The thesis focuses on analyzing copula-based measures of concordance as an important tool for quantifying and understanding relationships between random variables. We commence by defining copulas, mathematical structures that enable the description of the relationship between random variables independently of their marginal distributions. Through Sklar's theorem, we emphasize the link between copulas and cumulative distribution functions and elaborate on the copula-based concordance function Q. We define various measures of concordance, including Kendall's tau, Spearman's rho, Gini's gamma and Blomqvist's beta. For each measure, we present formulas, properties and connections with previously established concepts. Notably, we delve into the relationship between Kendall's tau and Spearman's rho, as well as Blomqvist's beta and the other three measures of concordance, which provide bounds for the values of one another when one measure is computed. We also provide an example of a weak measure of concordance. The thesis concludes by addressing a practical case study, where we compute values for all discussed measures and compare them through graphical representation. The final section explores the application of dependence measures in finance. This work offers an in-depth comprehension of dependence measures, their mathematical foundations and an idea of practical implementation in finance.
|
