The Log S-fBM model: Statistical analysis

Battle Count: 67

The paper is highly relevant to quantitative trading as it provides a unified framework for volatility modeling that bridges rough volatility (empirically observed with H≈0.1 for indices, H≈0.01 for individual stocks) and multifractal volatility. The hypothesis testing procedure directly addresses the critical question of which volatility regime applies to a given asset, which has direct implications for derivative pricing, risk management, and trading strategy design. The deviation inequalities provide rigorous bounds on extreme log-volatility fluctuations, essential for tail risk management. The scale invariance properties are fundamental for multi-timeframe trading strategies. However, the paper is primarily theoretical/statistical rather than directly implementing trading algorithms.

The mathematical framework is highly sophisticated, involving: (1) geometric construction via Takenaka time-scale domains (truncated cones in (t,h)-plane); (2) integration of non-homogeneous Gaussian white noise over these domains; (3) Gaussian multiplicative chaos theory; (4) Borel-TIS inequality, Sudakov minoration, and dyadic chaining for deviation bounds; (5) small intermittency perturbation expansions; (6) Monte Carlo simulation of the S-fBM process requiring careful discretization of the time-scale integral. The simulation of the S-fBM process itself is non-trivial due to the 2D integration over the cone domain. The hypothesis test implementation requires computing Vn(H) in closed form and estimating λ separately. The scale invariance formulas involve Gamma functions and power-law corrections.

The paper provides complete mathematical derivations with proofs in appendices, explicit formulas for all key quantities (covariance, variance Vn(H), test statistic, scale invariance formulas), and detailed Monte Carlo simulation parameters (n=5, Δ=1, L=2^12, T=2^12, λ²=0.02, various H values). However, no code repository or software implementation is provided. The theoretical framework is fully self-contained, but reproducing the numerical experiments would require implementing the S-fBM simulation algorithm from scratch. The geometric construction via the Takenaka time-scale domain and the Gaussian white noise integration are well-specified mathematically.