# 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?

• 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 Oct 15 '13 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). 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.

$$\frac{15}{\sqrt{12}} \approx 4.33$$

$$4.33 \times \sqrt{3} \approx 7.5$$

• Just added the volatility I was quoting. I guess the volatility figue I was quoting was the volatility of the SPX (the raw index) around it's 20 day moving average . Different from the volatility based on return percentages which I think your quoting. – jessica Oct 14 '13 at 23:15
• @jessica, no the issue is that you have several thought errors in your questions. First you (correctly) point out that volatility scales with the square root of time, then in your question you scale by time and wonder why the result is not anywhere near the quarterly vol you stated...btw...are none of the 15 (x number answers provided) out of your 20 questions worth marking as correct? Maybe you could work a bit on your accept rate? – Matt Oct 15 '13 at 5:36