We are especially grateful to P. Glasserman for many insightful discussions. We thank P. Carr, M. Dempster, T. Hurd, D. Hobson, M. Jeanblanc, G. Petrella, and J. Zhu for helpful comments. We thank three referees and the editor R. Jarrow for their suggestions. We are also grateful to many people who offered insights into this work, including seminar participants at Columbia University, Princeton University, New York University, University of Cambridge, University of Oxford, University of Essex; and conference participants at INFORMS annual meeting, 2004, and Financial Modeling Conference at University of Montreal, 2005. Part of this research is conducted when SK visited the Issac Newton Institute, University of Cambridge, under a research fellowship from the European Commission.
CREDIT SPREADS, OPTIMAL CAPITAL STRUCTURE, AND IMPLIED VOLATILITY WITH ENDOGENOUS DEFAULT AND JUMP RISK
Article first published online: 26 JUN 2009
© Copyright the Authors. Journal Compilation © 2009 Wiley Periodicals, Inc.
Volume 19, Issue 3, pages 343–378, July 2009
How to Cite
Chen, N. and Kou, S. G. (2009), CREDIT SPREADS, OPTIMAL CAPITAL STRUCTURE, AND IMPLIED VOLATILITY WITH ENDOGENOUS DEFAULT AND JUMP RISK. Mathematical Finance, 19: 343–378. doi: 10.1111/j.1467-9965.2009.00375.x
- Issue published online: 26 JUN 2009
- Article first published online: 26 JUN 2009
- Manuscript received July 2005; final revision received December 2007.
- credit risk;
- yield spreads;
- capital structure;
- implied volatility;
- jump diffusion;
- smooth pasting
We propose a two-sided jump model for credit risk by extending the Leland–Toft endogenous default model based on the geometric Brownian motion. The model shows that jump risk and endogenous default can have significant impacts on credit spreads, optimal capital structure, and implied volatility of equity options: (1) Jumps and endogenous default can produce a variety of non-zero credit spreads, including upward, humped, and downward shapes; interesting enough, the model can even produce, consistent with empirical findings, upward credit spreads for speculative grade bonds. (2) The jump risk leads to much lower optimal debt/equity ratio; in fact, with jump risk, highly risky firms tend to have very little debt. (3) The two-sided jumps lead to a variety of shapes for the implied volatility of equity options, even for long maturity options; although in general credit spreads and implied volatility tend to move in the same direction under exogenous default models, this may not be true in presence of endogenous default and jumps. Pricing formulae of credit default swaps and equity default swaps are also given. In terms of mathematical contribution, we give a proof of a version of the “smooth fitting” principle under the jump model, justifying a conjecture first suggested by Leland and Toft under the Brownian model.