  We introduce a minimal spin model for describing the magnetic properties of
$\mathrm{CuCrO_2}$. Our Monte Carlo simulations of this model reveal a rich
magnetic field induced phase diagram, which explains the measured field
dependence of the electric polarization. The sequence of phase transitions
between different mutiferroic states arises from a subtle interplay between
spatial and spin anisotropy, magnetic frustration and thermal fluctuations. Our
calculations are compared to new measurements up to 92 T.
