  We present precise values of the dipole polarizabilities ($\alpha$) of the
ground $\rm [4f^{14}6s] ~ ^2S_{1/2}$ and metastable $\rm [4f^{14} 5d] ~
^2D_{3/2}$ states of Yb$^+$, that are %vital {\bf important} in reducing
systematics in the clock frequency of the $\rm[4f^{14}6s] ~ ^2S_{1/2}
\rightarrow [4f^{14}5d] ~ ^2D_{3/2}$ transition. The static values of $\alpha$
for the ground and $\rm [4f^{14} 5d] ~ ^2D_{3/2}$ states are estimated to be
$9.8(1) \times 10^{-40} \,\,\rm Jm^2V^{-2}$ and $17.6(5) \times 10^{-40}\,\,
\rm Jm^2V^{-2}$, respectively, while the tensor contribution to the $\rm
[4f^{14} 5d] ~ ^2D_{3/2}$ state as $- 12.3(3) \times 10^{-40}\,\, \rm
Jm^2V^{-2}$ compared to the experimental value $-13.6(2.2) \times
10^{-40}\,\,\rm Jm^2V^{-2}$. This corresponds to the differential scalar
polarizability value of the above transition as $-7.8$(5)$\,\times\,
10^{-40}\,\rm Jm^2 V^{-2}$ in contrast to the available experimental value
$-6.9$(1.4)$\,\times\, 10^{-40}$\,\, $\rm Jm^2V^{-2}$. This results in the
black-body radiation (BBR) shift of the clock transition as $-0.44(3)$ Hz at
the room temperature, which is large as compared to the previously estimated
values. Using the dynamic $\alpha$ values, we report the tune-out and magic
wavelengths that could be of interest to subdue %major systematics due to the
Stark shifts and for constructing lattice optical clock using Yb$^+$.
