  Let $X$ and $Y$ be independent variance-gamma random variables with zero
location parameter; then the exact probability density function of the product
$XY$ is derived. Some basic distributional properties are also derived,
including formulas for the cumulative distribution function and the
characteristic function, as well as asymptotic approximations for the density,
tail probabilities and the quantile function. As special cases, we deduce some
key distributional properties for the product of two independent asymmetric
Laplace random variables as well as the product of four jointly correlated zero
mean normal random variables with a particular block diagonal covariance
matrix. As a by-product of our analysis, we deduce some new reduction formulas
for the Meijer $G$-function.
