Volatility smile is a graphical image that reflects the dependence of implied option volatility (with one expiry date) on the strike price and looks like a smile. This shape of the curve is observed in the Japanese stock market, i.e. Japanese investors do not demand a higher premium for put options than for call options.
A volatility smile shows the relationship between implied volatility and strike price. Unlike the other form of the volatility skew, the lowest point on the smile is associated with the forward price of the asset. This means that ATM options have the lowest implied volatility.
As the forward price of the underlying asset rises (or falls), the whole smile of volatility shifts to the right (or left).
The smile of volatility first appeared after the stock market fell in 1987. Before that, all options’ strikes were trading with the same implied volatility.