  We estimate the amplitude of the nano-Hz stochastic gravitational wave
background (GWB) resulting from an unresolved population of inspiralling
massive black hole binaries (MBHBs). To this aim, we use the L-Galaxies
semi-analytical model applied on top of the Millennium merger trees. The
dynamical evolution of MBHBs includes dynamical friction, stellar and gas
binary hardening, and gravitational wave feedback. At the frequencies proved by
the \textit{Pulsar Timing Array} experiments, our model predicts an amplitude
of ${\sim}1.2\,{\times}\,10^{-15}$ at ${\sim}\,3\,{\times}\,10^{-8}\, \rm Hz$
in agreement with current estimations. The contribution to the background comes
primarily from equal mass binaries with chirp masses above $\rm 10^{8}\,
M_{\odot}$. We then consider the recently detected common red noise in
NANOGrav, PPTA, and EPTA data, working under the hypothesis that it is indeed a
stochastic GWB coming from MBHBs. By boosting the massive black hole growth via
gas accretion, we show that our model can produce a signal with an amplitude
$A\approx 2-3 {\times}\,10^{-15}$. There are, however, difficulties in
predicting this background level without mismatching key observational
constraints such as the quasar bolometric luminosity functions or the local
black hole mass function. This highlights how current and forthcoming
gravitational wave observations can, for the first time, confront galaxy and
black hole evolution models.
