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Weak and Strong Taylor methods for numerical solutions of stochastic differential equations

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  • Maria Siopacha
  • Josef Teichmann
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    Abstract

    We apply results of Malliavin-Thalmaier-Watanabe for strong and weak Taylor expansions of solutions of perturbed stochastic differential equations (SDEs). In particular, we work out weight expressions for the Taylor coefficients of the expansion. The results are applied to LIBOR market models in order to deal with the typical stochastic drift and with stochastic volatility. In contrast to other accurate methods like numerical schemes for the full SDE, we obtain easily tractable expressions for accurate pricing. In particular, we present an easily tractable alternative to ``freezing the drift'' in LIBOR market models, which has an accuracy similar to the full numerical scheme. Numerical examples underline the results.

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    File URL: http://arxiv.org/pdf/0704.0745
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 0704.0745.

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    Date of creation: Apr 2007
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    Handle: RePEc:arx:papers:0704.0745

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    Web page: http://arxiv.org/

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    1. 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, American Finance Association, vol. 52(1), pages 409-30, March.
    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. Farshid Jamshidian, 1997. "LIBOR and swap market models and measures (*)," Finance and Stochastics, Springer, Springer, vol. 1(4), pages 293-330.
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    Cited by:
    1. Wolfgang Kluge & Antonis Papapantoleon, 2009. "On the valuation of compositions in L\'evy term structure models," Papers 0902.3456, arXiv.org.
    2. 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.
    3. Martin Keller-Ressel & Antonis Papapantoleon & Josef Teichmann, 2009. "The affine LIBOR models," Papers 0904.0555, arXiv.org, revised Jul 2011.

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