIBOR models," CREATES Research Papers 2011-22, Department of Economics and Business Economics, Aarhus University.
    2. Antonis Papapantoleon & John Schoenmakers & David Skovmand, 2011. "Efficient and accurate log-L\'evy approximations to L\'evy driven LIBOR models," Papers 1106.0866, arXiv.org, revised Jan 2012.
    3. Nicola Bruti-Liberati, 2007. "Numerical Solution of Stochastic Differential Equations with Jumps in Finance," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 1-2007, January-A.
    4. Zhanyu Chen & Kai Zhang & Hongbiao Zhao, 2022. "A Skellam market model for loan prime rate options," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 42(3), pages 525-551, March.
    5. Shin Kim, Young & Rachev, Svetlozar T. & Leonardo Bianchi, Michele & Fabozzi, Frank J., 2010. "Tempered stable and tempered infinitely divisible GARCH models," Journal of Banking & Finance, Elsevier, vol. 34(9), pages 2096-2109, September.
    6. Antonis Papapantoleon & Maria Siopacha, 2009. "Strong Taylor approximation of stochastic differential equations and application to the L\'evy LIBOR model," Papers 0906.5581, arXiv.org, revised Oct 2010.
    7. Antonis Papapantoleon, 2010. "Old and new approaches to LIBOR modeling," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 64(3), pages 257-275, August.
    8. Jean-Philippe Aguilar, 2021. "The value of power-related options under spectrally negative Lévy processes," Review of Derivatives Research, Springer, vol. 24(2), pages 173-196, July.
    9. Martin Keller-Ressel & Antonis Papapantoleon & Josef Teichmann, 2009. "The affine LIBOR models," Papers 0904.0555, arXiv.org, revised Jul 2011.
    10. Ming-Chieh Wang & Li-Jhang Huang, 2019. "Pricing cross-currency interest rate swaps under the Levy market model," Review of Derivatives Research, Springer, vol. 22(2), pages 329-355, July.
    11. Nicola Bruti-Liberati, 2007. "Numerical Solution of Stochastic Differential Equations with Jumps in Finance," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 1, July-Dece.
    12. Søren Asmussen, 2022. "On the role of skewness and kurtosis in tempered stable (CGMY) Lévy models in finance," Finance and Stochastics, Springer, vol. 26(3), pages 383-416, July.
    13. Szymon Borak & Adam Misiorek & Rafał Weron, 2010. "Models for Heavy-tailed Asset Returns," SFB 649 Discussion Papers SFB649DP2010-049, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
    14. Winston Buckley & Sandun Perera, 2019. "Optimal demand in a mispriced asymmetric Carr–Geman–Madan–Yor (CGMY) economy," Annals of Finance, Springer, vol. 15(3), pages 337-368, September.
    15. Chengwei Zhang & Zhiyuan Zhang, 2017. "Sequential Sampling for CGMY Processes via Decomposition of their Time Changes," Papers 1708.00189, arXiv.org, revised Aug 2018.
    16. Oleg Kudryavtsev, 2024. "A simplified Wiener–Hopf factorization method for pricing double barrier options under Lévy processes," Computational Management Science, Springer, vol. 21(1), pages 1-30, June.
    17. Arturo Kohatsu-Higa & Salvador Ortiz-Latorre & Peter Tankov, 2012. "Optimal simulation schemes for L\'evy driven stochastic differential equations," Papers 1204.4877, arXiv.org.
    18. Jorge Ignacio Gonz'alez C'azares & Aleksandar Mijatovi'c, 2021. "Monte Carlo algorithm for the extrema of tempered stable processes," Papers 2103.15310, arXiv.org, revised Dec 2022.
    19. Da Fonseca, José & Gnoatto, Alessandro & Grasselli, Martino, 2013. "A flexible matrix Libor model with smiles," Journal of Economic Dynamics and Control, Elsevier, vol. 37(4), pages 774-793.
    20. A. M. Ferreiro & J. A. Garc'ia & J. G. L'opez-Salas & C. V'azquez, 2024. "SABR/LIBOR market models: pricing and calibration for some interest rate derivatives," Papers 2408.01470, arXiv.org.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:spapps:v:120:y:2010:i:11:p:2258-2285. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/505572/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.