Is there any relationship between the Covariance(A, B) and the variance of the synthetic asset A/B?

Question

Let's say we have 2 pairs of currencies: EUR/USD and GBP/USD. The cross-asset (or synthetic asset) would be (EUR/USD) / (GBP/USD) = EUR/GBP.

Is there any relationship between the covariance(EUR/USD, GBP/USD) and the variance of EUR/GBP?

Answer 1

We have $$\ln(A/B)= \ln A - \ln B.$$ Hence $$\textrm{Var}(\ln(A/B))= \textrm{Var}(\ln A) + \textrm{Var}(\ln B) - 2\textrm{Cov}(\ln A,\ln B).$$

Hope that helps.