First observation of the decay $\overline{B}^0_s \to D^0 K^{*0}$ and measurement of the ratio of branching fractions $\frac{\mathcal{B}\left(\overline{B}^0_s \to D^0 K^{*0}\right)}{\mathcal{B}\left(\overline{B}^0_d \to D^0 \rho^{0}\right)}$

In $36\textrm{ pb}^{-1}$ of $pp$ collisions at a centre-of-mass energy $\sqrt{s}=7\textrm{ TeV}$, we observe for the first time the decay $\overline{B}^0_s \to D^0 K^{*0}$. A clear signal of $34.5 \pm 6.9$ events is obtained with a statistical significance over 9 standard deviations and we measure its branching fraction relative to that of $\overline{B}^0_s \to D^0 K^{*0}$: $\frac{\mathcal{B}\left(\overline{B}^0_s \to D^0 K^{*0}\right)}{\mathcal{B}\left(\overline{B}^0_d \to D^0 \rho^{0}\right)} = 1.39 \pm 0.31\pm 0.17\pm 0.18\pm$ ($f_d/f_s$), where the first uncertainty is statistical, the second systematic and the third is due to uncertainty in the hadronisation fraction $f_d/f_s$.