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.