Nonparametric estimation of quadratic variation using high‐frequency data

Abstract

Estimating the quadratic variation (QV) using high‐frequency financial data is studied in this article, and this work makes two major contributions: first, the fundamental Itô isometry is generalized to some extent, and as an application example, the widely known convergence property of realized volatility (RV) estimator of QV is analyzed by alternatively utilizing this generalized Itô isometry; second, we intuitively establish two types of new estimators of QV which permit volatility varying with time. To construct such estimators, the RV combined with realized bipower variation and RV with realized quarticity are employed, respectively. By applying the generalized Itô isometry, we further show that each of the proposed estimators can converge to QV, at a rate of O(n−1/2), almost surely and in mean square sense, both of which are stronger than the existing convergence in probability. Moreover, the error of approximation is also provided for each estimation. In addition, the obtained convergence property for both types of estimators is demonstrated by empirical investigations based on high‐frequency data of IBM stock.