Extreme Distribution of Realized and Range-Based Risk Measures

Abstract

The study of market volatility has been renewed by the introduction of high-frequency estimators. The popularity of the so-called realized volatility as a proxy of latent, unobservable volatility has been facilitated by the increased access to transaction data. Yet the recent studies do not totally agree on the empirical properties of that variable, especially its exact distribution.

While Andersen et al (2001) or Cizeau et al (2001) fit a log-normal to the volatility series, Liu et al (1999) find that the distribution is right-side asymmetric and heavy tailed and Thomakos and Wang (2003) report a low power of moment-based normality tests. In the current study, we apply several measures of risk on high-frequency data and test the goodness-of-it of possible candidate distributions for the volatility. Of particular interest is the estimation of the right-hand tail of the distribution, since it represents the periods of crisis, when the market is at its most volatile.

In order to include important shocks and crises, it is best to have a sample spanning a long period, where high-frequency data is not available. Thus we extend the study by using range-based estimators (Cf. Brandt and Diebold, 2003, and among others, Li and Weinbaum, 2000, Lebaron, 2001, Corrado and Miller, 2002, Alizadeh et al, 2002, Tims and Mahieu, 2003) and comparing them, whenever possible with high-frequency estimators.

Combining different risk measures (from classical variance to squared maximum drawdown) and distributional assumptions (from log-normal to inverse Gamma), we retrieve the likelihood of benchmarks market events such as the successive crises of the last 10 years (with high frequency data) and the famous historical crises from 1929 to 1987 (with daily data). We use extreme value theory to characterize the slope of the right hand tail of the volatility distribution and get characteristic return times for the market shocks. The result allows us to rank historical crises and to check whether the worst of them can fit in the general process or are in fact outliers as suggested for instance by Sornette et al (2003).

Keywords: Financial Crisis, Realized Volatility, Extreme Values, Range-based Volatility Estimators, High Frequency Data

JEL Classification: G10, G14