Why does Skew measure remain more-or-less constant for Listed Expiries?

Question

I have looked at the Variance Swap Papers published by GS-VarSwap and JPM-VarSWap where they talk about approximation to VarSwap strike using ATMF vol and Skew (slope of the volatility skew for 90-110 strikes).

But, I have also come across another 'skew measure' which is defined as $$ \mathrm{skew} = \frac{\sqrt{T}(\sigma_1 - \sigma_2)}{\log(K_1/K_2)}. $$ I understand that $\sqrt{T}$ makes the vol difference 'normalized' in maturity-space. And to my surprise, this measure remains almost constant for mid-to-long term maturities (>6M).

My questions are

1. What could be the assumption behind taking log-strikes instead of absolute strikes?
2. What is the intuition behind this measure being constant for different maturities for a given underlying?

Answer 1

This is a common convention. If your spot is $S$ and you're looking at options maturity in $T$, it is natural to look at the the strikes $S_\pm=S.exp^{-\frac12\sigma^2T\pm\alpha\sigma\sqrt T}$ for a fixed $\alpha$. So your skew measure will be something like $$ \frac{\sqrt T(\sigma_{S_+} -\sigma_{S_-} )}{\log (S_+/S_-)} = \frac{\sqrt T(\sigma_{S_+} -\sigma_{S_-} )}{2\alpha\sigma\sqrt T} = \frac{\sigma_{S_+} -\sigma_{S_-}}{2\alpha\sigma} $$

Which is as intrinsic as could be.

Think about it this way: 1 week before expiry a 110% call is way more OTM than a 110% call expiring in one year, just because 110% is very far if you've only got one week left. Another way to look at it is that you want to compare the difference in vol for a given change in delta rather than in % of the spot. And clearly the $\alpha$ is strongly tied to the delta.