Weak and Strong Taylor methods for numerical solutions of stochastic differential equations
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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- 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.
- Miltersen, K. & K. Sandmann & D. Sondermann, 1994. "Closed Form Solutions for Term Structure Derivatives with Log-Normal Interest Rates," Discussion Paper Serie B 308, University of Bonn, Germany.
- 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.
- Erik Schlögl, 2002. "A multicurrency extension of the lognormal interest rate Market Models," Finance and Stochastics, Springer, vol. 6(2), pages 173-196.
- Farshid Jamshidian, 1997. "LIBOR and swap market models and measures (*)," Finance and Stochastics, Springer, vol. 1(4), pages 293-330.
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