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.
Next, we analyze the possible dynamics of the thus defined market for portfolio credit derivatives and interest-rate derivatives. We give necessary conditions and sufficient conditions on the dynamics of the parametrization which ensure absence of dynamic arbitrage opportunities in the model. Similar to the HJM drift restrictions for default-free interest-rates, these conditions take the form of restrictions on the drifts of fn(t, T) and Fn(t, T), together with a set of regularity conditions on the stochastic characteristics of the parameterization.
The modelling framework presented in this paper can be applied very efficiently to the pricing of exotic portfolio credit-derivatives such as options on index CDS, leveraged super-senior tranches, options on STCDOs and in particular also hybrid derivatives on credit portfolios and interest-rates. Given the increased liquidity of the markets for index CDS and STCDOs, a particular advantage of this model over most competing models is that it can be calibrated very easily to a given set of prices for index CDS and STCDOs. Such a calibration only requires changing the initial conditions of the loss-contingent forward interest- and protection-rates fn(t, T) and Fn(t, T) but it does not require any changes to the dynamics of the model.
Empirical studies of the dependence between default rates and risk-free interest-rates usually find a negative dependence between interest-rates and defaults. Such results are found investigating the interest-rate correlations of credit spreads of corporate bonds (e.g. Duffee (1998), Duffee (1995), Collin-Dufresne et al. (2001)), and also when actual default event arrivals are investigated (see e.g. Duffie et al. (2006)). The most common explanation of this negative dependence is that high default rates and low interest-rates tend to coincide in recessions, while booms are usually characterized by low default rates and high interest-rates.
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Pricing Interest Rate-Sensitive Credit Portfolio Derivatives
