Scale, Shape, Location, and Speed: Decomposing Distributional Volatility Persistence

Abstract

Volatility persistence is usually represented by a scalar conditional variance, even though return distributions can remain abnormal through dispersion, center shifts, shape distortions, or continued movement. This paper develops a leakage-safe Wasserstein framework that separates these features into four rolling coordinates: scale, standardized shape, signed location, and distributional speed. Using daily S&P 500, FTSE 100, Nikkei 225, and Hang Seng returns from January 2000 to June 2026, the robust scale coordinate is strongly associated with 22-day realized volatility, whereas standardized shape is approximately orthogonal to the current volatility level. Signed location is negatively associated with realized volatility, and speed occupies an intermediate position because it measures one-step movement rather than state level. Scale is also the most synchronized coordinate across markets; location and speed are moderately synchronized, while shape is largely market-specific. Paired block-bootstrap tests, multiplicity corrections, block-length sensitivity checks, and a variance-matched heavy-tailed placebo support the separation of the coordinates. The results establish measurement nonredundancy rather than forecasting superiority or causal transmission. The framework therefore recasts volatility persistence as a multidimensional financial-risk state and provides testable inputs for subsequent state-model, recovery, and forecasting comparisons.

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