  In this paper we study short-time behavior of the at-the-money implied
volatility for Inverse and Quanto Inverse European options with fixed strike
price. The asset price is assumed to follow a general stochastic volatility
process. Using techniques of the Malliavin calculus such as the anticipating
Ito's formula we first compute the level of the implied volatility of the
option 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.
