strike → out of the money
expiry → longer
implied vol ↑
Helix  Terminal ← Exhibition
The smile at one expiry · Heston ● SVI —
—
Short-dated skew, ∂σ/∂k at the money
—
—
SVI, fitted to every expiry
—
—
Heston 1993 · Gatheral 2004 · Forde–Jacquier 2009

—The market's
smile.

Black–Scholes says every strike should trade at one volatility — a flat sheet. Real option prices bend it: downside puts cost more, and the bend is sharpest at short expiries. Let volatility itself wander, and the sheet curls into exactly this shape. Switch that off, and it goes flat.

Press Black–Scholes and the surface flattens · then drag ρ through zero and watch the skew change sign
ρ spot–vol−0.70
vol of vol ξ0.60
spot vol20%