Pricing kth-to-default swaps under default contagion: the matrix analytic approach

Abstract

We study a model for default contagion in intensity-based credit risk and its consequences for pricing portfolio credit derivatives. The model is specified through default intensities that are assumed to be constant between defaults, but that can jump at the times of defaults. The model is translated into a Markov jump process that represents the default status in the credit portfolio. This makes it possible to use matrix-analytic methods to derive computationally tractable closed-form expressions for single-name credit default swap (CDS) spreads and kth-to-default swap spreads. We “semicalibrate” the model for portfolios (of up to 15 obligors) against market CDS spreads and compute the corresponding kth-to-default spreads. In a numerical study based on a synthetic portfolio of 15 telecom bonds we study a number of questions: How do spreads depend on the amount of default interaction?; How do the values of the underlying market CDS prices used for calibration influence kth-th-to default spreads?; How does a portfolio with inhomogeneous recovery rates compare with a portfolio that satisfies the standard assumption of identical recovery rates?; How well can kth-thto default spreads in a non-symmetric portfolio be approximated by spreads in a symmetric portfolio?