Thesis (MSc (Mathematical Sciences))--University of Stellenbosch, 2011. / Imported from http://etd.sun.ac.za / ENGLISH ABSTRACT: In this thesis, we propose to use Levy processes to model the dynamics of asset prices. In
the first part, we deal with single asset options and model the log stock prices with a Levy
process. We employ pure jump Levy processes of infinite activity, in particular variance
gamma and CGMY processes. We fit the log-returns of six stocks to variance gamma and
CGMY distributions and check the goodness of fit using statistical tests. It is observed
that the variance gamma and the CGMY distributions fit the financial market data much
better than the normal distribution. Calibration shows that at given maturity time the
two models fit into the option prices very well.
In the second part, we investigate the effect of dependence structure to multivariate option
pricing. We use the new concept of Levy copula introduced in the literature by Tankov
[40]. Levy copulas allow us to separate the dependence structure from the behavior of
the marginal components. We consider bivariate variance gamma and bivariate CGMY
models. To model the dependence structure between underlying assets we use the Clayton
Levy copula. The empirical results on six stocks indicate a strong dependence between
two different stock prices. Subsequently, we compute bivariate option prices taking into
account the dependence structure. It is observed that option prices are highly sensitive to
the dependence structure between underlying assets, and neglecting tail dependence will
lead to errors in option pricing. / AFRIKAANSE OPSOMMING: In hierdie proefskrif word Levy prosesse voorgestel om die bewegings van batepryse te
modelleer. Levy prosesse besit die vermoe om die risiko van spronge in ag te neem, asook
om die implisiete volatiliteite, wat in finansiele opsie pryse voorkom, te reproduseer. Ons
gebruik suiwer–sprong Levy prosesse met oneindige aktiwiteit, in besonder die gamma–
variansie (Eng. variance gamma) en CGMY–prosesse. Ons pas die log–opbrengste van ses
aandele op die gamma–variansie en CGMY distribusies, en kontroleer die resultate met
behulp van statistiese pasgehaltetoetse. Die resultate bevestig dat die gamma–variansie en
CGMY modelle die finansiele data beter pas as die normaalverdeling. Kalibrasie toon ook
aan dat vir ’n gegewe verstryktyd die twee modelle ook die opsiepryse goed pas.
Ons ondersoek daarna die gebruik van Levy prosesse vir opsies op meervoudige bates.
Ons gebruik die nuwe konsep van Levy copulas, wat deur Tankov[40] ingelei is. Levy
copulas laat toe om die onderlinge afhanklikheid tussen bateprysspronge te skei van die
randkomponente. Ons bespreek daarna die simulasie van meerveranderlike Levy prosesse
met behulp van Levy copulas. Daarna bepaal ons die pryse van opsies op meervoudige bates
in multi–dimensionele exponensiele Levy modelle met behulp van Monte Carlo–metodes.
Ons beskou die tweeveranderlike gamma-variansie en – CGMY modelle en modelleer die
afhanklikheidsstruktuur tussen onderleggende bates met ’n Levy Clayton copula. Daarna
bereken ons tweeveranderlike opsiepryse. Kalibrasie toon aan dat hierdie opsiepryse baie
sensitief is vir die afhanlikheidsstruktuur, en dat prysbepaling foutief is as die afhanklikheid
tussen die sterte van die onderleggende verdelings verontagsaam word.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/6699 |
| Date | 03 1900 |
| Creators | Dushimimana, Jean Claude |
| Contributors | Ouwehand, Peter, University of Stellenbosch. Faculty of Science. Dept. of Mathematical Sciences. |
| Publisher | Stellenbosch : University of Stellenbosch |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | 140 p. : ill. |
| Rights | University of Stellenbosch |
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