If I know the price of $GBPUSD$ and $EURUSD$, I can retrive the $EURGBP$ price simple by $EURGBP = \frac{GBPUSD}{EURUSD}$.

Is there something equivalent to FX Volatility? Knowing the $\sigma_{GBPUSD}$, $\sigma_{EURUSD}$ and myabe $\rho[GBPUSD,EURUSD]$, is there a direct relationship or bounds to the value of $\sigma_{EURGBP}$?

I tried to think in terms of generic numeraie, like $USDX$ and $GBPX$, and take expectaions of the ratios, but I found no solution.

Any idea or clue would be helpful.


1 Answer 1


This is covered in the book "FX Derivatives Trader School" (and in other places, too, e.g. "The shape of things in a currency trio", a paper by Walter and Lopez, CS and FRB SF link).

  • $\begingroup$ @noob2 - thanks for adding the link! $\endgroup$
    – user42108
    Nov 19, 2021 at 12:57
  • $\begingroup$ Thanks. It's simple a the volatility of a 'portfolio' invested 100% in both pairs. I should've seen that coming! $\endgroup$ Nov 23, 2021 at 13:17

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