Statistical rehabilitation of improper correlation matrices

The simplest way to describe the dependence for a set of financial assets is their correlation matrix. This correlation matrix can be improper when it is specified element-wise. We describe a new method for obtaining a positive definite correlation matrix starting from an improper one. The expert's opinion and trust in each pairwise correlation is described by a beta distribution. Then, by combining these individual distributions, a joint distribution over the space of positive definite correlation matrices is obtained using Cholesky factorization, and its mode constitutes the new proper correlation matrix. The optimization is complemented by a visual representation of the entries that were most affected by the legalization procedure. We also sketch a Bayesian approach to the same problem.