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

Abstract

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 κα=t01-α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa _{\alpha }=t_0^{1-\alpha }$$\end{document}, and prices are reported across four values of t0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t_0$$\end{document}. 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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha = 1$$\end{document}; 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.