Not yet reviewed by Pith; the record is open.

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.