Measuring Integrated Variance with Extreme-Value Based Estimators

Abstract

Driven by the rise in computational power, it has become popular to measure integrated variance with high-frequency squared returns. Though the squared return is a natural choice as a variance estimate, it is not the most efficient one for a given interval length. Extreme-value based estimators incorporate the information of the first, high, low and last price observation of a price path interval and are several times more efficient than the squared return estimate. Though these estimators have mainly been applied to daily data, they are defined over an arbitrary interval length, hence it is straightforward to apply these estimators to high-frequency data to get a non-parametric integrated variance estimate similar to realized variance. This study compares several extreme-value based estimators and the realized variance estimator of integrated variance within a high-frequency framework in terms of bias and efficiency with an extensive Monte Carlo study for various assumptions about the underlying price process including large drifts, stochastic volatility with leverage effects and also an orthogonal market microstructure noise. The simulation results strongly suggest the use of extreme-value based estimators over the common practice of summed high-frequency squared returns to measure the integrated variance. All extreme-value based estimators are superior both interms of bias and efficiency for any given sampling frequency for all considered price process definitions. The efficiency of the estimators is confirmed empirically on the basis of 10 years of S&P 500 futures data sampled at a 5-minute frequency. A state-space framework reveals the latent stochastic volatility process and allows for a comparison of the error of the considered estimators.