A geometric framework for covariance dynamics

Employing methods of differential geometry, we propose a new framework for covariance dynamics modeling. Our approach respects the intrinsic geometric properties of the space of covariance matrices and allows their natural evolution. We develop covariance models that exploit either asset returns or realized covariances and propose a new estimation method that minimizes the length of the geodesic between the forecast and the realization. The geodesic length is equivalent to the Fisher information metric under the Gaussian assumption and is deemed a proper measure of similarity between two covariance matrices. Empirical studies involving three data samples and various performance metrics suggest that our models outperform existing ones.