Sensitivity analysis for jump-diffusions

Abstract

The objective of this thesis is to provide formulas for the calculation of sensitivities of the price of a contingent claim - also referred to as the Greeks - where the driving process follows a jump-diffusion. For this reason we show the existence of Malliavin weights under general hypoellipticity assumptions. We only apply Malliavin calculus for Brownian motion and do not have to establish any Malliavin-type calculus with respect to jump processes. In order to construct the above mentioned Malliavin weights and to derive explicit formulas for the Greeks, we have to prove the invertibility of the Malliavin covariance matrix. This approach extends the results by Davis and Johansson, as we do not have to impose any separability assumptions and are moreover able to consider hypoelliptic diffusion terms.<br />Furthermore, we present some numerical implementations of the Merton model and a stochastic volatility model driven by a jump-diffusion through Monte Carlo methods. We show that already for quite simple discontinuous payoff functions (e.g., digital options), sensitivities obtained by Malliavin-Monte-Carlo methods perform much better than those derived from finite difference approximations.