The goal of this Bachelor's Thesis is to examine nested Archimedean copulas. In particular, an effcient simulation scheme based on Levy subordinators is presented. As an application, a pricing model for collateralized debt obligations ("CDO") is developed. To understand the theory about nested Archimedean copulas, an introduction to copulas - distribution functions with standard uniform univariate margins -, including the definition, Sklar's Theorem - the main theorem in copula theory -, and some important measures of association are presented. Additionally, Archimedean copulas, a popular class of copulas that naturally extends to high dimensions, are explored. These copulas are explicit and can be expressed in terms of a one-dimensional function called the generator of the Archimedean copula. However, Archimedean copulas are permutation invariant with respect to their arguments, which is considered to be a drawback, especially in large dimensions. To circumvent this rather strong assumption, the class of nested Archimedean copulas can be used. Since members of this exible class are built by nesting Archimedean copulas, they allow for partial asymmetry and, at the same time, remain tractable as Archimedean copulas. More clearly, a probabilistic model for nested Archimedean copulas based on Levy subordinators is constructed. Independent exponential random variables are divided by group-specifc Levy subordinators which are evaluated at a common random time. The resulting random vector has a nested Archimedean survival copula. Therefore, the approach yields an ecient sampling algorithm. Finally, the construction of the random vector is used to understand the induced dependence structure. CDO's are asset-backed securities whose value and payment streams depend on a portfolio of fixed-income underlying assets. The portfolio is subdivided in classes of different risk characteristics, called tranches, which bear losses according to their seniority. CDOs can be interpreted as an insurance against defaults in the underlying portfolio. The protection buyer makes regular premium payments to the protection seller, who, in return, compensates for losses. The goal is to determine the premium such that the contract is fair for both parties. Since companies in the same industry sector are exposed to the same macroeconomic effects or political decisions, they are more correlated than firms in different sectors. Moreover, recovery rates have not been constant over the last 20 years and empirical studies suggest that default rates and recovery rates are negatively correlated. Therefore, a framework based on nested Archimedean copulas with two nesting levels is presented. The model is able to precisely capture the structure of the underlying portfolio and accounts for reverse correlation of default and recovery rates. However, the portfolio-loss distribution is not known explicitly when using nested Archimedean copulas. Thus, the fair spread of a CDO tranche has to be obtained by means of a Monte Carlo simulation. Hence, fast sampling algorithms are necessary. Finally, a calibration to CDO tranche spreads of the iTraxx Europe portfolio is performed to test thetting capability of the model.      «

The goal of this Bachelor's Thesis is to examine nested Archimedean copulas. In particular, an effcient simulation scheme based on Levy subordinators is presented. As an application, a pricing model for collateralized debt obligations ("CDO") is developed. To understand the theory about nested Archimedean copulas, an introduction to copulas - distribution functions with standard uniform univariate margins -, including the definition, Sklar's Theorem - the main theorem in copula theory -, and som...     »