Finding the Nearest Valid Covariance Matrix: a FX Market Case

Journal of Computational Finance. 2020. Vol. 24. No. 2. P. 103–127.

Minabutdinov A., Bouev M., Manaev I.

We consider the problem of finding a valid covariance matrix in the foreign
exchange market given an initial non-PSD estimate of such a matrix. The common
no-arbitrage assumption imposes additional linear constraints on such matrices,
whereby 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 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 its solution can be effectively found by application of virtually any numerical
method.

Keywords: foreign exchange, dimensionality reduction, nearest covariance matrix