Before revision: We study the shapes of the implied volatility when the underlying distribution has an atom at zero . We show that the behaviour at small strikes is uniquely determined by the mass of the atom up to the third asymptotic order, under mild assumptions on the remaining distribution on the positive real line.
After revision: We study the shapes of the implied volatility when the underlying distribution has an atom at zero and analyse the impact of a mass at zero on at-the-money implied volatility and the overall level of the smile. We further show that the behaviour at small strikes is uniquely determined by the mass of the atom up to the third asymptotic order, under mild assumptions on the remaining distribution on the positive real line.