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What are the conditions for the volatility of an asset to be zero?

In my opinion, the only condition is that the return on the asset needs to be constant.

On the web, some people imply that the price of the asset needs to be constant, meaning the return is not just constant but zero. I don't agree with this but wanted someone else to confirm my thinking.

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If you take a look at the formula of the volatility:

$\sigma = \sqrt{ \frac{1}{N-1} \sum_{i=1}^{N} ( x_i -\bar x )^2}$

Then you will realize that in order for the volatility of your data set to be equal to zero, all of its values must be identical (since $ x_i -\bar x = 0 $ for every $i$).

Now, if you are measuring the volatility of the return of an asset, then it simply depends on what data you are measuring. For instance, if your data consists of the returns calculated using the closing prices of a stock, then indeed, the return can be zero (if all closing prices are equal), whereas the price of the asset can vary inbetween. Thus, in this case, the price of the asset can have a volatility $\neq 0$ and while the return (of closing prices) is $=0$.

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  • $\begingroup$ As a further comment to support this answer, if a price was forecast to rise by x per day for the next T days, and every day that expectation was realised the asset could be argued to have zero volatility (deviation from expectation is zero everyday), even though its absolute price level is rising. The return could be zero relative to a positive cost of carry, for instance or indeed offer the same expected return as was initially forecast. $\endgroup$ – Attack68 Jun 27 '18 at 18:23

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