Portfolio credit risk with predetermined default orders

Portfolio credit risk models can be distinguished by the use of a top-down approach or a bottom-up one. The main difference between these two approaches is the information of default identities. In this paper, we propose a conditional top-down approach which models the default times with a predetermined default order of identities. Thus conditioned on the default order, the default times of a bottom-up model can be constructed simply using a top-down approach. We use the tool of assumptions to separate the information of default orders from the ordered default times. The predetermined assumption (a special assumption) introduced here allows that the construction of the loss process relates to a probability on permutations. We can derive the probabilities on default orders from the known bottom-up models satisfying the predetermined assumption (e.g. Jarrow-Yu’s contagion model), and obtain new choices of probability on default orders based on some simple and interesting indices of permutations such as the inverse index. Furthermore, under the predetermined assumption, some generic pricing problems of the bottom-up models can be simplified to the special case of the conditional Markov loss model. We then apply these results to Jarrow-Yu’s contagion model and give an efficient approach to the pricing problem of CDO tranches, where new expansions of the loss distributions are derived by the random matrix exponential.