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

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%.

• Thanks! This is much clearer. A couple of questions/clarifications: Could you explain a bit how do you get those weights through the triangular calculation? Not sure I follow you there. Also, in the quote of the OP you mention I meant the prices of EURGBP are not known but the prices of EURUSD and USDGBP yes. I edited the OP to be clearer about it. And lastly, how do you get the volatility of 9.37% using those weights? Might be because it's too early in the morning for me but I don't see it. Really appreciate your help! Feb 2 '21 at 9:23
• Sure - so basically, if EURUSD is 1.2073 then USDEUR is 0.8283, so selling 1 USD will deliver you 0.8283 EUR. To buy this 1 USD you sold, you'd need to spend 0.7304 GBP. Hence, EURGBP in an arbitrage-free FX spot market (ignoring bid-ask spreads) would need to be 1.2073 x 0.7304 = 0.7304 / 0.8283 = 0.8818. This is the price of 1 EUR expressed in GBP. So buying 0.8283 EURUSD and buying 0.7304 USDGBP will be the same as buying 1 EURGBP, which is the portfolio you're interested in. Feb 2 '21 at 9:53
• Regarding the portfolio variance consisting of $w_A$ % of asset $A$ and $w_B$ % of asset $B$, you have $\sigma^2_P = w_A^2 \sigma^2_A + w_B^2 \sigma^2_B + 2 \rho_{A, B} w_A w_B \sigma_A \sigma_B$. In your case, $\rho_{A, B} = 0$, and $w_A = 0.8283$, $w_B = 0.7304$ and the two vols with 10% and 6%. Plug this into the above formula and take the square root, then you should end up with 9.37%. Feb 2 '21 at 9:53
• Much appreciated! Feb 2 '21 at 10:11
• You're very welcome! Feb 2 '21 at 10:12

Vol3 = sqrt (vol1^2 + vol2^2 +2* vol1* vol2*correl)

Where Vol1 = vol eur/usd Vol2= vol usd/gbp Vol3 = vol eur/gbp

• Yes, exactly like that is how I get to that result, but intuitively I still can't see why the vol is higher if the 2 currencies are uncorrelated. Intuitively if correlation is 0 the vol of EUR/USD and EUR/GBP should be equal to each other no? Feb 1 '21 at 16:36
• No, if the correlation = 0, vol3 is always greater than vol1 and vol2. It’s straightforward from the formula, isn’t it?(sqrt(x) is square root of x)
– NN2
Feb 1 '21 at 16:39