We approximate normal implied volatilities by means of an asymp-
totic expansion method. The contribution of this paper is twofold:
to our knowledge, this paper is the rst to provide a uni ed approx-
imation method for the normal implied volatility under general local
stochastic volatility models. Second, we compared our method with
the Monte-Carlo simulations by using the parameters calibrated to the
actual market data and con rmed the accuracy.