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

TL;DR

This paper develops weak and strong Taylor expansion methods for solving stochastic differential equations, providing tractable formulas for LIBOR market models that improve pricing accuracy without complex numerical schemes.

Contribution

It introduces weight expressions for Taylor coefficients of SDE solutions and applies them to LIBOR models, offering a practical alternative to traditional drift freezing methods.

Findings

Accurate pricing formulas for LIBOR models derived from Taylor expansions.

Comparable accuracy to full numerical schemes with simpler expressions.

Numerical examples confirm the effectiveness of the proposed methods.

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.