How Does the Black-Scholes Model’s Assumption of Constant Volatility Fail to Capture the Volatility Smile?

The Black-Scholes model assumes that the underlying asset's volatility is constant across all strike prices and time to maturities. The existence of the volatility smile, which shows that implied volatility varies systematically with strike price, directly contradicts this assumption.

The model fails because it does not account for the market's perception of "fat tails," or the higher-than-normal probability of extreme price movements.