Pricing Exotic Derivatives Using Lévy Process Input

Sebastian Rasmus

Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology,
Lund University,

ISSN 1404-028X
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.