What are Volatility Skew and Smile?

By EC Assets Research Team · Published 15 May 2026 · Updated 15 October 2026

Volatility Skew and Smile: Skew means implied volatility differs by strike: equity index options price downside puts above at-the-money calls, a one-sided slope also called the smirk. A smile lifts both wings, the typical FX shape. Both patterns price the crash risk that a flat Black-Scholes surface ignores.

What Skew and Smile Measure

The Black-Scholes model assumes one volatility for all options on an underlying with the same expiry. Markets do not price them that way. Inverting the model strike by strike gives a different implied volatility for each strike, and plotting those numbers shows a characteristic shape.

A smile is the U-shape in which both out-of-the-money puts and out-of-the-money calls trade at higher implied volatility than at-the-money options, common in currencies. A skew is the tilted shape in which low strikes trade at much higher implied volatility than high strikes, typical of equity indices. Across expiries, these slices together form the volatility surface.

How It Works

Desks summarise the shape with two quotes, defined by option delta rather than strike so that they compare across markets and maturities:

$$\text{RR}{25\Delta} = \sigma{25\Delta,\text{call}} - \sigma_{25\Delta,\text{put}}, \qquad \text{BF}{25\Delta} = \frac{\sigma{25\Delta,\text{call}} + \sigma_{25\Delta,\text{put}}}{2} - \sigma_{\text{ATM}}$$

The risk reversal measures tilt; the butterfly measures curvature. For full surfaces, practitioners fit parametric forms such as SVI, which models total implied variance w = σ²T as a function of log-moneyness and can be constrained to rule out arbitrage (Gatheral and Jacquier, 2014).

Three forces produce the equity skew. Hedging demand: investors are structurally long equities and buy index puts, while call overwriting adds supply on the upside; Bollen and Whaley (2004) link the shape of index smiles to net buying pressure. The leverage effect: falling prices raise leverage and volatility, as Black (1976) observed, so prices and volatility are negatively correlated. Crash risk: Bates (2000) finds that index option prices since 1987 embed a premium for large downward jumps.

Stochastic volatility models capture the first two through a negative correlation between price and volatility; jump models add the steep short-dated wings.

Worked Example

S&P 500 options with 30 days to expiry, index at 5,000, interest and dividends ignored. The implied volatilities are illustrative of a calm market; the deltas are computed from them with Black-Scholes:

Each delta uses its own strike's implied volatility.

Interpolating linearly in delta, the 25-delta put trades at about 18.6 and the 25-delta call at about 14.0. The 25-delta risk reversal is therefore about -4.6 volatility points and the butterfly about 1.3.

The cost of the skew is clearest in the wing. The 4,500 put is worth 9.76 index points at its implied volatility of 24.5 percent. Priced at the at-the-money 15 percent it would be worth 0.48, so the skew makes this protection about 20 times as expensive as a flat-volatility model suggests.

The payoff builder shows how a risk reversal, long the call and short the put, turns this price difference into a position.

When It Applies (and Limitations)

No-arbitrage limits the shape. Call prices must be convex in strike, which rules out negative probability densities, and total implied variance must not fall with maturity. Naive fits to quoted volatilities can violate both.

Wing quotes are thin. Deep out-of-the-money options trade rarely, so wing volatilities often reflect indicative quotes. Surfaces calibrated to them can produce misleading risk numbers.

The surface moves with spot in different ways. Under a sticky-strike view each strike keeps its volatility as the index moves; under a sticky-delta view each moneyness keeps its volatility. Which one holds changes hedge ratios, and markets shift between them, especially in sharp selloffs.

The shape differs by market. Single stocks usually show flatter skews than indices, and stocks with takeover or speculative upside can show richer calls. Commodities exposed to supply shocks can tilt toward calls, while currency smiles lean toward whichever side investors fear, such as calls on safe-haven currencies. An index-style reading does not transfer automatically.

Term structure interacts. Short-dated skews are steeper than long-dated ones, so a trade across expiries carries exposure to both the level and the term structure of skew.

Why It Matters for Institutional Investors

Hedging cost. A fund protecting an equity book against a 10 percent fall buys options near the 90 percent strike, so the relevant price is the wing volatility, not the at-the-money level. Skew can steepen and raise the cost of protection even when the VIX barely moves.

Variance and volatility products. The replicating strip of a variance swap weights low strikes most heavily, so a steeper put skew lifts the variance strike above at-the-money implied volatility.

Risk beyond delta and vega. A book hedged for at-the-money vega still loses money when the shape changes. Vanna, the sensitivity of delta to volatility, and volga, the sensitivity of vega to volatility, measure that exposure and are monitored on any complex option book.

Stress signal. A steepening put skew shows that the market is paying more for tail protection, often before headline volatility indices react.

References

1. Rubinstein, M. (1994). Implied Binomial Trees. Journal of Finance, 49(3), 771-818.
2. Bates, D. S. (2000). Post-'87 Crash Fears in the S&P 500 Futures Option Market. Journal of Econometrics, 94(1-2), 181-238.
3. Bollen, N. P. B., & Whaley, R. E. (2004). Does Net Buying Pressure Affect the Shape of Implied Volatility Functions? Journal of Finance, 59(2), 711-753.
4. Black, F. (1976). Studies of Stock Price Volatility Changes. Proceedings of the 1976 Meetings of the American Statistical Association, Business and Economic Statistics Section, 177-181.
5. Gatheral, J., & Jacquier, A. (2014). Arbitrage-Free SVI Volatility Surfaces. Quantitative Finance, 14(1), 59-71.
6. Gatheral, J. (2006). The Volatility Surface: A Practitioner's Guide. Wiley.