Quadratic Models for Portfolio Credit Risk with Shot-Noise Effects

We propose a reduced form model for default that allows us to derive closed-form solutions to all the key ingredients in credit risk modeling: risk-free bond prices, defaultable bond prices (with and without stochastic recovery) and probabilities of survival. We show that all these quantities can be represented in general exponential quadratic forms, despite the fact that the intensity is allowed to jump producing shot-noise effects. In addition, we show how to price defaultable digital puts, CDSs and options on defaultable bonds. Further on, we study a model for portfolio credit risk where we consider both firm specific and systematic risks. The model generalizes the attempt from Duffie and Gârleanu (2001). We find that the model produces realistic default correlation and clustering of defaults. Then, we show how to price first-to-default swaps, CDOs, and draw the link to currently proposed credit indices.