
 Pricing Synthetic CDO Tranches in a Model with Default Contagion using the MatrixAnalytic Approach by Alexander Herbertsson of the University of Gothenburg September 10, 2008 Abstract: We value synthetic CDO tranche spreads, index CDS spreads, k^{ th }todefault swap spreads and tranchelets in an intensitybased 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 matrixanalytic approach to derive computationally tractable closedform 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 }todefault swap spreads for different subportfolios of the main portfolio. Studies of the implied tranchelosses and the implied loss distribution in the calibrated portfolios are also performed. We implement two different numerical methods for determining the distribution of the Markovprocess. These are applied in separate calibrations in order to verify that the matrixanalytic 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. JEL Classification: G33, G13, C02, C63, G32. AMS Classification: 60J75, 60J22, 65C20, 91B28. Keywords: Credit risk, intensitybased models, CDO tranches, index CDS, k^{ th }todefault swaps, dependence modelling, default contagion, Markov jump processes, Matrixanalytic methods. Books Referenced in this paper: (what is this?) 