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; Mandelbrot, Fisher & Calvet, 1997) to an implied volatility index. Using 9,118 daily observations of the CBOE VIX spanning January 1990 to March 2026—a period encompassing four economic cycles—we implement the complete MMAR estimation pipeline: partition functions, OLS-fitted scaling exponents, Legendre-transformed multifractal spectrum, lognormal cascade calibration, fractional Brownian motion generation, and Monte Carlo validation. VIX log-returns strongly reject Gaussianity (KS p<10⁻¹⁰, excess kurtosis 6.73). The scaling function τ^(q)=−0.022q²+0.230q−1.031 is strictly concave, confirming genuine multiscaling. The Hurst exponent H^=0.189 places VIX in the strongly subdiffusive regime (fractal dimension dH=1.811), consistent with its mean-reverting character and in sharp contrast with the persistent scaling of equity price indices. The most probable Hölder exponent α^₀=0.230 exceeds H^, rendering the lognormal cascade admissible: λ^=1.219 and σ^²=0.633. Formal validation via 10,000 Monte Carlo MMAR simulations and 1,000 Gaussian benchmarks reveals a partial but insufficient fit. MMAR produces mean excess kurtosis of 4.10 against the empirical 6.73—capturing 61% of tail mass but falling significantly short. Kolmogorov–Smirnov tests reject both models at all conventional significance levels. We interpret the residual kurtosis gap as evidence that the lognormal cascade underestimates the most extreme VIX spikes, with implications for volatility derivatives pricing.