Finding the nearest covariance matrix: the foreign exchange market case

We consider the problem of finding a valid covariance matrix in the foreign exchange market given an initial nonpositively semidefinite (non-PSD) estimate of such a matrix. The common no-arbitrage assumption imposes additional linear constraints on such matrixes, inevitably making them singular. As a result, even the most advanced numerical techniques will predictably balk at a seemingly standard optimization task. The reason is that the problem is ill posed, while its PSD solution is not strictly feasible. In order to deal with this issue we describe a low-dimensional face of the PSD cone that contains the feasible set. After projecting the initial problem onto this face, we come out with a reduced problem, which is both well posed and of a smaller scale. We show that, after solving the reduced problem, the solution to the initial problem can be recovered uniquely in one step. We run numerous numerical experiments to compare the performance of different algorithms in solving the reduced problem and to demonstrate the advantages of dealing with the reduced problem as opposed to the original one. The smaller scale of the reduced problem implies that its solution can effectively be found by the application of virtually any numerical method.