Ebook Dynamic hedging of portfolio credit derivatives
Static factor models, in particular the Gaussian Copula model, have been widely used for hedging portfolio credit derivatives such as collateralized debt obligations (CDOs). The risk of a CDO tranche has been often characterized in terms of sensitivities to shifts in risk factors in a copula-based framework. Accordingly, hedging practices have been typically based on such measure of sensitivity. The most common hedging approach has been to “delta-hedge” the changes due to spread fluctuations using credit default swaps (CDSs).
However, the recent turmoil in credit derivatives markets shows that these commonly used hedging approaches are inefficient. One of the main problems has been the lack of well-defined dynamics for risk factors in such static model, which prevents any model-based assessment of hedging strategies. In particular, delta-hedging of spread risk is loosely justified using a Black Scholes analogy which does not necessarily hold and the corresponding hedge ratios, the spread-deltas, are in fact computed from a static model without spread risk. Furthermore, delta-hedging of spread risk ignores default risk and jumps in the spreads which appeared to be critical for risk management during the difficult market environment in 2008. Although gamma hedging (hedging of small movements in spread-delta) can improve the performance slightly, it is not sufficient to solve these issues.
Realizing the deficiencies of copula-based hedging methods, alternatives have been proposed to tackle the problem of hedging portfolio credit derivatives. Durand and Jouanin describe hedging practices for credit derivatives and provide an approximation of the P&L of a portfolio hedged with misspecified parameters under the real-world probability measure. They correctly point out the inconsistency between most of the pricing models, where the risk is occurrence of defaults, and the real hedging strategy, where the trader will protect his portfolio against small CDS spreads movements. Bielecki, Jeanblanc and Rutkowski show that, in a bottom-up hazard process framework driven by a Brownian motion, perfect replication is possible by continuous trading in a sufficient number of liquidly traded CDS contracts.
However, these strategies are not tested on real data to assess their actual performance. Laurent, Cousin and Fermanian study hedging of synthetic CDO with the underlying index default swap in a Markovian contagion framework without spread risk. They show that the market is complete in their framework and CDO payoffs can be replicated by a self-financing portfolio consists of the index default swap and a risk-free bond. As we will see in Section 5, spread movements are a major source of risk even there in absence of defaults in the underlying portfolio so failure to incorporate spread risk in their model can be a major disadvantage. Moreover, the numerical results in are not obtained from models calibrated to the market data which lead to inconclusive comparisons.
Contents
1 Introduction
2 Portfolio credit derivatives
- 2.1 Index default swaps
2.2 Collateralized debt obligations
3 Top-down models for portfolio credit derivatives
- 3.1 Homogeneous Gaussian Copula model
3.2 Local intensity model
3.3 Bivariate spread-loss model
3.4 Compound Poisson model
4 Hedging strategies
- 4.1 Delta-hedging of spread risk
4.2 Hedging of default risk
4.3 Variance-minimizing strategy
5 Empirical studies
- 5.1 Before the storm: 20 September 2006 - 8 March 2007
5.2 Onset of the subprime crisis: 20 September 2007 - 20 March 2008
5.3 Impact of defaults on CDS index spreads
6 Conclusion
A Variance-minimizing hedge and Galtchouk-Kunita-Watanabe decomposition
B Derivation of the variance-minimizing hedge ratio
- B.1 Proof of Proposition 1
B.2 Proof of Proposition 2
B.3 Proof of Proposition 3
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