We investigate the statistical evidence for the use of `rough' fractional processes with Hurst exponent $H< 0.5$ for the modeling of 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. We investigate the finite sample performance of our estimator for measuring the roughness of sample paths of stochastic processes using detailed numerical experiments based on sample paths of fractional Brownian motion and other fractional 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 $\hat{H}<0.5$. These results suggest that the origin of the roughness observed in realized volatility time-series lies in the microstructure noise rather than the volatility process itself.

翻译：摘要：我们采用无模型方法，系统研究了将赫斯特指数$H<0.5$的“粗糙”分数过程用于金融资产波动率建模的统计证据。基于规范化$p$次变差沿分割序列的概念，我们提出了一种非参数方法，用于根据离散样本估计函数的粗糙程度。通过基于分数布朗运动及其他分数过程样本路径的详细数值实验，我们研究了所提估计量在测量随机过程样本路径粗糙度方面的有限样本性能。随后，我们将该方法应用于基于高频观测数据估计已实现波动率信号的粗糙度。基于随机波动率模型的详细数值实验表明：即使瞬时波动率具有与布朗运动相同粗糙度的扩散动力学，其对应的已实现波动率仍表现出赫斯特指数显著小于$0.5$的粗糙行为。对不同赫斯特指数分数波动率模型中已实现波动率与瞬时波动率粗糙度估计值的比较显示：无论瞬时波动率过程的粗糙程度如何，已实现波动率始终呈现表观赫斯特指数$\hat{H}<0.5$的“粗糙”特征。这些结果表明，已实现波动率时间序列中观察到的粗糙性根源在于微观结构噪声，而非波动率过程本身。