Hedging default risks of CDOs in Markovian contagion models

Abstract

We describe a replicating strategy of CDO tranches based upon dynamic trading of the corresponding credit default swap index. The aggregate loss follows a homogeneous Markov chain associated with contagion effects. Default intensities depend upon the number of defaults and are calibrated onto an input loss surface. Numerical implementation can be carried out thanks to a recombining tree. We examine how input loss distributions drive the credit deltas. 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.

Keywords:

• Actuarial science
• Asset management
• Defaultable securities
• Correlation modelling

Acknowledgements

The authors thank Salah Amraoui, Matthias Arnsdorf, Fahd Belfatmi, Tom Bielecki, Xavier Burtschell, Rama Cont, Stéphane Crepey, Michel Crouhy, Rüdiger Frey, Kay Giesecke, Emmanuel Gobet, Michael Gordy, Jon Gregory, Alexander Herbertsson, Steven Hutt, Monique Jeanblanc, Vivek Kapoor, Andrei Lopatin, Pierre Miralles, Franck Moraux, Marek Musiela, Thierry Rehmann, Marek Rutkowski, Antoine Savine, Olivier Vigneron and the participants at the Global Derivatives Trading and Risk Management conference in Paris, the Credit Risk Summit in London, the 4th WBS fixed income conference, the International Financial Research Forum on Structured Products and Credit Derivatives, the Universities of Lyon and Lausanne joint actuarial seminar, the credit risk seminar at the university of Evry, the French finance association international meeting and the doctoral seminars of the University of Dijon and ‘séminaire Bachelier’ for useful discussions and comments. We also thank Fahd Belfatmi, Marouen Dimassi and Pierre Miralles for very useful help regarding implementation and calibration issues. All remaining errors are ours. This paper has an academic purpose and may not be related to the way BNP Paribas hedges its credit derivatives books.

Notes

†Let us stress that the computed exposure at default is not equal to the usual ‘value on default’ or iOmega. In our model, the arrival of a default is associated with a shift in credit spreads and in base correlations due to contagion effects, while the value on default is usually computed under the assumption of constant spreads and correlations.

†In the general case where multiple defaults could occur, we have to consider possibly 2 ⁿ states, and we would require non-standard credit default swaps with default payments conditionally on all sets of multiple defaults to hedge CDO tranches.

†Note that the instantaneous credit default swaps are not exposed to spread risk but only to default risk.

‡We remark that the assumption of no simultaneous defaults also holds for Q.

§Note that . As a consequence, we readily obtain , which provides the time t replication price of M. We also remark that, for a small time interval dt, , which is consistent with market practice and regular rebalancing of the replicating portfolio. An investor who wants to be compensated at time t against the price fluctuations of M during a short period dt has to invest V _t in the risk-free asset and take positions in the n instantaneous digital credit default swaps. Recall that there is no initial charge to enter into a credit default swap position.

¶After the default of name i, the intensity is equal to zero: on .

⊥This Markovian assumption may be questionable, since the contagion effect of a default event may vanish as time goes by. The Hawkes process, which was used in the credit field by Giesecke and Goldberg (2006) and Errais et al. (2007), provides such an example of a more complex time dependence. Other specifications with the same aim are discussed by Lopatin and Misirpashaev (2007).

∥This means that the pre-default intensities have the same functional dependence to the default indicators.

†We remark that, on , , so that the pre-default intensity of name i actually only depends on the credit status of the other names.

‡Ding et al. (2006) consider the case where the intensity of the loss process is linear in the number of defaults. The loss distribution is then negative binomial.

§According to Feller's terminology, we should speak of a pure death process. Since we later refer to Karlin and Taylor (1975), we prefer their terminology.

¶Regarding the assumption of no simultaneous defaults, we also refer to Putyatin et al. (2005), Brigo et al. (2007) and Walker (2007b). Allowing for multiple defaults could actually ease the calibration onto senior CDO tranche quotes.

⊥Since is a martingale and using Ito–Doeblin's formula, it can be seen that V solves for the backward Kolmogorov equations:

†At this stage, for notational simplicity, we assume that there are no intermediate payments. This corresponds, for instance, to the case of zero-coupon CDO tranches with up-front premiums. The more general case is considered in section 4.

‡As above, in order to ease the exposition, we neglect at this stage actual payoff features such as premium payments, amortization schemes, etc. This is detailed in section 4.

§In the case where , there are no contagion effects and default dates are independent. We still have since is linear in the number of surviving names.

†Clearly, this involves more information that one could directly access through the quotes of liquid CDO tranches, especially with respect to small and large numbers of defaults. As for the computation of the number of default probabilities from quoted CDO tranche premiums, we refer to Krekel and Partenheimer (2006), Galiani et al. (2006), Meyer-Dautrich and Wagner (2007), Parcell and Wood (2007), Walker (2007a) and Torresetti et al. (2007). Practical issues related to the calibration inputs are also discussed by van der Voort (2006).

‡Therefore, the pre-default name intensity is such that . Let us recall that λ (t, n) = 0.

§In both approaches, there are as many unknown parameters as available market quotes.

¶Actually, the credit deltas at inception are the same whatever the choice.

⊥For such approaches, we refer to Moler and Van Loan (2003) and Herbertsson (2008a) regarding the numerical issues.

†We consider the value of the default leg immediately after t _i . Thus, we do not consider a possible default payment at t _i in the calculation of d (i, k).

‡As for the default leg, we consider the value of the premium leg immediately after t _i . Thus, we do not take into account a possible premium payment at t _i in the calculation of r(i, k) either.

§This is an approximation of the index spread since, according to market rules, the first premium payment is reduced.

¶If , the premium payment is the same whether the number of defaults is equal to k or k + 1. Therefore, it does not appear in the computation of the credit delta.

†ρ is the correlation between default events in a one-factor homogeneous Gaussian copula model where the time t conditional default probability (the probability that a name defaults before t given the latent factor V) is defined by

where Φ is the cumulative standard Gaussian density and is the time t marginal default probability. In former versions of the paper, ρ was associated with a conditional default probability defined by

‡, and .

§We checked that various choices of loss intensities for a large number of defaults had no effect on the computation of the deltas. We stress that this applies for the Gaussian copula case since the loss distribution has thin tails. For the market case example, we proceeded differently.

†For simplicity, we neglected the compounding effects over this short period.

‡We remark that the larger the correlation, the larger the change in market value of the default leg of the equity tranche at the arrival of the first default. Indeed, in a high correlation framework, this default means a relatively greater likelihood of default for the surviving names. This is not inconsistent with the previous results showing a decrease in credit deltas when the correlation parameter increases. The credit delta is the ratio of the change in value in the equity tranche and of the change in value in the credit default swap index. For a larger correlation parameter, the change in value in the credit default swap index is also larger due to magnified contagion effects.

†We recall that, in option pricing, the vanna is the sensitivity of the delta to a unit change in volatility.

‡We also computed the number of defaults distribution using entropic calibration. Although we could still compute loss intensities, the pattern with respect to the number of defaults was not monotonic. Depending on market inputs, direct calibration onto CDO tranche quotes can lead to shaky figures.

†Contrary to the Gaussian copula example, we computed the complete set of loss intensities using the procedure described in appendix C.

†Thus, given a recovery rate of 40%, the maximum expected loss is 60%.

†Due to the last assumption, the described calibration approach is not highly regarded by numerical analysts (see Moler and Van Loan (2003) for a discussion). However, it is well suited in our case studies.

‡Since λ _n = 0, p(t, n) takes a slightly different form. Its detailed expression is useless here since we only need to deal with to calibrate . Let us also remark that p(t, n) can equally be recovered from or from .