|dc.description.abstract||Our research falls into a broad area of pricing and hedging of contingent claims in
incomplete markets. In the rst part we introduce the L evy processes as a suitable
class of processes for nancial 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 L evy 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 integrodi
erential equation. We illustrate this in the last section by considering the
variance-optimal hedge of the European put option, and nd the solution numerically
by applying nite di erence method.||en_US