Taylor expansions of option prices by means of Malliavin Calculus

Abstract

In this thesis, we apply methods from Malliavin calculus to study the approximation of option prices. In financial mathematics, European-type option prices are given as solutions to Kolmogorov's backward equation. We are interested in stochastic differential equations depending on a parameter a, such that at a=0 the model is reduced to a well-known, simple model. In Fournie et al.<br />methods to deal with such situations have been developed for elliptic models. Our approach is built in the hypoelliptic framework. Similar expansions are described in Malliavin and Thalmaier, however, our results seem to be slightly more general.<br />Our main theorem is the existence of weak Taylor approximation schemes of order n under the assumption that there is a strong scheme of the same order with appropriate integrability assumptions and a hypoellipticity assumption.<br />We then concentrate on the applications of these concepts. We apply our methodology to the extensively studied and applied LIBOR market model. Drift terms, that drive LIBOR rates under the terminal measure, depend on LIBOR rates with longer maturities. In practice this stochastic drift is often 'frozen' in order to obtain tractable models, which causes an uncontrolled error. We show that through our strong and weak Taylor approximations we are able to obtain equally tractable models as with the frozen drift, which incorporate the stochasticity of the drift appropriately.<br />Especially the strong correction method yields results which not only perform very well, but also provide a way to solve the LIBOR equation analytically. Numerical experiments underline this feature.<br />Secondly, we develop a stochastic volatility LIBOR market model. We provide tractable formulas for a variety of options, such as plain-vanilla, digital and in-arrears caplets, spread options and swaptions. Moreover, we simultaneously calculate the sensitivity of a given option with respect to the volatility of the volatility parameter, a quantity which we call the partial vega. Our results on the construction of tractable approximations are supported by numerical evidence.<br />This thesis contains non technical introductions to the fundamentals of the theories in question, i.e. interest rate theory and Malliavin calculus. Numerical experiments, which evidence the results and should not be considered as perfect pieces of numerical mathematics, have been performed with MATLAB 7.1.<br />