Given $X_1$ and $X_2$ normal random variables with correlation coefficient $\rho$, how do I find the correlation between following lognormal random variables $Y_1$ and $Y_2$?
$Y_1 = a_1 \exp(\mu_1 T + \sqrt{T}X_1)$
$Y_2 = a_2 \exp(\mu_2 T + \sqrt{T}X_2)$
Now, if $X_1 = \sigma_1 Z_1$ and $X_2 = \sigma_1 Z_2$, where $Z_1$ and $Z_2$ are standard normals, from the linear transformation property, we get:
$Y_1 = a_1 \exp(\mu_1 T + \sqrt{T}\sigma_1 Z_1)$
$Y_2 = a_2 \exp(\mu_2 T + \sqrt{T}\sigma_2 (\rho Z_1 + \sqrt{1-\rho^2}Z_2)$
Now, how to go from here to compute correlation between $Y_1$ and $Y_2$?