Derivative Prices for Models using Lévy Processes and Markov Switching
Centre for Mathematical Sciences
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
(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 ?
8. 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 ? 8. 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. The results are
tested in the 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