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

Abstract

We propose a simple computational method for constructing an arbitrage-free CDO pricing model which matches a pre-specified set of CDO tranche spreads. The key ingredient of the method is a formula for computing the local default intensity function of a portfolio from its expected tranche notionals. This formula can be seen as an analog, for portfolio credit derivatives, of the well-known Dupire formula. Together with a quadratic programming method for recovering expected tranche notionals from CDO spreads, our inversion formula leads to an efficient non-parametric method for calibrating CDO pricing models.

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 default intensities implied by different credit portfolio models reveals that apparently very different models such as static Student-t copula models and reduced-form affine jump-diffusion models, lead to similar marginal loss distributions and tranche spreads.

Keywords: Portfolio credit derivatives, collateralized debt obligation, inverse problem, local intensity, default intensity, expected tranche notionals, calibration, CDO tranche

JEL Classification: G13, G12