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 which are assumed to be constant between defaults, but which can jump at the times of defaults. The model is translated into a Markov jump process which 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 spreads and kth-to-default swap spreads. We semi-calibrate 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 spreads depend on the amount of default interaction; how the values of the underlying market CDS-prices used for calibration influence kth-to-default spreads; how a portfolio with inhomogeneous recovery rates compares with a portfolio which satisfies the standard assumption of identical recovery rates; and, finally, how well kth-to-default spreads in a nonsymmetric portfolio can be approximated by spreads in a symmetric portfolio.

Keywords: Portfolio credit risk, intensity-based models, default dependence modelling, default contagion, CDS, kth-to-default swaps, Markov jump processes, Matrix-analytic methods

JEL Classification: G33, G13, C02, C63, G32