Volatility Smile and Surface Dynamics
Black-Scholes assumes one volatility for an underlying. If that held, every strike and expiry would imply the same .
They do not. Plotting implied volatility against strike gives a smile or skew, and the shape is stable, well understood, and full of information.
The shapes
Smile: implied vol higher at both wings than at the money. Common in FX, where large moves in either direction are plausible.
Skew (or smirk): implied vol falling steadily as strike rises. The standard shape in equity indices, where low-strike puts carry much higher implied vol than high-strike calls.
Why equity skew exists
Three reinforcing explanations, all partly true.
Real return distributions are skewed. Equity markets fall faster than they rise. Crashes are sharper than rallies, so the left tail is genuinely fatter than a log-normal model allows, and options must price it.
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