From fair price to fair volatility: Towards an Efficiency-Consistent Definition of Financial Risk

Volatility, as a primary indicator of financial risk, forms the foundation of classical frameworks such as Markowitz's Portfolio Theory and the Efficient Market Hypothesis (EMH). However, its conventional use rests on assumptions-most notably, the Markovian nature of price dynamics-that often fail to reflect key empirical characteristics of financial markets. Fractional stochastic volatility models expose these limitations by demonstrating that volatility alone is insufficient to capture the full structure of return dispersion. In this context, we propose pointwise regularity, measured via the Hurst-Holder exponent, as a complementary metric of financial risk. This measure quantifies local deviations from martingale behavior, enabling a more nuanced assessment of market inefficiencies and the mechanisms by which equilibrium is restored. By accounting not only for the magnitude but also for the nature of randomness, this framework bridges the conceptual divide between efficient market theory and behavioral finance.