Default Intensities implied by CDO Spreads: Inversion Formula and Model Calibration

We propose a simple computational method for constructing an arbitrage-free collateralized debt obligation (CDO) pricing model which matches a prespecified set of CDO tranche spreads. The key ingredient of the method is an inversion formula for computing the aggregate default rate in a portfolio, as a function of the number of defaults, from its expected tranche notionals. This formula can be seen as an analogue of the Dupire formula for portfolio credit derivatives. Together with a quadratic programming method for recovering expected tranche notionals from CDO spreads, our inversion formula leads to an efficient nonparametric method for calibrating CDO pricing models. Contrarily to the base correlation method, our method yields an arbitrage-free model. Comparing this approach to other calibration methods, we find that model-dependent quantities such as the forward starting tranche spreads and jump-to-default ratios are quite sensitive to the calibration method used, even within the same model class. On the other hand, comparing the local intensity functions implied by different credit portfolio models reveals that apparently different models, such as the static Student-t copula models and the reduced-form affine jump-diffusion models, lead to similar marginal loss distributions and tranche spreads.