Implied Volatility & the Smile

Implied volatility

The σ that, when input into the Black–Scholes formula, reproduces the market price of a European option. Found by numeri

Volatility smile

The U-shaped pattern of implied volatilities plotted against strike (or moneyness) at a fixed maturity: deep ITM and OTM

Volatility smirk / skew

An asymmetric smile where OTM puts (low-strike calls) trade at higher implied vols than OTM calls. Dominant in equity-in

Volatility surface

The two-dimensional map σ_imp(K, T) of implied volatility as a function of strike K and maturity T. Slices in K give the

Why Black–Scholes fails empirically

BS assumes constant σ and log-normal returns. Real returns show fat tails, jumps, volatility clustering, and the leverag