Hedging portfolio loss derivatives with CDSs∗

In this paper, we consider the hedging of portfolio loss derivatives using single-name credit default swaps as hedging instruments. The hedging issue is investigated in a general pure jump dynamic setting where default times are assumed to admit a joint density. In a first step, we compute default intensities adapted to the global filtration of defaults. In particular, we stress the impact of a default event on the price dynamics of non-defaulted names. In a two defaults setting, we also fully describe the hedging of a loss derivative with single name instruments. The methodology can be applied recursively to the case of a multidefault setting. We completely characterize the hedging strategies for general n-dimensional credit portfolios when default times are assumed to be ordered. The computation of the hedging strategies does not require any Markovian assumption. Introduction The hedging of loss derivatives such as CDO tranches or basket default swaps is a prominent risk-management issue especially given the recent revision of the Basel II regulation on calculation of trading book capital requirement. Indeed, according to Basel Committee Guidelines for computing capital for incremental risk in the trading book, July 2009, “for trading book risk positions that are typically hedged via dynamic hedging strategies, a rebalancing of the hedge within the liquidity horizon of the hedged position may be recognized as a risk mitigation. Moreover, any residual risks resulting from dynamic hedging strategies must be reflected in the capital charge.” As a result, the performance and efficiency of underlying hedging methods is going to have a direct impact on the amount of capital required for loss derivatives. Cousin and Laurent (2010) discuss various issues related to the use of models in designing hedging strategies for CDO tranches and back-testing or assessing hedging performance. In this paper, we consider the hedging of loss derivatives using single-name credit default swaps as hedging instruments. The hedging issue is investigated in a general pure jump setting where default times are assumed to admit a joint density which is the only input of the model – so that our results can be considered as model independent – and we compute default intensities adapted to the global filtration of defaults. We check that, if CDSs on each default are traded, the market is complete. The hedging strategies can be found by identifying the terms associated with the fundamental default martingales. We extend some recent results by Laurent, Cousin and Fermanian (2007) and Cousin, Jeanblanc and Laurent (2009). In particular, we stress the impact of a default event on the price dynamics of non-defaulted names. Moreover, in a two defaults setting, we fully describe the hedging of a loss derivative with single name instruments. The generalization to a multidefault setting can be done following the same methodology. Furthermore, we are able to completely characterize the hedging strategies in single-name CDS for general n-dimensional credit portfolios when default times are assumed to be ordered. The computation of the hedging strategies ∗This research is a part of CRIS programm †Département de Mathématiques, Equipe analyse et probabilité, Université d’Evry ‡Département de Mathématiques, Equipe analyse et probabilité, Université d’Evry; Institut Europlace de Finance