The Volatility Smile & Skew
Implied vol varies by strike because the market prices crashes — 1987 created the skew, and fat tails created the smile.
Before October 1987, desks quoted options through Black–Scholes with volatility roughly flat across strikes. The crash changed the picture overnight: out-of-the-money puts began trading at much higher implied volatilities than at-the-money options, producing the downward equity skew, while currency options kept a more symmetric smile around the forward. Read economically, the skew is the price of crash insurance and the smile is compensation for fat-tailed moves in either direction. Both falsify the lognormal backbone that constant-volatility models assume. Every model later in this track — local, stochastic, SABR — exists to reproduce this stubborn empirical fact.