Approximating the S&P 500 Volatility Smile from VIX and SKEW
Summary
The document considers whether public volatility indexes can be used to approximate the S&P 500 implied volatility smile. One proposed approach uses VIX as the volatility level and an expansion in standardized log-moneyness, with skew and kurtosis terms shaping implied volatility away from the center. The response relates SKEW to a skew parameter and suggests VVIX as a possible, indirect source of information about kurtosis or volatility of volatility.
Another route is to download closing option prices and estimate the smile directly. If those prices are unavailable or excluded, the response suggests fitting a stochastic volatility model to VIX and SKEW, with VVIX potentially improving the fit by informing volatility-of-volatility. These are suggestions rather than a validated reconstruction procedure: the expansion's higher-moment inputs and scaling need care, and fitting a model from a small set of index observations is described as difficult to do precisely.
Key ideas
- An implied volatility expansion can express smile variation using standardized log-moneyness and distribution moments.
- VIX supplies a volatility level, while SKEW may inform the skew component.
- Option closing prices provide a direct source for estimating the implied volatility smile.
- A stochastic volatility model fitted to volatility indexes offers an indirect approximation, with VVIX potentially informing volatility of volatility.
- The proposed index-based methods are approximate and lack a reported empirical validation.
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Full text
# Constructing an approximation of the S&P 500 volatility smile with publicly available data
# 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)?
Addendum: I created a follow up question here.
## Answer by onlyvix.blogspot.com (score 4, accepted)
https://quant.stackexchange.com/a/3183
There is a known expansion of implied volatility in moments (I'll find the reference)
\begin{equation} \textrm{IV} = \textrm{vol} * (1 + \frac{\textrm{skew}}{6} * \textrm{LMM} + \frac{\textrm{kurt}}{24}*(\textrm{LMM}^2-1)) \end{equation}
where log-moneyness is
\begin{equation} \textrm{LMM} = \frac{\log{\frac{\textrm{strike}}{\textrm{forward}}}}{\textrm{vol} * \sqrt{T}}. \end{equation}
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.
## Answer by Brian B (score 4)
https://quant.stackexchange.com/a/3147
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.)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.