Asymptotic approximations for the distribution of the product of correlated normal random variables

We obtain asymptotic approximations for the probability density function of the product of two correlated normal random variables with non-zero means and arbitrary variances. As a consequence, we deduce asymptotic approximations for the tail probabilities and quantile functions of this distribution, as well as an asymptotic approximation for the widely used risk measures value at risk and tail value at risk.