  We show how Adjoint Algorithmic Differentiation (AAD) allows an extremely
efficient calculation of correlation Risk of option prices computed with Monte
Carlo simulations. A key point in the construction is the use of binning to
simultaneously achieve computational efficiency and accurate confidence
intervals. We illustrate the method for a copula-based Monte Carlo computation
of claims written on a basket of underlying assets, and we test it numerically
for Portfolio Default Options. For any number of underlying assets or names in
a portfolio, the sensitivities of the option price with respect to all the
pairwise correlations is obtained at a computational cost which is at most 4
times the cost of calculating the option value itself. For typical
applications, this results in computational savings of several order of
magnitudes with respect to standard methods.
