Sovereign risk modeling and applications.

Summary This thesis deals with the mathematical modeling of sovereign risk and its applications.In the first chapter, motivated by the Eurozone sovereign debt crisis, we propose a model of sovereign default risk. This model takes into account both the movement of sovereign creditworthiness and the impact of critical political events, adding an idiosyncratic credit risk. We focus on the probabilities of default occurring on the dates of critical political events, for which we obtain analytical formulas in a Markovian framework, where we carefully deal with some unusual features, among them the CEV model when the elasticity parameter β >1. We explicitly determine the compensating process of the defect and show that the intensity process does not exist, which contrasts our model with classical approaches. In the second chapter, by examining some hybrid models from the literature, we consider a class of random times with discontinuous conditional distributions for which the classical assumptions of filtrations magnification are not satisfied. We extend the density approach to a more general setting, where Jacod's assumption relaxes, afin order to deal with such random times in the universe of progressive magnification of filtrations. We also study classical problems: the computation of the compensator, the decomposition of the Azema surmartingale, and the characterization of martingales. The decomposition of martingales and semimartingales in the extended filtration affirms that the H' hypothesis remains valid in this generalized setting. In the third chapter, we present applications of the models proposed in the previous chapters. The most important application of the sovereign default model and the generalized density approach is the valuation of securities subject to default risk. The results explain the large negative jumps in the actuarial yield of the Greek long-term bond during the sovereign debt crisis. Greece's creditworthiness tends to worsen over the filter years and the bond yield has negative jumps during critical political events. In particular, the size of a jump depends on the severity of an exogenous shock, the time since the last political event, and the value of the recovery. The generalized density approach also makes it possible to model simultaneous defaults which, although rare, have a severe impact on the market.