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 obviously
By 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 di = displacement during the ith interval, di = x(i) - x(i-1), i = 1,2,...,n, we have
as 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 (xmax - xmin) ≤ l during time interval t, we have
where 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
and
Given a set (l1, l2, ..., ln) of observed l values over n unit time intervals, we can write the extreme value estimate for D as follows:
Comparison of Dx and Dl Estimates for D: Computing the variance of Dx and Dl, we find that
and
where N is the number of observations. Thus, to obtain the same amount of variance using the two methods, we need Nx ≈ 5Nl. Clearly, the extreme value method is far superior to the traditional method and will be much more sensitive to variations in D.Mahalanobis - am 2006-12-03 23:26 - Rubrik: Finance
stxx meinte am 4. Dec, 03:27:
Comparison
I have recently compared the 30 day volatility estimate of close-close prices, Parkinson, Garman Klass (and exponentiall weighted Garman Klass), Rogers Satchell and Yang and Zhang. When testing them on the EURUSD and the S&P 500 one observes that the various estimators can vary substantially. Because volatility is unobservable (only its realization can be measured ex post) and not constant over time one would generally prefer an estimator which uses as few datapoints as possible, especially in trading. This however causes instability in the estimate.
The whole dilemma can be described as follows: The estimated volatility should be stable if an outlier occurs that causes a large price change. On the other hand it should include the large price movement if the price movement is based on increased volatility and immediately adjust the estimated volatility to the upside. The same problem occurs for "older" data points which should have only a limited influence on the estimate overall.
Other models employ REGARCH or high frequency data for accurate estimates. But a much bigger dataset is required for such estimation procedures.
Independent of the volatility estimate the question whether market implied volatility through options prices is too high or low or can be forecastable or is a good volatility forecast is another story. So, now having an efficient estimate for volatility based on high-low prices I still have to think of how to put the number at use ;)
The Unknown Professor (anonymous) meinte am 4. Dec, 04:30:
My first assignment as a Ph.D. student was to do some simulations in SAS for a couple of profs who were testing the Parkinson estimator versus a couple of alternate ones they'd devised. It wasn't my cup of tea, but it did get me up and running in SAS. So, I wrote it off as something I'd never see again, and I ended up working in empirical Corporate Finance. Thanks for the trip down memory lane.
nnyhav (anonymous) meinte am 5. Dec, 23:12:
You been reading Pynchon's Against the Day?
mike ward (anonymous) meinte am 7. Dec, 20:02:
why not treat this as a filtering problem, with volatility as the hidden state, and squared returns as the noisy measurment?ps. the hidden word i had to type to post this comment was "priing", which, modulo a vowel, is the name of a semi-famous author on technical analysis. is that an in-joke, michael? ;)
