The option pricing theory set forth by Black and Scholes assumes that the
underlying asset can be modeled by Geometric Brownian motion, with the
Brownian motion being the driving force of uncertainty. Recent empirical
studies, Dotsis, Psychoyios & Skiadopolous (2007) [17], suggest that the
use of Brownian motion alone is insufficient in accurately describing the
evolution of the underlying asset. A more realistic description of the underlying
asset’s dynamics would be to include random jumps in addition to
that of the Brownian motion.
The concept of including jumps in the asset price model leads us naturally
to the concept of a L'evy process. L'evy processes serve as a building
block for stochastic processes that include jumps in addition to Brownian
motion. In this dissertation we first examine the structure and nature of an
arbitrary L'evy process. We then introduce the stochastic integral for L'evy
processes as well as the extended version of Itˆo’s lemma, we then identify
exponential L'evy processes that can serve as Radon-Nikod'ym derivatives
in defining new probability measures.
Equipped with our knowledge of L'evy processes we then implement
this process in a financial context with the L'evy process serving as driving
source of uncertainty in some stock price model. In particular we look
at jump-diffusion models such as Merton’s(1976) [37] jump-diffusion model
and the jump-diffusion model proposed by Kou and Wang (2004) [30]. As
the L'evy processes we consider have more than one source of randomness
we are faced with the difficulty of pricing options in an incomplete market.
The options that we shall consider shall be mainly European in nature,
where exercise can only occur at maturity. In addition to the vanilla calls
and puts we independently derive a closed form solution for an exchange
option under Merton’s jump-diffusion model making use of conditioning
arguments and stochastic integral representations. We also examine some
exotic options under the Kou and Wang model such as barrier options and
lookback options where the solution to the option price is derived in terms
of Laplace transforms. We then develop the Kou and Wang model to include
only positive jumps, under this revised model we compute the value of a
perpetual put option along with the optimal exercise point.
Keywords
Derivative pricing, L'evy processes, exchange options, stochastic integration. / Thesis (M.Sc.)-University of KwaZulu-Natal, Westville, 2009.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:ukzn/oai:http://researchspace.ukzn.ac.za:10413/8196 |
| Date | January 2009 |
| Creators | Essay, Ahmed Rashid. |
| Contributors | O’Hara, J. G. |
| Source Sets | South African National ETD Portal |
| Language | en_ZA |
| Detected Language | English |
| Type | Thesis |
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