The variance-gamma ratio distribution

Let $X$ and $Y$ be independent variance-gamma random variables with zero location parameter; then the exact probability density function of the ratio $X\mathrm{/}Y$ is derived. Some basic distributional properties are also derived, including identification of parameter regimes under which the density is bounded, asymptotic approximations of tail probabilities, and fractional moments; in particular, we see that the mean is undefined. In the case that $X$ and $Y$ are independent symmetric variance-gamma random variables, an exact formula is also given for the cumulative distribution function of the ratio $X\mathrm{/}Y$.