Hedging default risks of CDOs in Markovian contagion models

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. Moreover, 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.

On Break-Even Correlation: The Way to Price Structured Credit Derivatives by Replication

“…as above, then the pricing factors ρ i that will be put in formula (27) are now given by the relations…”

“…stock returns 7 ρ ij should be invoked in the pricing formula (27). Nonetheless, it is not the case and the pricing formula (27) is still used, then these correlations should be different.…”

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

“…On practical grounds, another nice feature of the model concerns the estimation of model parameters from CDO tranche market quotes. As described in Laurent et al (2007), the knowledge of upfront premiums of equity CDO tranches with different maturities and detachment points (and given some recovery rate) is equivalent to the knowledge of marginal distributions of the number of defaults at different time horizons. Thanks to the forward Kolmogorov equations, one can then perfectly compute the intensities of the aggregate loss process or the pre-default intensities.…”

“…The numerical implementation of hedging strategies can be achieved in a more realistic case through a binomial tree as detailed in Laurent et al (2007) or by means of Markov chain techniques.…”

An extension of Davis and Lo's contagion model

“…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. As a first step among possible applications of the Davis and Lo's multi-period model, we illustrate the model tractability in terms of the pricing of synthetic CDO tranches.…”