  The modeling of the probability of joint default or total number of defaults
among the firms is one of the crucial problems to mitigate the credit risk
since the default correlations significantly affect the portfolio loss
distribution and hence play a significant role in allocating capital for
solvency purposes. In this article, we derive a closed-form expression for the
probability of default of a single firm and the probability of the total number
of defaults by any time $t$ in a homogeneous portfolio of firms. We use a
contagion process to model the arrival of credit events that causes the default
and develop a framework that allows firms to have resistance against default
unlike the standard intensity-based models. We assume the point process driving
the credit events to be composed of a systematic and an idiosyncratic
component, whose intensities are independently specified by a mean-reverting
affine jump-diffusion process with self-exciting jumps. The proposed framework
is competent of capturing the feedback effect, an empirically observed
phenomenon in the default events. We further demonstrate how the proposed
framework can be used to price synthetic collateralized debt obligation (CDO)
and obtain a closed-form solution for tranche spread. Finally, we present the
sensitivity analysis to demonstrate the effect of different parameters
governing the contagion effect on the spread of tranches and the expected loss
of the CDO.
