Before revision:  These formulae, by definition arbitrage-free , allow us to quantify the impact of the mass at zero on currently used implied volatility expansions. In particular we discuss how much those expansions become erroneous .
After revision:  These formulae, by definition arbitrage-free , allow us to quantify the impact of the mass at zero on existing implied volatility approximations, and in particular how correct/erroneous these approximations become .