Volatility Surface, Smile & Skew

A first options class often quotes one implied volatility, as in “this option is 30%.” A listed chain is a set of contracts. In a classroom chain the 90 strike might be 34%, the 100 strike 28%, and the 110 strike 25%. Change the expiration and those three numbers move again.

The market can assign a different implied volatility to each mix of strike, moneyness, expiration, and option type. The collection of those values is the volatility surface. One volatility cannot stand in for the whole chain.

Realized volatility asks how much the underlying moved. It is measured from historical returns over a window you choose: 5 days, 20 days, 60 days, or 252 days. A path such as $100, $102, $99, $103, $101 is the raw material. The window changes the answer. Annualized, that figure is the standard deviation of returns on the Return risk lesson.

Implied volatility runs the other way. Start from the option’s market price, put it through a pricing model, and solve for the volatility that makes the model print that price. If 25% produces $5.20 and the option trades at $5.80, the volatility consistent with $5.80 on these classroom numbers is about 29%.

That 29% is an input consistent with the quote under the model’s assumptions. It does not mean the underlying is guaranteed to realize 29%. An option marked at 30% implied volatility can later show 24% realized over the relevant window. The gap can come from a risk premium, supply and demand, hedging demand, uncertainty, liquidity, event risk, positioning, or demand for tail protection.

Implied volatility is not required to sit above realized volatility. The relationship changes through time.

Hold the underlying fixed and read implied volatility by expiration. A classroom term structure: 7 days 35%, 30 days 29%, 60 days 27%, 90 days 26%, 1 year 28%. Expiration changes the number even though the stock is the same. That relationship is the volatility term structure.

The shape has a few common forms.

• Upward sloping: longer-dated options carry higher implied volatility.
• Downward sloping: shorter-dated options carry higher implied volatility.
• Humped: an intermediate maturity sits above its neighbors.
• Event-driven: one expiration jumps because it contains a known event. Earnings in 20 days can leave 30-day implied volatility at 35% while 90-day implied volatility sits at 27%, because the short option is closer to the event.

Hold expiration near 30 days and change the strike. A classroom smile: $80 at 34%, $90 at 30%, $100 at 25%, $110 at 30%, $120 at 35%. Plot that and the wings sit above the middle. The curve is a volatility smile.

Different strikes can carry different implied volatilities. A simplified pricing framework that starts from one constant volatility is a teaching model. Listed markets price a curve.

Skew is the same strike axis when the curve is not symmetric. A classroom equity skew: $80 at 38%, $90 at 32%, $100 at 27%, $110 at 24%, $120 at 22%. Implied volatility falls as the strike rises.

In many equity and equity-index markets, downside puts trade at higher implied volatility than comparable upside calls. Portfolio hedges, market-maker inventory, supply and demand, leverage effects, crash-risk pricing, liquidity, and institutional positioning can all show up in that shape. One of those stories does not explain every market or every period.

Smile names the curvature across strikes. Skew names the slope, the lack of symmetry. A listed curve can carry both: wings above the middle, and the low-strike wing above the high-strike wing.

Put the two axes together. Strike runs one way, expiration the other, and height is implied volatility. That landscape is the volatility surface. Twenty strikes and ten expirations are two hundred combinations, and each one can have its own implied volatility.

A classroom stock at $100, three expirations and three strikes. There is no single correct implied volatility in the grid. The cells are the surface.

A pricing system has to pick a volatility for the contract in front of it. A 90-day $90 put on the classroom grid is 29%. Saying the stock’s volatility is 27% skips the strike and the expiration. The surface is the lookup: this strike, this expiration, this moneyness, in this market.

Vega measures how an option’s value responds when implied volatility changes. A book that is long short-dated puts, long long-dated calls, long at-the-money options, and long deep out-of-the-money puts does not have one uniform volatility exposure. “Vega is +500” still leaves the question of where on the surface that vega sits.

Desks often bucket that exposure.

• By maturity: 1 week, 1 month, 3 months, 6 months, 1 year.
• By moneyness: deep in the money, in the money, at the money, out of the money, deep out of the money.

Tomorrow’s surface can differ from today’s in level, slope, curvature, term structure, and skew. Portfolio profit and loss can change when the underlying barely moves, because the map itself moved.

Level is a broad shift, such as 25% to 30% across much of the surface. Slope is how steeply volatility changes from a low strike to a high strike, for example 40% down to 25%. Curvature is the smile’s bend: if the wings richen relative to at-the-money options, curvature has changed.

A book concentrated in downside puts can see put prices rise when downside implied volatility rises, even while the stock is quiet. Volatility risk is read off the surface, not only off the underlying price.

Calibration fits a model, or a surface, to observed option quotes. The path is market quotes, then cleaning, then implied volatility, then a surface, then a fit, then pricing, Greeks, and risk. Chains are messy: bid and ask spreads, missing strikes, illiquid names, stale prices, corporate actions, dividend uncertainty, and uneven liquidity across expirations. A raw quote is not automatically a clean surface point.

Interpolation estimates a missing point between observations. If the 90 strike is 30% and the 100 strike is 27%, a 95 strike with no reliable quote needs an estimate between them. Extrapolation estimates beyond the listed range. Strikes observed from $70 to $130 do not contain a $150 wing. That estimate is more sensitive to the model, and the wings are where it matters most.

Data quality is part of the surface. An option bid at $1.00 and offered at $3.00 does not hand the desk one obvious price. The choice of quote is part of building the map.

1. Realized volatility is what the underlying already did, over a chosen window.
2. Implied volatility is the volatility consistent with an option’s market price under a pricing model.
3. Term structure is how that implied volatility changes with expiration.
4. A smile is curvature across strikes.
5. Skew is asymmetry across strikes.
6. The surface is strike, expiration, and implied volatility together.
7. One volatility number does not describe the chain. The institutional question is this strike, this maturity, and the surface around it.

The chain of ideas runs in this order:

1. Option price
2. Implied volatility
3. Smile and skew
4. Term structure
5. Volatility surface
6. Calibration
7. Pricing, Greeks, and risk

The risk-neutral pricing lesson keeps the shorter sketch of the same surface, next to Black–Scholes and calibration. These pictures stay classroom examples.

The surface tells you what the chain is quoting by strike and expiration. It does not say how volatility will move tomorrow, and it is not yet a local-vol or Heston process.

Next: local and stochastic volatility, then calibration. Dupire and Heston live on those pages, not here.