FRM Exam Part II · Volatility Smiles and Volatility Surfaces

Volatility Smiles for Foreign Currency Options Explained

Updated 11 October 2026 · Fact-checked

A currency option volatility smile is a U-shaped plot of implied volatility against strike, lowest near at-the-money and higher for deep in- and out-of-the-money options. It is roughly symmetric. It shows the true exchange rate distribution has fatter tails than lognormal, because of volatility changes and jumps.

Understand Volatility Smiles for Foreign Currency Options

Black-Scholes (Garman-Kohlhagen for currencies) assumes the exchange rate is lognormal with constant volatility. If that were true, implied volatility would be the same for every strike. In real markets it is not. When you plot implied volatility against strike, you get a volatility smile.

For currencies the smile is roughly symmetric and U-shaped. Implied volatility is lowest for at-the-money options and rises for both high and low strikes. Compare this with equity options, where the pattern is a downward-sloping skew.

The smile tells you what the market believes about the distribution. High implied volatility for deep out-of-the-money options means the market gives those options more value than lognormal would. Both tails are therefore heavier than lognormal. Compared with a lognormal distribution with the same mean and standard deviation, the implied distribution has a higher peak and fatter tails. This is excess kurtosis. It also assigns less probability to intermediate moves (the shoulders), so mass moves from the shoulders to the centre and the tails.

There are two main reasons the true distribution is not lognormal. First, volatility is not constant. It is stochastic. If volatility is uncorrelated with the exchange rate, a random volatility mixes many normal-like distributions with different widths. The mix has fat tails, and this effect is relatively greater for longer-dated options. Second, the exchange rate makes jumps. Central bank interventions or political events cause sudden moves. Jumps matter most for short-dated options, and the effect fades as maturity grows because many small jumps average out.

The smile is symmetric for currencies because there is no strong link between volatility and the level of the rate, unlike equities, where volatility tends to rise as prices fall. Jumps can go either way. The smile generally becomes less pronounced as maturity increases. Jumps dominate the smile at short maturities, and stochastic volatility has a relatively greater effect on longer-dated options.

Key formulas to remember

Lognormal assumption: dS = (r_d − r_f) S dt + σ S dz; Black-Scholes-type model for exchange rate S under risk-neutral measure. σ is constant, so implied volatility would be flat across strikes.
Smile shape for currencies: Implied volatility is lowest at-the-money and rises for both low and high strikes; Roughly symmetric U shape. State this directly in answers.
Implied vs lognormal distribution: Implied distribution: higher peak, fatter left and right tails than lognormal with the same mean and standard deviation; Both tails are heavier, so deep OTM calls and puts are priced above Black-Scholes at at-the-money volatility.
Reasons for fat tails: Fat tails = stochastic volatility + jumps; Effect of both is that the smile exists. Jumps matter more for short maturities; stochastic volatility has a relatively greater effect on longer-dated options. The smile generally becomes less pronounced as maturity increases.
Put-call parity for implied volatility: Implied volatility of a European call = implied volatility of a European put with the same strike and maturity; Use this to read one side of the smile from the other. It holds only at the same strike. A 25-delta call and a 25-delta put have different strikes, so this equality does not apply between them.

How to solve Volatility Smiles for Foreign Currency Options questions

Use this method for any question on currency smiles, shapes or implied distributions.

1Identify the asset class. Currency options point to a symmetric U-shaped smile. Equity options point to a downward skew.
2Note the strike and maturity in the question. Decide whether the option is near the money, or deep in or out of the money.
3Compare to flat Black-Scholes. If implied volatility is above the at-the-money level, the market prices that tail above lognormal.
4Translate the smile into a distribution. U-shaped means both tails fatter than lognormal and a higher peak.
5Link the cause. Jumps affect short-dated options most. Stochastic volatility affects long-dated options most.
6Check the direction of any pricing statement. Higher implied volatility means a higher option price, all else equal.
7Pick the answer that matches the symmetry, the tails and the cause without overstatement.

Quickest way: Three-point shortcut for currency smiles

When to use it: Use when a multiple-choice question asks for the shape, the implied distribution or the cause of the smile and time is short.

Currency means symmetric U shape, minimum at the money.
U shape means both tails heavy and the peak higher than lognormal.
Cause is jumps (short-dated) and stochastic volatility (long-dated). Remove options that say constant volatility or a one-sided tail.

Common mistakes in Volatility Smiles for Foreign Currency Options

Saying the currency smile is a downward-sloping skew
Equity index skew is taught first and is mixed up with currencies.
Fix: Link currencies to symmetric U shape and equities to downward skew. The reason is that equity volatility rises as prices fall, but currencies lack that link.
Claiming the implied distribution has thinner tails than lognormal
Students think a higher volatility at the wings reduces probability.
Fix: Higher implied volatility at extreme strikes means the market assigns more probability to extreme moves. Tails are fatter.
Thinking the implied distribution has a lower peak only
Fat tails are mistaken for a flatter distribution overall.
Fix: With the same mean and standard deviation, fatter tails come with a higher peak and thinner shoulders. Mass moves from the shoulders to the centre and the tails.
Assigning jumps to long maturities and stochastic volatility to short ones
The two effects are memorised without their reasoning.
Fix: Jumps average out over long horizons, so they matter most for short maturities. Stochastic volatility accumulates, so it matters more for long maturities.
Saying the smile appears because exchange rates are normal rather than lognormal
Confusing the distribution assumption with the constant volatility assumption.
Fix: Lognormality with constant volatility gives a flat smile. The smile arises because the real distribution departs from that.
Treating a call and a put at the same strike as having different implied volatilities
Forgetting put-call parity.
Fix: For European options with the same strike and maturity, the implied volatilities are equal under put-call parity.

Worked examples

Example 1

The implied volatilities for 3-month EUR/USD options are 9.5% at-the-money, 10.4% for a 25-delta call and 10.5% for a 25-delta put. What does this pattern tell you about the market's implied distribution compared with lognormal?

Show the solution

The volatilities at both wings, 10.4% and 10.5%, are above the at-the-money level of 9.5%.
The 25-delta call and the 25-delta put have different strikes, so put-call parity does not force their implied volatilities to be equal. They are close because the currency smile is roughly symmetric.
The pattern is therefore U-shaped and nearly symmetric, which is typical for currency options.
Higher implied volatility at both wings means out-of-the-money calls and puts are priced above the flat-volatility Black-Scholes level.
So the market assigns more probability to large moves in either direction than a lognormal distribution does.
The implied distribution has fatter left and right tails and a higher peak than lognormal with the same mean and standard deviation.

Answer: Symmetric U-shaped smile: both tails are fatter than lognormal and the peak is higher.

Example 2

Which statement is correct about why currency options show a smile? A) Volatility is constant and the rate is normal. B) Exchange rates jump and volatility is stochastic, so the true distribution has fatter tails than lognormal. C) Volatility rises when the currency falls, giving a downward skew. D) The smile arises only because of put-call parity.

Show the solution

Option A describes constant volatility, which would give a flat smile, so it is wrong.
Option C describes the equity-style skew, not the symmetric currency smile.
Option D is wrong because put-call parity only links calls and puts at one strike. It does not create a smile.
Option B gives the two standard causes, jumps and stochastic volatility, and the correct result of fat tails.

Answer: B

Exam tips

Link currency to symmetric, equity to skew. Many questions test only this contrast.
When a question gives implied volatilities by strike, compare each to the at-the-money value before saying anything about tails.
Remember the maturity link: jumps for short dated, stochastic volatility for long dated.
Describe the implied distribution fully: higher peak, fatter tails, same mean and standard deviation.
Reject any option that says constant volatility, thinner tails or a one-sided tail for currencies.

Practice questions from Volatility Smiles and Volatility Surfaces

Volatility Smiles for Foreign Currency Options in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Volatility Smiles for Foreign Currency Options: frequently asked questions

Why do currency options have a volatility smile?

The Black-Scholes style model assumes constant volatility and a lognormal exchange rate. In reality volatility changes and the rate jumps. This makes the true distribution fat-tailed, so deep out-of-the-money options are worth more than the model says and their implied volatility is higher.

Why is the currency smile symmetric?

There is no strong link between the exchange rate level and its volatility, and jumps can occur in either direction. So both tails are heavy to a similar degree. Equities differ because volatility tends to rise when prices fall.

Is the implied distribution for currencies lognormal?

No. Compared with a lognormal distribution with the same mean and standard deviation, it has a higher peak and fatter tails. That is the information the smile carries.

How do jumps and stochastic volatility differ in their effect on the smile?

Jumps have the larger effect on short-dated options because they average out over long horizons. Stochastic volatility has a larger effect on long-dated options because its impact builds up over time.