CDO Models – Towards the Next Generation: Incomplete Markets and Term Structure

This article describes a new approach to the risk-neutral valuation of CDO tranches, based on a general specification of the tranche loss distributions and the index default distribution. The new model is a term-structure model, and the generality with which the basic distributions are specified allows it to be perfectly calibrated to any set of market prices (for any number of tranches and maturities) that is arbitrage-free. The use of the new model is illustrated by testing market prices for the standardized iTraxx index tranches (for all marketed tranches and maturities) to see if they are arbitrage-free. Other examples include the determination of the arbitrage-free range of prices allowed for an unmarketed standardized tranche and the determination of the cost of exiting a tranche position. For the latter example, both arbitrage-free price ranges, and a preferred price, are obtained. Prices for unmarketed maturities and unmarketed non-standard tranches are also obtained by an interpolation and extrapolation procedure. Because the model is an incomplete-market model characterized by many more parameters than market prices, it was essential to develop an efficient optimization approach to valuation. The article also makes use of a new approach to the problem of unequal recovery rates and notionals. Department of Physics, University of Toronto, Toronto, ON M5S 1A7, CANADA; email: [email protected]; telephone: (416) 978-3821. I thank David Beaglehole, Mark Davis, Julien Houdain, John Hull, Alex Kreinin, Dan Rosen and Alan White for discussions and encouragement; Efim Katz and the Theory Group of the Institute Laue-Langevin, where some of this work was carried out, for their hospitality; and Julien Houdain and Fortis Investments for providing the market prices used in the examples of this article. The support of the Natural Sciences and Engineering Research Council of Canada is acknowledged.