Before revision:  at no point do we need knowledge of the (explicit, but cumbersome) closed form expression of the Fourier transform of \log S_{T}  (equivalently: Mellin transform of S_{T} \% ) . Secondly, our analysis reveals a new parameter (" \textit{critical slope " \% ), defined in a model free manner, which drives the second and higher order terms in tail- and implied volatility expansions.
After revision:  at no point do we need knowledge of the (explicit, but cumbersome) closed form expression of the Fourier transform of \log S_{T} (equivalently: Mellin transform of S_{T} ); what matters is that these transforms satisfy ordinary differential equations of Riccati type . Secondly, our analysis reveals a new parameter (" \textit{critical slope " \% ), defined in a model free manner, which drives the second and higher order terms in tail- and implied volatility expansions.