What is a Volatility Smile?
A volatility smile is the U-shape of implied volatility plotted against strike for a single expiry — wings higher than ATM. It reflects the fat-tailed reality of return distributions.
Definition
A vol smile is the pattern in which both out-of-the-money puts AND out-of-the-money calls trade at higher implied vol than at-the-money options. Plotted, the curve looks like a smile (concave up). On FX, smiles are typically symmetric. On equity indices, the smile is asymmetric — puts have a higher wing than calls — and is called a "smirk" or "skew." On single stocks, the shape depends on the underlying: some show a smile (where both tails are bid), some show a reverse skew (calls more expensive on M&A-prone names). The presence of a smile is the market's acknowledgement that the lognormal assumption underlying basic pricing models is false — actual returns have fat tails, and the wings deserve more vol.
Why it matters & how it's calculated
A volatility smile shape is constrained by no-arbitrage: the second derivative of the call-price curve with respect to strike must be non-negative (butterfly arbitrage), and total variance must be non-decreasing in expiry (calendar arbitrage). These constraints rule out wild smile shapes, but leave a lot of freedom. Common parametric fits include stochastic-vol-inspired smile parameterisations, their arbitrage-free variants, and direct local-vol calibrations. Each gives a smile that prices the listed strikes correctly and interpolates to non-listed strikes via a model that may or may not match the dynamics you care about. The "smile dynamics" — how the smile moves when spot moves — is the actual P&L driver: a "sticky-strike" smile (each strike's IV stays constant when spot moves) gives different deltas than a "sticky-delta" smile (each delta's IV stays constant). Most real markets are between the two, with the actual sticky behaviour shifting by regime. A vol book that delta-hedges with the wrong sticky assumption bleeds steadily through the smile dynamics.
Worked example
SPX 1-month surface: 4,800 put IV = 22%, 4,900 put IV = 19%, 5,000 ATM = 16%, 5,100 call IV = 14%, 5,200 call IV = 16%. Plot it: U-shaped with the left wing higher than the right — a "smirk." The 4,800/5,200 risk reversal (downside more expensive than upside by 6 vol points) is the structural equity skew, the market price of crash insurance.
Related concepts
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