Left-Wing Asymptotics Of The Implied Volatility In The Presence Of Atoms

The paper considers 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 (CEV) 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. De Marco, Hillairet, and 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. In the paper, we show how to derive the De Marco–Hillairet–Jacquier formula from the new formula, and compare the performance of the two formulas in the case of the CEV model. The graphs included in the paper show that the new asymptotic formula provides a notably better approximation to the implied volatility at small strikes in the CEV model than the De Marco–Hillairet–Jacquier formula.