Linear and nonlinear optical susceptibilities of the ferroelectromagnet ${\mathrm{YMnO}}_{3}$ below its N\'eel temperature ${(T}_{N})$ have been studied by using the first-principles full-potential linearized augmented plane-wave (FPLAPW) method within the local density approximation (LDA) plus the multi-orbital mean field Hubbard model (LDA+$U),$ in which the on-site Coulomb interaction corrections are applied on the Mn $3d$ states. All independent components of the dielectric function $\ensuremath{\epsilon}(\ensuremath{\omega})$ and the second harmonic generation susceptibility ${\ensuremath{\chi}}^{(2)}(\ensuremath{-}2\ensuremath{\omega},\ensuremath{\omega},\ensuremath{\omega})$ are evaluated over a broad frequency range. The calculated static value of the dielectric function is close to the experimental observation data. For the nonlinear optical response, an alternative type second harmonic generation (SHG) response is obtained, which is caused by the coexistence of the noncentrosymmetric ferroelectric ordering of charges and the centrosymmetric spin ordering below ${T}_{N}.$ The calculated results are quite consistent with the experimental observation and show clearly the contributions to the nonlinear optical response from the transitions between the Mn $3d$ and O $2p$ bands.