Credit risk in the pricing and hedging of derivatives

Credit risk is present under various forms in any derivative products. However, the fact is that credit risk is not, in general, priced in the various models commonly used for equity, FX, IR of commodity derivatives. This note is aimed at presenting a few guidelines for the modelling of default risk and spread risk in the context of derivatives, thereby treating any derivative as a genuine credit hybrid. I shall emphasize, using simple examples where the credit risk is described by a reduced-form model, the impact of credit risk in pricing and hedging. It is quite enlightening to revisit the classical strategy of dynamic replication via a “delta” hedge, and understand how it is modified by the credit risk. I shall first use a trivial jump-to-zero model for a risky bond (or a CDS contract) in order to explain in a simplified setting the credit risk effect. Then I will consider a more realistic stochastic default intensity model. Finally, I will briefly present a practical method for the calibration of the joint dynamics. As regards the more theoretical aspects of credit risk modelling, I will rely on the reducedform models such as described in great details in e.g. [BJR] or [SS]. The reason for that choice is that, in capital markets environment, derivative traders are confronted to a host of (more or less) liquid products: stock, vanilla options, variance swaps, CDS, credit index,... and their main objective is the projection of the risks of a given, exotic instrument on existing market instruments. Therefore, one of the main prerequisites for a useful model is that it has good replication properties for the price and the dynamics of the liquid instruments. A typical structural approach for the credit risk, although interesting in itself as it gives rise to an interpretation of the credit event happenstance and therefore, to a better understanding of its governing mechanisms, has the strong disadvantage of making the models and the calibration extremely complicated, with poor performances. Interestingly enough, there are much fewer references on the hedging of credit risk than there are on its pricing... The pricing by replication with CDS, however natural an extension of the Black-Scholes-Merton theory to the world of credit as it may be, is not commonly used nor exposed in the core of the credit literature. The most relevant references in this direction are [BJR] and [JR]. It is noteworthy that this approach is not well spread in the pure credit derivatives business, where the hedging practices in the industry tend to consider the “spread” risk rather than the actual “default” risk, mostly because of the cost of a full hedging portfolio. However, in the context of credit-contingent options, or in that of counterparty risk management, it is essential to properly measure and eliminate the default risk which explains why I think the examples below are of interest.