TALK by B. Øksendal

June 5, 2008

An Introduction to Malliavin Calculus for Lévy Processes and Applications to Finance

Bernt Øksendal (University of Oslo)

The purpose of this lecture is to give a non-technical, yet rigorous introduction to Malliavin calculus for Lévy processes and its applications to finance. The lecture consists of two parts: Part 1 deals with the Brownian motion case. We first use the Wiener- Itô chaos expansion theorem to define the Mallavin derivative in this context and then study some of its fundamental properties, including the chain rule and the duality property (integration by parts). Then we apply it to finance. Examples of applications are (i) the hedging formula in complete markets provided by the Clark- Ocone theorem, (ii) "parameter sensitivity results", e.g. a numerically tractable computation of the "delta-hedge" and other "greeks" in finance. Part 2 deals with the general Lévy process case. To some extent a similar presentation of the Malliavin derivative can be given here as in Part 1, but there are also basic differences, for example regarding the chain rule. Examples of applications to finance are (i) optimal hedging in incomplete markets (based on the Clark-Ocone formula for Lévy processes), (ii) optimal consumption and portfolio with partial information in a market driven by Lévy processes. The presentation is mainly based on the forthcoming book G. Di Nunno, B. Øksendal and F. Proske: "Malliavin Calculus for Lévy Processes and Applications to Finance". Springer 2008/2009 (to appear).