Model-Based Measurement of Actual Volatility in High-Frequency Data

In this chapter, we aim to measure the actual volatility within a model-based framework using high-frequency data. In the empirical finance literature, it is widely discussed that tick-by-tick prices are subject to market micro-structure effects such as bid-ask bounces and trade information. These market micro-structure effects become more and more apparent as prices or returns are sampled at smaller and smaller time intervals. An increasingly popular measure for the variability of spot prices on a particular day is realised volatility that is typically defined as the sum of squared intra-daily log-returns. Recent theoretical results have shown that realised volatility is a consistent estimator of actual volatility, but when it is subject to micro-structure noise and the sampling frequency increases, the estimator diverges. Parametric and nonparametric methods can be adopted to account for the micro-structure bias. Here, we measure actual volatility using a model that takes account of micro-structure noise together with intra-daily volatility patterns and stochastic volatility. The coefficients of this model are estimated by maximum likelihood methods that are based on importance sampling techniques. It is shown that such Monte Carlo techniques can be employed successfully for our purposes in a feasible way. As far as we know, this is a first attempt to model the basic components of the mean and variance of high-frequency prices simultaneously. An illustration is given for three months of tick-by-tick transaction prices of the IBM stock traded at the New York Stock Exchange.