مشتقات اعتباری با مسائل مربوط به بدهی های متعدد

We evaluate the most actively traded types of credit derivatives within a unified pricing framework that allows for multiple debt issues. Since firms default on all of their obligations, total debt is instrumental in the likelihood of default and therefore in credit derivatives valuation. We use a single factor interest rate model where the exponential default frontier is based on total debt and is made coherent with observed bond prices. Analytical formulae are derived for credit default swaps, total return swaps (both fixed-for-fixed and fixed-for-floating), and credit risk options (CROs). Price behaviors and hedging properties of all these credit derivatives are investigated. Simulations document that credit derivatives prices may be significantly affected by terms of debt other than those of the reference obligation. The analysis of CROs indicates their superior ability to fine-tune the hedging of magnitude and arrival risks of default.

Credit derivatives are financial contracts that allow one party to transfer credit risk to another party. They permit the trading of credit risk separately from other sources of risk. The market for credit derivatives, with New York and London being its most active places, is estimated between $400 billion and $1 trillion notional outstanding according to Hargreaves (2000). Its exponential growth within the last ten years has retained the attention of many researchers in credit risk pricing. Contributions are commonly classified the same way as the literature on corporate bond pricing. “Reduced form” models (or “intensity” models) directly specify the dynamics of the bond price and view default as an unpredictable event (e.g. the first jump of a Poisson process). This approach, initiated by Jarrow and Turnbull (1995), has been extended to credit derivatives by Flesaker et al. (1994), Kijima and Komoribayashi (1998), Duffie and Singleton (1999), or Chen and Sopranzetti (2002). By contrast, “structural” models (or “firm value” models) specify the dynamics of the firm assets value and use contingent claims analysis to price the securities issued by this firm. This is the approach we follow here. The seminal reference is Merton (1974) yielding a solution to the price of a corporate discount bond. Extensions of this work evaluate more sophisticated debt contracts (see e.g. Black and Cox (1976) for debt with a safety covenant; Geske (1977) for coupon-bearing debt, or Brennan and Schwartz (1980) for convertible debt). Other extensions evaluate debt contracts under more complex default rules (see e.g. Leland (1994) where the decision to default is driven by the capital structure static trade-off; Anderson and Sundaresan (1996) where debt is subject to strategic default, or François and Morellec (2002) where default leads to a Court-supervised reorganization procedure). Applications of structural models to credit derivatives include the works of Longstaff and Schwartz (1995a), Das (1995), Pierides (1997) and Ammann (2001). All these papers mostly study credit risk options (referred to as CROs). The model of Longstaff and Schwartz (1995a) relies on the empirically supported assumption that credit spreads are mean-reverting. Their model values European CROs that pay the credit spread of a bond at a given date. However, this pay-off is not triggered by the event of default, and the option is therefore inadequate for hedging purposes. Das (1995) derives a closed-form formula in continuous time for the value of a CRO written on a bond paying a continuous coupon. Interest rate is assumed constant. He further develops a discrete time model with a Heath–Jarrow–Morton term structure model, but no analytical result is available. Das and Sundaram (2000) work in that same framework, but directly model the credit spread. Hence they value the same kind of CROs as in Longstaff and Schwartz (1995a). Ammann (2001) adopts a compound option approach to value CROs written on discount bonds. In these three papers, default is exogenous. By contrast, Pierides (1997) values CROs with an endogenous default rule at the cost of assuming that interest rates are constant. Recently, Bélanger et al. (2001) have developed a general pricing framework for credit derivatives that embed both structural and reduced form models. With respect to the literature on contingent claims models for credit derivatives, our contribution consists of four points. First, we provide closed form formulae not only for CROs but also for all other most actively traded credit derivatives within a unified pricing framework. Specifically, we study credit default swaps (CDS), fixed-for-fixed total return swaps (fixed TRS), and fixed-for-floating TRS (floating TRS), and we make a distinction between default CRO and downgrading CRO which respectively compensate for the total default loss or for a pre-specified downgrading of the reference obligation. Second, consistent with CROs as hedging instruments, our model makes their pay-off contingent upon the default or more generally upon the downgrading event. Third, our model incorporates stochastic interest rates and a default policy that is made coherent with observed bond prices. Fourth, our model makes a clear distinction between the reference obligation underlying the credit derivative and the issuer's total debt. The issuer is allowed to have multiple classes of debt outstanding with different nominal rates, maturities and seniorities. This specification enables to capture the whole information contained in the debt structure about the default likelihood. In particular, it measures the impact of a marginal change in the debt structure on credit derivative pricing. In line with “structural models”, we characterize the event of default using the first passage time approach. This default modeling is pioneered by Black and Cox (1976) where risk free rate is assumed constant. Extensions are provided in two notable directions. On one hand, Kim et al. (1993) or Longstaff and Schwartz (1995b) introduce interest rate risk but use constant barriers. Though a constant barrier may be justified by a particular debt provision (such as a solvency constraint or a safety covenant), the general case is that shareholders have the unrestricted (American) option to default, which implies a default boundary that is better approximated by an exponential function for finitely lived debt. On the other hand, some papers like Saà-Requejo and Santa-Clara (1999) assume a stochastic default boundary. Though more comprehensive, this setting complicates model calibration by introducing additional parameters. Moreover, it generally admits no analytical solution and requires the use of numerical methods. An exception is Finkelstein et al. (2002) who obtain a closed-form formula for CDS at the cost of assuming a constant risk free rate. We adopt the in-between approach followed by Briys and de Varenne (1997) where the exogenous default boundary is related to default-free bond prices. In extension of their work however, we explicitly link the boundary to the firm capital structure. This view is consistent with a shareholders' decision to default based on total debt. Our main findings may be classified in two categories, one regarding hedging properties of credit derivatives and the other regarding term structures of credit derivatives premia. Hedging properties of CDS and fixed TRS are very similar. Interest rate risk as well as credit risk hedging performance of TRS vary with the issuer's leverage. CROs are the most effective credit risk hedging tools. By contrast with other credit derivatives, their maturity date is an additional degree of freedom that allows to fine-tune the hedge between credit risk intensity and magnitude. Term structures of CDS and TRS premia are similar in shape as those of corporate yield spreads. Term structures of CRO premia exhibit a bell shape. All term structures are affected in a non-trivial way by marginal changes in the issuer's debt structure. Section 1 presents the pricing framework. Section 2 derives pricing formulae for credit derivatives and conducts simulations analysis. Section 3 concludes. Technical proofs are gathered in Appendix A.

Our modeling of bond default risk with a structural form approach departs from a large body of the credit derivatives literature. However, the results proposed in this paper are obtained by relaxing some typical, yet disturbing assumptions of firm value models. We show that the focus on the broader corporate debt structure does not preclude analytical formulations for the value and behavior of the most standard credit derivative contracts. The pricing and hedging implications originating from this key element allow to emphasize the importance of several dimensions. Even for simple contracts like CDS or TRS, a contract designed on a particular corporate bond experiences very different price sensitivities with respect to the total interest payout, debt principal or corporate leverage. This finding proves to be crucial when considering any change in the liability structure that would leave the QDR unaffected. In particular, the very same phenomenon may lead to drastically opposite results on the credit derivative depending on the maturity of the underlying bond. The ignorance of terms of debt contracts other than the reference obligation would completely offset these issues. Rather, thanks to our integrative approach, such behavior differences outline the clear dependence of credit derivatives prices on broader corporate finance decisions. The ability to price more complex derivatives in the very same framework provides specific hedging properties associated with each of them. Of remarkable interest is the apparent complementarity of the fixed TRS, floating TRS and default CRO contracts. The first one can be viewed as a hybrid between the other two in terms of interest-rate risk and credit risk hedging properties. However, the option maturity of the default CRO provides an additional degree of freedom whose main consequence is the completely adverse influences of arrival risk and magnitude risk of default. A further look at term structures of downgrading CRO premia confirms that this type of contract enables to fine-tune hedging properties between default risk components. Our approach opts for a parsimonious modeling with closed-form expressions for credit derivatives prices and a limited number of parameters to estimate. A possible extension of our work could therefore give up on this analytical tractability in order to better track interest rate dynamics. If a two-factor model of interest rates were to be used, it would certainly be best to use the volatility of the instantaneous risk free rate as the second factor, i.e. σr in our model, as in the stochastic volatility term structure (SVTS) model proposed by Fong and Vasicek (1991) where the instantaneous risk free rate variance σ2r follows a positive mean-reverting process. In this realistic setting however, closed-form expression for default probabilities are no longer available. Thus, theoretical resolution of the model would require numerical methods to obtain credit derivatives prices. It is possible nevertheless to figure out some of the effects that a SVTS model would introduce on our results. Indeed, Fong and Vasicek (1991) show the volatility exposure for a zero-coupon bond is an increasing and concave (almost everywhere) function of maturity. Fundamentally, credit derivatives compensate for a possible default loss and their prices behave like the price difference between a riskless bond and its risky counterpart. Fong and Vasicek (1991) show that long-term and short-term riskless and risky bonds are affected by volatility risk in a comparable manner. Hence, the corresponding credit derivatives prices should be little affected by volatility risk. However, for middle-term maturities, the bond price sensitivity to volatility heavily depends on maturity. This is where the duration differential implies a high volatility exposure. Thus, credit derivatives prices may incorporate a non-negligible volatility premium for these maturities. As a further step for future research, the extent of this contribution has to be examined empirically. In this respect, the paper contains closed-form formulae and term structures analyses that are testable. The implementation of the Fong and Vasicek (1991) model would require the estimation of additional parameters (including the market prices of interest rate and volatility risks), which is certainly a relevant but challenging direction of further empirical investigation.