I think the question here is what we know about $\mathrm{Var}\left(\frac1X\right)$. Is this the right question to ask, and if so is there anything that can be said?
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2$\begingroup$ Related questions quant.stackexchange.com/questions/39737/… quant.stackexchange.com/questions/10993/fx-rate-dynamics $\endgroup$– nbbo2Jul 9, 2021 at 4:31
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2 Answers
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The trick here is that you're not asking about $\mathrm{Var}\left(\frac1X\right)$.
Imagine one currency is a stock $S$ and the other a stock $Q$. Then the volatility of the exchange is the square root of: $$\mathrm{Var}\left( \ln\left(\frac SQ\right) \right) = \mathrm{Var}\left( -\ln\left(\frac QS\right) \right) = \mathrm{Var}\left( \ln\left(\frac QS\right) \right)$$
So the volatility of pounds/euro is the same as the volatility of euros/pound.
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Some additional comments to @Jeremy909. That "problem" is one reason why logs are so useful. See Reason 2: The log difference is independent of the direction of change.
However, I think it is not clear from your question if you think about volatility of the price series itself or volatility of returns. There are a bunch of ways historical volatilities can be estimated. The most common is the (annualized) standard deviation (SD) of log returns. Other methods are for example Parkinson, Garman-Klass and Rogers-Satchell.
Using prices is very misleading as the same SD in USD (or any currency) has a vastly different meaning depending on the price level. Below is some simulated data (in Julia) which has the same SD but different mean (Avg). I guess it is clear which one fluctuates more in terms of returns.
Variance and SD are indeed almost identical for both series (simulated that way).
Normalizing the two series to be equal at the start reveals how little the second one moves relative to the first one.
This is also visible in terms of returns of the two time series.
If we compute the inverse of stock A and compute SD of log returns we see how the two match (and that flipping the values in the log differences only changes the sign, but not the value itself as @jeremy909 showed).
This quora answer illustrates another interesting issue with price data, namely that the order of occurrence does not matter. The screenshot is from the link and both series have a SD of ~USD 6.2.
Something that can be quickly checked with this series as well.
SD of log returns is vastly different as the chart suggests.
