A Structural Model with Unobserved Default Boundary

Part of this work was done while the first author stayed at the Isaac Newton Institute in Cambridge. Financial support from Isaac Newton Institute and Deutsche Forschungsgemeinschaft is gratefully acknowledged. The authors would like to thank M. Davis, L.C.G. Rogers, K. Giesecke and T. Bjrk for their inspiring comments and Ling Xu for her excellent help with the simulations. Moreover, the authors are grateful to N. Kordzakhia and two anonymous referees for their suggestions which helped to improve the paper considerably.

Abstract

A firm‐value model similar to the one proposed by Black and Cox (1976) is considered. Instead of assuming a constant and known default boundary, the default boundary is an unobserved stochastic process. This process has a Brownian component, reflecting the influence of uncertain effects on the precise timing of the default, and a jump component, which relates to abrupt changes in the policy of the company, exogenous events or changes in the debt structure. Interestingly, this setup admits a default intensity, so the reduced form methodology can be applied.

Part of this work was done while the first author stayed at the Isaac Newton Institute in Cambridge. Financial support from Isaac Newton Institute and Deutsche Forschungsgemeinschaft is gratefully acknowledged. The authors would like to thank M. Davis, L.C.G. Rogers, K. Giesecke and T. Bjrk for their inspiring comments and Ling Xu for her excellent help with the simulations. Moreover, the authors are grateful to N. Kordzakhia and two anonymous referees for their suggestions which helped to improve the paper considerably.

Keywords:

• Structural model
• equity default swaps
• default boundary
• jump‐diffusion

Notes

Part of this work was done while the first author stayed at the Isaac Newton Institute in Cambridge. Financial support from Isaac Newton Institute and Deutsche Forschungsgemeinschaft is gratefully acknowledged. The authors would like to thank M. Davis, L.C.G. Rogers, K. Giesecke and T. Bjrk for their inspiring comments and Ling Xu for her excellent help with the simulations. Moreover, the authors are grateful to N. Kordzakhia and two anonymous referees for their suggestions which helped to improve the paper considerably.

1. For an introduction into credit risk we refer to the surveys by Giesecke (2004), Schmidt and Stute (2004) or one of the excellent textbooks Lando (2004), Schönbucher (2003), McNeil, Frey, and Embrechts (2005).

2. See, e.g. Shiryaev (1996, Theorem VII.4.3).

3. See Rolski, Schmidli, Schmidt, and Teugels (1999), p.502. The η _{i :k} denote the order statistics of η _i , that is the η _i are ordered, such that η_1:k η_2:k …η _{k :k} . denotes the law of a random variable.