Credit Spreads, Optimal Capital Structure, and Implied Volatility With Endogenous Default and Jump Risk

Abstract

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.

Optimal Capital Structure With Scale Effects Under Spectrally Negative Lévy Models

“…This model was first studied by [30,31] where they assumed a geometric Brownian motion for the firm's asset value, and was later extended to a Lévy model, among others, by [13,14,22,27,29]. By introducing jumps, it allows the value of bankruptcy costs to be stochastic, and more importantly resolves the contradictory conclusion under the continuous diffusion model that the credit spreads go to zero as the maturity decreases to zero.…”

“…Kyprianou and Surya [27] later showed for a general spectrally negative process that the optimal bankruptcy level exists and is explicitly determined by applying continuous and smooth fit when X is of bounded and unbounded variation, respectively. Regarding the cases with both positive and negative jumps, Chen and Kou [13] and Dao and Jeanblanc [14] solved for double exponential jump diffusion and Le Courtois and Quittard-Pinon [29] solved for stable processes.…”

“…While it is fairly well-known that the continuous and/or smooth fit can be applied for the exponential Lévy model as in [13,22,27], their results rely particularly on the form of the bankruptcy cost as an exponential function of the Lévy process; once the function is generalized, it is no longer clear if the same principle is directly applicable. However, thanks to the recent advances in the spectrally negative Lévy process and its fluctuation theories (e.g.…”

“…exponential jumps. We compute the optimal bankruptcy levels and the corresponding equity/debt/firm values as well as the optimal leverage ratios as solutions to the two-stage problem considered in [13,30,31]. We conduct a sequence of numerical experiments to analyze the impacts of the scale effects on the bankruptcy strategy and the capital structure.…”

“…The problem reduces to a non-standard optimal stopping problem, and the optimal bankruptcy level can be obtained via the continuous/smooth fit principle. It was solved, in particular, for the double exponential jump diffusion process [13], for stable processes [29], and for a general spectrally negative Lévy process [22,27].…”