Beyond the Gaussian copula: stochastic and local correlation

Abstract

We consider stochastic correlation models that account for the correlation smile in the pricing of synthetic CDO tranches. These can be viewed as tractable extensions of the one factor Gaussian copula model. We analyse these models through their conditional default probability distributions. We also give some examples of using a three states stochastic correlation model to fit the market and discuss some risk management issues. We provide some analytical computations within the large homogeneous portfolio approximation. Eventually, we compare the stochastic correlation model with another popular state dependent correlation model, namely the random factor loading model. JEL Classification: C 31, G 13

Pricing of CDOs based on the multivariate Wang transform

“…First, contrary to the criticism, we show that the one-factor Gaussian copula model is consistent with Bühlmann's equilibrium pricing model (1980), 4 whence it has a sound economic interpretation. Second, the Gaussian copula model is extended within the Bühlmann's framework to fit market prices of CDO tranches better by taking the well-recognized facts in the credit derivatives market into consideration.…”

“…and White (2006) and Burtschell, Gregory and Laurent (2007) for such extensions. 4 Bühlmann's model (1980) has been developed for the pricing and hedging of insurance risk.…”

“…4 Bühlmann's model (1980) has been developed for the pricing and hedging of insurance risk. Insurance market is incomplete, in the sense that risks in the market cannot be replicated by other assets in the market, and so is the CDO market.…”

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

“…For instance, when looking at an equity tranche, the names with the highest credit spreads have a delta equal to one 38 , while the remaining names have a delta equal to zero. Such a phenomenon also occurs in the stochastic correlation model described by Burtschell et al (2007). The bumps in Figure 2 are related to the comonotonic (perfect dependence) state and the heterogeneity amongst credit spreads.…”

“…This feature should be captured by the hedging strategies computed in this framework. 14 Let us note that this theorem is usually written for martingales adapted to a filtration generated by a standard Brownian motion with independent components. However, as an intermediary step, the correlated Brownian motion W ɶ can be expressed as the product of the square root of its correlation matrix with a standard ndimensional Brownian motion with independent components.…”