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FRM Exam Part II · Volatility Smiles and Volatility Surfaces

Implied Distributions and Detecting Mispricing from the Volatility Smile

Updated 11 October 2026 · Fact-checked

A volatility smile contains the market's risk-neutral distribution of the future asset price. Call prices across strikes give the density: g(K) = e^(rT) × the second derivative of the call price with respect to K. Compare this density with a lognormal one. Where the market gives more or less weight than your view, options look rich or cheap.

Understand Determining Implied Distributions and Detecting Mispricing

Black-Scholes assumes the asset price is lognormal at expiry, so one volatility would price every strike. Real markets quote different implied volatilities by strike. That pattern is the smile or skew. It means the market is not using a lognormal distribution.

The link is this. The price of a European call at each strike depends on the risk-neutral probabilities of finishing above that strike. If you know call prices at all strikes for one maturity, you can recover the whole risk-neutral density of the price at that maturity. This is the Breeden-Litzenberger result. In practice you use a butterfly spread: buy a call at K − δ, buy a call at K + δ, sell two calls at K. Its payoff is a small triangle centred on K. Its price is the discounted probability mass around K, so it measures the density at K.

The shapes are well known. For equity options the implied distribution has a heavier left tail and a thinner right tail than the lognormal. That is why low-strike implied volatilities are high. For foreign currency options the smile is more symmetric, with both tails heavier than lognormal and a higher peak. In both cases the implied distribution is not lognormal.

Mispricing is relative. The smile is a risk-neutral view. It is not automatically wrong. If your own model or belief about the true distribution differs from the implied one, you can trade the difference. If you think the true left tail is thinner than the market implies, deep out-of-the-money puts look expensive and you would sell them, hedged. If you think tails are heavier than implied, you would buy them. Be careful: risk-neutral and real-world distributions differ by risk premiums, so a gap is not proof of arbitrage. A negative implied density (call prices not convex in strike) is the real arbitrage signal.

Key formulas to remember

Risk-neutral density from call prices
g(K) = e^(rT) × ∂²c/∂K²
c is the European call price as a function of strike K, for one maturity T. g(K) is the risk-neutral density of the price at T.
Finite-difference estimate
g(K) ≈ e^(rT) × [c(K − δ) + c(K + δ) − 2c(K)] ÷ δ²
Strikes must be equally spaced by δ. The bracket is the cost of the butterfly spread.
Butterfly price and density
Butterfly price ≈ e^(−rT) × g(K) × δ²
The payoff is δ at S_T = K and falls to zero at K ± δ. This is the same relation rearranged.
Approximate probability in a band
P(K − δ/2 < S_T < K + δ/2) ≈ g(K) × δ
Use for a rough risk-neutral probability around K. It is an approximation.
No-arbitrage check
c(K − δ) + c(K + δ) − 2c(K) ≥ 0
If negative, the implied density is negative and a butterfly arbitrage exists.
Smile shape versus lognormal
Equity: heavier left tail, thinner right tail. Currency: both tails heavier, higher peak.
Heavier tail means higher implied volatility for strikes in that tail.

How to solve Determining Implied Distributions and Detecting Mispricing questions

Use this order for any question on implied distributions or smile-based mispricing.

  1. 1Identify what is given: call prices at equally spaced strikes, or a smile (implied volatility by strike), plus r and T.
  2. 2If given a smile, note the shape: downward sloping skew suggests equity, a U-shape suggests currency. Match to the tail description.
  3. 3For a density, compute the butterfly bracket: c(K − δ) + c(K + δ) − 2c(K).
  4. 4Multiply by e^(rT) and divide by δ² to get g(K). Multiply by δ for the probability near K if asked.
  5. 5Compare with the lognormal or the trader's view at the same K. Higher implied density means the market gives that region more weight.
  6. 6Translate to pricing: more weight in a tail means higher implied volatility and richer options there. Less weight means cheaper options.
  7. 7State the trade or conclusion: sell what is rich, buy what is cheap relative to the view, hedged. Add that the gap is relative to a view, not a guaranteed profit.
  8. 8Check for a negative density. If found, call it an arbitrage, not just a view.

Quickest way: Butterfly shortcut and tail logic

When to use it: Use when the question gives three call prices or asks which tail is heavier or which option is rich or cheap.

  1. Numeric: bracket = c(low) + c(high) − 2 × c(mid). Then × e^(rT) ÷ δ².
  2. Check the sign first. Negative means arbitrage.
  3. Conceptual: high implied volatility at a strike means the market's density in that tail is above the lognormal.
  4. Equity: left tail heavy, so low-strike puts are rich. Currency: both wings rich.
  5. Rich versus your view means sell. Cheap means buy. Mention hedging and that risk premiums can explain gaps.

Common mistakes in Determining Implied Distributions and Detecting Mispricing

  • Forgetting to multiply by e^(rT) when converting the butterfly price to a density.

    The butterfly price is a present value, and students read it as the probability directly.

    Fix: Remember that prices are discounted. Density = e^(rT) × bracket ÷ δ². With a short maturity or low rate the factor is close to 1, but still show it.

  • Dividing by δ instead of δ².

    Mixing this up with a first-derivative estimate.

    Fix: It is a second derivative, so it is δ². Check units: density is probability per unit of price.

  • Saying the equity smile has a heavier right tail.

    Confusing high implied volatility at low strikes with upside risk.

    Fix: Equity skew: high implied volatility at low strikes means a heavier left tail. The right tail is thinner than lognormal.

  • Calling any gap between implied and lognormal an arbitrage.

    Treating the lognormal as the true distribution.

    Fix: The gap is a relative-value view. Real-world and risk-neutral distributions differ. Only a negative density or violated convexity is a true arbitrage.

  • Reading the density value as a probability.

    The number g(K) looks like a percentage.

    Fix: g(K) is a density per unit of price. Multiply by the band width δ to get an approximate probability.

  • Using unequally spaced strikes in the finite-difference formula.

    Strikes taken from a quote screen without checking spacing.

    Fix: The simple formula needs equal spacing. Otherwise interpolate the smile to equal strikes first.

Worked examples

Example 1

A one-year European call on an index has prices of $12.00 at K = 95, $9.00 at K = 100 and $6.50 at K = 105. The risk-free rate is 5% (continuous). Estimate the risk-neutral density at K = 100.

Show the solution
  1. δ = 5, T = 1, r = 5%.
  2. Butterfly bracket = 12.00 + 6.50 − 2 × 9.00 = 18.50 − 18.00 = 0.50.
  3. The bracket is positive, so there is no butterfly arbitrage here.
  4. e^(0.05) ≈ 1.0513.
  5. g(100) ≈ 1.0513 × 0.50 ÷ 25 = 1.0513 × 0.02 ≈ 0.0210.

Answer: The risk-neutral density at 100 is about 0.0210 per $1 of price. That is about 2.1% per dollar, or roughly 10.5% for a $5 band around 100.

Example 2

Six-month calls on EUR/USD-style underlying have prices of $14.20 (K = 90), $8.60 (K = 100) and $4.40 (K = 110). The rate is 3% (continuous). A trader who believes in a lognormal model says the true density at 100 is 0.0170 per $1. Find the implied density at 100, the approximate risk-neutral probability of S_T between 95 and 105, and the trade implied by the trader's view.

Show the solution
  1. δ = 10, T = 0.5, so e^(rT) = e^(0.015) ≈ 1.0151.
  2. Bracket = 14.20 + 4.40 − 2 × 8.60 = 18.60 − 17.20 = 1.40.
  3. g(100) ≈ 1.0151 × 1.40 ÷ 100 = 1.4212 ÷ 100 ≈ 0.0142.
  4. Band 95 to 105 has width 10, so probability ≈ 0.0142 × 10 ≈ 0.142, about 14.2%.
  5. The implied density 0.0142 is below the trader's 0.0170. The market gives less weight to prices near 100 than the trader does.
  6. So the butterfly centred at 100 is cheap relative to the trader's view. The trader would buy it.

Answer: Implied density at 100 ≈ 0.0142 per $1. Risk-neutral probability of 95 to 105 ≈ 14.2%. Relative to the trader's lognormal view, the 90/100/110 butterfly is cheap, so the trader buys it. This is a view-based trade, not an arbitrage, because the risk-neutral and real-world distributions can differ.

Exam tips

  • Expect the butterfly calculation with equally spaced strikes. Write the bracket first, then apply e^(rT) and δ².
  • Know the tail shapes cold: equity left tail heavy and right tail thin, currency both tails heavy.
  • When asked about trading, say which options are rich or cheap relative to the view and mention hedging.
  • Watch for the word arbitrage. It needs a negative density or violated convexity, not just a smile.
  • Options are priced as present values, so check whether the question wants a density, a probability or a discounted price.

Practice questions from Volatility Smiles and Volatility Surfaces

Determining Implied Distributions and Detecting Mispricing: frequently asked questions

How do you get a risk-neutral distribution from a volatility smile?

Convert the smile into call prices for many strikes at one maturity. Then take the second derivative of the call price with respect to strike and multiply by e^(rT). In practice use the butterfly spread estimate with equally spaced strikes.

How does the implied distribution compare with the lognormal?

For equity options it has a heavier left tail and a thinner right tail. For currency options both tails are heavier and the peak is higher. The lognormal is the Black-Scholes case with a flat smile.

Does a smile that differs from the true distribution mean arbitrage?

No. It signals possible relative mispricing against your view. Risk premiums mean the risk-neutral and real-world distributions normally differ. A negative implied density is the clear arbitrage signal.

Why is the butterfly spread used?

Its payoff is a narrow triangle around the middle strike, so its price measures the discounted probability near that strike. Divided by δ² and scaled by e^(rT), it gives the density.