Converting 30day annualized vol to 2day annualized vol

Question

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.

Answer 1

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.

Answer 2

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.