Recovering Portfolio Default Intensities Implied by Cdo Quotes

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.

On the drawdowns and drawups in diffusion-type models with running maxima and minima

“…In this respect, the coefficients of the diffusion-type model considered in this paper can be interpreted as the Markovian projection of a continuous semimartingale conditioned on the current state of the associated five-dimensional Markov process with four path-dependent components. Other similar Markovian projections were studied by Bremaud [4] for queues and by Cont and Minca [7] for marked point processes with path-dependent intensities. Calibrational aspects of such models with path-dependent distributional characteristics were recently studied by Hambly, Mariapragassam, and Reisinger [22] among others.…”

Dynamic Hedging of Synthetic CDO Tranches: Bridging the Gap between Theory and Practice

“…This is not however the common way to appraise the Gaussian copula. If one were to observe the path of assets associated with different names instead associated with the multivariate structural model are well understood and lead to perfect replication of CDO tranches when concentrating on spread risk (defaults are predictable), we can think of deltas coming out of the Gaussian copula model to have some economic significance 23 . As discussed above, Fermanian and Vigneron (2009) propose a different approach.…”

“…The culmination of the inverse problem of model design from market prices leads to some nonparametric approaches such as the local volatility model in the equity field and its local intensity counterpart in the credit domain (see Cont and Minca (2008), Cont, Deguest. and Kan (2009)).…”

“…We would definitely need a more systematic investigation of this point. 23 In theory, one should start with the computation of the derivatives of the value of the CDO tranche and of hedging CDS with respect to underlying state variables, here the asset values at the current time. Then, the amount of hedging CDS is the ratio of the above derivatives.…”

An extension of Davis and Lo's contagion model

“…All along years, the contagion phenomena was modeled using Bernoulli random variables (Davis and Lo (2001)), copula functions (Schönbucher and Schubert (2001)), interacting particle systems (Giesecke and Weber (2004)), incomplete information models (Frey and Runggaldier (2010)) or Markov chains (see for example Schönbucher (2006), Graziano and Rogers (2009) or Kraft and Steffensen (2007)). As far as the risk management of synthetic CDO tranches is concerned, Markov chain contagion models have also been investigated by several papers such as Van der Voort (2006), Herbertsson and Rootzén (2006), Herbertsson (2007), Frey and Backhaus (2010), Frey and Backhaus (2008), De Koch, Kraft and Steffensen (2007), Epple, Morgan and Schloegl (2007), Lopatin and Misirpashaev (2007), Arnsdorf and Halperin (2008), Cont and Minca (2008), Cont, Deguest and Kan (2010) among others. The hedging issue for CDO tranches is also addressed by Laurent, Cousin and Fermanian (2010) and Cousin, Jeanblanc and Laurent (2010) in the class of Markovian contagion models.…”