I am trying to solve $\mathbb E[\exp(abXY)]$ where $a$ and $b$ are parameters, and $X$, $Y$ are zero-mean Gaussian random variables with variance $\sigma_x$ and $\sigma_y$ respectively. I suppose that they are independent.
For example, for the univariate case we have:
$$\mathbb E[\exp(a b X)] = \exp (a^2 b^2 \sigma_x^2 / 2),$$
from the Bolton contract theory, but I want some similar expression for the bivariate case, How can I solve that?
