I would like to convert 30-day annualized volatility $\sigma_{30d}^a$ to 2-day annualized volatility $\sigma_{2d}^a$.
Am i right to say:
$$\sigma_{2d}^a = \sqrt{\frac{2}{30}} \cdot \sigma_{30d}^a$$
I don't have the returns but only 30-day annualized vol.
I assume with 30 day annualized volatility you mean, you calculated the volatility from the 30 day returns and then annualized it by multiplying with $\sqrt{252/30}$.
You can calculate the 2 day volatility only if you assume independence of the returns. In that case though all annualized volatilities are identical, especially $\sigma_{2d}^a = \sigma_{30d}^a$.
If you want to capture the effects of correlation in your data, there is no way to calculate the 2 day correlations from the 30 day correlations and hence there is no way to scale $\sigma_{30d}^a$ to get to $\sigma_{2d}^a$ apart from the first approximation that they are the same.
Just to expand, let's say the assumptions needed for the variances to be identical and additive are satisfied, so the annual variance will be the sum of monthly variances. I am going to assume 360 days in a year but you will have to change it to reflect the local holidays/weekends etc. So the annual variance will be the sum of 12 monthly variances:
$\sigma^2_{360d}=12 \times \sigma^2_{30d}$
$\sigma^2_{360d}=\sigma^2_{30d}\frac{360}{30}$
You can similarly write for the 2-days horizons:
$\sigma^2_{360d}=\sigma^2_{2d}\frac{360}{2}$
Hence,
$\sigma^2_{360d}=\sigma^2_{30d}\frac{360}{30}=\sigma^2_{2d}\frac{360}{2}$
And if you rearrange you get:
$\sigma^2_{2d}=\sigma^2_{30d}\frac{2}{30}$
The square root of which is the relationship you have, but this is the relationship between the volatility of return over 30 days and the volatility of returns over 2 days as @Ami44 explained above. If the assumptions for variances to be additive are satisfied then both shall give the same annual volatility when annualised.
