We present a comprehensive theory of homogeneous volatility (and variance) estimators of arbitrary stochastic processes that fully exploit the OHLC (open, high, low, close) prices. For this, we develop the theory of most efficient point-wise homogeneous OHLC volatility estimators, valid for any price processes. We introduce the "quasi-unbiased estimators", that can address any type of desirable constraints. The main tool of our theory is the parsimonious encoding of all the information contained in the OHLC prices for a given time interval in the form of the joint distributions of the high-minus-open, low-minus-open and close-minus-open values, whose analytical expression is derived exactly for Wiener processes with drift. The distributions can be calculated to yield the most efficient estimators associated with any statistical properties of the underlying log-price stochastic process. Applied to Wiener processes for log-prices with drift, we provide explicit analytical expressions for the most efficient point-wise volatility and variance estimators, based on the analytical expression of the joint distribution of the high-minus-open, low-minus-open and close-minus-open values. The efficiency of the new proposed estimators is favorably compared with that of the Garman-Klass, Roger-Satchell and maximum likelihood estimators.
Volatility of volatility estimation: central limit theorems for the Fourier transform estimator and empirical study of the daily time series stylized facts
Asymptotic Normality for the Fourier spot volatility estimator in the presence of microstructure noise
本文的主要贡献是证明了在[Malliavin and Mancino，2002]中引入的Fourier现货波动率估计在价格过程被微观结构噪声污染的情况下是一致的和渐近有效的。具体来说，在存在添加剂的微观结构噪声，我们证明了一个中心极限定理的最佳收敛速度$n^{1/8}$。该结果是在不需要对原始数据进行任何处理或偏差校正的情况下获得的。此外，我们通过推导具有最佳收敛速度$n^{1/4}$的中心极限定理，完成了最初在[Mancino and Recchioni，2015]中提出的无噪声条件下傅立叶现货波动率估计量的渐近理论。最后，我们提出了一种新的可行的自适应方法的最佳选择的参数的傅立叶现货波动率估计器与噪声的高频数据的实施，并提供支持，其准确性的数值和经验。
Efficient asymptotic variance reduction when estimating volatility in high frequency data
本文展示了在存在随机波动率和微观结构噪声的情况下，如何使用来自[Barndorff-Nielsen et al.，2008]和[Xiu，2010]中研究的拟最大似然估计（QMLE）。为了获得这样的减少，我们将数据分割成B个块，计算每个块的RK（或QMLE），并聚合块估计。当B增加到问题的参数版本中的比率时，渐近方差与渐近效率的界限的比率收敛，即在最快的RK Tukey-Hanning 16和QMLE的情况下为1.0025。仔细研究了随机抽样时间和价格过程中的跳跃的影响。两种估计的有限样本性能进行了研究，在模拟中，而实证工作说明了在实践中的增益。
