Volatility smile

Implied volatility patterns, known as volatility smiles, deviate from the Black-Scholes model's expectation of a flat surface when plotted against strike prices. These patterns arise because the Black-Scholes formula assumes constant volatility, which does not align with market observations. The volatility smile indicates that options with strike prices far from the underlying asset's forward price exhibit different implied volatilities.

The volatility smile is a graphical representation of how implied volatility changes with strike price. For a given expiration date, options with strike prices significantly different from the underlying asset's forward price show varying implied volatilities. This results in a skewed pattern rather than the expected flat surface predicted by the Black-Scholes model.

Empirical evidence suggests that the standard Black-Scholes model's assumption of constant volatility and log-normal distributions of underlying asset returns does not hold true in practice. The observed volatility smiles across different markets, such as equity options in American markets post-Crash of 1987, indicate that market participants reassess probabilities of fat-tail events, leading to higher prices for out-of-the-money options. This anomaly highlights the deficiencies of the Black-Scholes model in accurately pricing options.