Finding the Nearest Valid Covariance Matrix: An FX Market Case

We consider the problem of finding a valid covariance matrix in the FX market given an initial non-PSD estimate of such a matrix. The standard no-arbitrage assumption implies additional linear constraints on such matrices, which automatically makes them singular. As a result, one cannot just take the given estimate plug it into the standard optimization problem and solve it by applying even the most advanced numerical techniques developed recently. The reason is that such a problem is not well-posed while the PSD-solution is not strictly feasible. In order to deal with this issue, we described 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 turns out to be well posed and of a smaller scale. We show that after solving the reduced problem the solution to the initial problem can be uniquely recovered in one step. We run numerous numerical experiments to compare 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 virtually any numerical method can be applied effectively to find its solution.