Geometric Deep Learning for Realized Covariance Matrix Forecasting

TL;DR

This paper introduces a Riemannian-geometry-aware deep learning framework for forecasting realized covariance matrices, effectively capturing their geometric structure and improving prediction accuracy in high-dimensional financial data.

Contribution

It presents a novel deep learning approach that incorporates Riemannian geometry for covariance matrix forecasting, extending the HAR model to matrix-variate data.

Findings

Outperforms traditional methods in predictive accuracy

Handles high-dimensional covariance matrices efficiently

Demonstrates effectiveness on S&P 500 data

Abstract

Traditional methods employed in matrix volatility forecasting often overlook the inherent Riemannian manifold structure of symmetric positive definite matrices, treating them as elements of Euclidean space, which can lead to suboptimal predictive performance. Moreover, they often struggle to handle high-dimensional matrices. In this paper, we propose a novel approach for forecasting realized covariance matrices of asset returns using a Riemannian-geometry-aware deep learning framework. In this way, we account for the geometric properties of the covariance matrices, including possible non-linear dynamics and efficient handling of high-dimensionality. Moreover, building upon a Fr\'echet sample mean of realized covariance matrices, we are able to extend the HAR model to the matrix-variate. We demonstrate the efficacy of our approach using daily realized covariance matrices for the 50 most capitalized companies in the S&P 500 index, showing that our method outperforms traditional approaches in terms of predictive accuracy.