Credit derivatives have attracted much attention in recent years. A number of models have been developed aimed at the pricing of these instruments. In this paper, we offer a framework for modelling risky debt and valuing credit derivatives that is flexible and simple to implement, and that is, to the maximum extent possible, based on observables. We utilise the risk0neutral pricing methodology, providing an arbitrage0free model for valuing credit derivatives.
Two distinct approaches are visible in the literature to the modelling of credit risk. One, following the lead of Merton [34], views debt as a contingent claim written on the assets on the firm. The typical model here posits a process for the evolution of firm value, and specifies the conditions leading to bankruptcy as well as the payoffs to various parties in the event of bankruptcy. The value of debt is then derived as a consequence.
An appealing feature of this approach is that once default regions and the firm value process are specified, the stochastic process driving default occurrence may be endogenously determined from the composition of securities in the firm capital structure. The recovery rate in default may also be endogenized by assuming as in Merton [34], for instance, that the absolute priority rule holds in the event of default; alternatively, it may be specified exogenously as in Longstaff and Schwartz [31]. However, there are also important practical weaknesses. First, since many of the firm's assets are typically not traded, the firm's value process is fundamentally unobservable; this makes these models difficult to implement. Second, in valuing a particular tranche of corporate debt in this approach, one also has to simultaneously value all debt senior to it, thus increasing computational complexity significantly.
An alternative that has gained in popularity over the past few years is to take a educed form approach and directly model the default process of risky debt. Combining this with a term structure model and assumptions concerning the recovery rate in default, the value of risky debt may be determined. One set of models along these lines employs a credit rating based approach in which default is depicted through a gradual change in ratings driven by a Markov transition matrix (see, e.g., Das and Tufano [9], Jarrow, et al [23], or Lando [29]). Others, such as Duffie and Singleton [17], and Madan and Unal [32],[33], model the default process without reference to a credit rating scheme. The models differ in their assumptions concerning the recovery rate. For example, Jarrow, et al [23] use a Recovery of Treasury assumption (the terminology is from Duffie and Singleton [17]) that upon default a zero coupon risky bond trades for the same price as units of a default0risk0free zero coupon bond with the same maturity, where is an exogenously given constant. Duffie and Singleton [17] employ instead what they term a Recovery of Market Value (RMV) condition that upon default a zero coupon risky bond trades for a fraction a of its market value.
In this paper, we offer a discrete time reduced0form model for valuing risky debt. Our model possesses the advantage of simple implementation mechanics and requires as inputs only easily available information. Our approach is based on the term structure model of Heath, Jarrow, and Morton [21](hereafter HJM). We extend the HJM model to include risky debt by adding a forward spread process to the forward rate process for default risk0free bonds. No restrictions are placed on the correlation between the two processes. The probability of default at any point in time is allowed to depend on the entire history of the process to that point. Upon default, we assume that the RMV condition of Duffie and Singleton [17] applies.
Download
A Discrete Time Approach to Arbitrage Free Pricing of Credit Derivatives
