Volatility Surface

The volatility surface is a three dimensional picture of implied volatility for one underlying: strike (or moneyness) along one axis, time to expiry along another, and implied volatility as the height. Each slice at a fixed expiry is a smile or skew; each slice at a fixed strike is a term structure. Option traders, risk managers and pricing models all rely on the surface, because it summarises how the market prices every option on that underlying at once.

The two slices#

A sample equity index surface#

Illustrative figures. Typical features:

• Skew is steepest at short expiries and flattens with time.
• At the money volatility rises with expiry in calm markets (upward term structure).
• Long dated volatility is more stable than short dated.

How a surface is built#

1. Collect option quotes across strikes and expiries; use mid prices and remove stale or illiquid quotes.
2. Compute implied volatility for each option using a suitable model (Black Scholes, Black 76, or a tree for American options).
3. Use out of the money options on each side, since they are usually more liquid and less affected by dividends and early exercise.
4. Fit a smooth model to each expiry, such as SVI or SABR.
5. Interpolate across time, usually in total variance.
6. Check for arbitrage: no negative butterfly prices and no falling total variance across expiries.

See Volatility Interpolation and Extrapolation.

What the surface is used for#

• Pricing any option, including strikes and expiries that are not quoted.
• Marking portfolios to market consistently.
• Risk management: measuring exposure to shifts, tilts and twists of the surface.
• Exotic pricing: local volatility models are calibrated directly to the surface. See Local Volatility.
• Relative value: finding options that look expensive or cheap compared with neighbours.

Ways the surface moves#

Vega by bucket#

Because different parts of the surface move differently, traders measure vega for each expiry and strike bucket, not just in total. A book can be vega flat overall while being long short dated volatility and short long dated volatility, a position that can still lose money if the term structure twists. See Managing Portfolio Greeks.

Surfaces in other markets#

• FX: quoted in delta terms with at the money, risk reversal and butterfly at each expiry.
• Interest rates: a cube of expiry, underlying swap tenor and strike, often called the swaption volatility cube. See SABR Model.
• Commodities: each futures contract month has its own smile, linked to its own underlying price.

Frequently asked questions#

What is a volatility surface?#

A map of implied volatility for every strike and expiry of options on one underlying, combining the smile and the term structure.

Why is the volatility surface important?#

It lets traders price any option consistently, measure detailed volatility risk and calibrate models for exotic options.

How does the volatility surface change in a selloff?#

Usually short dated volatility rises most, the term structure inverts and put skew steepens.

Next, look closely at the time dimension in Volatility Term Structure.