What happens after a default: the conditional density approach

TL;DR

This paper introduces a density approach for modeling default times, providing a comprehensive characterization of the default process before and after default, and exploring its implications for pricing and martingale representation.

Contribution

It proposes a novel density process framework that captures the full dynamics of default times, including post-default information, extending existing intensity-based models.

Findings

Density process characterizes default time before and after default

Girsanov transformation impacts density, intensity, and immersion property

Full filtration martingales can be described via reference filtration martingales

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.