  Except for certain parameter values, a closed form formula for the mode of
the generalized hyperbolic (GH) distribution is not available. In this paper,
we exploit results from the literature on modified Bessel functions and their
ratios to obtain simple but tight two-sided inequalities for the mode of the GH
distribution for general parameter values. As a special case, we deduce tight
two-sided inequalities for the mode of the variance-gamma (VG) distribution,
and through a similar approach we also obtain tight two-sided inequalities for
the mode of the McKay Type I distribution. The analogous problem for the median
is more challenging, but we conjecture some monotonicity results for the median
of the VG and McKay Type I distributions, from we which we conjecture some
tight two-sided inequalities for their medians. Numerical experiments support
these conjectures and also lead us to a conjectured tight lower bound for the
median of the GH distribution.
