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