Volatility smile, skew and implied volatility surface

The implied volatility surface represents the IVs recovered from options on the same underlying at a common timestamp. Its two main contractual coordinates are strike K and expiry T:

σ_imp = σ_imp(t₀; K, T | model, inputs, price)

Strike is often normalized against the forward for that expiry:

k = ln(K / F₀,T)

or replaced by a delta coordinate. The choice changes the appearance and dynamics of the surface. An image without axes, timestamp and conventions may look attractive, but it is not a reproducible measure.

Smile, skew and smirk

For one expiry, a slice is observed across strike or moneyness.

• A smile is a curved shape in which the wings have higher IV than the central area, with some degree of symmetry.
• Skew describes a slope or asymmetry: one side of the strike distribution quotes higher IV than the other.
• Smirk is an informal term for an asymmetric shape that combines slope and curvature.

Terminology is not entirely uniform across markets and data vendors. “Negative skew” may mean IV rises toward lower strikes; another report may define the sign through call IV − put IV, yielding a negative number for the same pattern. Formula, delta, option type and sign must accompany the label.

Volatility skew is not the same as the statistical skewness of a return series. The former is a relationship among option-implied prices; the latter is a moment or coefficient estimated from observations. Nor is the Cboe SKEW Index a synonym for any skew: it is an index with its own underlying, methodology and horizon.

Coordinates: strike, moneyness and delta

Comparing the same numerical strike across different underlyings or expiries may convey little. Common coordinates answer specific questions:

A “25-delta” surface should specify whether delta is spot or forward, premium-adjusted or not, and which model computes it. The strike associated with 25 delta can change as spot, time and IV change. Sticky-strike and sticky-delta are therefore scenario assumptions, not market laws.

The other fundamental slice is the term structure: hold a consistent coordinate and move across expiries. Smile and term structure are two cuts through the same surface.

The observed surface is sparse

Exchanges quote discrete contracts, not a continuous sheet. Each series may have a bid, ask, size, last price and settlement. Many far-from-the-money nodes trade infrequently, have no bid or have a wide bid-ask spread. IV is calculated only after a price has been selected and a model has been inverted.

A documented pipeline should retain:

1. the universe of strikes and expiries;
2. timestamp and source of each quote;
3. bid, ask, mid and the rule for last trades;
4. forward, rates, dividends and calendar;
5. model for each contract type and exercise style;
6. filters for zero bids, crossed markets, stale quotes and evident arbitrage;
7. interpolation method and extrapolated regions;
8. surface version and quality controls.

Mid is not always an executable price. Bid and ask surfaces can delimit the economic uncertainty more faithfully than one mid surface. Interpolation fills gaps between nodes; it does not create traded information or eliminate model risk.

Consistency before smoothness

No-arbitrage conditions are checked first in price space. For comparable European options, a call price across strike must satisfy monotonicity and convexity under the relevant assumptions. Across expiry, prices and total variance are examined while accounting for carry, style and settlement. A mathematically smooth fit can still violate these relationships or cross non-executable quotes.

Put-call parity links a call and put with the same strike and expiry. Under consistent European conditions, call and put do not provide two independent IVs at the same K,T: material differences may indicate inconsistent carry inputs, asynchronous quotes, American exercise or data problems, not two separate “opinions” about volatility.

Local measures of slope and curvature

Two common conventions, which must be stated explicitly, are:

25Δ risk reversal = 25Δ call IV − 25Δ put IV

25Δ butterfly = 0.5 × (25Δ call IV + 25Δ put IV) − ATM IV

Risk reversal summarizes asymmetry between the wings; butterfly compares the average of the wings with the center. Neither describes the entire surface, and product conventions may differ.

Neutral example

For a 45-day expiry at the same timestamp:

Under the formulas above, risk reversal is 20% − 28% = −8 volatility points; butterfly is 0.5 × (20% + 28%) − 22% = 2 points. The slice slopes toward the put wing and has curvature relative to ATM. It does not follow that the underlying will fall or that buying puts is attractive: the numbers describe relative prices under the selected model, coordinates and bid-ask.

If the 120-day risk reversal is −4 points, the slope also varies across expiry. Referring only to “the underlying's skew” hides this third dimension.

Surface and position risk

The vega of one leg measures a local response to a shock in its IV. A position spanning strikes and expiries is exposed to non-parallel moves: level, slope, curvature and term structure can change together. A scenario that adds two points to the whole surface is a useful check, but it does not cover rotations, wing compression or a shock concentrated in one expiry.

Delta and gamma also depend on the surface used for repricing. When the underlying moves, it is not known in advance whether the surface will remain fixed by strike, by delta or follow another dynamic. Observed P&L combines the underlying move, passage of time, surface, carry, hedging and execution.

Common errors

1. Calling one number “the stock's IV” without strike and expiry.
2. Confusing volatility skew with return skewness. They are different measures.
3. Treating the interpolated surface as fully traded. Actual nodes are discrete and have bid-ask spreads.
4. Mixing coordinates. Fixed strike, fixed delta and fixed moneyness create different slices.
5. Giving skew sign a universal meaning. Conventions can be opposite.
6. Using asynchronous or stale quotes. The resulting shape may never have existed in the market.
7. Turning the more expensive wing into a directional forecast. The price of protection, premia and demand do not equal a future outcome.

The surface is therefore not a directional forecast of the underlying.

Checklist

1. Which underlying, style, settlement and multiplier define the universe?
2. Do all points belong to the same timestamp?
3. Which coordinate is used on each axis?
4. Which model and inputs transform price into IV?
5. Do bid, ask and mid produce materially different surfaces?
6. Which quotes were filtered, and why?
7. Do interpolation and extrapolation preserve coherent pricing relations?
8. Is the sign convention for risk reversal and skew stated?
9. Do scenarios include non-parallel surface moves?

Sources

• CME Group, Understanding Options: a real-world example on the impact that skew can have on an options position — IV differences across strikes and put-call relationships.
• CME Group, Learn about Convexity — smile, skew and IV curvature.
• Options Industry Council, Understanding Volatility and Options Skew — delta coordinates, term structure and skew.
• Cboe, The Cboe SKEW Index — methodology and history of the SPX implied volatility curve.
• Cboe, S&P 500 SMILE Index Methodology — put, call and delta conventions in a published methodology.
• Cboe, The S&P 500 Spot Volatility Index — an example of measures tied to specific regions of the surface.
• Fischer Black and Myron Scholes, The Pricing of Options and Corporate Liabilities, Journal of Political Economy, 1973 — the classic model and replication principle.
• The Options Clearing Corporation, Characteristics and Risks of Standardized Options — features and risks of standardized contracts.

Related topics

• Options and volatility
• Implied and realized volatility
• Volatility term structure
• Options
• Option vega
• Option delta
• Option gamma
• Statistical skewness of returns
• Bid-ask spread
• Model risk