Pricing Synthetic CDO Tranches in a Model with Default Contagion Using the Matrix-Analytic Approach

Abstract

We value synthetic CDO tranche spreads, index CDS spreads, k-th-to-default swap spreads and tranchelets in an intensity-based credit risk model with default contagion. The default dependence is modelled by letting individual intensities jump when other defaults occur. The model is reinterpreted as a Markov jump process. This allows us to use a matrix-analytic approach to derive computationally tractable closed-form expressions for the credit derivatives that we want to study. Special attention is given to homogenous portfolios. For a fixed maturity of five years, such a portfolio is calibrated against CDO tranche spreads, index CDS spread and the average CDS spread, all taken from the iTraxx Europe series. After the calibration, which renders perfect fits, we compute spreads for tranchelets and k-th-to-default swap spreads for different subportfolios of the main portfolio. Studies of the implied tranche-losses and the implied loss distribution in the calibrated portfolios are also performed. We implement two different numerical methods for determining the distribution of the Markov-process. These are applied in separate calibrations in order to verify that the matrix-analytic method is independent of the numerical approach used to find the law of the process. Monte Carlo simulations are also performed to check the correctness of the numerical implementations.

Keywords: Credit risk, intensity-based models, CDO tranches, index CDS, k(th)-to-default swaps, dependence modelling, default contagion, Markov jump processes, Matrix-analytic methods

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