Volatility is a directionless measure for the dispersion of a variable within a certain time horizon. Although it seems quite obvious what a volatility measure should do, the estimation of volatility requires many decisions and is thus exposed to subjectivity. The following points out the ways in which volatility estimators can differ from each other.
The clearest difference is thegeneral method: One can broadly distinguish between non-parametric estimators based on historical prices, non-parametric estimators based on historical prices, and implied volatility estimators.8 The most widely used representatives of the first two methods are realized volatilities and GARCH model-based volatilities, respectively.
Since my later analysis focuses only on several forms of realized and GARCH model-based volatilities, I use these terms in the following instead of parametric and non-parametric.9 The main difference between realized and implied estimators is the information on which the estimator is based. Realized volatility uses only price information within a certain period to estimate the volatility of exactly this period. Unlike this, implied estimators extract market participants’ volatility expectations from the prices of currently traded options. These estimators thus include all price information, from the past—theoretically infinitely far back—to the beginning of the period for which volatility is to be estimated, that might be relevant for future price movements, as well as all non-price information that might affect future price movements, such as information about stock levels or weather.
Due to the different information with regard to content and to the temporal frame, the implied estimators are often called ex ante estimators because they use information up to a certain point in time to estimate the volatility after that point. The realized estimators are primarily ex post estimators because they can only estimate the volatility of a period
8In addition, among implied volatilities, one can distinguish between parametric and non-parametric estimators. I address this point later, in the description of the estimators.
9See Andersen, Bollerslev, and Diebold (2009) for other forms of parametric and non-parametric estimators based on historical data.
at the end of that period. This shows that the methods mainly serve different purposes. If one wants to forecast volatility in future periods, the implied methods directly provide an estimator. On the contrary, if one wants to use historical realized volatilities to create a
“realized forecast”, one needs a model based on assumptions on how historical volatility behavior will be transferred to future periods. However, to analyze volatility in retrospect, realized volatility has the major advantage of using data within the period of interest, which is not incorporated by the implied methods. GARCH volatility is more difficult to classify because the information used in the volatility estimation model depends on the specific method applied. Similar to realized volatility, GARCH volatility is based only on price data and not on other market-relevant information.10 Unlike realized and implied volatility estimators, GARCH model-based estimators allow for more possibilities regarding the period used for estimation. GARCH volatility for a certain period could contain price information up to that period, ending afterward or much later. Again, different objectives are satisfied. If data up to that period are used, the GARCH model fit to the data allows forecasting the volatility of the next period. If the data of the relevant period are included, it allows for an ex post estimation of volatility. Like the end of the data period, its beginning is also flexible in this approach. One can use either the full available data period for all volatility estimations, a rolling window, or a window that has a fixed beginning and end after the period for which the volatility is to be estimated. After deciding on a general concept of volatility estimation, more issues—partially depending on the general concept—must be taken into account, as discussed below.
The time horizon is the period over which the volatility is estimated. Which horizon is relevant depends on the purpose of the analysis. In the food price volatility literature, typical time horizons are a week or a month. For realized estimators, the time horizon directly defines the length of the period from which data are used. For GARCH model-based estimators, the volatility of a time horizon can be estimated either directly or model-based on volatilities estimated for smaller horizons. The precise application depends on the data frequency, discussed in the next paragraph. Implied estimators are based on currently
10An exception is the GARCH-X models, which are an extension of the standard GARCH model and allow for additional variables in the variance equation (see, e.g., the model of Brenner, Harjes, and Kroner (1996)).
traded options and reflect the expected average volatility from that time until the maturity of the option. Therefore, time to maturity must be equivalent to the time horizon to extract the appropriate implied information. Volatilities are often annualized with the square root of time rule; for example, the monthly volatility is multiplied by √
12. This makes volatilities more comparable but these must not be confused with the time horizon on which they are based. Although the square root of time rule is often applied, it is only appropriate if returns are independent and identically distributed (Diebold, Hickman, Inoue, and Schuermann (1997)). Several studies indicate that this assumption is invalid for the stock market and also for commodity futures.11
Depending on the estimation method, thedata frequency must be chosen. This is especially important for realized volatility, since the frequency must be higher than the time horizon.
Often daily price data are used if the time horizon is a week or a month. If the horizon is longer, for example, one year, data at a weekly or monthly frequency could be used instead of daily data. Lower frequencies are often necessary due to data limitations. In addition, different data frequencies are possible for GARCH models. The frequency does not necessarily have to be higher than the horizon but could also be equal. The frequency does not matter for implied volatilities because those volatilities only use the information at one point in time.
A more general question is how volatility is exactlydefined. Almost all papers dealing with the volatility of returns define it as the standard deviation of relative price changes (log-returns). While this is a well-known statistical measure, the experiment of Goldstein and Taleb (2007) shows that even financial professionals misinterpret the standard deviation as the average absolute deviation from the mean. Although the average absolute deviation is hardly considered in the literature for describing volatility in financial or commodity markets, it nevertheless seems to be a moreintuitive measure of volatility. While standard GARCH models are only designed to estimate the standard deviation, realized and implied volatility estimation methods can easily contribute to both definitions.
11See, for example, Lo and MacKinlay (1988) for the stock market and Gordon (1985) for several agricultural futures markets.