  We investigate the excitonic spectrum of MoS$_2$ monolayers and calculate its
optical absorption properties over a wide range of energies. Our approach takes
into account the anomalous screening in two dimensions and the presence of a
substrate, both cast by a suitable effective Keldysh potential. We solve the
Bethe-Salpeter equation using as a basis a Slater-Koster tight-binding model
parameterized to fit ab initio MoS$_2$ band structure calculations. The
resulting optical conductivity is in good quantitative agreement with existing
measurements up to ultraviolet energies. We establish that the electronic
contributions to the C excitons arise not from states in the vicinity of the
$\Gamma$ point, but from a set of $k$-points over extended portions of the
Brillouin zone. Our results reinforce the advantages of approaches based on
effective models to expeditiously explore the properties and tunability of
excitons in TMD systems.
