Looking for example at this image from bloomberg of the OMX volatility surface, there is only a faint resemble of a smile at the shortest tenors that quickly dissipates as maturity is increased. I find that this is true for all equity surfaces. It seems there is just a very distinct skew, where the implied volatility is higher for lower strike values. Looking at it from the perspective that people value downside protection this pattern makes sense to me, since high demand for OTM puts would make them more expensive and increase the IV, but then why was the smile ever a thing (assuming that it is in face gone)?

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    $\begingroup$ Just a comment: I think the expression 'smile', as in: a nearly symmetrical shape with a 'trough' in the middle section, always relates to specific markets, e.g. FX markets. Especially for large Stock Market Indices, a 'smile' was never the thing, it was always rather a 'smirk' or whatever you want to call it: IVols are rather higher for OTM puts and decrease towards ITM calls. $\endgroup$ – Kermittfrog Oct 22 '20 at 13:49
  • $\begingroup$ @kermittfrog I think you mean "otm calls" $\endgroup$ – will Oct 25 '20 at 12:53
  • $\begingroup$ Yes. Thanks. I meant OTM calls $\endgroup$ – Kermittfrog Oct 25 '20 at 13:34

It's probably important that we're talking about IV of an index. From "Volatility Trading" by Euan Sinclair:

In equity indexes the skew will be more pronounced than in the individual stocks that make up the index. The volatility of an index, $σ$, is related to the volatility of the components, $σ_i$, by: $$σ^2=\sum_{i=1}^N w_i^2 σ_i^2+2\sum_{i=1}^{N-1}\sum_{j>i} w_i w_j ρ_{ij} σ_i σ_j$$ where $w_i$ are the component weights and $ρ_{ij}$ are the correlations between the components.

So we can see that there are two ways the index volatility can increase: Either the component volatilities can increase or the correlations can increase. Equation above is equally applicable to realized volatility and correlation and to implied volatility and correlation. So the implied volatility of an index also contains an implied correlation effect. Even if all the components have flat-implied volatility surface, the index can exhibit a smile if correlation is expected to increase as the underlying moves. And it is a generally held belief that correlation between stocks increases in crashes or sharp downward moves.


"why was the smile ever a thing"

There are periods where there's a 'spot up, vol up' regime, as recently seen in SPX and NDX. Likely true for NASDAQ/tech stocks in the late '90s and in 2012/2013 for NKY. Latter is easier to check as Bloomberg and other vendors probably have data for that period.


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