Cont, Rama, Andreea Minca, "Recovering Portfolio Default Intensities Implied by CDO Quotes ", Mathematical Finance, Vol. 23, No. 1, (January 2013), pp. 94–121.
Abstract: We propose a stable nonparametric algorithm for the calibration of “top-down” pricing models for portfolio credit derivatives: given a set of observations of market spreads for collateralized debt obligation (CDO) tranches, we construct a risk-neutral default intensity process for the portfolio underlying the CDO which matches these observations, by looking for the risk-neutral loss process “closest” to a prior loss process, verifying the calibration constraints. We formalize the problem in terms of minimization of relative entropy with respect to the prior under calibration constraints and use convex duality methods to solve the problem: the dual problem is shown to be an intensity control problem, characterized in terms of a Hamilton–Jacobi system of differential equations, for which we present an analytical solution. Given a set of observed CDO tranche spreads, our method allows to construct a default intensity process which leads to tranche spreads consistent with the observations. We illustrate our method on ITRAXX index data: our results reveal strong evidence for the dependence of loss transitions rates on the previous number of defaults, and offer quantitative evidence for contagion effects in the (risk-neutral) loss process.
Keywords: collateralized debt obligation, duality, portfolio credit derivatives, reduced-form models, default risk, intensity control, top-down credit risk models, relative entropy, inverse problem, model calibration, stochastic control.