FRM Exam Part II · Volatility Smiles and Volatility Surfaces

Volatility Term Structure and Volatility Surfaces Explained

Q: What is a volatility surface?

It is a three-dimensional plot of implied volatility against strike and time to maturity. Traders use it to see how the market prices options across strikes and maturities. It is built from quoted option prices.

Q: What does an upward-sloping volatility term structure mean?

Long-dated implied volatility is higher than short-dated. This usually happens when current volatility is below its long-run level, so the market expects it to rise. It reflects mean reversion.

Q: When is the volatility term structure downward sloping?

It is typically inverted when current volatility is high, for example in a market crisis. Short-dated options carry higher implied volatility because volatility is expected to fall back toward its long-run average.

Q: How do you use the volatility surface to price a nonstandard option?

Find the implied volatility for the option's strike and maturity by interpolating between quoted grid points. Then use that volatility in Black-Scholes-Merton. This keeps the price consistent with the market's quoted options.

Updated 11 October 2026 · Fact-checked

A volatility surface plots implied volatility against both strike and maturity. The term structure is the slice across maturities at one strike. To price an option with a nonstandard strike or maturity, interpolate implied volatility from the surface, then put that volatility into Black-Scholes-Merton.

Understand The Volatility Term Structure and Volatility Surfaces

Black-Scholes-Merton assumes one constant volatility. Real option prices disagree. When you back out implied volatility from market prices, it changes with strike (the smile or skew) and with maturity (the term structure).

The volatility term structure shows implied volatility for options of different maturities, usually at the same strike or moneyness. It is upward sloping when long-dated volatility is above short-dated volatility. It is downward sloping when short-dated volatility is higher.

The usual reason for the shape is mean reversion. Volatility tends to return to a long-run level. If current volatility is low, expected future volatility is higher, so the term structure slopes up. If current volatility is high, as in a crisis, the structure slopes down (inverts). Short-dated implied volatility moves more than long-dated.

The volatility surface combines both dimensions. One axis is strike (or K/S, or delta), the other is time to maturity, and the height is implied volatility. Traders quote it on a grid, for example at a few strikes and maturities.

Smile effects are usually larger for short maturities and flatten for long maturities. So the surface is not a flat sheet. You use it as a lookup table. For a strike or maturity not on the grid, you interpolate between nearby grid points. Then you price with Black-Scholes-Merton using that implied volatility.

Key formulas to remember

Implied volatility: σ_imp solves: Market price = BSM(S, K, T, r, q, σ_imp); Each strike and maturity has its own σ_imp. The BSM price is a quoting tool, not a belief in constant volatility.
Moneyness: K ÷ S (or K ÷ F for forward moneyness); Surfaces are often indexed by moneyness so that the grid stays relevant as the spot price moves.
Linear interpolation in strike: σ(K) = σ₁ + (K − K₁) ÷ (K₂ − K₁) × (σ₂ − σ₁), for K₁ ≤ K ≤ K₂; Use it for a strike between two quoted strikes at the same maturity.
Interpolation in maturity (total variance): σ²(T) × T = linear in T between T₁ and T₂, using σ₁²T₁ and σ₂²T₂; Interpolating total variance is a common refinement. Simple linear interpolation of σ is also accepted in basic exam questions. Follow the method the question states.
Term structure shape rule: σ(long) > σ(short): upward sloping. σ(long) < σ(short): downward sloping (inverted); Usually: low current volatility gives upward slope, high current volatility gives downward slope.

How to solve The Volatility Term Structure and Volatility Surfaces questions

Use this method for any question on volatility term structure or surfaces.

1Identify what is given: strikes, maturities and implied volatilities, as a table or grid.
2Decide what is asked: the shape, an interpolated volatility, or an option price.
3For shape, compare long-dated and short-dated volatility at the same strike or moneyness. Higher long end means upward sloping.
4For interpretation, link the shape to current volatility versus its long-run level (mean reversion).
5For a nonstandard option, locate the two grid points that bracket its strike and maturity.
6Interpolate. Do strike first at each bracketing maturity, then maturity. Use the method the question names.
7Put the interpolated volatility into Black-Scholes-Merton and price the option.
8Sanity check: the volatility should lie between the bracketing values, and the price should be sensible.

Quickest way: Bracket, interpolate, then price

When to use it: Use when a grid is given and you need a volatility for a strike or maturity that is not on it.

Circle the two nearest strikes and two nearest maturities.
If only one dimension is off the grid, do a single linear interpolation.
Compute the weight w = (target − low) ÷ (high − low).
Volatility = low vol + w × (high vol − low vol).
Eliminate any option that gives a result outside the bracketing values.

Common mistakes in The Volatility Term Structure and Volatility Surfaces

Saying an upward-sloping term structure means volatility is rising over time for sure.
The slope is read as a forecast.
Fix: It shows market-implied expected average volatility for each horizon. It signals current volatility below the long-run level, not a certainty.
Thinking a high short-term volatility and an inverted term structure are unrelated.
Shape and level are studied separately.
Fix: After shocks, short-dated volatility jumps above long-dated volatility because of mean reversion, so the term structure inverts.
Using a single flat volatility to price every option on the same underlying.
BSM is taught with one σ.
Fix: Use the implied volatility for that strike and maturity from the surface.
Extrapolating the interpolated volatility outside the quoted grid without comment.
The formula still runs.
Fix: Interpolation is reliable inside the grid. Flag extrapolation as less reliable.
Interpolating between the wrong grid points, such as non-adjacent strikes.
Rushing through a table.
Fix: Always pick the two points that directly bracket the target.
Assuming the smile is equally pronounced at all maturities.
The surface is treated as a uniform sheet.
Fix: The smile is usually stronger at short maturities and flatter at long maturities.

Worked examples

Example 1

A 3-month option has implied volatility 24% and a 12-month option has 18%, both at the money on an index. Describe the term structure and give the likely market condition.

Show the solution

Compare long and short: 18% (12-month) is below 24% (3-month).
Long-dated volatility is lower than short-dated, so the term structure is downward sloping (inverted).
Mean reversion implies current volatility is above its long-run level.

Answer: The term structure is downward sloping, consistent with a high-volatility, stressed market where volatility is expected to fall back.

Example 2

At a 6-month maturity, the 100 strike has implied volatility 20% and the 110 strike has 17%. Estimate the implied volatility for a 6-month 104 strike using linear interpolation.

Show the solution

Weight w = (104 − 100) ÷ (110 − 100) = 4 ÷ 10 = 0.4.
Volatility difference = 17% − 20% = −3%.
σ(104) = 20% + 0.4 × (−3%) = 20% − 1.2% = 18.8%.
Check: 18.8% lies between 17% and 20%.

Answer: 18.8%, which you then use in Black-Scholes-Merton to price the 104-strike option.

Exam tips

Read the question for which dimension is off the grid: strike, maturity or both.
Link shape to mean reversion: low current volatility gives upward slope, high gives downward slope.
Remember that smiles are more pronounced at short maturities and flatten at long ones.
If a multiple-choice option says BSM is invalid because volatility varies, prefer the answer that treats implied volatility as a quoting convention.
Compute weights carefully and test the answer against the bracketing values.

Practice questions from Volatility Smiles and Volatility Surfaces

The Volatility Term Structure and Volatility Surfaces: frequently asked questions

What is a volatility surface?

It is a three-dimensional plot of implied volatility against strike and time to maturity. Traders use it to see how the market prices options across strikes and maturities. It is built from quoted option prices.

What does an upward-sloping volatility term structure mean?

Long-dated implied volatility is higher than short-dated. This usually happens when current volatility is below its long-run level, so the market expects it to rise. It reflects mean reversion.

When is the volatility term structure downward sloping?

It is typically inverted when current volatility is high, for example in a market crisis. Short-dated options carry higher implied volatility because volatility is expected to fall back toward its long-run average.

How do you use the volatility surface to price a nonstandard option?

Find the implied volatility for the option's strike and maturity by interpolating between quoted grid points. Then use that volatility in Black-Scholes-Merton. This keeps the price consistent with the market's quoted options.