Why Inverse Currency Pairs Have the Same Log-Return Volatility
Summary
The document explains why the volatility of pounds per euro matches the volatility of euros per pound when volatility is measured using log returns. Reversing the exchange-rate quote negates each log return, and negation changes its sign but leaves its variance and standard deviation unchanged. This means the question is not about the variance of the reciprocal price level.
It also distinguishes return volatility from the standard deviation of raw prices, which can be misleading because price levels and units affect the result. It notes that historical volatility can be estimated in several ways, including standard deviation of log returns and range-based estimators such as Parkinson, Garman-Klass, and Rogers-Satchell. The examples described illustrate that price series with similar price-level dispersion can have quite different return behavior. The conclusion applies to log-return volatility; it does not imply that reciprocal price levels have the same variance or that all volatility estimators produce identical results.
Key ideas
- Reversing a currency quote negates log returns but preserves their variance and standard deviation.
- The reciprocal price series can have a different variance from the original price series.
- Volatility comparisons should generally use returns rather than raw price levels.
- Historical volatility can be estimated with standard deviation of log returns or range-based methods.
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Full text
# If the volatility of pounds/euros = .2 do we know anything about the volatility of euros/pounds?
# If the volatility of pounds/euros = .2 do we know anything about the volatility of euros/pounds?
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?
## Answer by jeremy909 (score 18, accepted)
https://quant.stackexchange.com/a/65938
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.
## Answer by AKdemy (score 4)
https://quant.stackexchange.com/a/65943
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.