Drivers of Implied Volatility Skew and Its Term Structure

# How does volatility skew change with underlying spot?


# 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)?

## Answer by SI7 (score 3, accepted)

https://quant.stackexchange.com/a/51569

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

## Answer by Stéphane (score 3)

https://quant.stackexchange.com/a/51553

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.