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.

Purpose: This thesis investigates the relationship concerning traditional safe haven assets, gold and US 10-year treasury bonds during periods of market instability, specifically during the economic concerns raised by the COVID-19 pandemic. It assesses the hedging and safe haven properties of these assets and their dynamic nature throughout two periods of unconventional monetary and fiscal policy measures by the Federal Reserve & US Congress respectively. Furthermore, the study explores a unique divergence between the price movements of the two assets, as well as potential changes in their properties and relationships. Theoretical Perspective: The study is anchored in theoretical concepts based on previous research such as Modern Portfolio Theory, Safe Haven Theory and Hedging Theory. These theories explain asset behaviours during financial turmoil and the relationship between gold and US 10-year treasury bonds during financial crises. The research gap and research questions were formulated based on the information gathered. Methodology: The research employs a quantitative, explanatory approach, anchoredin objectivism and realism, focusing on testing established theories through empirical data. Using a deductive methodology, it investigates potential changes in the dynamic between traditional safe haven assets, gold and US 10-year treasury bonds. Empirical Foundation: Based on a thorough literature review, this study integrates insights from past research and with new data emerging from the pandemic's influence on financial markets and subsequent policy action. The empirical evidence is integrated through quantitative analysis, leveraging ARCH/GARCH models and quantile regression to understand asset performance amid market shocks and policy changes. Conclusion: The findings indicate that gold did not initially act as a hedge against bonds but did so against other assets such as Oil, USD, and BTC during the height of COVID-19. In the recovery phase, this relationship shifted, with gold emerging as a hedge against bonds while its hedging capacity against Oil and Real Yield was negated. Additionally, gold's role as a safe haven against bonds was consistently unsupported across both periods studied. Furthermore, a portfolio analysis revealed a shift in investment strategy, from a balanced gold-bonds mix during the crisis to a sole preference for gold in the recovery phase, adapting to the evolving market conditions and policy changes.