D.Comm. / Copulas provide a useful way to model different types of dependence structures explicitly. Instead of having one correlation number that encapsulates everything known about the dependence between two variables, copulas capture information on the level of dependence as well as whether the two variables exhibit other types of dependence, for example tail dependence. Tail dependence refers to the instance where the variables show higher dependence between their extreme values. A copula is defined as a multivariate distribution function with uniform marginals. A useful class of copulas is known as Archimedean copulas that are constructed from generator functions with very specific properties. The main aim of this thesis is to construct multivariate Archimedean copulas by nesting different bivariate Archimedean copulas using the vine construction approach. A characteristic of the vine construction is that not all combinations of generator functions lead to valid multivariate copulas. Established research is limited in that it presents constraints that lead to valid multivariate copulas that can be used to model positive dependence only. The research in this thesis extends the theory by deriving the necessary constraints to model negative dependence as well. Specifically, it ensures that the multivariate copulas that are constructed from bivariate copulas that capture negative dependence, will be able to capture negative dependence as well. Constraints are successfully derived for trivariate copulas. It is, however, shown that the constraints cannot easily be extended to higher-order copulas. The rules on the types of dependence structures that can be nested are also established. A number of practical applications in the financial markets where copula theory can be utilized to enhance the more established methodologies, are considered. The first application considers trading strategies based on statistical arbitrage where the information in the bivariate copula structure is utilised to identify trading opportunities in the equity market. It is shown that trading costs adversely affect the profits generated. The second application considers the impact of wrong-way risk on counterparty credit exposure. A trivariate copula is used to model the wrong-way risk. The aim of the analysis is to show how the theory developed in this thesis should be applied where negative correlation is present in a trivariate copula structure. Approaches are considered where conditional and unconditional risk driver scenarios are derived by means of the trivariate copula structure. It is argued that by not allowing for wrong-way risk, a financial institution’s credit pricing and regulatory capital calculations may be adversely affected. The final application compares the philosophy behind cointegration and copula asset allocation techniques to test which approach produces the most profitable index-tracking portfolios over time. The copula asset allocation approach performs well over time; however, it is very computationally intensive.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:uj/uj:7351 |
| Date | 05 November 2012 |
| Source Sets | South African National ETD Portal |
| Detected Language | English |
| Type | Thesis |
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