  The optical response functions and band structures of LiCoO$_2$ are studied
at different levels of approximation, from density functional theory (DFT) in
the generalized gradient approximation (GGA) to quasiparticle self-consistent
QS$GW$ (with $G$ for Green's function and $W$ for screened Coulomb interaction)
without and with ladder diagrams (QS$G\hat W$) and the Bethe Salpeter Equation
(BSE) approach. The QS$GW$ method is found to strongly overestimate the band
gap and electron-hole or excitonic effects are found to be important. They
lower the quasiparticle gap by only about 11~\% but the lowest energy peaks in
absorption are found to be excitonic in nature. The contributions from
different band to band transitions and the relation of excitons to band-to-band
transitions are analyzed. The excitons are found to be strongly localized. A
comparison to experimental data is presented.
