On absolutely continuous compensators and nonlinear filtering equations in default risk models

TL;DR

This paper investigates the structure of default risk models with incomplete information, demonstrating that default indicators have absolutely continuous compensators and exploring their implications for pricing defaultable assets.

Contribution

It establishes the absolute continuity of default indicator compensators and analyzes their role in nonlinear filtering equations for default risk modeling.

Findings

Default indicator processes have absolutely continuous compensators in general Markov models.

The paper provides formulas for pricing defaultable assets using these compensators.

Alternative pricing formulas are proposed based on the compensator analysis.

Abstract

We discuss the pricing of defaultable assets in an incomplete information model where the default time is given by a first hitting time of an unobservable process. We show that in a fairly general Markov setting, the indicator function of the default has an absolutely continuous compensator. Given this compensator we then discuss the optional projection of a class of semimartingales onto the filtration generated by the observation process and the default indicator process. Available formulas for the pricing of defaultable assets are analyzed in this setting and some alternative formulas are suggested.