We consider the problem of estimating the roughness of the volatility process in a stochastic volatility model that arises as a nonlinear function of fractional Brownian motion with drift. To this end, we introduce a new estimator that measures the so-called roughness exponent of a continuous trajectory, based on discrete observations of its antiderivative. The estimator has a very simple form and can be computed with great efficiency on large data sets. It is not derived from distributional assumptions but from strictly pathwise considerations. We provide conditions on the underlying trajectory under which our estimator converges in a strictly pathwise sense. Then we verify that these conditions are satisfied by almost every sample path of fractional Brownian motion (with drift). As a consequence, we obtain strong consistency theorems in the context of a large class of rough volatility models, such as the rough fractional volatility model and the rough Bergomi model. We also demonstrate that our estimator is robust with respect to proxy errors between the integrated and realized variance, and that it can be applied to estimate the roughness exponent directly from the price trajectory. Numerical simulations show that our estimation procedure performs well after passing to a scale-invariant modification of our estimator.

翻译：我们研究了在随机波动率模型中估计波动率过程粗糙度的问题，该模型源于带漂移项的分形布朗运动的非线性函数。为此，我们提出了一种新的估计量，用于度量连续轨迹的所谓粗糙指数，该估计量基于其原函数的离散观测值。该估计量形式简洁，能够在大规模数据集上高效计算。其推导不依赖于分布假设，而完全基于路径层面的考量。我们给出了底层轨迹所需满足的条件，在此条件下我们的估计量在严格的路径意义上收敛。随后我们验证了这些条件几乎被所有带漂移项的分形布朗运动样本路径所满足。因此，我们在包括粗糙分形波动率模型与粗糙Bergomi模型在内的一大类粗糙波动率模型中获得了强一致性定理。我们还证明了该估计量对积分方差与实际方差之间的代理误差具有鲁棒性，并且可直接应用于价格轨迹的粗糙指数估计。数值模拟表明，在采用我们估计量的尺度不变修正形式后，该估计程序表现出良好的性能。