FX Cross Volatility Depends on Exchange Rate Weights and Correlation
Summary
The document explains why an FX cross rate's volatility is not generally the simple result a trader might expect from the volatilities of its two component pairs. It frames the cross as exposure to a portfolio of EUR/USD and USD/GBP positions. With zero correlation, an equally weighted combination has variance equal to the sum of the component variances, so its volatility can exceed either component's volatility.
The key correction is that the actual exposures needed to construct EUR/GBP depend on exchange rates. The accepted answer illustrates the point by deriving weights from sample spot rates and reports a cross volatility below the equally weighted portfolio example. A separate reply gives the familiar variance formula with a correlation term. The numerical illustration is tied to quoted rates and assumptions, including fair triangular pricing; it is not a universal volatility estimate. The broader lesson is to account for position units or portfolio weights, as well as covariance, when translating component FX volatility into cross-rate volatility.
Key ideas
- An FX cross can be represented as positions in its two component currency pairs.
- Cross-rate volatility depends on the exchange-rate-based weights assigned to those positions.
- With zero correlation, the variance of an equally weighted combination is the sum of the component variances.
- The covariance term must be included when component returns are correlated.
- The example relies on its stated exchange rates and triangular pricing assumption.
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Full text
# Answer by KevinT (score 2, accepted) # If I have 2 uncorrelated currencies, why is the volatility of their product higher than either of the volatilities? (better explanation inside) Let's say we have 3 currencies: - EUR/USD - USD/GBP - EUR/GBP For a minute let's assume that we calculated the EUR/USD-USD/GBP correlation for the last N days and the result was 0.0 (I know this is not realistic but please bear with me). Other assumptions: - EUR/USD volatility = 0.1 - USD/GBP volatility = 0.06 - GBP/USD was fairly priced during the whole period (as in EUR/USD*USD/GBP = EUR/GBP) We want to calculate what has been the volatility of EUR/GBP without knowing the prices (of EUR/GBP, but we know the prices of EUR/USD and USD/BP). Using the triangulation method I get a volatility for the EUR/GBP of 0.117, which sounds weird. My intuition tells me that if the correlation between EUR/USD-USD/GBP was 0.0 during that period, the volatilities of EUR/USD and EUR/GBP should be equal. Could somebody please explain if this makes sense? EDIT to add: From my calculations, to get a volatility of 0.1 for the EUR/GBP, the correlation between EUR/USD-USD/GBP should be -0.3 ## Answer by KevinT (score 2, accepted) https://quant.stackexchange.com/a/60878 I refer to @NHN's answer here, which is correct in the formula and its interpretation: for a Vol of 10% and 6%, respectively, the equally-weighted portfolio of 1 `EURUSD` + 1 `USDGBP` gives you a Vol of 11.6%. However, the bold-faced words - equally-weighted portfolio - are key here; you need to consider that this FX cross is simply a portfolio of x `EURUSD` and y `USDGBP`, and you/NHN implicitly assigned a weight of 100% to the `EURUSD` (buy the `EUR`, sell the `USD`) and 100% to the `USDGBP` (buy the `USD` sell the `GBP`). You can easily check that the variance of this very portfolio (1.36%) is just the sum of the two individual variances (1% and 0.36%, respectively); and this should make sense considering you assume a correlation of zero between the two. Now I highlight this because of your statement in the question: > We want to calculate what has been the volatility of `EURGBP` without knowing the prices. If you do not know the prices of `EURUSD` and `USDGBP`, we (a) either stop here, or (b) simply assume 1 `EURUSD` + 1 `USDGBP` = 1 `EURGBP` like in the above calculation. But this is clearly not the case in reality; in fact, as of yesterday (rough numbers): - 1 `EUR` buys you 1.2073 `USD` - 1 `USD` buys you 0.7304 `GBP` So to construct the vol of 1 `EURGBP`, you should be using weights of 0.8283 and 0.7304 instead of 100% and 100% (you should be able to verify these numbers very easily through a triangular calculation). Using these weights, you'd get a portfolio (= `EURGBP`) volatility of 9.37%. ## Answer by NN2 (score 1) https://quant.stackexchange.com/a/60868 Vol3 = sqrt (vol1^2 + vol2^2 +2* vol1* vol2*correl) Where Vol1 = vol eur/usd Vol2= vol usd/gbp Vol3 = vol eur/gbp
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