# Volatility Scaling

Since the VIX is an annualized volatility, to convert it into other frequencies we must divide by the square root of time. So to convert a VIX of 15 into daily volatility, we would need to divide

$$\frac{15}{\sqrt{252}} = .94$$

Monthly volatility is

$$\frac{15}{\sqrt{12}} = 3.4$$

and a quarterly volatility is

$$\frac{15}{\sqrt{4}} = 7.5$$

Two questions:

1. Why is it that if I multiply a monthly vol of 3.4 by 3 to convert it to quaretly, I do not get 7.5?

2. 20-day (i.e. monthly) realized volatility on the SPX is 17.29%. How is it that the monthly VIX is only 3.4%? Is the options marketing trading at such a low premium to realized vol?

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Your question contains contradictions. Once you (correctly) point out that volatility scales with the square root of time, then in your question you do not apply the same. –  Matt Wolf Oct 15 at 6:09

With regards to part 2, SPX monthly realized volatility is not 17% (I think what you're looking at is the last 20 days worth of data annualized - and probably not current). Annualized realized for the last 20 days worth of data is around 13% which means that monthly is around 3.75%. Thus, monthly VIX is above SPX realized which is normal in a low vol environment.

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$$\frac{15}{\sqrt{12}} \approx 4.33$$
$$4.33 \times \sqrt{3} \approx 7.5$$