Portfolio Losses and the Term Structure of Loss Transition Rates : A New Methodology for the Pricing of Portfolio Credit Derivates

In this paper, we present a model for the joint stochastic evolution of the cumulative loss process of a credit portfolio and of its probability distribution. At any given time, the loss distribution of the portfolio is represented using forward transition rates, i.e. the transition rates of a hypothetical time-inhomogeneous Markov chain which reproduces the desired transition probability distribution. This approach allows a straightforward calibration of the model (e.g. to a full initial term- and strike structure of synthetic CDOs including the correlation smile) and it is shown that (except for regularity restrictions) every arbitrage- free loss distribution admits such a representation with forward transition rates. To capture the stochastic evolution of the loss distribution, the transition rates are then equipped with stochastic dynamics of their own, and martingale / drift restrictions on these dynamics are derived which ensure absence of arbitrage in the model. Furthermore, we analyze the dynamics of spreads and STCDO-prices that are implied by the model and show that the input parameters can be viewed as spread move parameters and correlation move parameters. We also show how every dynamic model for correlated individual defaults can be cast into this framework.