  In this paper we study the short-time behavior of the at-the-money implied
volatility for European and arithmetic Asian call options with fixed strike
price. The asset price is assumed to follow the Bachelier model with a general
stochastic volatility process. Using techniques of the Malliavin calculus such
as the anticipating It\^o's formula we first compute the level of the implied
volatility when the maturity converges to zero. Then, we find a short maturity
asymptotic formula for the skew of the implied volatility that depends on the
roughness of the volatility model. We apply our general results to the SABR and
fractional Bergomi models, and provide some numerical simulations that confirm
the accurateness of the asymptotic formula for the skew.
