Geometric turbulence: a geodesic-regression crisis indicator for equity covariance dynamics, with evidence from African markets

[2608.15205] Geometric turbulence: a geodesic-regression crisis indicator for equity covariance dynamics, with evidence from African markets Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Differential Geometry arXiv:2608.15205 (math) [Submitted on 15 Aug 2026] Title: Geometric turbulence: a geodesic-regression crisis indicator for equity covariance dynamics, with evidence from African markets Authors: E. B. Camara , Y. U. Gaba , I. G. M. Nsiloulou View a PDF of the paper titled Geometric turbulence: a geodesic-regression crisis indicator for equity covariance dynamics, with evidence from African markets, by E. B. Camara and 2 other authors View PDF HTML (experimental) Abstract: The covariance matrix of a basket of assets is a symmetric positive-definite object that evolves on a curved manifold, not on a flat vector space. Standard linear regression on its vectorised entries ignores that geometry and, in periods of market stress, can return fits that fail to be positive definite. We study geodesic regression on the symmetric positive-definite cone $\SPD(n)$ under the log-Euclidean and affine-invariant metrics, with the rolling sample covariance of a basket of log-returns as data. The fitted geodesic's velocity norm is proposed as a geometry-aware turbulence indicator. On daily data for four equity baskets- a ten-stock historical S\&P sub-basket (2006-2024), a current S\&P~100 mega-cap basket, the top ten listings of the Johannesburg Stock Exchange (JSE), and the ten most-traded Egyptian Exchange (EGX) tickers the indicator peaks cleanly on known stress episodes (the 2008 global financial crisis, the 2020 COVID crash, the 2022 Fed rate-hike cycle, and the October-November 2019 Egyptian political-economic tension episode) without any calibration. On hold-out evaluation in crash windows, log-Euclidean regression achieves 150 to 300 times smaller geodesic mean squared error than Euclidean ordinary least squares, because the latter produces non-SPD forecasts that blow up under the intrinsic metric. A naive velocity-aware de-risking strategy reduces maximum drawdown by 2.5 percentage points at the cost of 1.4 percentage points of annual return, consistent with a slow-moving indicator that signals stress reliably but translates imperfectly into tactical trading. All data, code, and experiments are reproducible from the companion repository. Comments: 22 pages, 8 figures Subjects: Differential Geometry (math.DG) Cite as: arXiv:2608.15205 [math.DG] (or arXiv:2608.15205v1 [math.DG] for this version) https://doi.org/10.48550/arXiv.2608.15205 Focus to learn more arXiv-issued DOI via DataCite Submission history From: El Hadji Baye Camara [ view email ] [v1] Sat, 15 Aug 2026 12:36:49 UTC (1,387 KB) Full-text links: Access Paper: View a PDF of the paper titled Geometric turbulence: a geodesic-regression crisis indicator for equity covariance dynamics, with evidence from African markets, by E. B. Camara and 2 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.DG < prev | next > new | recent | 2026-08 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... 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