Measuring volatility with the realized range
Introduction
Measuring and forecasting volatility of financial asset returns is important for portfolio management, risk management and option pricing. By now it is well established that volatility is both time-varying and, to a certain extent, predictable. An important issue is how to measure ex-post volatility, which is necessary for proper evaluation of competing volatility forecasts, among other purposes. Recently, much research has been devoted to the use of high-frequency data for measuring volatility. In particular, the sum of squared intra-day returns, called realized variance, is rapidly gaining popularity for estimating daily volatility. In theory, the realized variance is an unbiased and highly efficient estimator, as illustrated in Andersen et al. (2001b), and converges to the true underlying integrated variance when the length of the intra-day intervals goes to zero, see Barndorff-Nielsen and Shephard (2002). In practice, market microstructure effects such as bid–ask bounce pose limitations to the choice of sampling frequency. Returns at very high frequencies are distorted such that the realized variance becomes biased and inconsistent, see Bandi and Russell, 2005, Bandi and Russell, 2006, Aı¨t-Sahalia et al. (2005) and Hansen and Lunde (2006b). Popular choices in empirical applications are the 5- and 30-min intervals, which are believed to strike a balance between the increasing accuracy of higher frequencies and the adverse effects of market microstructure frictions, see e.g. Andersen and Bollerslev (1998), Andersen et al., 2001a, Andersen et al., 2003, and Fleming et al. (2003).
An alternative way of measuring volatility is based on the difference between the maximum and minimum prices observed during a certain period. Parkinson (1980) shows that the daily (log) high–low range, properly scaled, not only is an unbiased estimator of daily volatility but is five times more efficient than the squared daily close-to-close return. Correspondingly, Andersen and Bollerslev (1998) and Brandt and Diebold (2006) find that the efficiency of the daily high–low range is between that of the realized variance computed using 3- and 6-h returns.
This paper starts from the crucial observation that Parkinson's result concerning the relative efficiency of the high–low range applies to any interval, in particular also to the intra-day intervals employed by the realized variance. That is, in theory, for each intra-day interval the high–low range is a more efficient volatility estimator than the squared return over that interval. We therefore suggest to measure daily volatility by the sum of high–low ranges for intra-day intervals. The resulting estimator, which we dub ‘realized range’, should be more efficient than the realized variance based on the same sampling frequency. Indeed, in concurrent independent work, Christensen and Podolskij (2005) derive the theoretical properties of the realized range, similar to Barndorff-Nielsen and Shephard (2002) for realized variance. In an ideal world (continuous trading, no market frictions) the realized range is five times more efficient than the corresponding realized variance, and converges to the integrated variance at the same rate. At the same time, in such an ideal world there seems to be no need to consider the realized range, as the true daily volatility can be approximated arbitrarily closely by the realized variance using higher and higher frequencies. However, it is often claimed that the daily range is more robust against the effects of market microstructure noise than the realized variance, see Alizadeh et al. (2002) and Brandt and Diebold (2006), among others. Obviously, the realized range will be affected more heavily by microstructure noise as each of the intra-day ranges is contaminated. Nevertheless, in realistic settings it is an open question as to whether the realized variance or the realized range renders a superior measure of daily volatility. In this paper we attempt to shed light on this question.
Our approach is based upon Monte Carlo simulation and an empirical analysis for S&P500 index-futures and the individual stocks in the S&P100 index. The simulation experiments reveal that both realized range and realized variance are upward biased in the presence of bid–ask bounce. We find that in fact the realized range is affected more than the realized variance at the same sampling frequency. Infrequent trading induces a downward bias in the realized range, while it does not affect the realized variance. In case the price path is not observed continuously the observed minimum and maximum price over- and underestimate the true minimum and maximum, respectively, such that the observed range underestimates the true range.
We consider a bias-adjustment procedure for the realized range estimator, which involves scaling the realized range with the ratio of the average level of the daily range and the average level of the realized range. This is based upon the idea that the daily range is (almost) not contaminated by microstructure noise and thus provides a good indication of the true level of volatility. In the simulation experiments, we find that the scaled realized range is more efficient than the (scaled) realized variance estimator based on returns sampled at considerably higher frequencies. Comparing the scaled realized range with popular corrections of the realized variance for microstructure noise, we find that it outperforms kernel-based estimators, as considered by Barndorff-Nielsen et al. (2004) and Hansen and Lunde, 2005, Hansen and Lunde, 2006b, and is competitive to the two time-scales (TTS) estimator of Zhang et al. (2005).
The empirical analysis confirms that for both measuring and forecasting volatility the (scaled) realized range can compete with and often improves upon realized variance estimators at popular sampling frequencies, including the versions that correct for microstructure noise. This is slightly more so for more actively traded stocks.
The remainder of this paper is organized as follows. Section 2 defines the realized range, including the suggested bias-adjustment procedure to counter the adverse effects of market microstructure noise. Section 3 describes the design and results of the simulation experiments that illustrate the properties of the realized range in the presence of market microstructure frictions. Section 4 presents the empirical results for the S&P500 index-futures, both concerning basic properties of the realized range and a comparison between realized range and realized variance. Section 5 extends the comparison to the constituents of the S&P100 index. Finally, Section 6 concludes.
Section snippets
The realized range estimator
Let the security price at time t follow the geometric Brownian motionwhere denotes the drift term, is the constant volatility parameter and is a standard Brownian motion. By Ito's lemma, the log price process follows a Brownian motion with drift and volatility . For ease of notation we normalize the daily interval to unity. Then for the ith interval of length on day t, for with assumed to be integer, we observe the last price in that
Simulation
We simulate prices for 24-h days, assuming that trading can occur round the clock. For each day t, the initial price is set equal to 1, and subsequent log prices are simulated usingwhere J is the number of prices per day. To approximate the ideal situation with continuous trading and no market frictions we simulate 100 prices per second, such that as there are 86,400 s in a 24-h day. The shocks are independent and normally distributed
S&P500 index futures
The S&P500 index-futures contract is the largest equity futures contract in the world. It is trading virtually round the clock, with floor trading on the Chicago Mercantile Exchange (CME) from 9:30 to 16:15 Eastern Standard Time (EST), and electronic trading on GLOBEX almost 24 h a day apart from Friday evening to Sunday evening. Our sample contains transaction prices and bid and ask quotes running from 4 January 1999 to 23 February 2004. The S&P500 futures contract has maturities in March,
S&P100 constituents
Obviously, the analysis in the previous section only considers 1 particular asset, the S&P500 index-futures, which is an extremely liquid asset with a small bid–ask spread. With the main issue at stake being whether the realized range better deals with market microstructure than the realized variance this could provide a biased picture. For that reason we also consider the individual stocks in the S&P100 index (constituents in June 2004), where the data consist of open, high, low and close
Concluding remarks
In this paper we studied the properties and merits of the ‘realized range’, a measure of daily volatility computed by summing high–low ranges for intra-day intervals. In theory the high–low range is a more efficient estimator of volatility than the squared return. Hence, just like the daily high–low range improves over the daily squared return, the ‘realized range’ should improve over the ‘realized variance’ obtained by summing squared returns for intra-day intervals. Theoretically that is. In
Acknowledgements
We thank Peter Hansen, participants at the International Conference on Finance in Copenhagen (September 2–4, 2005), the special issue guest editors Herman van Dijk and Philip Hans Franses, and two anonymous referees for useful comments and suggestions. Any remaining errors are ours.
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