We investigate the statistical evidence for the use of ‘rough’ fractional processes with Hurst exponent H < 0.5 for modeling the volatility of financial
assets, using a model-free approach. We introduce a non-parametric method
for estimating the roughness of a function based on discrete sample, using the
concept of normalized p-th variation along a sequence of partitions. Detailed
numerical experiments based on sample paths of fractional Brownian motion
and other fractional processes reveal good finite sample performance of our
estimator for measuring the roughness of sample paths of stochastic processes. We then apply this method to estimate the roughness of realized
volatility signals based on high-frequency observations. Detailed numerical
experiments based on stochastic volatility models show that, even when the
instantaneous volatility has diffusive dynamics with the same roughness as
Brownian motion, the realized volatility exhibits rough behaviour corresponding to a Hurst exponent significantly smaller than 0.5. Comparison
of roughness estimates for realized and instantaneous volatility in fractional
volatility models with different values of Hurst exponent shows that, irrespective of the roughness of the spot volatility process, realized volatility
always exhibits ‘rough’ behaviour with an apparent Hurst index H < 0.5.
These results suggest that the origin of the roughness observed in realized
volatility time series lies in the estimation error rather than the volatility
process itself.