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I saw a chart which showed implied skew and spot/vol covariance (I assume) and I was wondering what these terms actually mean and how to "back them out" of Option prices or vols?

Here is the chart:

Chart

And the caption along with it was:

this is for the S&P500, for example (term structure of implied covariance dividend by implied variance)

Now to elaborate based off my minimal understanding, I assume this chart is not showing the more simply comprehended Implied Volatility Skew when it mentions Implied Skew.

Doing a search on "implied skew" yielded almost zero results other than tons of results talking about "implied volatility skew" which is what we are assuming is not shown here on the chart and has a different meaning I suppose.

Although there was only one website showing a definition I found that could be what's being shown here:

Implied Skew is the change in implied volatility that is priced into today’s surface assuming perfect foresight by the market of what the return is going to be in the future.

I'm not sure if that's correct though or how that could be backed out of Options. It could be but my knowledge is lacking.

And here when "spot-volatility beta" is shown, I assume that means correlation between the spot price and implied volatility and not realized volatility. Because you can easily find out the what correlation between the spot price and realized volatility is.

Having said that, the spot & vol relationship shown here is I'm assuming a reference to dVol/dSpot a.k.a how implied vol changes for a change in the spot price.

And also, as there are tenors in the future being shown on the X-axis, it would further confirm that spot-vol beta here is most definitely referring to implied vol and not realized vol. Because how could it be possible that spot/realized vol was extracted from the future (Option prices) if it's "realized"?

To summarize and finalize my questions:

  • What is this chart supposed to be showing exactly?
  • What do the terms "Implied Skew" and "Spot/Vol Beta" really mean in this context?
  • How can you back them out of Options in future expirations? Would you use some type of heuristic?

Thank you in advance.

EDIT:

I was able to find a statement by the same creator of the chart (Benn P. Eifert) in some conversations that are separate from discussion here:

A common heuristic is that skew tells you the implied move in the ATM vol as the forward moves away from the current level. this is directionally true but not literally true, at least the way derivatives traders typically mean "implied". In principle we could talk about higher order moments here, but lets just take the simple case of a linear inverted skew curve in equity index, where implied vanna (spot* / fixed strike vol covariance is negative). *technically forward but leave that aside for now When we say "the implied move in ATM vol as spot moves", we are inherently referring to spot/vol covariance. "Implied" means that if you have a pure vanna position, the vanna PnL from that level of realized spot vol covariance will break even against the theta paid for that vanna And if floating strike vol were to just move exactly along the skew curve as the spot price moved, by definition fixed strike implied volatility would be staying the same, and realized vanna PNL would be zero. For a pure vanna position to break even, floating strike vol has to outperform the skew curve enough on average to generate enough realized vanna PnL to offset the theta paid (which is positive if the skew curve is downward sloping) Which means that the implied move in vol as spot moves is actually somewhat steeper than the skew curve; as she said, you have to calculate the breakevens.

Maybe this could possibly help.

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For what date is the chart derived? One definition for implied volatility skew is: (25 delta put implied volatility - 25 delta call implied volatility) / 50 delta. Can you test to see if this calculation for the options maturities is consistent with the values on your graph for the date in question?

With respect to the vol beta, this appears to be a measure of the implied vol sensitivity of options with that maturity to spot vol. This can be defined in many ways and would probably be very specific to the generator of the chart. The definition would depend on 1) How is spot vol calculated (what period is used to calculate spot vol, what prices, implied or realized?) 2) what is the moneyness of options for which the sensitivities are generated? Here it looks like they do the calc for both fixed and floating strikes. It looks like for each date they are using the implied vols for a particular strike (fixed strike, perhaps the strike of the 25 delta options in the skew calc) and comparing the vol change against some spot vol calculation. Also, it looks like they are doing the same for strikes as a % of the spot price (floating strike).

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  • $\begingroup$ The chart was posted on August 9. Sorry but it's not mine and I don't have access to historical Option prices to test it out. I've also seen Ernest Chan (famous quant) define implied skew as you do. Would you have an idea on how to define implied spot/vol covar or beta? Thanks. $\endgroup$ Sep 28 '20 at 22:51
  • $\begingroup$ @JackBueller Just edited answer with my thoughts on the vol beta calc. Perhaps you can get an updated report so you can verify the calcs. I imagine the updated report would also define these for you. $\endgroup$
    – AlRacoon
    Sep 29 '20 at 0:04
  • $\begingroup$ I think the spot/implied vol covar in this context could mean Vanna of said fixed strike or floating strike Option? So this is a chart that could possibly be "Implied Skew" / Vanna? I have added a statement I found from the creator that could possibly help, although I still don't comprehensively understand what's exactly going on. Maybe my edit could provide further information to help us out. By the way I thought "fixed strike" meant a strike that stays "fixed" and can always be referred to without confusion e.g. 300 strike on SPY? And floating strike was by delta? Thanks. $\endgroup$ Sep 29 '20 at 1:04
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    $\begingroup$ Yes, fixed is fixed. I am suggesting that the strike they used was the 25 delta strike (for example 300) and used the implied vol for this particular (300) strike for the relationship to changes in spot vol (same option for every data point). For the floating strike, they might have used the 25 delta strike or say for example the 75% strike in reference to the spot--so the strike changes every time the spot changes to the then 25 delta or 75% of spot (so potentially different option for every data point. $\endgroup$
    – AlRacoon
    Sep 29 '20 at 1:42
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Maybe you would like to take a look at Managing forward volatility and skew risk for a direct and robust relation between spot-volatility correlation/covariance and the implied vol skew in the context of (fractional) stoch vol models. Although the result is for forward start case, by letting the forward start date equal spot date it is valid for spot starting case as well.

There are many definitions and measures for spot vol correlation, this is another one. Of course for you to decide which of the multitude of definitions appeals most to you and is of most practical value.

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