The volatility smile

The Black–Scholes model implies that all options on the same asset have identical
implied volatilities, regardless of time to maturity and moneyness. However, there are systematic ‘biases’ in the implied volatilities of options on most assets. This is a
convenient if somewhat misleading label, since the phenomena in question are biased
only from the point of view of the Black–Scholes model, which neither dealers nor
academics consider an exact description of reality:
Ω Implied volatility is not constant but changes constantly.
Ω Options with the same exercise price but different tenors often have different
implied volatilities, giving rise to a term structure of implied volatility and
indicating that market participants expect the implied volatility of short-dated
options to change over time.
Ω Out-of-the money options often have higher implied volatilities than at-the-money
options, indicating that the market perceives asset prices to be kurtotic, that is,
the likelihood of large moves is greater than is consistent with the lognormal
distribution.
Ω Out-of-the money call options often have implied volatilities which differ from
those of equally out-of the money puts, indicating that the market perceives the
distribution of asset prices to be skewed.
The latter two phenomena are known as the volatility smile because of the
characteristic shape of the plot of implied volatilities of options of a given tenor
against the delta or against the exercise price.