# Triangular Arbitrage In FX Volatility

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.