Explicit Computations for a Filtering Problem with Point Process Observations with Applications to Credit Risk

TL;DR

This paper develops explicit filtering methods for a Cox process-based credit risk model with unobserved stochastic intensity, enabling calculation of survival probabilities and default bond prices.

Contribution

It introduces an explicit filtering solution for Cox-Ingersoll-Ross intensity models with unobserved Brownian motion, using Gamma initial distributions and deriving the conditional moment generating function.

Findings

Explicit filtering dynamics for the intensity process.

Conditional moment generating function as a mixture of Gamma distributions.

Application to default probability and bond pricing calculations.

Abstract

We consider the intensity-based approach for the modeling of default times of one or more companies. In this approach the default times are defined as the jump times of a Cox process, which is a Poisson process conditional on the realization of its intensity. We assume that the intensity follows the Cox-Ingersoll-Ross model. This model allows one to calculate survival probabilities and prices of defaultable bonds explicitly. In this paper we assume that the Brownian motion, that drives the intensity, is not observed. Using filtering theory for point process observations, we are able to derive dynamics for the intensity and its moment generating function, given the observations of the Cox process. A transformation of the dynamics of the conditional moment generating function allows us to solve the filtering problem, between the jumps of the Cox process, as well as at the jumps. Assuming that the initial distribution of the intensity is of the Gamma type, we obtain an explicit solution to the filtering problem for all t>0. We conclude the paper with the observation that the resulting conditional moment generating function at time t corresponds to a mixture of Gamma distributions.