The aggregation law: why an index is less intermittent than its stocks, and why it cannot be why the index is rough

Abstract

Stock indices are less intermittent than their constituent stocks and, at the same time, rougher: the log-volatility of an index has a smaller intermittency coefficient λ² and a larger Hurst exponent H than the log-volatility of the individual names it is built from. Two mechanisms have been proposed for the intermittency half — aggregation destroying log-correlation, or a realized-volatility measurement artifact — and a nested factor model with a rough common factor has been fit to reconcile both halves at once. This note reports a sharper result than a reconciling fit: a closed-form aggregation law with a necessity structure. Analytically, then on synthetic aggregates of known cascades, and then estimator-free on a committed 180-stock S&P-500 panel, aggregating D imperfectly-correlated H = 0 cascades into an equal-weight aggregate variance drives the intermittency down by λ²_agg = ρλ²₀ + (1−ρ)λ²₀/D_eff — so the lower intermittency of an index is aggregation — but cannot raise H: a weighted sum of H = 0 fields is again H = 0, because linear mixing preserves the correlation exponent. The roughness half is therefore not aggregation, not iid microstructure noise (excluded separately), and not constituent-level liquidity (excluded on the own panel): a genuinely rough common volatility factor is necessary, not merely one sufficient modelling choice.

The structural argument is first-order in the (weak) intermittency, and the real panel prices its own second-order term: the covariance-mixture law holds estimator-free to [0.958, 1.016] at the six dyadic lags the slopes are read on and [0.894, 1.046] across all integer lags 1…32. The calibration-validated slope-ratio is ρ ≈ 0.58 (calibrated 0.587) with a moving-block-bootstrap 95% interval [0.43, 0.65]; converged to 2,000 replicates at the declared block length the interval excludes the synthetic comparator 0.65, so the honest statement is that 0.65 sits on the boundary of the interval, and the boundary moves ±0.03 with the resampling choices — and the comparator is itself cross-measurement. The single-common-cascade approximation is measurably violated: ρ(τ) falls with lag, the fingerprint of exactly the rough, shorter-scale common factor the necessity argument forces. The delta over the nested-factor fit is thus a necessity argument where the prior work supplied a sufficiency fit — established structurally and on synthetic aggregates, and consistent with, but not powered to exclude, an aggregate as rough as the index on the real panel at its 1,259-day length.