CFA Level II Exam · Valuation of Contingent Claims

Implied Volatility and the Volatility Smile Explained

Updated 7 October 2026 · Fact-checked

Implied volatility is the volatility input that makes a model price, usually Black-Scholes-Merton, equal the observed option price. You find it by iteration, since the model cannot be inverted. If implied volatility differs across strikes or maturities, you get a smile, skew or term structure, and together they form the volatility surface.

Understand Implied Volatility and Volatility Smile

A model such as Black-Scholes-Merton (BSM) takes in the underlying price, strike, time, risk-free rate and volatility, and gives an option price. Everything except volatility can be observed. So when you see a market price, you can ask: what volatility would the model need to produce this price? That answer is the implied volatility. It is forward-looking and reflects what the market is pricing, unlike historical volatility, which looks backward.

You cannot rearrange BSM to solve for volatility directly. Instead you use trial and error (or a numerical method). Guess a volatility, compute the price, and adjust. Option price rises with volatility, so a price too low means raise the guess. This works for European options under BSM because vega is positive for them.

If BSM were perfectly true, implied volatility would be the same for every strike and maturity on the same underlying. In practice it is not. Plotting implied volatility against strike for a single maturity gives the volatility smile (U-shaped) or the volatility skew (sloping, usually downward for equity indexes, with low strikes having higher implied volatility). This tells you the market does not believe returns are lognormal with constant volatility. Real returns have fatter tails and negative skewness, and volatility itself changes.

Plotting implied volatility against time to expiration for a given moneyness gives the term structure of implied volatility. Combining strike and maturity gives the volatility surface, a three-dimensional picture. Traders use it to price options that are not quoted, to spot rich or cheap options, and to gauge market sentiment. A steep skew means investors pay up for downside protection.

Key formulas to remember

Implied volatility definition: Find σ such that BSM(S, X, T, r, σ) = observed market price; Solved by iteration. For European options under BSM, value increases with σ (vega > 0), so the solution is unique for a given price within no-arbitrage bounds.
Direction of adjustment: Model price < market price → increase σ; model price > market price → decrease σ; Works for European calls and puts under BSM because vega > 0.
Volatility smile / skew: Implied σ plotted against strike (or moneyness X ÷ S) for one maturity; Under BSM assumptions this would be flat. A non-flat shape signals departures from BSM.
Term structure of implied volatility: Implied σ plotted against time to expiration for the same strike or moneyness; Upward-sloping, flat or downward-sloping. Short-dated volatility often exceeds long-dated after market stress.
Volatility surface: Implied σ as a function of strike and maturity; Used to price options with strikes and maturities not directly quoted, by interpolation.
Linear interpolation on the surface: σ at K = σ1 + (K − K1) ÷ (K2 − K1) × (σ2 − σ1); A simple way to estimate implied volatility between two quoted strikes.

How to solve Implied Volatility and Volatility Smile questions

Use this method for any question on implied volatility, smile, skew or term structure.

1Read the vignette and identify what is given: market option prices, strikes, maturities, and any implied volatilities or a table or chart of them.
2Decide what is asked: calculating or estimating implied volatility, interpreting a shape, or using the surface to price or compare options.
3For estimation, compare model price with market price. Higher market price means higher implied volatility, all else equal.
4For interpretation, state the shape (flat, smile, skew, or term structure slope) and what it says: fat tails, negative skewness, or demand for protection.
5Compare to BSM: constant volatility across strikes and maturities is the BSM assumption. Any pattern is a departure.
6If asked to price at a strike not quoted, interpolate implied volatility between neighbouring points, then use it in the model.
7Check that your conclusion has the right direction before choosing an option.

Quickest way: Read the shape, then the direction

When to use it: Use when the item set gives a table or chart of implied volatilities and asks for an interpretation or a comparison of option values.

Find the row or column with the strike or maturity in the question.
Higher implied volatility means a higher option price for that option, same other inputs.
Low strikes with higher implied volatility than high strikes means a downward skew, so the market prices downside risk richly.
If both low and high strikes exceed the at-the-money level, it is a smile, which implies fat tails on both sides.
For interpolation, apply the linear formula to implied volatility, not to option price.

Common mistakes in Implied Volatility and Volatility Smile

Treating implied volatility as a forecast of realized volatility that must be correct.
The word forward-looking suggests accuracy.
Fix: Implied volatility is the market's pricing input and includes a risk premium. It can differ from later realized volatility.
Saying BSM predicts a smile.
Students confuse the model with market behaviour.
Fix: BSM assumes constant volatility, so it implies a flat line. The smile is observed in markets and shows BSM is imperfect.
Interpolating option prices instead of implied volatilities.
Prices are the visible numbers.
Fix: Interpolate volatility, then recompute the price with the model. Price is not linear in strike.
Reading the skew direction backwards.
Mixing up strikes with moneyness for puts and calls.
Fix: For equity indexes, low strikes (out-of-the-money puts) usually have the highest implied volatility, reflecting demand for downside protection.
Thinking implied volatility can be solved with an algebraic formula.
Other BSM inputs are solved directly.
Fix: You find it by iteration. In exams you usually compare given values rather than solve exactly.

Worked examples

Example 1

An analyst observes one-month implied volatilities on an equity index: 80% strike 28%, 90% strike 24%, 100% strike 20%, 110% strike 18%, 120% strike 17%. (1) What is the shape? (2) What does it suggest about the market's view of returns? (3) Which strike has the higher implied volatility, the 90% strike or the 110% strike?

Show the solution

Implied volatility falls steadily as strike rises, so it is a downward-sloping skew, not a symmetric smile.
BSM assumes constant volatility, so a skew shows the market does not price returns as lognormal. Higher implied volatility at low strikes indicates a greater probability of large downward moves (negative skewness, fat left tail) or strong demand for downside protection.
Compare 24% at the 90% strike with 18% at the 110% strike. The 90% strike has the higher implied volatility.

Answer: (1) Downward skew. (2) The market prices more downside risk than BSM with constant volatility implies. (3) The 90% strike has the higher implied volatility (24% vs 18%).

Example 2

Three-month implied volatility is 21% at strike 95 and 25% at strike 105. The three-month at-the-money (strike 100) implied volatility is not quoted. The six-month at-the-money (strike 100) implied volatility is quoted at 26%. Estimate the three-month implied volatility at strike 100 by linear interpolation, then compare it with the six-month at-the-money value to state whether the at-the-money term structure slopes up or down.

Show the solution

Use the three-month values at strikes 95 and 105: σ at 100 = 21% + (100 − 95) ÷ (105 − 95) × (25% − 21%).
Compute: 5 ÷ 10 = 0.5, and 0.5 × 4% = 2%.
Estimated three-month σ at strike 100 = 21% + 2% = 23%.
Compare on the same strike: three-month at-the-money is about 23% and six-month at-the-money is 26%. Volatility rises with maturity, so the term structure is upward sloping.

Answer: Interpolated three-month implied volatility at strike 100 is 23%. Compared with the quoted six-month at-the-money value of 26%, the at-the-money term structure is upward sloping.

Exam tips

Expect interpretation questions: name the shape and say what it implies about tail risk or demand for protection.
Remember BSM implies a flat surface. Any non-flat shape means BSM assumptions fail.
If a table is given, check whether it is by strike (smile or skew) or by maturity (term structure) before answering.
Higher implied volatility means a higher option price, all else equal. Use that to rank options without calculating.
Interpolate volatility, not price. If the question asks for an approximate value, linear interpolation is enough.

Implied Volatility and 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.

Implied Volatility and Volatility Smile: frequently asked questions

What is implied volatility in options?

It is the volatility value that, when put into an option pricing model such as Black-Scholes-Merton, gives the option's current market price. It is found by iteration. It reflects what the market is pricing in, not a guaranteed forecast.

What is the difference between a volatility smile and a volatility skew?

Both plot implied volatility against strike. A smile is U-shaped, with higher volatility for both low and high strikes. A skew slopes in one direction, typically with higher implied volatility at low strikes for equity indexes.

Why does the volatility smile exist?

Real returns have fatter tails and often negative skewness, and volatility changes over time. BSM assumes constant volatility and lognormal returns, so market prices imply different volatilities at different strikes. Demand for protection also plays a role.

What is the volatility surface?

It is implied volatility shown as a function of both strike and time to expiration. Traders use it to price options that are not quoted and to compare relative value across options on the same underlying.