AN ANALYSIS OF DEPENDENCE STRUCTURES IN CREDIT RISK MODELS
The subject of this thesis is an analysis of sorne models for portfolios of credit risks, with a special attention to the tools used in them to model the dependence between individuai defaults, which is one of the main concerns in credit risk modeling and rnanagement. We will especially focus our attention to son1e rnodels for several groups of exchangeable risks; in particular we will tackle the problerns related to the rnaxirnum likelihood estimation of the pararneters involved, and to this purpose we introduce an approach based on the utilization of the Expetation-Maximization algoritlun. In the first chapter we introduce the rnain rnathernatical tools used in the the thesis. We describe briefly the theory of exchangeable sequences of randorn variables and we show a proof, for the case of randorn vectors, of the main result about them, the De Finetti's theoren1. Then we introduce the concept of copula and we present the Sklar's theorem, showing how it works in the case of continuous distribution functions. Through sorne examples we illustrate how copulas can be used to describe the dependence structures in rnultivariate distributions. We also show how it is possible to express in a natural way the coefficient of tail dependence, an alternative measure of dependence between random variables, by mean of copulas. In the second chapter we review the two most cornrnon classes of rnodels for dependent credit risks, the latent variables rnodels and the Bernoulli rnixture models. We see how essentially alllatent variables models are based on the Gaussian copula, and we ci te an example of a generalization to a more generai dependence structure. Then we show some exarnples of Bernoulli rnixture models, and we present a simple result that allows to rewrite latent variable models as Bernoulli mixture ones. At the end of the chapter we introduce the first model for several exchangeable groups of risks that we will study in detail in the following. In the third chapter we analyze the two rnodels on which we have devoted the rnajor attention: the multidimensional and iterative urn schernes. They are two examples of rnulti factors Bernoulli mixture models whose dependence structure is introduced by generalizations of the Pòlya urn scheme, in such a way that it allows for dependence both in the same rating group and in different rating groups, introducing then some form of contagion between defaults. For both of them, by using the tools of the theory about exchangeable sequences of random variables, we derive the expression of the joint default probability for the number of defaults in the different rating groups. The cornplexity involved in the statistica! estimation of the parameters of these models lead us to introduce the Expectation-11axirnization algorithm for iterative rnaximum likelihood estimations, which is the subject of chapter four. We briefly present the generai theory about it, and then show how to apply it in the case of the three models considered. In the fifth chapter we face the technical aspects of the irnplernentation of the algorithm and present the results obtained, discussing about advantages and disadvantages of the use of the algorithrn. In the last chapter we plot sorne graphics to compare qualitatively the rnodels and to analyze their capacity of expressing the dependence between defaults in different rating classes.