Pricing Credit Derivatives with Rating Transitions

Abstract

The pricing of credit derivatives is reaching some level of modeling maturity. In particular, "reduced form" models that directly specify the default process or the credit spread have resulted in successful conjoint implementations of term structure models with default models. We contribute to this literature by presenting a discrete-time reduced-form model for valuing risky debt based on the term-structure model of Heath, Jarrow, and Morton (1990).

We extend the HJM model to include risky debt by adding a "forward spread" process to the forward rate process for default risk-free bonds as in Das and Sundaram (2000). Instead of modeling the movement of the spread itself, the engineering of our model focuses on the stochastic process for inter-rating spreads. Working with inter-rating spreads provides any credit spread as the sum of higher rated inter-rating spreads. This approach offers analytical tractability. No restrictions are placed on the correlation between these stochastic processes. The probability of default at any point in time is allowed to depend on the entire history of the process to that point, and is determined from rating transition matrices, exogenously supplied. The model is flexible to incorporate any specification for the recovery process that is consistent with the default process and the spread processes.

In Das and Sundaram (2000), the pricing lattice was developed by computing a no-arbitrage tree embedding the riskless term structure and the term structure of credit spreads. While this tree considered the modeling of only a single rating category at a time, this paper extends that model by calibrating all rating classes jointly on the same pricing lattice. Embedding all rating categories on one pricing lattice requires a set of conditions ensuring consistency across all classes of debt. The additional information required to engineer this comes from the introduction of the rating transition matrix. Thus, in our model, we are now able to price credit derivatives based on multiple classes of debt, which was not possible using simpler models.

To understand the consistency conditions across rating classes, note that the credit rating of a corporate borrower can improve or deteriorate during the life of its issued debt. Thus, the credit spread on its debt contains valuable information about the future credit spreads on debt of all possible rating classes that the borrower could migrate to. This is true for a corporate borrower with any given rating at a point of time. This interdependence of spreads across rating classes immediately implies that calibration of the forward spread process for a given rating class must be undertaken simultaneously with the calibration of the forward spread processes for all other rating classes. Formalizing this interdependence and characterizing the joint calibration process (Proposition 3.2) is the primary contribution of this paper.

Our model requires as input the government yield curve. In addition, it also uses the term structures of credit spreads for each rating class, available from providers such as Bloomberg. The same source delivers required interest rate and spread volatilities. The model can be efficiently implemented and lends itself most appropriately to pricing of credit derivatives such as credit sensitive notes where the coupon payments are linked to credit quality of the underlying corporate borrower. We provide a numerical example to illustrate the calibration of the model and its use to price credit sensitive notes.

Keywords: Risky Debt, Rating Transitions, Credit Derivatives, Credit Sensitive Note, HJM Model

JEL Classification: G12, G13