Pricing of Credit Risk Derivatives with Stochastic Interest Rate

Abstract

This paper deals with a credit derivative pricing problem using the martingale approach. We generalize the conventional reduced-form credit risk model for a credit default swap market, assuming that the firms’ default intensities depend on the default states of counterparty firms and that the stochastic interest rate follows a jump-diffusion Cox–Ingersoll–Ross process. First, we derive the joint Laplace transform of the distribution of the vector process (rt,Rt) by applying piecewise deterministic Markov process theory and martingale theory. Then, using the joint Laplace transform, we obtain the explicit pricing of defaultable bonds and a credit default swap. Lastly, numerical examples are presented to illustrate the dynamic relationships between defaultable securities (defaultable bonds, credit default swap) and the maturity date.

Keywords

• reduced-form model
• jump-diffusion CIR (JCIR) model
• piecewise deterministic Markov process theory
• martingale theory
• defaultable bonds
• credit default swap (CDS)