Implied Volatility & the Smile

The σ that, when input into the Black–Scholes formula, reproduces the market price of a European option. Found by numerical inversion (e.g., Newton on vega) of C_BS(σ) = C_market.

The U-shaped pattern of implied volatilities plotted against strike (or moneyness) at a fixed maturity: deep ITM and OTM options trade at higher implied vols than at-the-money options.

An asymmetric smile where OTM puts (low-strike calls) trade at higher implied vols than OTM calls. Dominant in equity-index markets since the 1987 crash; reflects crash-risk premium and leverage effects.

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 smile; slices in T give the term structure.

BS assumes constant σ and log-normal returns. Real returns show fat tails, jumps, volatility clustering, and the leverage effect — producing non-flat implied-vol surfaces that BS cannot match.

ν = ∂C/∂σ — sensitivity of option price to implied volatility. For a BS European option: ν = S · φ(d₁) · √T, where φ is the standard normal PDF. Peaks near ATM.