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

by Alexander Herbertsson of Göteborg University

March 15, 2007

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 then reinterpreted as a Markov jump process. This allow 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 a homogenous portfolio. For a fixed maturity of five years, such a portfolio is calibrated against CDO tranche spreads, index CDS spread and the average CDS and FtD spreads, all taken from the iTraxx Europe series. After the calibration, which render perfect fits, we compute spreads on tranchelets and k^th-to-default swap spreads for different subportfolios in the main portfolio. We also investigate implied tranche-losses and the implied loss distribution in the calibrated portfolios.

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

AMS 2000 Classification: 60J75, 60J22, 65C20, 91B28.

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.

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