Hedging Default Risks of CDOs in Markovian Contagion Models
by Jean-Paul Laurent of the Université Lyon 1 & BNP Paribas
April 8, 2008
Abstract: We describe a hedging strategy of CDO tranches based upon dynamic trading of the corresponding credit default swap index. We rely upon a homogeneous Markovian contagion framework, where only single defaults occur. In our framework, a CDO tranche can be perfectly replicated by dynamically trading the credit default swap index and a risk-free asset. Default intensities of the names only depend upon the number of defaults and are calibrated onto an input loss surface. Numerical implementation can be carried out fairly easily thanks to a recombining tree describing the dynamics of the aggregate loss. Both continuous time market and its discrete approximation are complete. The computed credit deltas can be seen as a credit default hedge and may also be used as a benchmark to be compared with the market credit deltas. Though the model is quite simple, it provides some meaningful results which are discussed in detail. We study the robustness of the hedging strategies with respect to recovery rate and examine how input loss distributions drive the credit deltas. Using market inputs, we find that the deltas of the equity tranche are lower than those computed in the standard base correlation framework. This is related to the dynamics of dependence between defaults. We can think of our model as a "sticky implied tree" while the hedge ratios computed by market participants correspond to "sticky strike" deltas, following the terminology of Derman (1999).
Keywords: CDOs, hedging, complete markets, contagion model, Markov chain, recombining tree.
Forthcoming in: Quantitative Finance.