SuperDerivatives founder is catching up. Very mathematically convoluted version that is hard to generalize for higher order corrections/jumps and build the rest of the machinery upon, but correct intuition and interesting perspective.
"We show that the three quantities that determine the volatility smile include the “pivot” volatility for the expiration (which can be thought of as the At The Money (ATM) volatility), the expected variance of the pivot volatility from inception to the expiration, and the expected covariance of the pivot volatility and the underlying asset/rate from inception to the expiration. After testing against a wide selection of asset across all asset classes, we conclude that the options market considers all liquid financial assets as though they obey the same type of new probability density function generated by this model. Moreover, we see that different asset classes are governed by different regions of the parameter zone leading to the varying shape of volatility smiles across asset classes."