When is Volatility Fair? Hölder Regularity and Financial Risk

Battle Count: 69

The paper provides a novel theoretical framework for understanding volatility as a function of market regularity rather than mere dispersion. The 'fair volatility' concept offers a benchmark for identifying when markets are in momentum (H_t > 1/2) or mean-reversion (H_t < 1/2) regimes, which is directly relevant to regime-switching trading strategies. However, the paper explicitly notes that deviations from H_t=1/2 do not imply exploitable arbitrage due to transaction costs, liquidity constraints, and the distinction between statistical and economic predictability. The framework is more suited for risk management and market state classification than for direct alpha generation. The MPRE-based volatility outperforms GARCH(1,1) and HAR-RV in estimation accuracy, suggesting practical utility for volatility forecasting in trading systems.

Implementation requires: (1) understanding of fractional calculus and Hölder regularity theory, (2) implementing the MPRE simulation (available via FracLab Toolbox), (3) coding the rolling-window Hurst exponent estimator with bias correction (Figure 3 pipeline), (4) performing Monte Carlo validation with 500+ replications, (5) computing the analytical constant A_H involving Gamma functions and trigonometric terms, (6) inverting the sd-H relationship to extract ν_t, and (7) constructing fair volatility confidence intervals. The mathematical sophistication is high, but the computational steps are well-defined. The main challenge is the estimator implementation and ensuring numerical stability of the Hölder exponent estimation.

The paper uses the FracLab Toolbox 2.2 from INRIA (MATLAB) for MPRE simulation and references the estimator from Pianese, Bianchi, and Palazzo (2018). However, no dedicated code repository is provided. The estimator pipeline is described in detail (Figure 3), and Monte Carlo parameters are specified (500 replications, δ=20 trading days). The 14 equity indices are standard Bloomberg tickers. Reproduction would require implementing the Hölder exponent estimator and MPRE simulation framework.