Why Equity Index Volatility Skew Reflects Implied Correlation
Summary
The discussion asks why equity index implied volatility surfaces often show a pronounced downside skew rather than a smile, and why smiles may have appeared historically. One explanation is that index volatility combines the volatilities of its constituents with their pairwise correlations. As a result, index options embed expectations about correlation as well as individual stock volatility.
Even if constituent stocks have flat implied volatility surfaces, an index can show a smile or skew when expected correlation changes with the index level. If correlations are expected to rise during sharp market declines, downside index options can carry higher implied volatility. The discussion also points to periods described as “spot up, vol up,” which can produce a different surface shape. These are conceptual explanations and illustrative historical references, not a systematic study: no surface data, time series analysis, or evidence is provided to establish that smiles have disappeared across equity markets.
Key ideas
- Index volatility depends on both constituent volatility and cross-stock correlation.
- Index options therefore reflect implied correlation as well as component-level volatility.
- Rising expected correlations during market declines can contribute to a downside volatility skew.
- Periods in which spot and volatility rise together may produce a smile-like pattern.
- The discussion offers explanations and examples but no systematic historical test.
Tags
Full text
# Is the volatility smile a thing of the past?
# Is the volatility smile a thing of the past?
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)?
## Answer by Adam N. (score 7)
https://quant.stackexchange.com/a/58916
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.
## Answer by user42108 (score 0)
https://quant.stackexchange.com/a/58929
"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.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.