Derivative Prices for Models using Levy Processes and Markov Switching

Abstract: This thesis contributes to mathematics, finance and computer simulations. In terms of mathematics this thesis concerns applied probability and Lévy processes and from the financial point of view the thesis concerns derivative pricing. Within these two areas several simulation techniques are investigated. The thesis is organized as follows. The first two chapters are to be considered as reviews on derivative pricing (Chapter 1) and Lévy processes (Chapter 2). Chapter 3 concerns simulation techniques for general Lévy process and the techniques are implemented and evaluated for the normal inverse Gaussian ( extsf{NIG}) Lévy process. The first algorithm deals with the generation of sample paths of a Lévy process. The idea behind the algorithm has been known for a while, but has been theoretically motivated in the resent paper Asmussen and Rosinski (2001). Then two variance reduction techniques are considered. The first is importance sampling for the barrier option and the second is stratification for subordinated Brownian motion. Chapter 4 considers asymptotics for derivative prices as the maturity$T oinfty$. The motivation for this investigation is that the implied volatility smile in most theoretical markets tend to fade out as the maturity grows. This can be interpreted as the prices converge to the price in the Black & Scholes market as $T oinfty$. For some derivatives we are able to show that this actually is the case. We further find that for other derivatives the prices diverge from the price in the Black & Scholes market. These results are tested in the extsf{NIG} market, where we can quantify the divergence and convergence. The final Chapter 5 considers financial markets based on Markov modulated Lévy processes. That is, the parameters in the Lévy process change according to a finite state Markov jump process. This is also often called regime switching markets or hidden Markov markets and the most common choice is the regime switching Brownian motion. In this thesis the arbitrage theory for a general regime switching Lévy market is developed. The chapter also contain several pricing algorithms for different derivatives and it ends with a section that investigate the volatility smile in these markets.