The Empirical Smile — Fat Tails, Skew, and the Limits of Black-Scholes

Three things are established here. First, the log-normal distribution that Black-Scholes assumes is demonstrably wrong: equity returns have fat tails, and volatility clusters. Second, the implied volatility surface $\Sigma(K,T)$ is the market's pricing summary — a grid of single-parameter fits to observed option prices — and its shape encodes the collective market assessment of tail and skew risk. Third, any model for the options market must reproduce these features; a model that fails to match the smile will systematically misprice exotic options even when it correctly prices the vanilla hedge instruments.

• state the four stylised facts of equity return distributions and explain which each implies about the inadequacy of log-normal GBM;
• define implied volatility precisely and explain why it is a "wrong number that gives the right answer";
• describe the qualitative shape of the SPX implied vol surface (skew, curvature, term structure) and its historical evolution;
• explain why the smile forces exotic desk models beyond Black-Scholes;
• calculate a one-period implied vol from an option price in a worked example.

1. 1.1 The four stylised facts
2. 1.2 Implied volatility: a market language, not a model
3. 1.3 The shape of the smile: what the surface reveals
4. 1.4 Why the smile breaks exotic pricing
5. 1.5 Worked example: computing an implied vol