Realized range-based estimation of integrated variance

TL;DR

The paper develops realized range-based variance ($RRV$) as a nonparametric estimator of the integrated variance ($IV$) for continuous semimartingales, addressing the inconsistency of realized variance ($RV$) under microstructure noise. It proves consistency of $RRV$ and establishes a mixed normal central limit theorem with a substantial efficiency gain over $RV$, using both fully observed paths and discretized high-frequency data through the discretized estimator $RRV_{m}^{\Delta}$ and its feasible inference via $RRQ^{\Delta}$. Monte Carlo simulations and an empirical application to General Motors show that $RRV$ achieves narrower confidence bands and smoother path estimates than $RV$, supporting its practical advantage in high-frequency volatility estimation. The results highlight the potential of intraday high-low ranges to extract richer information about volatility and point to future work on jumps, microstructure noise, and multivariate extensions.

Abstract

We provide a set of probabilistic laws for estimating the quadratic variation of continuous semimartingales with realized range-based variance -- a statistic that replaces every squared return of realized variance with a normalized squared range. If the entire sample path of the process is available, and under a set of weak conditions, our statistic is consistent and has a mixed Gaussian limit, whose precision is five times greater than that of realized variance. In practice, of course, inference is drawn from discrete data and true ranges are unobserved, leading to downward bias. We solve this problem to get a consistent, mixed normal estimator, irrespective of non-trading effects. This estimator has varying degrees of efficiency over realized variance, depending on how many observations that are used to construct the high-low. The methodology is applied to TAQ data and compared with realized variance. Our findings suggest that the empirical path of quadratic variation is also estimated better with the realized range-based variance.

Figures (7)

• Figure 1: The distribution of the absolute return and range of a standard Brownian motion over an interval of unit length.
• Figure 2: $\lambda_{2, m}$ against $m$ on a log scale. All estimates are from a simulation with 1,000,000 repetitions and the dashed line represents the asymptotic value.
• Figure 3: We plot $\Lambda_{m}$, the variance factors appearing in the CLT of $RRV_{m}^{ \Delta}$, which are estimated from a simulation with 1,000,000 repetitions. The dashed line is the asymptotic value.
• Figure 4: Asymptotic normality for the standardized realized range-based statistic in estimating $IV$. The figure plots kernel densities of the sampling errors of $RRV_{m}^{ \Delta}$ for the small sample settings $n = 10$, $50$, $100$ and $m = 10$. All plots are based on a simulation with 1,000,000 repetitions from a log-normal diffusion for $\sigma$, as explained in the main text. The upper panel depicts t-statistics of the feasible CLT for $RRV_{m}^{ \Delta}$, while the lower panel is the corresponding log-based version. The solid line is the N(0,1) density.
• Figure 5: The time series of $RV^{ \Delta}$ and $RRV_{m}^{ \Delta}$ for GM are shown through the sample period January 3, 2000 to December 31, 2004; or 1,255 trading days in total. These are constructed from 5-minute midquote returns or ranges for data extracted from TAQ.

Theorems & Definitions (9)

• Theorem 1
• Definition 1
• Theorem 2
• Remark 1
• Corollary 1
• Remark 2
• Theorem 3
• Remark 3
• Lemma 1