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
There is a known expansion of implied volatility in moments (I'll find the reference)
IV=vol∗(1+skew6∗LMM+kurt24∗(LMM2−1))
where log-moneyness is
LMM=logstrikeforwardvol∗√T.
Use VIX for vol.
If I remember correctly SKEW index is 100−100∗skew, so skew=100−SKEW100. 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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