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Strong Taylor approximation of stochastic differential equations and application to the L\'evy LIBOR model

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  • Antonis Papapantoleon
  • Maria Siopacha
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    Abstract

    In this article we develop a method for the strong approximation of stochastic differential equations (SDEs) driven by L\'evy processes or general semimartingales. The main ingredients of our method is the perturbation of the SDE and the Taylor expansion of the resulting parameterized curve. We apply this method to develop strong approximation schemes for LIBOR market models. In particular, we derive fast and precise algorithms for the valuation of derivatives in LIBOR models which are more tractable than the simulation of the full SDE. A numerical example for the L\'evy LIBOR model illustrates our method.

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    File URL: http://arxiv.org/pdf/0906.5581
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 0906.5581.

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    Date of creation: Jun 2009
    Date of revision: Oct 2010
    Publication status: Published in Proceedings of the Actuarial and Financial Mathematics Conference, pp. 47-62, 2011
    Handle: RePEc:arx:papers:0906.5581

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    Web page: http://arxiv.org/

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    1. Mark Joshi & Alan Stacey, 2008. "New and robust drift approximations for the LIBOR market model," Quantitative Finance, Taylor & Francis Journals, Taylor & Francis Journals, vol. 8(4), pages 427-434.
    2. Erik Schlögl, 1999. "A Multicurrency Extension of the Lognormal Interest Rate Market Models," Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney 20, Quantitative Finance Research Centre, University of Technology, Sydney.
    3. Maria Siopacha & Josef Teichmann, 2007. "Weak and Strong Taylor methods for numerical solutions of stochastic differential equations," Papers 0704.0745, arXiv.org.
    4. Tim Dunn & Erik Schlögl & Geoff Barton, 2000. "Simulated Swaption Delta-Hedging in the Lognormal Forward Libor Model," Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney 40, Quantitative Finance Research Centre, University of Technology, Sydney.
    5. Ernst Eberlein & Fehmi Özkan, 2005. "The Lévy LIBOR model," Finance and Stochastics, Springer, Springer, vol. 9(3), pages 327-348, 07.
    6. Black, Fischer, 1976. "The pricing of commodity contracts," Journal of Financial Economics, Elsevier, Elsevier, vol. 3(1-2), pages 167-179.
    7. Paul Glasserman & S. G. Kou, 2003. "The Term Structure of Simple Forward Rates with Jump Risk," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 13(3), pages 383-410.
    8. Nicolas Merener & Paul Glasserman, 2003. "Numerical solution of jump-diffusion LIBOR market models," Finance and Stochastics, Springer, Springer, vol. 7(1), pages 1-27.
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