Flexing the default barrier

Abstract

The paper introduces a Black–Cox-type structural model for credit default swaps (CDS). The existing literature on structural CDS pricing is extended by allowing a general functional form for the default barrier specified without reference to asset volatilities, dividend yields or interest rates. We develop a fast and robust algorithm to compute survival probabilities numerically. An empirical application suggests that the market-implied barrier is stable over time, with a possibly hump-shaped term structure. The implied barrier can be used for computing survival probabilities consistent with objective expectations of asset evolution, for pricing under counterparty risk, and for determining optimal corporate bond covenants.

Keywords:

• Credit default swap
• Structural model
• Default boundary
• Green's function
• Calibration

Acknowledgement

Our thanks go to Hannelore De Silva and Klaus Pötzelberger for mathematical assistance.

Notes

†An estimate of objective survival probabilities consistent with CDS premia using a reduced-form model demands knowledge of the drift in the default intensity under the physical measure. This drift can only be inferred from long time series of CDS premia, which do not exist in this relatively young market.

‡Additional technical requirements are integrability of functions r and q as well as square integrability of σ.

†Our valuation approach ignores the default risk of the two counterparties to a CDS contract and only accounts for the default risk of the underlying obligor. More precisely, the assumption is that during the life of the contract the counterparties either maintain the credit rating underlying generic (e.g. A-rated) CDS or have symmetric default probabilities (credit quality) (Duffie and Singleton 1997). We presume that this aspect has a relatively low impact on the spreads of typical CDS contracts.

‡Various refinements of this basic idea are possible, e.g. accounting for bankruptcy costs or assuming a stationary leverage ratio as in Collin-Dufresne and Goldstein (2001).

†Similarly as the Merton (1974) model, first-passage-time models generally also produce spreads too close to zero for short maturities to be consistent with market data. The reason for such behavior is that the default time is predictable with respect to the natural filtration of the asset value process. The predictable-default setup used in this paper can be extended to produce inaccessible default times by allowing investors to observe only incomplete information on the asset value process and/or the default boundary. A random default barrier unobserved by investors is introduced by Giesecke and Goldberg (2004), Giesecke (2006) and Schmidt and Novikov (2008), while Brigo and Tarenghi (2005a) and Brigo and Morini (2006) suggest a scenario-dependent default barrier modeling the uncertain level of liabilities.

†More precisely, the Bloomberg item for the debt-to-market-cap ratio is used, which is calculated as the sum of short-term and long-term debt over market capitalization.

†Up to the logarithm, which transforms a geometric barrier into an arithmetic one.

†The values are obtained from Vazza et al. (2007).