The Three-Dimensional Decomposition of Volatility Memory

TL;DR

This paper introduces a novel three-dimensional framework for decomposing volatility memory into level, shape, and tempo, unifying various models and revealing state-dependent effects in financial markets.

Contribution

It develops a unified canonical representation of volatility memory, integrating regime-switching, fractional-integration, and business-time approaches, with theoretical conditions and empirical validation.

Findings

Volatility memory varies with market state and asset class.

Regime and tempo effects dominate in equities.

Fractional-memory effects are prevalent in foreign exchange.

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.