Paper: arXiv 2512.02166

Authors: Ziyao Wang, A. Alexandre Trindade, Svetlozar T. Rachev

Abstract

This paper develops a three-dimensional decomposition of volatility memory into orthogonal components of level, shape, and tempo. The framework unifies regime-switching, fractional-integration, and business-time approaches within a single canonical representation that identifies how each dimension governs persistence strength, long-memory form, and temporal speed. We establish conditions for existence, uniqueness, and ergodicity of this decomposition and show that all GARCH-type processes arise as special cases. Empirically, applications to SPY and EURUSD (2005–2024) reveal that volatility memory is state-dependent: regime and tempo gates dominate in equities, while fractional-memory gates prevail in foreign exchange. The unified tri-gate model jointly captures these effects. By formalizing volatility dynamics through a level–shape–tempo structure, the paper provides a coherent link between information flow, market activity, and the evolving memory of financial volatility.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 8.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a rigorous theoretical foundation by unifying regime-switching, long-memory (FIGARCH), and business-time volatility models into a 3D canonical level-tempo-shape decomposition, backed by complete mathematical proofs. The empirical evaluation is extensive, spanning 20 years of SPY and EURUSD data using out-of-sample forecasting, density loss metrics, and modern VaR/ES backtesting methodologies.

Research Flowchart

  flowchart TD
    A[Research Goal: Unify Volatility Memory via 3D Decomposition] --> B[Theory: Formulate Level, Shape & Tempo Tri-Gate Model]
    B --> C[Proof: Establish Existence, Ergodicity & GARCH Special Cases]
    D[(Data: SPY & EURUSD High-Frequency 2005-2024)] --> E[Empirical Estimation & Cross-Asset Calibration]
    B --> E
    E --> F[Findings: Equities Dominated by Regime/Tempo Gates]
    E --> G[Findings: FX Dominated by Fractional-Memory Gates]