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Vocabulary question

Hi David,

I need to apologize in advance for a somewhat confusing question.

I realize that volatility (according to GARCH, EWMA, SMA) is the quare root of the weighted average of the squared realized returns over a certain time period. Does the term "realzied volatility" mean anything? In other words, if the volatility forecast for today was .0005 and the return was .02 (squared would be .000004), does .000004 actually represent anything in terms of a "realized volatility"? Obviously, it would be used in one of the above methods for finding the updated volatility, but, besides that, does the quantity have any real meaning?

Thanks,
Mike
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
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Hi Mike,

I think it is profound question; i.e., I might not be up to the task!. My swipe at at:

* On the surface, "realized" ~ "historical" and we typically contrast realized/historical volatility with implied (forward-looking) volatility. In usage, i think often it is used to simply distinguish from implied; where both are (somewhat arbitrarily) getting a current volatility and the difference is the model; realized = f[historical model], compare to implied = f[analytical (BSM) model]
* It gives me a chance to copy/paste my favorite volatility quote, by Carol Alexander, see below
* I suppose you could argue that, in your example, the most recent square-return ("innovation") is itself a one-day variance,
* But i prefer to think i'm following Carol to say that: as there is no really no such that as instantaneous volatility (specifically, what i mean is, there is a current stock price which can be observed, but there is not really a current volatility in the same way), neither is the new squared-return any kind of variance unless we are re-defining terms, it is just a new piece of information for the volatility model. So, i would say, no it has no real meaning
* I think it's involved with this tricky point that, realized/historical volatility requires a time frame, which is pretty near to saying it needs a series ... which clearly makes it very different than a variable like market price.

I hope that's a start! David

From MRA Vol II (emphasis mine):
"Volatility is unobservable . We can only ever estimate and forecast volatility, and this only within the context of an assumed statistical model. So there is no absolute ‘true’ volatility: what is ‘true’ depends only on the assumed model. Even if we knew for certain that our model was a correct representation of the data generation process we could never measure the true volatility exactly because pure volatility is not traded in the market. Estimating volatility according to the formulae given by a model gives an estimate of volatility that is ‘realized’ by the process assumed in our model. But this realized volatility is still only ever an estimate of whatever volatility had been during the period used for the estimate.

Moreover, volatility is only a sufficient statistic for the dispersion of the returns distribution when we make a normality assumption. In other words, volatility does not provide a full description of the risks that are taken by the investment unless we assume the investment returns are normally distributed. In general, we need to know more about the distribution of returns than its expected return and its volatility. Volatility tells us the scale and the mean tells us the location, but the dispersion also depends on the shape of the distribution. The best dispersion metric would be based on the entire distribution function of returns."

Source: Carol Alexander's MRA, Volume II: http://www.amazon.com/dp/0470998016/?tag=bionturt-20
 
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