Estimating covariance matrices in high-dimensional settings is a challenging problem central to modern finance. The sample covariance matrix is well-known to give poor estimates in high dimensions with insufficient samples, and may cause severe risk underestimates of optimized portfolios in the Markowitz framework. In order to provide useful estimates in this regime, a variety of improved covariance matrix estimates have been developed that exploit additional structure in the data. Popular approaches include low-rank (principal component and factor analysis) models, banded structure, sparse inverse covariances, and parametric models. We investigate a novel nonparametric prior for random vectors which have a spatial ordering: we assume that the covariance is monotone and smooth with respect to this ordering. This applies naturally to problems such as interest-rate risk modeling, where correlations decay for contracts that are further apart in terms of expiration dates. We propose a convex optimization (semi-definite programming) formulation for this estimation problem, and develop efficient algorithms. We apply our framework for risk measurement and forecasting with Eurodollar futures, investigate limited, missing and asynchronous data, and show that it provides valid (positive-definite) covariance estimates more accurate than existing methods.