# What are popular metrics for Option Skew?

What are popular metrics to track skew? Would it be the difference between OTM option and ATM option IV? Would it be a percentage difference in IV?

Also, if both are valid, would a % change be better or absolute change be better?

My goal is to track how skew changes over time, obviously in relation to ATM volatility.

Thanks!

My preferred measure for skew would be the difference between the implied volatilities corresponding to the strikes where the Black-Scholes $$d_2 = 0$$ and $$d_1 = 0$$.

The reason I prefer this to others you often come across is that when the smile is symmetric the $$d_2=0$$ implied volatility equals the $$d_1 = 0$$ implied volatility.

Furthermore, the implied volatility corresponding to $$d_2=0$$ is quite insensitive to correlation, whereas the $$d_1 = 0$$ implied vola is very sensitive to correlation.

Also, the $$d_2=0$$ implied volatility is related to the volatility swap strike and corresponds to the the mid-point of the Black-Scholes log-normal distribution. Hence in a sense you can regard the $$d_2=0$$ strike as the pivot point about which the skew "rotates" as correlation changes, and correlation of course impacts skewness.

But as far as I know there is no generally accepted measure for skew, I am just giving my opinion.

EDIT:

I suspect even that the $$d_1 = 0$$ implied volatility is the most correlation sensitive point on the skew, but I cannot prove that (yet).

• Interesting answer. Why is, let's call it $\Sigma_{d_2=0}$, less sensitive to correlation that $\Sigma_{d_1=0}$? – Daneel Olivaw Nov 28 '19 at 14:37
• Hi @DaneelOlivaw, because as shown in that paper $\Sigma_{d_2}$ is approximately the volatility swap strike for any stochastic volatility model of the form $\sigma_t = a(t, \sigma_t) dt + b(t, \sigma_t) dW$ and for any correlation value. Since $\sigma_t$ does not depend on the asset $S_t$ any volatility derivative, including the volatility swap, and hence also $\Sigma_{d_2}$ canot be sensitive to correlation. Also take a look at "It takes three to smile" at SSRN. There you will see that $\Sigma_{d_1}$ and any other implied vol is sensitive to correlation. – ilovevolatility Nov 28 '19 at 15:08
• I'll be happy to answer any questions on model-free vanna pricing of volswaps if a new question about it is posted. – ilovevolatility Nov 28 '19 at 15:09

Volatility Skew is generally quoted in terms of Risk Reversals. I know that for FX products and because of Delta stickiness, the quoted Risk reversals are regarding 10% and 25% Delta.

Edit: To complete my answer, Skew results from a difference in terms of offer and demand for calls/Puts. So the best way to track it's changes over time would be by an absolute change of Risk Reversals. This absolute change is for instance what the Fundemantal Review Of the Trading Book (FRTB) requires to monitor...