FRM Exam Part II · Volatility Smiles and Volatility Surfaces

Models for Volatility Smiles: Jumps, Stochastic Volatility and Alternatives

Updated 11 October 2026 · Fact-checked

Black-Scholes assumes constant volatility and a lognormal price, which gives a flat smile. Jumps and stochastic volatility create fat tails and skew in the implied distribution, so implied volatility varies with strike. Jumps matter most for short maturities. Stochastic volatility matters more for longer maturities. Correlation between price and volatility sets the skew direction.

Understand Models for Smiles: Jumps, Stochastic Volatility and Alternatives

Black-Scholes assumes the asset price follows a lognormal process with constant volatility. If that were true, every option on the same asset and maturity would have the same implied volatility, and the smile would be flat. Real markets show smiles and skews. So the real distribution must differ from lognormal.

Jumps are sudden large price moves. A jump-diffusion model adds random jumps to the usual smooth diffusion. Jumps fatten the tails of the return distribution. Fatter tails make deep out-of-the-money and deep in-the-money options worth more than Black-Scholes says, so their implied volatilities rise. That gives a smile. Jumps have a big effect over short horizons. Over longer horizons, many small and large jumps average out, so the return distribution moves closer to normal and the smile flattens as maturity grows.

Stochastic volatility models let volatility itself move randomly, often with mean reversion. Randomly changing volatility also fattens tails, because returns are a mix of calm and turbulent periods. The effect builds up over time, so the smile from stochastic volatility is usually more persistent at longer maturities than the smile from jumps.

The correlation between the asset price and its volatility controls the skew. Negative correlation (volatility rises when prices fall, as in equities) gives a heavy left tail and a downward-sloping skew, with low strikes having higher implied volatility. Zero correlation gives a symmetric smile. Positive correlation gives a skew where high strikes have higher implied volatility, as seen in some commodities.

An alternative is the local volatility model, where volatility is a deterministic function of the asset price and time. It can be fitted to match today's observed option prices exactly. But it describes how the smile should evolve in a rigid way, so it is often judged less realistic for future smile dynamics than stochastic volatility. Stochastic volatility models have their own parameters, such as mean reversion speed, long-run variance and volatility of variance, which are calibrated to market prices.

Key formulas to remember

Black-Scholes benchmark: dS = μ S dt + σ S dz, with σ constant; Gives a lognormal price and a flat implied volatility across strikes.
Jump-diffusion process: dS ÷ S = (μ − λk) dt + σ dz + dp; λ is the average number of jumps per year, k is the average proportional jump size, dp is the jump component. The −λk term compensates for the expected effect of jumps on the drift.
Stochastic volatility process (Heston-type form): dS ÷ S = μ dt + √V dz₁ ; dV = a(V_L − V) dt + ξ V^α dz₂; a is the mean reversion rate, V_L the long-run variance, ξ the volatility of variance. The correlation ρ between dz₁ and dz₂ drives skew.
Local volatility: dS ÷ S = (r − q) dt + σ(S, t) dz; Volatility is a deterministic function of price and time, so no extra random source is added.
Correlation and skew rule: ρ < 0: downward skew; ρ = 0: symmetric smile; ρ > 0: upward skew; Describes the direction of the skew under stochastic volatility.

How to solve Models for Smiles: Jumps, Stochastic Volatility and Alternatives questions

Use this sequence for any question that links a model to a smile shape.

1Identify the observed pattern: symmetric smile, downward skew, or upward skew, and the maturity (short or long).
2Identify the model feature named in the question: jumps, stochastic volatility, correlation, or local volatility.
3Decide the effect on the tails of the implied distribution: fatter tails raise implied volatility for far-from-the-money strikes.
4Apply the maturity rule: jump effects are strongest at short maturities and fade as maturity grows; stochastic volatility effects persist or build.
5Apply the sign rule: negative price-volatility correlation gives a left-heavy distribution and a downward skew; positive gives the reverse.
6Check whether the question is about fitting today's prices (local volatility can fit exactly) or about future smile dynamics (stochastic volatility is usually seen as more realistic).
7Eliminate options that claim Black-Scholes or constant volatility can generate a smile.

Quickest way: Three-question shortcut

When to use it: Use it for conceptual multiple-choice questions when time is short.

Ask: what is the maturity? Short points to jumps; long points to stochastic volatility.
Ask: what is the sign of the correlation or the jump direction? Negative or downward jumps mean a left skew.
Ask: is the goal exact fit to current prices or realistic dynamics? Exact fit points to local volatility.

Common mistakes in Models for Smiles: Jumps, Stochastic Volatility and Alternatives

Saying jumps create a smile that gets stronger as maturity increases.
Students mix up jumps with stochastic volatility, where the effect accumulates.
Fix: Remember that with jumps, longer horizons average out toward normality, so the smile flattens with maturity.
Assuming zero correlation between price and volatility gives no smile.
Students link smile only to skew.
Fix: With zero correlation, stochastic volatility still fattens both tails and gives a symmetric smile.
Getting the skew direction wrong for equities.
Students forget that volatility tends to rise when equity prices fall.
Fix: Equities: negative correlation, heavy left tail, low strikes have higher implied volatility.
Treating local volatility as the same as stochastic volatility.
Both let volatility change.
Fix: Local volatility is a deterministic function of price and time with one random source. Stochastic volatility has its own random driver.
Claiming Black-Scholes can produce a smile if volatility is high enough.
Confusing the level of volatility with its variation across strikes.
Fix: Constant volatility gives a flat smile at any level. A smile needs non-lognormal features.

Worked examples

Example 1

An equity index has a downward-sloping implied volatility skew at all maturities. A risk manager fits a stochastic volatility model. What sign of correlation between the index return and its volatility would fit, and why?

Show the solution

A downward skew means low strikes have higher implied volatility than high strikes.
That means the implied distribution has a heavier left tail than lognormal.
Under stochastic volatility, volatility must rise when the price falls to make large falls more likely.
Rising volatility with falling prices means negative correlation.

Answer: Negative correlation between price and volatility.

Example 2

A currency option desk sees a pronounced smile for 1-week options that is much flatter for 2-year options. Which model feature best explains the pattern, and why?

Show the solution

The smile is strong at short maturity and fades at long maturity.
Jumps have a large effect on short-horizon returns.
Over longer horizons, the effect of many jumps averages out and returns look closer to normal.
Stochastic volatility effects usually persist or build with maturity, so they fit this pattern less well.

Answer: Jumps best explain it, because their smile effect is strongest at short maturities and flattens as maturity grows.

Exam tips

Questions often test the maturity pattern: jumps short, stochastic volatility long.
Memorise the correlation sign rule and link it to equities versus some commodities.
Be ready to contrast local volatility with stochastic volatility on fit versus dynamics.
Reject any option saying a flat smile arises under constant volatility with jumps or stochastic volatility.

Practice questions from Volatility Smiles and Volatility Surfaces

Models for Smiles: Jumps, Stochastic Volatility and Alternatives: frequently asked questions

How do jumps create a volatility smile?

Jumps make large price moves more likely than a lognormal model allows. This fattens the tails of the distribution. Far out-of-the-money options become worth more, so their implied volatilities rise.

What is the difference between local volatility and stochastic volatility?

In a local volatility model, volatility is a deterministic function of the asset price and time. In a stochastic volatility model, volatility has its own random process. Local volatility can fit today's smile exactly. Stochastic volatility is usually seen as more realistic for how the smile evolves.

Does stochastic volatility always create a skew?

No. The skew depends on the correlation between price and volatility. With zero correlation you get a symmetric smile. Negative correlation gives a downward skew.

Why does the smile flatten for longer maturities under jumps?

Over longer horizons, many jumps combine and their effect on the return distribution moves it closer to normal. So the smile from jumps fades as maturity grows.