Free PDF Ebooks Files @AcrobatPlanet.Com

In this paper we present a modelling framework for portfolio credit risk which incorporates a new methodology to model the dependence between risk-free interest-rates and the default loss process, allowing direct dependence between interest-rates and the loss process. We provide a stochastic and arbitrage-free framework for the evolution of the prices of a set of contingent-claims on the credit portfolio’s loss distribution. This set of contingent claims is complete in the sense that it spans all European contingent claims on the loss process L(t). In particular, the prices of index credit default swaps (CDS) and single tranche collateralized debt obligations (STCDO) of all maturities and attachment points can be easily constructed from these contingent claims. This allows a straightforward calibration of the model to the so-called “correlation smile”. In contrast to Schönbucher (2005), though, these prices can not be interpreted as probabilities under the spot martingale measure as we allow dependence between the loss process and risk-free interest-rates.

The prices of the basic contingent claims are parameterized using a set of loss-contingent forward interest-rates fn(t, T) and loss-contingent forward credit protection rates Fn(t, T). The forward interest-rates fn(t, T) must be loss contingent in order to allow us to capture its credit dependence. These rates can be viewed as the interest-rates of forward-rate agreements that are contingent on a certain number of losses L(T) = n. Clearly, if there is dependence between the loss process and the default-free interest-rates, the loss-contingent forward rates fn(t, T) must differ over different values of n. We show that (up to weak regularity conditions), existence of such a parametrization is necessary and sufficient for the absence of static arbitrage opportunities in the underlying assets, i.e. the parameterization fully describes the set of arbitrage-free price systems in this model.