Pricing Exotic Derivatives Using Lévy Process Input
Centre for Mathematical Sciences
Lund Institute of Technology,
The main contributions in this thesis are twofold. First an approximation
of the sample paths of a Lévy process is investigated. This is done
by the implementation of a simulation algorithm for the Normal Inverse Gaussian
(NIG) Lévy process. The second contribution is a comparison between
derivative prices in the NIG market and the Black & Scholes market.
The general idea in the simulation algorithm is based on the Lévy-Ito
decomposition of the Lévy process, that is to generate the linear
drift, the Brownian motion and the jumps separately. This thesis focus on
the jumps. The challenge are the jumps smaller than some positive constant,
which may be infinitely many and thus impossible to simulate exactly. For
some Lévy processes, like the NIG case, the small jumps may be
approximated with a Brownian motion. The novelty in this thesis is not the
method itself but the implementation for the NIG Lévy process and
the investigation of the performance of the approximation. It is also shown
how the approximation may be used to reduce the discretization error in the
simulation of some functionals, like for instance hitting a barrier or reflecting
the sample paths.
With the simulated sample paths of the NIG Lévy process exotic derivative
prices in the NIG market may be estimated. In this thesis the price of the
Asian call option, the barrier option and the Russian option are estimated.
As a byproduct also the European call option is considered. Two different
risk neutral measures are considered, namely the Esscher measure and the
minimal entropy measure. In the Black & Scholes market there are closed
form pricing formulas for all derivatives mentioned above, which makes it
easy to compare the prices between the two markets. The main question is
to relate the prices as maturity grows. This question is considered both
analytically and by using simulation. For some derivatives the prices tend
to the same limit, other have different asymptotes. The prices under the
two different risk neutral measures in the NIG market are however very close
in all cases.