Lie Symmetry Methods for Time-Fractional Option Pricing: Similarity Reductions, Mittag–Leffler Propagators, and Rough-Volatility Diagnostics

We formulate a dimensionally consistent time-fractional Black–Scholes model through inverse-stable subordination. A product pricing measure fixes the clock law and operational numeraire, while clock risk remains unspanned. The physical memory intensity is κ α=t₀1-α κ α = t 0 1 - α , and prices are reported across four values of t₀ t 0 . The Lie analysis distinguishes an equivalence scaling of the independent-coefficient family, a gauge-augmented transformation that preserves the Black–Scholes coefficient constraint, and fixed-parameter scaling of the diffusion core. The Erdélyi–Kober reduction is proved under domination and trace conditions. A graded-mesh L1 solver and an independent two-million-draw subordination calculation place the benchmark crossover near 1.095 years. A transform-curvature argument separates inverse-stable time-changed Heston from rough Heston, and the rough-Heston solver is refined over 36 damped-contour cases. The empirical study uses 17,520 hourly Bitcoin observations, four volatility proxies, three scaling estimators, block-bootstrap ranges, and three ACF models. In a chronologically held-out Deribit test, the baseline and three tighter filters select α = 1 α = 1 ; the baseline test error is 7.30 volatility points. Neither the scaling estimates nor the option test supports a universal identification of Caputo order with a rough-volatility Hurst exponent.