Spectral Expansions for Credit Risk Modelling with Occupation Times

Abstract

We study two credit risk models with occupation time and liquidation barriers: the structural model and the hybrid model with hazard rate. The defaults within the models are characterized in accordance with Chapter 7 (a liquidation process) and Chapter 11 (a reorganization process) of the U.S. Bankruptcy Code. The models assume that credit events trigger as soon as the occupation time (the cumulative time the firm’s value process spends below some threshold level) exceeds the grace period (time allowance). The hazard rate model extends the structural occupation time models and presumes that other random factors may also lead to credit events. Both approaches allow the firm to fulfill its obligations during the grace period. We derive new closed-from pricing formulas for credit derivatives containing the (risk-neutral) probability of defaults and credit default swap (CDS) spreads as special cases, which are derived analytically via a spectral expansion methodology. Our method works for any solvable diffusion, such as the geometric Brownian motion (GBM) and several state-dependent volatility processes, including the constant elasticity of variance (CEV) model. It allows us to write the pricing formulas explicitly as infinite series that converges rapidly. We then calibrate our models (assuming that GBM governs the firm’s value) to market CDS spreads from the Total Energy company. Our calibration results show that the computations are fast, and the fit is near-perfect.

Keywords

• credit risk models
• occupation time
• spectral expansions
• default probability
• credit default spread
• hazard rate function