When is volatility fair? Hölder regularity and financial risk

Positing that the real nature of financial risk is not variability itself, but rather irregularity – i.e. unpredictability – of price dynamics, we introduce a framework for measuring financial risk based on the local regularity of log -returns, captured by the time-varying Hurst-Hölder exponent. The paradigm shift resulting from disentangling variability and irregularity naturally leads to define the notion of fair volatility, i.e. the volatility level consistent with efficient market and martingale dynamics. In this view, fair volatility describes the maximum level of financial risk, which does not coincide with the higher volatility. Within the very large class of Multifractional Processes with Random Exponent (MPRE), we establish an analytical relation between regularity and scale of price increments. Applying existing estimators, we compare MPRE-implied volatility with realized volatility across fourteen international equity indices. The results reveal coherent volatility patterns across markets and time, and highlight phases in which temporary inefficiencies occur that are corrected by types of opposite behavioral schemes.