Chapter 2 provides a basic introduction to copula theory. Various approaches to the construction of copulas are considered. Sklar’s theorem makes use of the relationship between the multivariate distribution function and the copula function which joins the marginal distribution functions. There are also constructions which are geometric where no reference is made to distribution functions or random variables. The focus of this thesis is Archimedean copulas. Archimedean copulas form a different family of copulas in that they are not defined in terms of Sklar’s theorem, but are derived using generator functions that have very specific properties.
The various approaches available to estimate the copula parameters are considered in Chapter 2 as well as the implicit assumptions under each approach.
Four different goodness-of-fit tests are considered that can be used to test whether a specific copula adequately models the dependence structure of a set of variables. The
goodness-of-fit tests entail comparing the empirical copula function with the fitted para- metric copula function, or comparing the copula distribution function with its empirical counterpart. Approaches like Akaike’s Information Criteria are derived from the likeli- hood function and are typically used to compare different copula functions to choose the copula that shows the best fit.
The chapter concludes with a discussion on the different sampling techniques that are relevant to Archimedean copulas.
In Chapter 3 the properties of various one-parameter copulas are examined. Although the main focus of the thesis is Archimidean copulas, some of the best-known bivariate copulas are briefly mentioned to gain a better understanding of the types of dependence structures that the various copula families allow for. The link between dependence measures such as Kendall’s tau and the copula parameter is also considered. Another important aspect is to understand whether the copula captures tail dependence. Tail dependence is a very important property in areas such as financial risk management when the dependence between risk drivers such as equity prices or foreign exchange prices is considered.
Bivariate copulas form the building blocks of the multivariate copulas considered in this thesis. Chapter 3 provides a comprehensive examination of the copula properties in preparation for the analyses in Chapters 5 and 6.
In the second part of the chapter approaches to extend the copula dependence structure are considered. Extending the dependence structure of a specific copula implies that where the copula perhaps could only be used to model negative dependence, it will be able to also model positive dependence or tail dependence after the transformation. This obviously extends the usefulness of the specific copula function. Empirical analyses are performed to understand specifically the power transform and its effect on the copula dependence structure.
A practical use of bivariate copulas in the financial industry is examined in Chapter 4. A new approach is developed to identify trading opportunities in the equity market by making use of bivariate copulas to model the dependence structure of equity-pairs.
The first strategy considered is known as pairs-trading. Pairs-trading is a strategy based on the relative mispricing between a pair of correlated stocks. The strategy involves taking a long-short position on the pair of stocks when they diverge from their historical relationship – that is, buying the undervalued stock and selling the overvalued stock. The position is then reversed when the two stocks revert to their historical relationship, and at that time a profit should be locked in. The main assumption is that the stocks will always revert to their historical relationship.
1 Introduction
The second strategy that is considered is referred to as the single stock futures (SSF) trading strategy. The approach to identify a trading opportunity is the same as that used with the pairs-trading strategy; however, in this case only a single SSF contract is added when a trading signal is triggered. It is shown how this approach can be used to construct a portfolio.
It is argued that the trading strategies are shortterm in nature and need active man- agement. There are various subtleties that have to be considered. These include, but are not limited to, how to define the trading signals, drifts in the equity prices, liquidity of the instruments as well as trading costs.
The methodologies in this chapter are illustrated by focusing on the equity market; however, the techniques can easily be extended to other asset classes.
Chapter 5 considers different ways in which to extend the bivariate Archimedean copula function to more than two dimensions. The idea is to generate multivariate Archimedean copulas by nesting various bivariate Archimedean copulas in different structures. A lot of research is available on the topic, but the research is focused on positive dependence. In this chapter the research is extended by establishing the necessary constraints to incorporate negative dependence in the vine structure.
The chapter only considers the case where Archimedean copulas from the same family
are nested to generate the trivariate copula. Analysing the constraints necessary to
ensure the validity of the trivariate copula is extremely intensive mathematically, because it involves analysing the first three derivatives of the trivariate copula function. The complexity of the derivatives leads to formulae that cannot easily be solved analytically, therefore numerical differentiation is used to establish the constraints.
The results obtained in Chapter 5 is further extended in Chapter 6 by establishing the constraints when Archimedean copulas from different families are nested. The constraints derived from the generator functions affect the area over which the copula function is defined.
It is important to understand whether it is possible to nest two copulas that capture very different types of dependence structures. The analyses examine the possibility of nesting a copula that does not allow for any type of tail dependence with a copula that does capture tail dependence, or nesting a copula with only upper-tail dependence with a copula that only captures lower-tail dependence. Similarly, the analyses will consider whether it is possible to nest a copula that only captures negative dependence with a copula that only captures positive dependence.
The final set of analyses investigate whether it is possible to extend the results obtained for trivariate copulas in a simple way to four-dimensional copulas.
Chapter 7 considers a way to allow for wrong-way risk when deriving counterparty credit exposure. The intention of the analysis is to illustrate the theory developed in Chapters 5 and 6 for trivariate copulas using a practical example.
The analyses show how conditional and unconditional scenarios for the underlying risk drivers can be generated and how the exposure profile of a counterparty can be derived with Monte Carlo simulation techniques.
It is argued that very specific rules have to be followed when incorporating negative dependence into a trivariate copula structure and in cases where the copula is only defined over a subarea, a tranformation is necessary to ensure that the area under the density function still adds up to one. These transformations are not considered in this thesis and is proposed as a topic for future research.
Chapter 8 considers the differences between cointegration and copula asset allocation approaches. Cointegration allows simple estimation methods to capture dependencies between non-stationary time series. Copulas are correlation-based and fitted to station- ary price-return series.
Patton (2004) argues that stock price returns are more highly correlated in market downturns than in market upturns, a phenomenon that is referred to as asymmetric dependence. He models the asymmetric dependence with a bivariate copula and uses the copula structure to estimate index-tracking portfolio weights by maximising the excess returns earned over a benchmark. In this chapter Patton’s approach is extended to a multivariate case. Another simulation-based copula approach is also developed.
The focus of the chapter is to compare the cointegration and copula-based asset al- location approaches. The two approaches are based on very different assumptions and theories. Empirical analyses are performed using different trading strategies and sets of selected stocks over various economic conditions to test the performance of the two ap- proaches over time. Transaction costs are not taken into account in this chapter, because it is assumed that the transaction costs will have equivalent effects on the generated returns of the index-tracking portfolios.
The thesis concludes in Chapter 9 where the main results are summarised. Conclusions are derived and a number of recommendations are made for future research.
2 Copula Theory
2.1 Overview
In this chapter the basic copula theory is covered. In Sections 2.2 and 2.3 various defi- nitions and constructions of copulas are discussed, as are some of the properties of the copula function that are important in Chapters 5 and 6. In Section 2.4 how to estimate the parameter of the bivariate copula function is shown, and goodness-of-fit testing is discussed in Section 2.5. The chapter concludes with algorithms that can be used when sampling from copulas in Section 2.6.