How does volatility skew change with underlying spot?

We know that generally ATM implied vol is negatively correlated with the underlying spot for equity indices, i.e. implied vol goes up when spot moves down. Therefore I wonder if there are any relations between the implied volatility skew and the underlying (spot, historical volatility, etc.)?

If not, are there any other factors that can to some degree "explain" the evolution of the implied volatility skew (for example 90-110 implied vol difference)?

• Is you question about historical relations (i.e. how have past vol skews $\{\mathcal{S}_t\}_t$ evolved with respect to the underlying?) or about how the static vol skew $\mathcal{S}$ at some fixed time $t$ depends on the underlying value at $t$ (in which case a model for the local volatility is needed, in which local vol depends on the underlying)? – Daneel Olivaw Mar 7 '20 at 15:26
• Hi Daneel, my question is about historical relations. Thanks! – CABLE Mar 7 '20 at 18:01
• If you do PCA on the vol surface you get about 3 major components that evolve over time: level, skew, and curvature. You can look how those change over time after doing a PCA. Or, for a quick and dirty solution, you could just take the 90-110 (or some other spread around ATM) implied volatility spread and just look how that evolves over time. Regress it on spot, etc. – roz Mar 9 '20 at 18:02
• Hi roz, I am exactly trying to find out what would be the explanation variable. – CABLE Mar 10 '20 at 2:09
• Its up to you. A simple place to start would be for each time interval you look at the spot price, then you take call that is x% out of the money and a put that is x% out of the money. Then you calculate the implied volatility on each of these options. Then you would take the difference of these two implied violates. Then for each time period you would have a spot value and a value that is the difference between an otm call and put (which is one way to measure skew). Then you can run the regression of that skew variable on spot. – roz Mar 10 '20 at 16:07

This is a very interesting question and obviously (as almost any reasonable question w.r.t to spot-vol-skew correlation) does not have a unique answer. Rather, different regimes may exist.

In one of your comments you said "if spot does not quantify this 'fear', then what would?". Potential answers may be (but are not limited to)

• absence or illiquidity of CDS market: If for a single stock underlying there is not a liquid CDS market (or non at all), then market participants will try to hedge their positions via downside options. In this case, there is high demand for OTM puts even though if spot starts to rally, keeping the skew stressed. Currently, there are examples of these kind of single stocks

• Hedging activities by large banks. Large banks issuing exotic products on famous underlyings (like EuroStoxx) have a strong directional risk position. Once the market starts to trend, Banks needs to hedge their Vega positions due to their internal risk limits. It is well known in the market that this may often create positive correlation between spot and vols but also between spot and skew. As an example, a few years ago banks started to sell risk reversals on the Eurostoxx in order to limit their vega exposure. In this case, as the index started to fall and large banks started to hedge their positions, the skew got flatter (positive spot-skew correlation).

My answer is surely not complete and there are many other reasons for different dynamics observed in the market. Besides the mentioned examples, I just want to stress that when analyzing these kind of correlations you at least have to separate between:

• Index vs. Single Stocks dynamics
• Short term skew vs. long term skew (skew often flattens in the long term)
• Correlations to other asset classes (check the CDS example above)
• Literature and empirical results pre- and post exotics world (auto callable issues from large banks). Much of the cited literature today w.r.t spot-vol-skew correlation is of less use as these results stem from the pre-exotics world or their scope is limited to just short term options
• Hi SI7, the question is about skew on index. I totally agree with the fact that the structured products have sometimes created positive spot-vol correlations, especially in Asia. However, this is the first time I hear about banks selling risk reversals. – CABLE Mar 11 '20 at 1:37
• Do you have any materials/slides on this activity? Thanks a lot in advance! – CABLE Mar 11 '20 at 1:39
• If I understand you correctly, the skew change is primarily driven by dealers' positioning? – CABLE Mar 11 '20 at 4:41
• predominantly yes, but again, I'm not talking about the short term options but rather about the middle part of the term structure (corresponding to typical auto callable maturities and call schedules). There is high pressure on the Eurostoxx vol term structure coming from auto callable books. Regarding risk reversals. Autocallable issuers have a challenging negative Vanna exposure (each Bank has the same directional risk there) forcing them to sell vega on the downside (that's where positive spot-vol correlation comes from). One possible hedge for this negative Vanna is to use risk reversal – SI7 Mar 11 '20 at 18:44
• These risk reversals were the main reason why the skew got significantly flatter once the Index was stressed (positive spot-skew correlation). It is still a very important and famous hedging tool but liquidity constraints and the trade size you have to execute is challenging. Finally, one book where these historical observations are covered, at least to some sufficient general overview, is NEIL C. SCHOFIELD, EQUITY DERIVATIVES, CORPORATE AND INSTITUTIONAL APPLICATIONS – SI7 Mar 11 '20 at 18:51

I'm surprised that no one has yet mentionned the Christoffersen, Heston and Jacobs paper entitled "The Shape and Term Structure of the Index Option Smirk: Why Multifactor Stochastic Volatility Models Work so Well" published in 2009 in Management Science. You can find it here.

They actually look into the evolution of implied volatility surface of (European) option contracts written on the S\&P500. They filter out "odd" contracts as is routinely done in the literature (basically, they apply the Bakshi, Cao and Chen (1997) filters and other consistency checks). Then, they fit a quadratic polynomial in both moneyness and maturity on the implied volatility surface by ordinary least squares. The idea is that this gives you a statistically sound way of spanning the whole surface. Once this is done, they generate a fitted value surface for a large range of maturity and moneyness that they want to look into. Finally, they apply principal component analysis to extract latent factors from this surface and find that 2 factors explain most of the variance present in the data across time.

If you dig into their paper, you will find that they perform regressions in the moneyness space with these two factors and it seems that one of the factor captures the shifting "levels" of the smile, while the other captures the "slope." So, if your interest lies in thinking about the evolution of the (risk-neutral) volatility, what this tells you is that at least two sources of variations are required in the volatility process of your model because your smile jumps up and down, as well moves between being more and less "smirky" for a given maturity.

Now, going back to your concern about volatility and skewness, your hesitation might stem from missing an important detail. We can produce realtively reliable estimates of stock market volatility using high frequency data -- that is, we can get estimated time series of volatility under the physical measure. Neglecting concerns over jumps, say you believe realized volatility is good enough. Well, that's negatively correlated with returns on the index. The same thing would happen if you used a model to filter out a conditional volatility time series under the physical measure. In other words, under the physical measure, you have a negative correlation. You would expect that for most maturities and moneyness, you'd find the same thing under the risk-neutral measure.

• Hi Stephane, I am afraid I could not find the paper you mentioned on the internet. But according to your description, the meanings (whether they are related to some kind of transformation of something that we know (spot, historical vol)) of the those 2 factors is what I am looking for. – CABLE Mar 10 '20 at 2:08
• The slope of the smirk usually is generated using conditional non-normality (jumps with negative average, or an odd distribution for schocks in GARCH models) and a negatively correlated conditional volatility process (stochastic volatility or GARCH models). You can have both things combined, obviously. You need more mass on the left tail of the risk-neutral distribution. So, we have these explanations -- but they are filtered through models. Intuitively it's something like a combination of fearing huge drops and very real changes of seeing them. – Stéphane Mar 10 '20 at 17:40
• Cont'd. You also have to think it's a RELATIVE commentary: the smile says Black-Scholes-Merton underprices some OTM options, for example, because it spits out high implied volatilities for these options. Usually, this means you need a negative variance premia: under the risk neutral distribution, variance is higher. Under BSM, per Girsanov's theorem, volatility under both measures should be identical. If it is not, ALL options become more valuable. So, to get changing level, you can use a time-varying Variance premium; to get changing slopes, you need time-varying (cond.) skewness or kurtosis. – Stéphane Mar 10 '20 at 17:46
• If you wonder why most models admit a negative variance premia, think about when variance is high: if you have a positive exposure to variance, you have an insurance when the market drops. If this changes over time, your level changes over time -- plus, you solve the variance puzzle on why IV produces a biased forecast of realized variance. The answer is the bias is a premium. – Stéphane Mar 10 '20 at 17:49
• I just added a link to the paper, taken from Christoffersen's website. I confused the title of a subsection in my course notes with the title of the paper. Sorry. I corrected the mistake. – Stéphane Mar 10 '20 at 17:52