Paper: arXiv 2503.16200

Authors: Piotr Chmielowski

Abstract

This note outlines an approach to stress testing of covariance of financial time series, in the context of financial risk management. It discusses how the geodesic distance between covariance matrices implies a notion of plausibility of covariance stress tests. In this approach, correlation stress tests span a submanifold of constant determinant of the Fisher–Rao manifold of covariance matrices. A parsimonious geometrically invariant definition of arbitrarily large correlation stress tests is proposed, and a few examples are discussed.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a rigorous geometric framework for correlation stress testing using the Fisher-Rao manifold and geodesic distance. However, it is primarily a theoretical and methodological note with minimal empirical validation or backtesting on real historical market datasets in the excerpt.

Research Flowchart

  flowchart TD
    A[Research Goal: Plausible & Invariant Correlation Stress Testing] --> B[Inputs: Multi-Asset Financial Time Series & Base Covariance Matrix Σ]
    B --> C[Map Covariance Matrices to Fisher-Rao Riemannian Manifold]
    C --> D[Constrain Stress Tests to Submanifold of Constant Determinant]
    D --> E[Compute Geodesic Distances to Quantify Test Plausibility]
    E --> F[Key Outcome: Geometrically Invariant Framework for Large Stress Tests]

Cited References (18)

  1. The Riemannian Geometry of the Space of Positive-Definite Matrices and Its Application to the Regularization of Positive-Definite Matrix-Valued Data (2011) — link
  2. Supervisory Stress Testing of Central Counterparties.
  3. Rates correlations break down amid volatility surge. (2022)
  4. Severe but Plausible — or Not?
  5. IMF Survey: What Is a Bank Stress Test? (2010) — link
  6. On closed-form expressions for the Fisher–Rao distance (2024) — link
  7. Information and the accuracy attainable in the estimation of statistical parameters (1945)
  8. Correlation scenarios and correlation stress testing (2022) — link
and 10 more
  1. RiskMetrics: Technical Document (1996) — link
  2. Killing vector fields of constant length on Riemannian manifolds (2006) — link
  3. Approximation and bounding techniques for the Fisher-Rao distances between parametric statistical models (2024) — link
  4. Correlation scenarios and correlation stress testing (2023) — link
  5. The Fisher-Rao geometry of CES distributions (2023) — link
  6. Structural Scenario Analysis with SVARs (2021)
  7. A Comprehensive Introduction to Differential Geometry (1999)
  8. A factor-model approach for correlation scenarios and correlation stress testing (2019) — link
  9. Correlation scenarios and correlation stress testing (2023) — link
  10. Reprint of: Mahalanobis, P.C. (1936) “On the Generalised Distance in Statistics.” (2018) — link

0 of 18 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.