In a lot of my work, including my new book, I use two different ways of measuring standard deviation. The first method, which most people are familiar with, is to use some series of recent *percentage* returns. Given a series of prices p_t you might imagine the calculation would be something like this:

Sigma_% = f([p_t - p_t-1]/p_t-1, [p_t-1 - p_t-2]/pt-2, ....)

NOTE: I am not concerned with the form that function *f* takes in this post, but for the sake of argument let's see it's a simple moving average standard deviation. So we would take the last N of these terms, subtract the rolling mean from them, square them, take the average, and then take the square root.

For futures trading we have two options for p_t: the 'current' price of the contract, and the back adjusted price. These will only be the same in the days since the last roll. In fact, because the back adjusted price can go to zero or become negative, I strongly advocate using the 'current' price as the denominator in the above equation, and the changein back adjusted price as the numerator. If we used the change in current price, we'd see a pop upwards in volatility every time there was a futures roll. So if p*_t is the current price of a contract, then:

Sigma_% = f([p_t - p_t-1]/p*_t-1, [p_t-1 - p_t-2]/p*t-2, ....)

The alternative method, is to use some series of *price differences*:

Sigma_d = f([p_t - p_t-1], [p_t-1 - p_t-2], ....)

Here these are all

If I wanted to convert this standard deviation into terms comparable with the % standard deviation, then I would divide this by the current price (*not* the backadjusted price):

Sigma_d% = Sigma_d / p*_t

Now, clearly these are not going to give exactly the same answer, except in the tedious case where there has been no volatility (and perhaps a few, other, odd corner cases). This is illustrated nicely by the following little figure-ette (figure-ine? figure-let? figure-ling?):

`import pandas as pd`

perc =(px.diff()/pxc.shift(1)).rolling(30, min_periods=3).std()

diff = (px.diff()).rolling(30, min_periods=3).std().ffill()/pxc.ffill()

both = pd.concat([perc,diff], axis=1)

both.columns = ['%', 'diff']

The two series are tracking pretty closely, except in the extreme vol of late 2008, and even they aren't that different.

Here is another one:

That's WTi crude oil during COVID; and there is quite a big difference there. Incidentally, the difference could have been far worse. I was trading the December 2020 contract at the time... the front contract in this period (May 2020) went below zero for several days.**price difference standard deviation is far more important**.

Sigma_d% = Sigma_d / p*_t

**standard deviation in price difference terms**. We can eithier estimate this directly, or as the equation suggests recover it from the standard deviation in percentage terms, which we then multiply by the current futures price.

**I recommend using price differences to estimate the standard deviation**.

**Footnote:**Shout out to the diff(log price) people. How did those negative prices work out for you guys?

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