Barycenters of Covariance Matrices and Riemannian Entropy for Analyzing Structural Risk in Financial Markets

Abstract

Introduction:

This paper offers a new geometric-entropic framework that identifies and assesses structural change points in financial markets, with a focus on the effects of the COVID-19 pandemic. Conventional econometric models continue to rely on using covariance matrices in Euclidean space, disregarding important geometric properties, such as the fact that covariance matrices are symmetric positive definite (SPD) matrices. This study emerged from the limitations of using SPD covariance matrices. The pandemic was an important systemic shock that altered correlations, volatility, and risk dependencies in the financial markets, thus allowing an ideal condition to apply and evaluate the new method.

Objectives:

The central aim is to develop and examine a new mathematical approach represent two original concepts:

• Capturing the central tendency of time-varying market covariances by locating their barycenter on the Riemannian manifold.


• Characterizing the structural disorder and uncertainty of the market using a recently introduced Riemannian entropy measure.


• Show that this dual-action approximation is superior to volatility- or correlation-based approaches in deriving timely and accurate indication of structural seams or systemic shocks, notably along with the evidence presented during the COVID-19 pandemic.

Methods:

The research employs the following methodology stages:

• Data: The analysis uses weekly log-returns for 17 major international stock market indexes ranging from January 2016 to January 2023.


• Covariance Modeling: Each rolling-window covariance matrix is estimated and treated as a point on the SPD manifold.


• Barycenter Calculation: The Riemannian barycenter for the covariance matrices is computed over time to present the evolving central dependency structure for the global markets.


• Entropy Measurement: A new Riemannian entropy measure is calculated in addition to Fréchet variance to measure disorder and dispersion of the covariance matrices about the barycenter.


• Change-Point Detection: Statistical procedures are applied to each trajectory of the barycenter and entropy to isolate distinct regime shifts associated with notable market events.

Results:

• After the outbreak of COVID-19, we observed a significant shift in market covariances’ Riemannian barycenter, indicating a shift in market risk geometry. During severe market stress, especially in March 2020 and again in some of the following waves of the pandemic, we saw marked and pronounced increases in Riemannian entropy.


• We found that the increase in the entropy (a measure of structural uncertainty) occurs in advance of sudden volatility, our alerts achieve early warning of systemic stress prior to the increase in volatility identified using traditional methods.


• We found that, compared to traditional methods, whether using the barycenter or entropy on its own approach, the joint approach of barycenter movement and entropy oscillation was an effective combination of two-dimension approach, for arguably diagnosing structural breakdowns.

Conclusions:

In conclusion, we have effectively developed and applied a new geometric-entropic framework that unifies Riemannian entropy with covariance barycenters. The analysis indicates that the COVID-19 pandemic has changed both risk levels and the structural geometry of financial markets. The methodology creates an institutional formal and intuitive way to analyze dynamic financial systems, and presents a unique and worthwhile contribution to mathematical finance. The methodology has improved the early detection of structural regime shifts and financial contagion, while allowing for the study of time-varying covariances from an entirely new perspective.