Let's say we have 3 currencies:
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