Log Normal Vol

If I have lognormal vol for security x and y as well as the correlation between the two securities. How to get the vol for a derivative that has the
payout of x*y and x/y?

log(x*y)=log(x)+log(y),
sigma(log(x*y)) = sqrt[sigma(log(x))^2+sigma(log(y))^2+2*corr*sigma(log(x))*
sigma(log(y))],
corr, sigma(log(x)), sigma(log(y)) are the known values.

log(x/y)=log(x)-log(y),
sigma(log(x/y)) =
sqrt[sigma(log(x))^2+sigma(log(y))^2-2*corr*sigma(log(x))*sigma(log(y))]

log(y)=-log(1/y), so they should have the same lognormal vol.