Credit risk and credit derivatives: mathematical and numerical modeling.

Summary This thesis deals with the modeling of credit derivatives and consists of two parts: The first part concerns the density model, recently proposed by El Karoui et al. where we make the assumption that the conditional law of default time knowing reference filtration is equivalent to its (unconditional) law. Under this assumption, we give different (and simpler) proofs to the already existing results in the theory of initial and progressive coarsening of filtrations. In addition, we present new results such as the predictable representation theorem for progressively coarsened filtration in the multidimensional case. We then propose several methods for constructing density models in both the one-dimensional and multi-dimensional cases. Finally, we show that the density model is an efficient approach for dynamic hedging of multi-name credit derivatives. In the second part, in order to study the counterparty risk in a CDS contract, we propose a Markov model in which simultaneous defaults are possible. The wrong-way risk is thus represented by the fact that, at the time of the counterparty default, there is a strictly positive probability that the reference entity will also default. We begin by considering a Markov chain with four states corresponding to two names. In this simple case, we obtain semi-explicit formulas for most important quantities, such as price, CVA, EPE, or hedge ratios. We then generalize this framework to account for spread risk by introducing stochastic factors. We treat a Markovian copula model with stochastic intensities. We also address the issue of dynamic CVA hedging with a written CDS on the counterparty. For the implementation of the model, we specify the intensities by affine processes, which given the dynamic copula property of the model, makes the calibration of this model efficient. Numerical results are presented to show the relevance of the CVA behavior in the model with the stylized market facts.