Valuation of a CDO and an n^th to Default CDS Without Monte Carlo Simulation

by John Hull of the University of Toronto, and
Alan White of the University of Toronto

September 2004

Abstract: In this paper we develop two fast procedures for valuing tranches of collateralized debt obligations and n^th to default swaps. The procedures are based on a factor copula model of times to default and are alternatives to using fast Fourier transforms. One involves calculating the probability distribution of the number of defaults by a certain time using a recurrence relationship; the other involves using a "probability bucketing" numerical procedure to build up the loss distribution. We show how many different copula models can be generated by using different distributional assumptions within the factor model. We examine the impact on valuations of default probabilities, default correlations, the copula model chosen, and a correlation of recovery rates with default probabilities. Finally we look at the market pricing of index tranches and conclude that a "double t-distribution" copula fits the prices reasonably well.

Published in: Journal of Derivatives, Vol. 12, No. 2, (Winter 2004), pp. 8-23.

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