  The existence of ferroelectricity in {$\mathrm{BiMnO}_3$} has been a
long-standing question for both experimentalists and theorists. In addition to
a highly distorted bulk structure, the ionic crystal planes cause a large
roughness in thin films that makes it extremely difficult to nail down the
physical mechanisms underlying a possible ferroelectric-ferromagnetic phase. We
approach the problem by including the substrate explicitly to study the
polarization. With this model, we investigate mono-, di-, and trilayer
{$\mathrm{BiMnO}_3$} thin films on $\mathrm{SrTiO}_3$ substrates. We find that
thin film systems have both strong ferromagnetism and strong ferroelectricity.
Substrate constraints weaken the competition between displacements induced by
stereochemically active Bi-$6s^2$ lone pairs and by Jahn-Teller distortions
around Mn ions found in the bulk, such that the sum of off-center displacements
of Bi ions in bulk {$\mathrm{BiMnO}_3$} nearly cancel. In {$\mathrm{BiMnO}_3$}
thin films, in contrast, all Bi ions displace roughly in parallel, resulting in
a strongly polar structure. We also find spontaneous charge disproportionation
of Mn ion pairs in {$\mathrm{BiMnO}_3$} thin films.
