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A closed-form extension to the Black-Cox model


  • Aurélien Alfonsi

    () (CERMICS - Centre d'Enseignement et de Recherche en Mathématiques et Calcul Scientifique - ENPC - École des Ponts ParisTech, MATHFI - Financial mathematics - Inria Paris-Rocquencourt - Inria - Institut National de Recherche en Informatique et en Automatique - ENPC - École des Ponts ParisTech - UPEC UP12 - Université Paris-Est Créteil Val-de-Marne - Paris 12)

  • Jérôme Lelong

    () (UMA - Unité de Mathématiques Appliquées - Univ. Paris-Saclay, ENSTA ParisTech - École Nationale Supérieure de Techniques Avancées, MATHFI - Mathématiques financières - LJK - Laboratoire Jean Kuntzmann - UPMF - Université Pierre Mendès France - Grenoble 2 - UJF - Université Joseph Fourier - Grenoble 1 - Institut Polytechnique de Grenoble - Grenoble Institute of Technology - CNRS - Centre National de la Recherche Scientifique - UGA - Université Grenoble Alpes)


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.

Suggested Citation

  • Aurélien Alfonsi & Jérôme Lelong, 2012. "A closed-form extension to the Black-Cox model," Post-Print hal-00414280, HAL.
  • Handle: RePEc:hal:journl:hal-00414280
    DOI: 10.1142/S0219024912500537
    Note: View the original document on HAL open archive server:

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

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