A Structural Credit-Risk Model based on a Jump Diffusion
by Matthias Scherer of the University of Ulm
December 2, 2005
Abstract: In this paper, we generalize the pure diffusion approach for structural credit risk modeling by including jumps in the firm-value process. In pure diffusion models, the probability for a solvent company to default within a small interval of time is negligible, whereas a real company may face sudden financial distress. Our generalization allows those unpredicted extremal events, raising the probability for a solvent company to default within a small interval of time to a realistic level. Compared to a pure diffusion model, including jump risk increases credit spreads especially for small maturities. The resulting term structure of credit spreads is extremely flexible, hence, our model provides a powerful tool to fit a real spread curve.
Evaluating bond prices in a jump-diffusion model is complicated, as the distribution of first-passage times is not available in closed form. We present two approaches to overcoming this problem. First, we derive a semi-analytical Monte Carlo simulation which is unbiased and efficient and allows all possible jump distributions. The algorithm only requires us to simulate the firm value at the times of jumps and not on a fine grid. Then, we analytically calculate bond prices conditioned on the simulated jumps. The second approach to first-passage times and bond pricing uses specific properties of jumps with two-sided exponential distribution. In this scenario it is possible to calculate the Laplace transform of survival probabilities. Those survival probabilities are then recovered numerically and used to price corporate bonds.
The last section of this paper presents a method of obtaining parameter estimates based on observed bond prices. Our first results indicate that the overall volatility of the firm-value process is explained to a large extent by jumps, supporting the need for unpredicted jump risk in a realistic firm-value model.
