# Constructing an approximation of the S&P 500 volatility smile with publicly available data

Besides of the VIX there is another vol datum publicly available for the S&P 500: the SKEW.

Do you know a procedure with which one can extrapolate other implied vols of the S&P 500 smile with these data (or with other publicly available vol data)?

I created a follow up question here.

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What exactly are you trying to create? 30-day implied vol curve? – onlyvix.blogspot.com Apr 1 '12 at 21:14
@onlyvix: Exactly! – vonjd Apr 2 '12 at 16:19
You should be aware that the vix is not the volatility, it's the sqrt of the 30 par varswap rate - the two are not the same. – will May 9 at 18:25
@will: What are you getting at? VIX stands for Volatility IndeX and it is built to measure expected volatility. – vonjd May 9 at 19:11
@vonjd It may have been, but read the spec, google it a bit, you'll see that it's not the vol. it's the sqrt of the par 30 day varswap rate - the two are not the same thing. – will May 10 at 10:17

There is a known expansion of implied volatility in moments (I'll find the reference)

$$\textrm{IV} = \textrm{vol} * (1 + \frac{\textrm{skew}}{6} * \textrm{LMM} + \frac{\textrm{kurt}}{24}*(\textrm{LMM}^2-1))$$

where log-moneyness is

$$\textrm{LMM} = \frac{\log{\frac{\textrm{strike}}{\textrm{forward}}}}{\textrm{vol} * \sqrt{T}}.$$

Use VIX for vol.

If I remember correctly SKEW index is $100-100*\textrm{skew}$, so $\textrm{skew} = \frac{100-\textrm{SKEW}}{100}$. Kurtosis is unknown, but you could try to use VVIX index and re-scale it in some way.

Or maybe another way would be to take the equation and regress for multipliers for VIX, VIX*SKEW, and VIX*VVIX using IV smile data.

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Thank you, I accepted your answer already, but could you please provide the reference? It would be super great if you could give an example calculation also :-) Another thing: Are kurtosis and VVIX equivalent? Thank you again! – vonjd Apr 6 '12 at 7:18
@vonjd: VVIX is more like the varvol (or vol of vol) parameter found in stochastic volatility models. What onlyvix provides here is a common parameterization of the skew curve with convenient simplicity. See this question for more on that. – Brian B Apr 10 '12 at 12:49
Brian B is correct, VVIX is definitely not kurtosis, but probably can be used as a proxy if properly scaled. The reference is here faculty.baruch.cuny.edu/lwu/papers/bias.pdf, see page 8, formula (16) – onlyvix.blogspot.com May 23 '12 at 19:12

Actually, closing options prices can be downloaded from the exchange, so the data necessary to get the skew is available.

If for some reason you don't want to use those closing prices, it is possible to obtain a vol skew from VIX and SKEW. You would need to fit the parameters of a stochastic volatility model (such as Heston's) to the VOL and SKEW data. It's hard to do carefully but easy to do approximately. Then the S&P skew is whatever has been implied by that model.

(Edit: if you are willing to include the VVIX, your fits will become much better. The VVIX tels you the size of the volatility-of-volatility parameter in a stochastic vol model.)

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Thank you Brian, where do you get these historic closing option prices? – vonjd Mar 30 '12 at 16:23
Updated with link to prices. It's not historic, but obviously you can download and form your own historic series. – Brian B Mar 30 '12 at 17:49