7 research outputs found
Sample-path Large Deviations in Credit Risk
The event of large losses plays an important role in credit risk. As these
large losses are typically rare, and portfolios usually consist of a large
number of positions, large deviation theory is the natural tool to analyze the
tail asymptotics of the probabilities involved. We first derive a sample-path
large deviation principle (LDP) for the portfolio's loss process, which enables
the computation of the logarithmic decay rate of the probabilities of interest.
In addition, we derive exact asymptotic results for a number of specific
rare-event probabilities, such as the probability of the loss process exceeding
some given function
Explicit Computations for a Filtering Problem with Point Process Observations with Applications to Credit Risk
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.