Volatility skew and smile

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If one pricing model fit the market perfectly, every strike on a stock would show the same implied volatility. In practice they do not. Plotting IV against strike for a single expiration produces a curve — a smile when both wings sit above the middle, a skew when one side is clearly higher.

In equities the usual pattern is a put skew: strikes below the current price carry higher IV than strikes above it. The option market, in effect, prices sharp drops as more likely — or more worth insuring — than rallies of the same size.

Options Band measures this with a 25-delta skew for each expiration, comparing an out-of-the-money put and call at matched moneyness.

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Delta is the standard yardstick for how far out-of-the-money an option sits, so skew is usually quoted between options of matched delta rather than matched dollar distance. A worked example for one expiration:

• 25-delta put IV: 35%
• At-the-money IV: 30%
• 25-delta call IV: 27%

The 25-delta skew is 35% − 27% = 8 vol points, put over call. The downside option is priced as if the stock were meaningfully jumpier on the way down than on the way up, even though both options sit a comparable probability-distance from the current price.

Why the tilt exists is one of the older questions in the field, and the usual explanations stack rather than compete:

• Hedging demand. Institutions hold stock and buy puts as insurance; persistent demand supports downside IV.
• The leverage effect. Falling prices raise a firm's debt-to-equity ratio, and equities have historically been more volatile in declines than in advances.
• Fat tails. Real return distributions produce large drops more often than the lognormal model assumes, and skew is how option prices absorb that.

Skew varies by name and by expiration, and it moves — typically steepening when markets fall. Reading a stock's 25-delta skew alongside its term structure on the volatility page gives a fuller picture than any single IV number, since two stocks with identical IV30 can price very different shapes of risk.

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Skew is commonly summarized by two quoted structures: the risk reversal (25-delta call IV minus 25-delta put IV, negative in most equity markets) and the butterfly (average wing IV minus at-the-money IV), capturing the smile's tilt and curvature respectively.

RR25 = IV(25Δ call) − IV(25Δ put); BF25 = (IV(25Δ call) + IV(25Δ put)) ÷ 2 − IV(ATM)

Through the Breeden-Litzenberger relationship, the full smile encodes the risk-neutral distribution of the underlying at expiration: a put skew corresponds to a left tail fatter than lognormal. The smile also raises a dynamics question — whether IV attaches to fixed strikes ("sticky strike") or to moneyness ("sticky delta") as the underlying moves — and the honest answer is regime-dependent.

Misconceptions: skew is not evidence that particular strikes are mispriced; it is the market declining to adopt the constant-volatility assumption of Black-Scholes. A steep skew is not a crash forecast either — persistent equity skew has been present through calm decades. And comparing skew across names or tenors requires matched delta conventions, since fixed-strike comparisons confound skew with moneyness.