Scaling Annualized Volatility to Monthly and Quarterly Horizons

Summary

The document explains how to convert annualized volatility into volatility over shorter horizons using the square root of time. It initially gives an incorrect monthly conversion for a VIX reading of 15; the accepted answer corrects it to about 4.33 percent and shows that multiplying by the square root of three gives about 7.5 percent quarterly volatility. Monthly volatility does not scale linearly by multiplying by three, because volatility scales with the square root of elapsed time under the usual assumptions.

The discussion also clarifies a comparison between VIX and recent SPX realized volatility. The reported 20-day realized figure was apparently annualized, rather than a monthly volatility; the answer estimates annualized realized volatility near 13 percent, or monthly volatility near 3.75 percent. It concludes that monthly VIX above realized volatility can occur in a low-volatility environment. These are approximate figures, and the exchange does not establish that implied volatility will predict realized volatility.

Key ideas

• Convert annualized volatility to a shorter horizon by multiplying by the square root of the fraction of a year.
• Monthly volatility scaled to a quarter uses the square root of three, not a factor of three.
• A 20-day realized volatility figure may be annualized and should be rescaled before comparing it with monthly VIX.
• Implied volatility can exceed recent realized volatility, though the example does not establish a predictive relationship.

Tags

# Volatility Scaling


# 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:

- 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?

- 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?

## Answer by frickskit (score 2, accepted)

https://quant.stackexchange.com/a/9154

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.

## Answer by Yike Lu (score 4)

https://quant.stackexchange.com/a/9153

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

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

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.