  We consider the asymptotic behavior of the implied volatility in stochastic
asset price models with atoms. In such models, the asset price distribution has
a singular component at zero. Examples of models with atoms include the
constant elasticity of variance model, jump-to-default models, and stochastic
models described by processes stopped at the first hitting time of zero. For
models with atoms, the behavior of the implied volatility at large strikes is
similar to that in models without atoms. On the other hand, the behavior of the
implied volatility at small strikes is influenced significantly by the atom at
zero. S. De Marco, C. Hillairet, and A. Jacquier found an asymptotic formula
for the implied volatility at small strikes with two terms and also provided an
incomplete description of the third term. In the present paper, we obtain a new
asymptotic formula for the left wing of the implied volatility, which is
qualitatively different from the De Marco-Hillairet-Jacquier formula. The new
formula contains three explicit terms and an error estimate. We show how to
derive the De Marco-Hillairet-Jacquier formula from our formula, and compare
the performance of the two formulas in the case of the CEV model. The resulting
graphs show that the new formula provides a notably better approximation to the
smile in the CEV model than the De Marco-Hillairet-Jacquier formula.
