Credit risk and interdependence.

Summary The objective of this thesis is to study certain problems related to credit risk. These issues are divided into two main themes, namely the monotonicity of transition matrices, and the modeling of interdependence in credit risk. The first theme is motivated by the idealization of empirical transition matrices practiced by banks. We propose in this thesis an optimal solution that allows to approximate an empirical matrix by a monotonic matrix and thus to realize an idealization of the whole matrix. We also prove some theoretical results on the stability of monotonicity under two types of transformations. The second theme of the thesis concerns interdependence in credit risk in general, and we study contagion and its propagation as a special case. The idea is to see a credit portfolio as a network whose nodes are the entities of the portfolio connected via links. We therefore build a graphical Markov field model able to take into account both exogenous factors and interactions between entities. Under the formalism of this model, we study several aspects of interdependence in credit risk, including the occurrence of critical phenomena, the effect of network topology on risk factors and risk propagation. We have been able to make theoretical contributions on these topics by proving theorems and properties that are quite interesting and allow us to predict the behavior of the portfolio under certain conditions. On the other hand, we also propose several ways to solve the computational and calibration problems that made this type of model difficult to use in practice.