Single and joint default in a structural model with purely discontinuous asset prices

Abstract

Structural models of credit risk are known to present both vanishing spreads at very short maturities and a poor spread fit over longer maturities. The former shortcoming, which is due to the diffusive behaviour assumed for asset values, can be circumvented by considering discontinuous asset prices. In this paper the authors resort to a pure jump process of the Variance-Gamma type. First the authors calibrate the corresponding Merton type structural model to single-name data for the DJ CDX.NA.IG and CDX.NA.HY components. By so doing, they show that it also circumvents the diffusive structural models difficulties over longer horizons. Particularly, it corrects for the underprediction of low-risk spreads and the overprediction of high-risk ones. Then the authors extend the model to joint default, resorting to a recent formulation of the VG multivariate model and without superimposing a copula choice. They fit default correlation for a sample of CDX.NA names, using equity correlation. The main advantage of our joint model, with respect to the existing non-diffusive ones, is that it allows full calibration without the equicorrelation assumption, but still in a parsimonious way. As an example of the default assessments which the calibrated model can provide, the authors price an FtD swap.

Keywords:

• Credit risk
• Structural models
• Lévy asset prices
• Default probability
• Joint default

Acknowledgements

Patrizia Semeraro gratefully acknowledges financial support from Torino Finanza and its associates, in particular Unicredit and Toro Assicurazioni. The authors thank participants at the Actuarial and Financial Mathematics Conference, Brussels, 7–8 February 2008, and two anonymous referees for helpful comments and suggestions. Part of this research was conducted when Filippo Fiorani was with Aristeia Capital and was circulated under the title ‘Credit risk in pure jump structural models’. The authors thank Aristeia for having kindly provided them with the data; the views expressed in this paper are those of the authors and do not necessarily reflect those of Aristeia.

Notes

†A gamma process {G(t), t ≥ 0} with parameters (a, b) is a Lévy process so that the defining distribution of Y(1) is gamma with parameters (a, b) (shortly ℒ(Y(1)) = Γ(a, b)). The parameters a and b are restricted to be positive.

†As discussed in Luciano and Semeraro (2007) and Semeraro (2008) the linear correlation is zero in the symmetric case (θ _j = 0) although the assets are still dependent. The corresponding model, which will turn out to be irrelevant for the sample of this paper, can be calibrated via nonlinear dependence.