Valuing CDOs of Bespoke Portfolios with Implied Multi-Factor Models

by Dan Rosen of the Fields Institute and R2 Financial Technologies, and
David Saunders of the University of Waterloo

December 23, 2007

Abstract: This paper presents a robust and practical CDO valuation framework based on the application of multi-factor credit models in conjunction with weighted Monte Carlo techniques used in options pricing. The framework produces arbitrage-free prices and can be used to value consistently CDOs of bespoke portfolios, CDO-squared and cash CDOs. Multi-factor models allow us to model systematically heterogeneous portfolios, sector and geographical concentrations, deals which refer simultaneously to multiple indices and potentially other risk factors such as recoveries and prepayments. We demonstrate the practical advantages of working through multi-factor models, rather than directly on a common hazard rate (or a set of them).

The multi-factor credit models which determine the codependence of obligor defaults are defined generally within the mathematical construction of Generalized Linear Mixed Models (GLMMs). The implied copula approach can be seen as a special case of a GLMM, as are other common credit portfolio models. For a given model, the quoted prices of various credit portfolio instruments, such as CDO tranches, are used to imply the "risk-neutral" distributions (or processes) for the underlying systematic risk factors, which drive joint obligor defaults. We solve numerically the inverse problem of implying the factors' joint distribution, by creating first discrete scenarios on the factors. Although standard quadrature points may be used for low dimensions, more realistic problems require Monte Carlo simulation. We describe various numerical techniques for effectively sampling factor scenarios and obtaining well behaved factor distributions as the solutions from the optimization problem. While this paper focuses on a static version of the model (by defining the codependence of default times), the framework is general and can be potentially extended to a dynamic setting.

JEL Classification: G13, C44.

Keywords: CDOs, credit derivatives, factor models, implied distributions.

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Related reading: A Comparative Empirical Study of Asset Correlations

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