Fair Volatility in the Fractional Stochastic Regularity Model

Within the efficient markets framework, discounted stock prices are typically represented through Brownian martingales. The primary measure for evaluating risk is the volatility of log-returns, under the assumption that higher variability indicates greater associated risk. The theoretical foundation of this claim stems from the characterization of the path regularity of price process through the Lévy characterization theorem of Brownian motion. Since this explanation lacks a financial interpretation when considering more realistic models, such as stochastic volatility models, it is necessary to disentangle volatility and regularity. Replacing volatility by the Hölder regularity provides insights into market deviations from the equilibrium of the martingale model, and - within the Fractional Stochastic Regularity Model - contributes to identify the “fair” volatility aimed by the market.