FRM Exam Part II · Volatility Smiles and Volatility Surfaces

Alternative Ways to Characterize the Volatility Smile

Updated 11 October 2026 · Fact-checked

A volatility smile is implied volatility plotted against some measure of strike. You can use the strike K, K/S0 (spot moneyness), K/F0 (forward moneyness) or option delta. Each form is a relabelling of the x-axis. Convert the strike to the measure asked for, then read implied volatility off the curve.

Understand Alternative Ways to Characterize the Volatility Smile

Implied volatility is the volatility that makes the Black-Scholes-Merton price equal the market price. If you plot it against strike for options of one maturity, you get a smile or skew. The raw strike is awkward because it is tied to today's price level. If the underlying moves from 100 to 120, a strike of 100 means something very different.

So practitioners rescale the x-axis. K/S0 divides the strike by spot. At-the-money is 1.0 in spot terms. K/F0 divides the strike by the forward price for the option's maturity, where F0 = S0 × e^((r − q) × T). At-the-money-forward is then exactly 1.0. This is the more natural centre, because the forward is the risk-neutral expected price at maturity.

The third option is delta. You plot implied volatility against the option's delta, for example the 25-delta put, the 50-delta option and the 25-delta call. Delta is a standardised measure of how far in or out of the money an option is, and it also reflects volatility and time to maturity. Delta-based smiles are standard in foreign exchange markets. The same quotes work for different maturities without recomputing strikes.

The reading is the same in every form. A call and a put with the same strike and maturity have the same implied volatility, by put-call parity. Their deltas differ: put delta = call delta − e^(−qT). So the 25-delta put (delta −0.25) is the same option as the 75-delta call when q = 0.

One trap: delta depends on volatility, so a delta axis is not a simple rescaling of strike. You need the implied volatility to find the delta. In practice you solve for strike and volatility together.

Key formulas to remember

Forward price: F0 = S0 × e^((r − q) × T); r is the risk-free rate and q the dividend or foreign-currency yield, both continuous. For FX, q is the foreign rate.
Spot moneyness: K/S0; Equals 1 when the strike equals spot. Not the same as at-the-money-forward unless r = q.
Forward moneyness: K/F0; Equals 1 when the strike equals the forward. This is the at-the-money-forward point.
d1 in Black-Scholes-Merton: d1 = [ln(S0/K) + (r − q + σ²/2) × T] ÷ (σ × √T); If K = F0, d1 = σ√T ÷ 2.
Call and put delta: Call delta = e^(−qT) × N(d1); Put delta = e^(−qT) × [N(d1) − 1]; Put delta = call delta − e^(−qT). With q = 0, put delta = call delta − 1.

How to solve Alternative Ways to Characterize the Volatility Smile questions

Use this routine for any question that asks you to convert between smile forms or read one.

1Identify the axis the question uses: K, K/S0, K/F0 or delta.
2Write down S0, r, q, T and the strike or delta given.
3If a forward is needed, compute F0 = S0 × e^((r − q) × T) first.
4For a moneyness axis, divide the strike by S0 or F0 and compare with 1.
5For a delta axis, compute d1 from the stated volatility, then call delta = e^(−qT) × N(d1). Get put delta by subtracting e^(−qT).
6Link calls and puts: same strike means same implied volatility. Match a put delta to the call delta by put delta = call delta − e^(−qT).
7Interpret: is the option in, at or out of the money? Is volatility higher for low strikes (skew) or both wings (smile)?
8Check that the answer uses the axis the question asked for, and the right sign for put delta.

Quickest way: Centre the axis, then convert

When to use it: Use this when a multiple-choice question gives a strike and asks for moneyness, or gives a delta and asks for the matching option.

Compute F0 first. It is the true at-the-money-forward strike.
If K is above F0, a call is out of the money and a put is in the money. If K is below F0, the reverse holds.
Remember the anchor: at-the-money-forward call delta is a little above 0.5 when q = 0, since d1 = σ√T ÷ 2 is positive.
For call-put matching with q = 0, add 1 to the put delta: −0.25 matches a call delta of 0.75.
Eliminate options that use K/S0 where the question says forward.

Common mistakes in Alternative Ways to Characterize the Volatility Smile

Treating K/S0 = 1 as at-the-money-forward.
Spot and forward look similar when rates are low.
Fix: The forward is S0 × e^((r − q) × T). The two centres agree only when r = q.
Giving put delta as a positive number or mixing up its sign.
Quotes often say '25-delta put' without the minus sign.
Fix: Put delta is negative. A 25-delta put has a delta of −0.25 and equals a 75-delta call at the same strike when q = 0.
Saying the smile against delta is just a rescaled strike axis.
Delta looks like a function of strike alone.
Fix: Delta also depends on volatility and maturity, so the mapping must be solved with the volatility itself.
Assuming a call and put at the same strike have different implied volatilities.
Their prices and deltas differ.
Fix: By put-call parity they have the same implied volatility for the same strike and maturity. Only the delta label differs.
Forgetting the dividend or foreign-rate term when computing F0 or delta.
Students memorise the no-dividend versions.
Fix: Use r − q in the forward and d1, and e^(−qT) in delta. Set q = 0 only when told.

Worked examples

Example 1

An equity index has S0 = 100, a continuous risk-free rate of 5%, a dividend yield of 2% and a maturity of 1 year. A call has strike K = 110. Find K/S0 and K/F0, and say whether the call is out of the money.

Show the solution

F0 = 100 × e^((0.05 − 0.02) × 1) = 100 × e^0.03 = 100 × 1.030455 = 103.05.
K/S0 = 110 ÷ 100 = 1.10.
K/F0 = 110 ÷ 103.05 = 1.0675, about 1.07.
K is above F0, so the call is out of the money in forward terms. The at-the-money-forward strike is 103.05, not 100.

Answer: K/S0 = 1.10 and K/F0 ≈ 1.07. The call is out of the money, and the at-the-money-forward strike is about 103.05.

Example 2

S0 = 100, K = 100, r = 0, q = 0, T = 1 year, and implied volatility is 20%. Find the call delta and put delta, and state which call delta corresponds to this put on a delta-based smile.

Show the solution

d1 = [ln(100/100) + (0 + 0.2²/2) × 1] ÷ (0.2 × 1) = 0.02 ÷ 0.2 = 0.10.
N(0.10) = 0.5398, so call delta ≈ 0.54.
Put delta = 0.5398 − 1 = −0.4602, about −0.46.
Same strike and maturity, so both options have the same implied volatility of 20%. The put is the 46-delta put, which matches the 54-delta call.

Answer: Call delta ≈ 0.54 and put delta ≈ −0.46. The 46-delta put and the 54-delta call are the same strike and share the 20% volatility.

Exam tips

Read the axis in the question first. Many wrong answers come from using spot moneyness when the question says forward.
Always compute F0 with r − q before judging in or out of the money.
Write put delta with its negative sign and use put delta = call delta − e^(−qT) to match calls and puts.
Expect interpretation questions: delta axes keep quotes comparable across maturities, and the at-the-money-forward point sits at K/F0 = 1.
Do not claim delta is a pure rescaling of strike. It depends on volatility too.

Practice questions from Volatility Smiles and Volatility Surfaces

Alternative Ways to Characterize the Volatility Smile in other exams

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

CFA Level IIImplied Volatility and Volatility Smile

Alternative Ways to Characterize the Volatility Smile: frequently asked questions

Why plot the volatility smile against delta instead of strike?

Delta measures how far in or out of the money an option is in a standardised way. It also allows the same quote, such as the 25-delta put, to be compared across dates and maturities. This is why it is common in foreign exchange option markets.

What is the difference between K/S0 and K/F0?

K/S0 compares the strike with today's spot. K/F0 compares it with the forward price for the same maturity. They are equal only when the forward equals spot, which means r = q.

Do a call and a put with the same strike have the same implied volatility?

Yes, for the same maturity, by put-call parity. Their deltas differ, though. Put delta equals call delta minus e^(−qT).

Is the volatility smile against delta just the strike smile relabelled?

Not exactly. Delta depends on volatility, so the mapping between strike and delta uses the implied volatility itself. It is still the same information about option prices in a different form.