Optimal Martingale measures and hedging in models driven by Levy processes

Our research falls into a broad area of pricing and hedging of contingent claims in incomplete markets. In the first part we introduce the Levy processes as a suitable class of processes for financial modelling purposes. This in turn causes the market to become incomplete in general and therefore the martingale measure for the pricing/hedging purposes has to be chosen by introducing some subjective criteria. We study several such criteria in the second section for a general stochastic volatility model driven by Levy process, leading to minimal martingale measure, variance-optimal, or the more general q-optimal martingale measure, for which we show the convergence to the minimal entropy martingale measure for q ↓ 1. The martingale measures studied in the second section are put to use in the third section, where we consider various hedging problems in both martingale and semimartingale setting. We study locally risk-minimization hedging problem, meanvariance hedging and the more general p-optimal hedging, of which the meanvariance hedging is a special case for p = 2. Our model allows us to explicitly determine the variance-optimal martingale measure and the mean-variance hedging strategy using the structural results of Gourieroux, Laurent and Pham (1998) extended to discontinuous case by Arai (2005a). Assuming a Markovian framework and appealing to the Feynman-Kac theorem, the optimal hedge can be found by solving a three-dimensional partial integrodifferential equation. We illustrate this in the last section by considering the variance-optimal hedge of the European put option, and find the solution numerically by applying finite difference method.