Journal of Credit Risk

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Pricing synthetic CDO tranches in a model with default contagion the matrix analytic approach

Alexander Herbertsson

ABSTRACT

We value synthetic collateralized debt obligation (CDO) tranche spreads, index credit default swap (CDS) spreads, kth-to-default swap spreads and tranchelets in an intensity-based credit risk model with default contagion. The default dependence is modeled 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 homogeneous portfolios. For a fixed maturity of five years, such a portfolio is calibrated against CDO tranche spreads, index CDS spreads 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 kth-todefault 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. Both methods 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.

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