What happens after a default: The conditional density approach

Abstract

We present a general model for default time, making precise the role of the intensity process, and showing that this process allows for a knowledge of the conditional distribution of the default only "before the default". This lack of information is crucial while working in a multi-default setting. In a single default case, the knowledge of the intensity process does not allow to compute the price of defaultable claims, except in the case where immersion property is satisfied. We propose in this paper the density approach for default time. The density process will give a full characterization of the links between the default time and the reference filtration, in particular "after the default time". We also investigate the description of martingales in the full filtration in terms of martingales in the reference filtration, and the impact of Girsanov transformation on the density and intensity processes, and also on the immersion property. * This work has benefited from financial support by La Fondation du Risque et Fédération Bancaire Française.

Constructing Random Times with Given Survival Processes and Applications to Valuation of Credit Derivatives

“…From this perspective, it is important to observe that a random time τ constructed in this work has the same intensity under P and Q, but it has different conditional probability distributions with respect to F under P and Q. This illustrates the important fact that the default intensity does not contain enough information to price credit derivatives (in this regard, we refer to El Karoui et al [3]). In several papers in the financial literature, the modeling of credit risk is based on the postulate that the process…”

“…where the first equality holds under the H-hypothesis 1 (see Section 3.2 in El Karoui et al [3]) and the second follows from (ii). It thus suffices to show that…”

“…The following auxiliary result is based on El Karoui et al [3], in the sense that it can be seen as a consequence of Theorem 5.7 therein. For the sake of completeness, we provide a simple proof of Proposition 2.…”

“…However, if the intensity (λ u , u ∈ R + ) of τ under P exists then one can show that for any u ∈ R + the process (Z t (u)λ u G u , t ∈ [u, ∞)) is a (P, F)-martingale (see El Karoui et al [3]). This property implies in turn condition (B.2) provided that the intensity process λ does not vanish.…”

“…In other words, an equivalent change of a probability measure may result in a change of the decreasing component in the multiplicative decomposition as well. The interested reader is referred to Section 6 in El Karoui et al [3] for a more detailed analysis of the change of a probability measure in the framework of the so-called density approach to the modelling of a random time.…”