Subdiffusive Multifractal Scaling of Implied Volatility: Evidence from 36 Years of VIX Data Using the MMAR Framework

We present the first application of the Multifractal Model of Asset Returns (MMAR) to an implied volatility index, using 36 years of daily CBOE VIX observations spanning four economic cycles. Three general conclusions emerge. First, implied volatility is multifractal: its scaling function is strictly concave, and this curvature survives explicit comparison against monofractal, ARMA, and ARFIMA nulls fitted to the same data, so it cannot be reproduced by anti-persistence or short-range linear dependence alone. Second, unlike equity price indices which are persistent, the VIX is strongly subdiffusive (H^≈0.18, far below 12), which is the multifractal signature of its mean-reverting character; the lognormal cascade is nonetheless admissible, so the construction is internally consistent. Third, admissibility notwithstanding, the lognormal cascade is insufficient in the extreme tails. Across Monte Carlo validation, higher-moment and tail-risk (VaR/ES) comparisons, and a GARCH/EGARCH/FIGARCH benchmark, it captures the bulk of the distribution but systematically underestimates the most violent volatility spikes and does not reproduce VIX’s pronounced positive skewness. We quantify this: the admissible cascade recovers about 84% of the excess kurtosis and reproduces 95–99% Value-at-Risk and 95% Expected Shortfall almost exactly, but it understates the deepest Expected Shortfall, and, being symmetric, it cannot reproduce the positive skew, underpricing far-out-of-the-money option premia by up to 100%. The indicated direction is asymmetric, heavier-tailed cascade extensions. Beyond VIX, the analysis offers a reproducible template for distinguishing genuine multifractality from its linear imitators in any volatility series.