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CFA Level I Exam · Pricing and Valuation of Options

Implied Volatility, Smile and Skew for CFA Level I

Updated 7 October 2026

Implied volatility is the volatility input that makes a model price, such as Black-Scholes-Merton, equal the observed market option price. It reflects expected future volatility, not past volatility. Plotted against strike, it often forms a smile or skew, which shows the model's constant-volatility assumption does not hold.

Understand Implied Volatility and Option Strategies

An option's price depends on the underlying price, strike, time to expiry, risk-free rate, any income on the underlying, and volatility. Five of these can be observed. Volatility cannot. So you can use the model in two directions.

Forward direction: you estimate volatility, for example from past returns, and the model gives you a price. Reverse direction: you take the price the market is actually paying, and solve for the volatility that makes the model match. That solved value is the implied volatility. It is forward-looking. It is the market's consensus view of volatility over the option's life, expressed as an annualized standard deviation of returns.

Historical volatility is the standard deviation of past returns. It is backward-looking. Implied volatility comes from today's prices. If implied is above historical, options look expensive relative to past behaviour, which may mean the market expects turbulence or is paying a risk premium. Because option value rises with volatility, a higher implied volatility means a higher option price, all else equal.

There is no closed-form formula for implied volatility. It is found by iteration: try a volatility, compute the price, compare, adjust. Calculators and software do this.

If Black-Scholes-Merton were fully right, 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 gives a volatility smile (higher at both low and high strikes, lowest near at-the-money) or a volatility skew (for example, higher for low strikes than for high strikes, common in equity index options). Plotting against maturity gives the term structure of volatility. Together, strike and maturity give the volatility surface. These patterns show the market assigns more probability to large moves, or to downside moves, than a lognormal distribution with constant volatility implies.

Key formulas to remember

Implied volatility definition
Find σ such that Model price(σ) = Observed market price
Solved by iteration. Holds the other inputs (S, X, T, r, income) fixed.
Volatility and option value
∂(option value) ÷ ∂σ > 0 (vega is positive for both calls and puts; long positions gain when volatility rises)
Higher volatility raises both call and put values. Lower implied volatility means a cheaper option.
Annualizing historical volatility
σ(annual) = σ(periodic) × √(periods per year)
For example, daily × √252 (the number of trading days is usually given). Needed to compare with implied volatility.
Smile versus skew
Smile: IV high at low and high strikes, lowest near ATM. Skew: IV differs systematically across strikes, e.g. higher at low strikes.
Both contradict Black-Scholes-Merton's constant-volatility assumption.
Reading implied versus historical volatility
Implied > Historical: options priced rich. Implied < Historical: options priced cheap.
A heuristic comparison, not a guarantee of profit.

How to solve Implied Volatility and Option Strategies questions

Use this approach for any question on implied volatility, smile or skew.

  1. 1Identify what is given: a price and you must find volatility (reverse use), or a volatility and you must find price or direction (forward use).
  2. 2Remember that implied volatility is forward-looking and historical volatility is backward-looking. Many questions test only this.
  3. 3If two options are compared on the same underlying, hold other inputs equal and use the rule that higher price means higher implied volatility.
  4. 4If the question gives implied volatility at several strikes, look at the shape: U-shaped is a smile, sloping is a skew. State what it implies about the distribution of returns.
  5. 5Remember that higher volatility raises both call and put values, so vega is positive for both.
  6. 6Check whether the question is about volatility or about the direction of the underlying. A volatility view is not a direction view.
  7. 7Eliminate options that say implied volatility is observable directly, is backward-looking, or has a closed-form solution.

Quickest way: Price up, volatility up

When to use it: Use for comparison questions where you must rank options or judge cheap versus expensive without computing.

  1. Same underlying, strike, expiry and rates: higher market price means higher implied volatility.
  2. Compare implied volatility with the historical figure given. Implied above historical means options are relatively expensive.
  3. For smile or skew, ask which strikes have the highest implied volatility. That is where the market assigns more tail risk.
  4. For strategies, ask: does the position gain when volatility rises? If yes, it is long volatility.

Common mistakes in Implied Volatility and Option Strategies

  • Calling implied volatility a measure of past price movement.

    The word volatility makes students think of historical standard deviation.

    Fix: Implied volatility is extracted from current option prices and is forward-looking. Historical volatility uses past returns.

  • Saying implied volatility can be calculated with a simple formula.

    Students expect an inverse of Black-Scholes-Merton.

    Fix: There is no closed-form inverse. It is found by iteration or numerical search.

  • Thinking a volatility smile supports the Black-Scholes-Merton model.

    Students confuse a pattern in the data with a confirmation of the model.

    Fix: The model assumes constant volatility, so implied volatility should be flat across strikes. A smile or skew shows the assumption is violated.

  • Treating a long straddle as a bet on direction.

    Both a call and a put are held, so the direction seems unclear, and the position is confused with a one-sided bet.

    Fix: A long straddle profits from a large move either way or from a rise in implied volatility. It loses if the underlying stays near the strike.

  • Assuming high implied volatility always means options are overpriced.

    It is a tempting shortcut.

    Fix: Compare implied with expected future volatility. High implied volatility may be justified. It is only relatively rich against historical or forecast volatility.

Worked examples

Example 1

A call option on a stock trades at 6.20. Using Black-Scholes-Merton with a volatility input of 20%, the model gives 5.40. With 25% the model gives 6.60 (all other inputs unchanged). Using linear interpolation, which volatility is closest to the implied volatility? A. 20% B. 23% C. 25%

Show the solution
  1. Implied volatility is the input that makes the model price equal the market price. The market price is 6.20.
  2. At 20% the model gives 5.40, which is below 6.20. At 25% it gives 6.60, which is above 6.20. So the implied volatility lies between 20% and 25%.
  3. Neither 20% nor 25% reproduces 6.20, so A and C cannot be the implied volatility.
  4. Interpolate. The model price rises from 5.40 to 6.60, a gain of 1.20, over the 5-point volatility range. That is about 0.24 per 1% of volatility.
  5. The market price is 0.80 above the 20% model price. Fraction of the range: 0.80 ÷ 1.20 = 0.667. Implied volatility ≈ 20% + 0.667 × 5% = 23.3%.
  6. The market price is closer to the 25% price than to the 20% price, so the answer must sit in the upper part of the range. Of the three choices, 23% is the one nearest 23.3%.

Answer: B. Implied volatility is approximately 23%.

Example 2

Index options show implied volatility of 28% for a 90% strike, 21% for a 100% strike and 18% for a 110% strike, all with the same expiry. Which statement is most accurate? A. The pattern is a volatility skew, with the market pricing more downside risk than Black-Scholes-Merton implies. B. The pattern shows implied volatility is constant across strikes. C. The pattern is a volatility smile, symmetric about the at-the-money strike.

Show the solution
  1. List implied volatility by strike: 28%, 21%, 18%. It falls steadily as strike rises.
  2. It is not constant, so B is wrong.
  3. A smile would be higher at both low and high strikes with a minimum near the money. Here the highest strikes have the lowest volatility, so it is not a smile. C is wrong.
  4. A downward-sloping pattern is a skew. The low-strike puts, which protect against falls, carry higher implied volatility, so the market is pricing greater downside risk than a constant-volatility lognormal model.

Answer: A. It is a volatility skew with more downside risk priced in.

Exam tips

  • Expect conceptual questions: forward-looking versus backward-looking, and what smile or skew shows about Black-Scholes-Merton assumptions.
  • Before choosing, eliminate any option that calls implied volatility directly observable or backward-looking.
  • Know that higher volatility raises both call and put prices. Do not link volatility to direction.
  • You get no penalty for wrong answers, so always answer. Aim for about 90 seconds per question.

Practice questions from Pricing and Valuation of Options

Implied Volatility and Option Strategies: frequently asked questions

What is implied volatility in CFA Level I?

It is the volatility that, when put into an option pricing model such as Black-Scholes-Merton, gives the observed market price of the option. It is forward-looking and reflects the market's view of future volatility.

What is the difference between implied and historical volatility?

Historical volatility is the standard deviation of past returns. Implied volatility is backed out of current option prices. Implied is forward-looking, historical is backward-looking, and the two often differ.

What is the volatility smile and skew?

A smile is a plot of implied volatility against strike that is higher at both low and high strikes. A skew is a sloped pattern, often with higher implied volatility at low strikes. Both show that constant volatility in Black-Scholes-Merton does not hold in practice.

How do I calculate implied volatility from an option price?

There is no direct formula. You search for the volatility that makes the model price equal the market price, adjusting up or down by iteration. On the exam you are usually asked to interpret or compare rather than solve it by hand.