A straightforward consequence of Lee's lemma on large and small strike implied volatility bounds and Fukasawa's theorem on the normalizing transformations of implied volatility is that the zero vanna strike does not always exist

The Alos decomposition formula is written down, but making use of Bachelier price formula instead of the Black-Scholes price formula. Due to the linearity of the Bachelier formula in volatility at the at-the-money strike, the decomposition formula gives an exact expression for the price of a...