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The Lévy LIBOR model

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  • Ernst Eberlein

    ()

  • Fehmi Özkan

    ()

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    Abstract

    Models driven by Lévy processes are attractive because of their greater flexibility compared to classical diffusion models. First we derive the dynamics of the LIBOR rate process in a semimartingale as well as a Lévy Heath-Jarrow-Morton setting. Then we introduce a Lévy LIBOR market model. In order to guarantee positive rates, the LIBOR rate process is constructed as an ordinary exponential. Via backward induction we get that the rates are martingales under the corresponding forward measures. An explicit formula to price caps and floors which uses bilateral Laplace transforms is derived. Copyright Springer-Verlag Berlin/Heidelberg 2005

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    File URL: http://hdl.handle.net/10.1007/s00780-004-0145-4
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    Bibliographic Info

    Article provided by Springer in its journal Finance and Stochastics.

    Volume (Year): 9 (2005)
    Issue (Month): 3 (07)
    Pages: 327-348

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    Handle: RePEc:spr:finsto:v:9:y:2005:i:3:p:327-348

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    Related research

    Keywords: Lévy processes; LIBOR market model; forward process; instantaneous forward rates; caps; floors;

    References

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    Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
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    1. Farshid Jamshidian, 1997. "LIBOR and swap market models and measures (*)," Finance and Stochastics, Springer, vol. 1(4), pages 293-330.
    2. Giovanni Di Masi & Tomas Björk & Wolfgang Runggaldier & Yuri Kabanov, 1997. "Towards a general theory of bond markets (*)," Finance and Stochastics, Springer, vol. 1(2), pages 141-174.
    3. Alan Brace & Dariusz G�atarek & Marek Musiela, 1997. "The Market Model of Interest Rate Dynamics," Mathematical Finance, Wiley Blackwell, vol. 7(2), pages 127-155.
    4. 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.
    5. Black, Fischer, 1976. "The pricing of commodity contracts," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 167-179.
    6. Miltersen, Kristian R & Sandmann, Klaus & Sondermann, Dieter, 1997. " Closed Form Solutions for Term Structure Derivatives with Log-Normal Interest Rates," Journal of Finance, American Finance Association, vol. 52(1), pages 409-30, March.
    7. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-54, May-June.
    8. Ernst Eberlein & Sebastian Raible, 1999. "Term Structure Models Driven by General Lévy Processes," Mathematical Finance, Wiley Blackwell, vol. 9(1), pages 31-53.
    9. Lotz, Christopher & Schlogl, Lutz, 2000. "Default risk in a market model," Journal of Banking & Finance, Elsevier, vol. 24(1-2), pages 301-327, January.
    10. Erik Schlögl, 1999. "A Multicurrency Extension of the Lognormal Interest Rate Market Models," Research Paper Series 20, Quantitative Finance Research Centre, University of Technology, Sydney.
    11. Philipp J. Schönbucher, 2000. "A Libor Market Model with Default Risk," Bonn Econ Discussion Papers bgse15_2001, University of Bonn, Germany.
    12. Ernst Eberlein & Jean Jacod & Sebastian Raible, 2005. "Lévy term structure models: No-arbitrage and completeness," Finance and Stochastics, Springer, vol. 9(1), pages 67-88, January.
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    Citations

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    Cited by:
    1. 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.
    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. Marcel Ladkau & John G. M. Schoenmakers & Jianing Zhang, 2012. "Libor model with expiry-wise stochastic volatility and displacement," Papers 1204.5698, arXiv.org.
    4. Antonis Papapantoleon & David Skovmand, 2010. "Picard approximation of stochastic differential equations and application to LIBOR models," Papers 1007.3362, arXiv.org, revised Jul 2011.
    5. 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.
    6. Jos\'e Da Fonseca & Alessandro Gnoatto & Martino Grasselli, 2012. "A flexible matrix Libor model with smiles," Papers 1203.4786, arXiv.org.
    7. Martin Keller-Ressel & Antonis Papapantoleon & Josef Teichmann, 2009. "The affine LIBOR models," Papers 0904.0555, arXiv.org, revised Jul 2011.
    8. Ernst Eberlein & Wolfgang Kluge & Antonis Papapantoleon, 2006. "Symmetries In Lévy Term Structure Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 9(06), pages 967-986.
    9. Leippold, Markus & Strømberg, Jacob, 2014. "Time-changed Lévy LIBOR market model: Pricing and joint estimation of the cap surface and swaption cube," Journal of Financial Economics, Elsevier, vol. 111(1), pages 224-250.

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