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...

Using nine years of options data, we reconstruct a time-series for the at-the-money implied volatility skew of the SP500 index, and analyse its term structure. Though the well-known power-law dependence reported by several works appears to be satisfied for longer maturities, we observe a change...