  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
difference 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 tells that
implied variance is at most asymptotically linear in log-strike, other
celebrated results for exact smile asymptotics such as Benaim and Friz (09) or
Gulisashvili (10) do not apply in this setting--essentially due to the
breakdown of Put-Call duality.
