Using copulas in modeling nonlinear forms of dependence between random variables

Diploma thesis deals with theory of copulas and their practical use in the field of financial and non-financial data. Copula is a flexible tool for modeling multivariate joint probability distributions focusing on diverse forms of linear and nonlinear dependency structures. The main copula families are presented and graphically compared with the intention to emphasize their characteristic features. The information criteria and selected dependency measures used to choose the corresponding copula are also described. The most frequent methods used to estimate copula parameters are those based on maximum likelihood, which are explained in more detail in the thesis. The end of this chapter describes algorithms generating random values from selected copulas. Subsequently, the method of inverse transformation of randomly generated variables into values simulating the corresponding joint distribution is presented graphically.The first half of the practical part is the application of the copula to describe the mutual relationship of two stock indices S&P 500 and Euronext 100. At first, Student's distribution as the marginal distribution function of both indices is determined. Furthermore, based on the ability to capture tail dependence is selected Student's copula as a suitable function to describe the dependency structure of stock indices. Finally, a bivariate joint distribution is shown compared to the distribution based on Gaussian copula and also to empirical values. The second part of the practical application focuses on describing the relationship of relative humidity measured in the morning and afternoon of the same day. A beta distribution was chosen to describe the marginal distributions of both random variables. The dependence diagnosis is then performed to select the Gumbel-Clayton copula. The resulting joint distribution was able to capture the basic features of the analyzed data.