  The spin resonance observed in the inelastic neutron scattering data on
Fe-based superconductors has played a prominent role in the quest for
determining the symmetry of the order parameter in these compounds. Most
theoretical studies of the resonance employ an RPA-type approach in the
particle-hole channel and associate the resonance in the spin susceptibility
$\chi_S (\mathbf{q}, \omega)$ at momentum ${\bf Q} = (\pi,\pi)$ with the
spin-exciton of an $s^{+-}$ superconductor, pulled below $2\Delta$ by residual
attraction associated with the sign change of the gap between Fermi points
connected by ${\bf Q}$. Here we explore the effect of fluctuations in the
particle-particle channel on the spin resonance. Particle-particle and
particle-hole channels are coupled in a superconductor and to what extent the
spin resonance can be viewed as a particle-hole exciton needs to be addressed.
In the case of purely repulsive interactions we find that the particle-particle
channel at total momentum ${\bf Q}$ (the $\pi $channel) contributes little to
the resonance. However, if the interband density-density interaction is
attractive and the $\pi-$resonance is possible on its own, along with the
spin-exciton, we find a much stronger shift of the resonance frequency from the
position of the would-be spin-exciton resonance. We also show that the expected
double-peak structure in this situation does not appear because of the strong
coupling between particle-hole and particle-particle channels, and $\mathrm{Im}
\chi_S ({\bf Q}, \omega)$ displays only a single peak.
