Zero Return Volatility Depends on the Observed Return Series

Summary

The document distinguishes volatility in an asset’s price from volatility in its measured returns. Volatility, calculated as the standard deviation of observations, is zero when every observation in the chosen data series is identical. Thus, if the observations are returns, they must all be equal; they need not be zero.

The answer illustrates this distinction using returns calculated from closing prices. Identical consecutive closing prices produce zero measured returns, even if prices moved between those closes. In that case, the sampled return series has zero volatility while the asset’s price path may have varied. The result depends on what is sampled and how returns are defined, so zero volatility in one return series does not establish that the price was constant throughout the measurement intervals.

Key ideas

• A data series has zero standard deviation when all its observations are identical.
• A return series can have zero volatility when its returns are constant, including a nonzero constant return.
• Returns sampled from closing prices can be zero even when prices changed between closes.
• Volatility conclusions depend on the data series and sampling method being measured.

Tags

# Does the price of an asset need to be constant in order for its volatility to be zero?


# Does the price of an asset need to be constant in order for its volatility to be zero?












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.

## Answer by Andr (score 1)

https://quant.stackexchange.com/a/40522

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.