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 high asymptotic order, under mild assumptions on the remaining distribution on the positive
real line. We investigate the structural di erence with the no-mass-at-zero case, showing how one can{
theoretically{distinguish between mass at the origin and a heavy-left-tailed distribution. We numerically
test our model-free results in stochastic models with absorption at the boundary, such as the CEV process,
and in jump-to-default models. Note that while Lee's moment formula [
25
] tells that implied variance is at
most asymptotically linear in log-strike, other celebrated results for exact smile asymptotics such as [
3
,
17
]
do not apply in this setting{essentially due to the breakdown of Put-Call duality.