Risk-neutral and actual default probabilities with an endogenous bankruptcy jump-diffusion model

Abstract

This paper focuses on historical and risk-neutral default probabilities in a structural model, when the firm assets dynamics are modeled by a double exponential jump diffusion process. Relying on the Leland [(1994a) Journal of Finance, 49, 1213–1252; (1994b) Bond prices, yield spreads, and optimal capital structure with default risk. Working paper no. 240, IBER, University of California, Berkeley] or Leland and Toft [(1996) Journal of Finance, 51(3), 987–1019] endogenous structural approaches, as formalized by Hilberink and Rogers [(2002) Finance and Stochastics, 6(2), 237–263], this article gives a coherent construction of historical default probabilities. The risk-neutral world where evolve the firm assets, modeled by a class of geometric Lévy processes, is constructed based on the Esscher measure, yielding useful and new analytical relations between historical and risk-neutral probabilities. We do a complete numerical analysis of the predictions of our framework, and compare these predictions with actual data. In particular, this new framework displays an enhanced predictive power w.r.t. current Gaussian endogenous structural models. Copyright Springer Science+Business Media, LLC 2006Cumulative default probability, Structural model, Jump-diffusion, Endogenous capital structure, Esscher transform, Kou processes, G32, G33,

Pricing the Zero-Coupon Bond and its Fair Premium Under a Structural Credit Risk Model with Jumps

“…Then, the diffusion part contributes more to the default probability than the jump for relatively larger times t. We can also see that the default probability of credit risk process (1.1) with jumps is larger than that with no jumps over a short interval at the beginning. This fact was also noted in Zhou (2001), Le Courtois and Quittard-Pinon (2006), and Ramezani and Zeng (2007).…”

“…To overcome this shortcoming, many authors consider structural form credit risk models in which the firm's market value process contains jumps. See, for example, Zhou (2001), Hilberink and Rogers (2002), and Le Courtois and Quittard-Pinon (2006). Following the idea of these papers, we also consider a structural form model with jumps.…”

Fair Valuation of Participating Life Insurance Contracts with Jump Risk

“…These dynamics have already been used in the literature to tackle finance issues. For instance, Dao and Jeanblanc (2006) and Chen and Kou (2008) study the problem of the valuation of credit spreads, and Le Courtois and Quittard-Pinon (2006) the one of the prediction of default probabilities in such a context. With Kou processes, quasi-closed-form formulas are obtained for European options and an efficient procedure can be implemented for barrier options.…”

“…In general, these processes have the advantage, over other Le´vy processes such as NIG or CGMY processes, to allow for the obtention of semiclosed-form formulas (formulas than can be computed by performing a simple Laplace inversion). An example of application of these dynamics is the choice of a risk-neutral measure and of the calibration of ad hoc parameters: the reader can be referred to Le Courtois and Quittard-Pinon (2006) for the use of the Esscher measure with Kou processes, and the way Kou process parameters are changed when going from the historical world to the risk-neutral world, or conversely. The present section's contribution is to give the complete development of the LIC pricing formula.…”

The optimal capital structure of the firm with stable Lévy assets returns

“…In a recent paper, Le Courtois and Quittard-Pinon (2006) build a structural model that is well-designed for the computation of default probabilities. The latter paper uses a particular type of jump diffusion that is adapted to the problem they tackle, but that does not satisfy the criteria of Carr et al (2002), because of the presence of a diffusive component and of a finite arrival rate of jumps.…”