Pricing Credit Derivatives with Rating Transitions

We present a model for pricing risky debt and valuing credit derivatives that is easily calibrated to existing variables. Our approach expands a classical term-structure model to allow for multiple rating classes of debt. The framework has two salient features: (1) it uses a rating-transition matrix as the driver for the default process, and (2) the entire set of rating categories is calibrated jointly, which allows arbitrage-free restrictions across rating classes as a bond migrates among them. We illustrate the approach by applying it to price credit-sensitive notes that have coupon payments linked to the rating of the underlying credit. The pricing of credit derivatives is approaching 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 a classical term-structure model developed in 1990.We extend this model to include risky debt by adding a “forward spread” process to the forward-rate process for default-risk-free bonds. Instead of modeling the movement of the spread itself, however, 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 the forward-spread and the forward-rate 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 matrixes that are exogenously supplied. The model is flexible enough to incorporate any specification for the recovery process that is consistent with the default process and the spread processes.We extend an earlier pricing lattice that was developed by computing a no-arbitrage tree that embedded the riskless term structure and the term structure of credit spreads. This tree considered the modeling of only a single rating category at a time, but our model calibrates all rating classes jointly on the same pricing lattice. Embedding all rating categories in one pricing lattice requires a set of conditions that will ensure consistency across all classes of debt. The additional information comes from the introduction of the rating-transition matrix. Thus, our model can be used to price credit derivatives based on multiple classes of debt, which was not possible using simpler models.The consistency conditions across rating classes can be explained as follows: 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 condition is true for a corporate borrower with any given rating at a point in 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 is the primary contribution of this paper.Our model requires as inputs the government yield curve and the term structures of credit spreads for each rating class, which are available from a number of providers. The same sources deliver the required interest rate and spread volatilities. The model can be efficiently implemented and lends itself most appropriately to pricing credit derivatives, such as credit-sensitive notes, whose coupon payments are linked to the 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.