Recently, Teresa asked for my opinion regarding range-based volatility estimators because, for years, traders have incorporated Wells Wilders' Average True Range into many facets of their analysis and systems.

While I know nothing whatsoever about ATR, I had to figure out something to keep her happy (guys, you know how it is...): The most simple range-based volatility estimator is based on the difference between the maximum and minimum prices observed during a certain period. Parkinson [1980] showed that the daily high-low range, properly scaled, is also an unbiased estimator of daily volatility -- but five times more efficient than the squared daily close-to-close return when the underlying process is a random walk.

Many other estimators that include high, low, open, and close values have been developed (see Garman and Klass [1980], Rogers and Satchell [1991], Alizahdeh, Brandt and Diebold [2001] and Yang and Zhang [2002]). I think one has to play around with all these estimators to see how they perform in the wilderness. What I liked about Parkinson's paper was that it was a clear and easy-to-follow exposition, so I reprinted (slightly edited) the most interesting part for those of you who have never heard of range-based volatility estimators before and do not have acess to JSTOR:

Suppose a share price undergoes a continuous random walk with a diffusion constant D. Then, the probability of finding the share price in the interval (x, x + dx) at time t, if it started at point x₀ at time t = 0, is obviouslyBy comparison with the normal distribution, we see that D is the variance of the displacement x - x₀ after a unit time interval. This suggests the traditional way to estimate D: we measure x(t) for t = 1,2,...,n. Then, defining d_i = displacement during the ith interval, d_i = x(i) - x(i-1), i = 1,2,...,n, we haveas an estimate for D. However, instead of measuring x(n), for n = 0,1,2,..., suppose we have measured only the difference l between the maximum and minimum position during each time interval. These differences should be capable of giving a good estimate for D, for it is intuitively clear that the average difference will get larger or smaller as D gets larger or smaller.

A derivation of the probability distribution for l has already been published (see Feller 1951). Defining P(l,t) to be the probability that (x_max - x_min) ≤ l during time interval t, we havewhere erfc(x) = 1 - erf(x) and erf(x) is the error function. We can now show straightforwardly that(p real and ≥ 1), where ζ(x) is the Riemann zeta function. In particular, we have
andGiven a set (l₁, l₂, ..., l_n) of observed l values over n unit time intervals, we can write the extreme value estimate for D as follows:Comparison of D_x and D_l Estimates for D: Computing the variance of D_x and D_l, we find that
andwhere N is the number of observations. Thus, to obtain the same amount of variance using the two methods, we need N_x ≈ 5N_l. Clearly, the extreme value method is far superior to the traditional method and will be much more sensitive to variations in D.