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

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  • Maria Siopacha
  • Josef Teichmann

Abstract

We apply the results of Malliavin-Thalmaier-Watanabe for strong and weak Taylor expansions of solutions of perturbed stochastic differential equations (SDEs). In particular, we determine weight expressions for the Taylor coefficients of the expansion. The results are applied to LIBOR market models in order to find precise and quick algorithms. In contrast to methods such as Euler-Maruyama-Monte-Carlo for the full SDE, we obtain more tractable expressions for accurate pricing. In particular, we present a readily tractable alternative to 'freezing the drift' in LIBOR market models that has an accuracy similar to the Euler-Maruyama-Monte-Carlo scheme for the full LIBOR market model. Numerical examples underline our results.

Suggested Citation

  • Maria Siopacha & Josef Teichmann, 2010. "Weak and strong Taylor methods for numerical solutions of stochastic differential equations," Quantitative Finance, Taylor & Francis Journals, vol. 11(4), pages 517-528.
    • Handle: RePEc:taf:quantf:v:11:y:2010:i:4:p:517-528
    DOI: 10.1080/14697680903493573
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    References listed on IDEAS

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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. Tim Dun & Geoff Barton & Erik Schlögl, 2001. "Simulated Swaption Delta–Hedging In The Lognormal Forward Libor Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 4(04), pages 677-709.
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    Cited by:

    1. Akihiko Takahashi & Yoshihiro Yajima, 2011. "An Asymptotic Expansion with Push-Down of Malliavin Weights," CIRJE F-Series CIRJE-F-824, CIRJE, Faculty of Economics, University of Tokyo.
    2. Akihiko Takahashi & Toshihiro Yamada, 2011. "An Asymptotic Expansion with Push-Down of Malliavin Weights," CARF F-Series CARF-F-256, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo.
    3. Akihiko Takahashi & Toshihiro Yamada, 2009. "An Asymptotic Expansion with Push-Down of Malliavin Weights," CIRJE F-Series CIRJE-F-695, CIRJE, Faculty of Economics, University of Tokyo.
    4. 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.
    5. Akihiko Takahashi & Toshihiro Yamada, 2009. "An Asymptotic Expansion with Push-Down of Malliavin Weights," CARF F-Series CARF-F-194, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Apr 2011.
    6. Antonis Papapantoleon, 2010. "Old and new approaches to LIBOR modeling," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 64(s1), pages 257-275.

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