Necessary and Sufficient Conditions for the CDs Survival Curve to Exist

Abstract

The survival probability curve associated with a given underlying reference entity is a central quantity to price credit risk. In the current credit market, Credit Defaults Swaps(CDSs) are liquidly traded for a large range of obligors and maturities, such that the implied survival probabilities may be backed out from CDSs quotes. For doing, it is common to make use of bootstrapping approach, whose the idea relies on some iterative steps. However the procedure may fail at some step. No clear and explicit condition on the feasibility of such a procedure is available on financial literature. Therefore our purpose in this paper is to find necessary and sufficient conditions on the CDS term structure which ensure the existence (and uniqueness) of the hazard rate at each step. We do this in the setting of hazard term structure defined by piecewise flat function of maturity. Among our other contributions here is the derivation of low and high bound estimates for each hazard rate. The point here is that such bounds may be used to run efficiently a search root numerical approach when the hazard rate is assumed to exist. Both the existence/uniqueness conditions and the hazard rate bound estimates we propose are given in terms of analytical expressions such that they may be easily implemented for a practice purpose. Numerical considerations lead us to observe that a good approximation to the hazard rate under exploration is just the hazard low bound estimate we have found.