A closed-form extension to the Black-Cox model

Aurélien Alfonsi (1, 2) , Jérôme Lelong (3, 4)

1 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique
2 MATHFI - Financial mathematics
3 MATHFI - Mathématiques financières
4 OC - Optimisation et commande

In the Black-Cox model, a firm defaults when its value hits an exponential barrier. Here, we propose an hybrid model that generalizes this framework. The default intensity can take two different values and switches when the firm value crosses a barrier. Of course, the intensity level is higher below the barrier. We get an analytic formula for the Laplace transform of the default time. This result can be also extended to multiple barriers and intensity levels. Then, we explain how this model can be calibrated to Credit Default Swap prices and show its tractability on different kinds of data. We also present numerical methods to numerically recover the default time distribution.

Mots clés: Parisian options; hybrid model; Black-Cox model; structural model; Credit risk; intensity model; ParAsian options

Domaines: Probabilités [math.PR]; Finance quantitative [q-fin.CP]

Aurélien Alfonsi, Jérôme Lelong. A closed-form extension to the Black-Cox model. International Journal of Theoretical and Applied Finance, 2012, 15 (8), pp.1250053:1-30. ⟨10.1142/S0219024912500537⟩. ⟨hal-00414280v2⟩