FRM Exam Part I · The Black-Scholes-Merton Model

Historical vs Implied Volatility and the Volatility Smile

Updated 11 October 2026 · Fact-checked

Historical volatility is the annualized standard deviation of past log returns. Implied volatility is the σ that makes the Black-Scholes-Merton price equal the observed option price, found numerically. One looks backward, the other reflects the market's expectations and risk premia. Implied volatility that varies with strike is called the volatility smile.

Understand Volatility: Historical and Implied

Volatility is the standard deviation of a stock's continuously compounded return, quoted per year. In the Black-Scholes-Merton (BSM) model it is the one input you cannot observe directly. You have two ways to get it.

Historical volatility looks backward. Take daily prices, compute log returns ln(Sᵢ ÷ Sᵢ₋₁), find their sample standard deviation, then scale to a year. Hull-style convention uses about 252 trading days a year, because volatility seems to arise mainly when markets are open. So you multiply the daily figure by √252, not √365.

Implied volatility looks forward. Option prices are observed in the market. BSM gives price as a function of S, K, r, q, T and σ. You turn it around: find the σ that makes the model price equal the market price. There is no closed-form inverse, so you use a numerical search such as bisection or Newton-Raphson. Because the price of a European option rises steadily with σ (vega is positive), the answer is unique. European calls and puts with the same strike and maturity must have the same implied volatility, because of put-call parity.

If BSM were exactly right, implied volatility would be the same for every strike and maturity. In practice it is not. Plot implied volatility against strike and you get the volatility smile. For currency options it is U-shaped: both tails are fatter than lognormal. For equity options it is a downward skew (sometimes called a smirk): implied volatility is higher for low strikes and lower for high strikes. That means the market implies a heavier left tail and thinner right tail than the lognormal distribution. Leverage effects and crash fears are common explanations. Plotting implied volatility against strike and maturity gives the volatility surface.

The VIX index is the best-known implied volatility measure. It is built from a strip of out-of-the-money S&P 500 options across many strikes, not from one option and not from BSM inversion. It estimates the 30-day expected volatility, expressed as an annualized percentage. It is model-free: it weights option prices by 1 ÷ K², and it matches the fair strike of a variance swap.

Key formulas to remember

Log (continuously compounded) return: uᵢ = ln(Sᵢ ÷ Sᵢ₋₁); Use this for historical volatility. Ignore dividends unless the question says to adjust for them.
Sample standard deviation of returns: s = √[ Σ(uᵢ − ū)² ÷ (n − 1) ]; n is the number of returns, which is one fewer than the number of prices. Many questions set ū ≈ 0, which gives s = √[Σuᵢ² ÷ n]. Read the question to see which version is intended.
Annualizing volatility: σ per year = s × √252; Use the number of periods per year stated in the question. 252 trading days is the usual default, 12 for monthly returns, 52 for weekly.
Volatility over a horizon: Standard deviation over T years = σ × √T; Volatility scales with the square root of time, variance scales with time.
Standard error of a volatility estimate: Standard error ≈ σ̂ ÷ √(2n); Approximate, for n observations. More data gives a tighter estimate.
Implied volatility condition: c_BSM(S, K, r, q, T, σ_imp) = c_market; Solve for σ_imp numerically. Same for puts: p_BSM(σ_imp) = p_market.
Newton-Raphson update: σ_new = σ_old − (c_BSM(σ_old) − c_market) ÷ Vega; Vega is the derivative of the option price with respect to σ.
VIX (variance form): σ² = (2 ÷ T) Σ [ΔKᵢ ÷ Kᵢ²] e^(RT) Q(Kᵢ) − (1 ÷ T)(F ÷ K₀ − 1)²; VIX = 100 × σ; Q(Kᵢ) is the mid-quote of the out-of-the-money option at strike Kᵢ. K₀ is the first strike below the forward F. T is about 30 days. Know the structure, you are unlikely to compute it in full.

How to solve Volatility: Historical and Implied questions

Use this method for any question on historical volatility, implied volatility or the smile.

1Identify what is asked: a historical estimate, an implied value, or an interpretation of the smile or VIX.
2For historical volatility, convert prices to log returns ln(Sᵢ ÷ Sᵢ₋₁). Count the returns, which is one fewer than the prices.
3Compute the mean and the sum of squared deviations. Divide by n − 1 unless the question says to assume a zero mean and divide by n.
4Take the square root to get the periodic volatility, then multiply by √(periods per year) to annualize.
5For implied volatility, set the BSM price equal to the market price. Do not try to solve algebraically. Pick two σ values that bracket the market price and interpolate, or apply the Newton-Raphson step with vega.
6Check the answer is sensible: higher market price means higher implied volatility, and the answer should lie between your bracketing values.
7For smile questions, match the shape to the asset: currency gives a symmetric smile (fat tails both sides), equity gives a downward skew (fat left tail). Then state what the implied distribution looks like compared with lognormal.

Quickest way: Bracket, interpolate and sanity-check

When to use it: Use when a question gives BSM prices at two volatilities, or asks which direction implied volatility moves. It takes under a minute.

Note that price rises with σ, so a market price above the model price means the implied σ is higher than the one used.
If two model prices are given, interpolate: σ ≈ σ₁ + (σ₂ − σ₁) × (c_mkt − c₁) ÷ (c₂ − c₁).
For historical volatility with a stated zero mean, compute √(Σu² ÷ n) directly, then multiply by √252.
For a smile question, eliminate any option that says implied volatility is flat across strikes or that equity skew rises with strike.
On a financial calculator, use the √ key and store the daily figure before multiplying by √252.

Common mistakes in Volatility: Historical and Implied

Annualizing with √365 instead of √252, or multiplying by 252 instead of √252.
Students mix up variance scaling and volatility scaling, or forget the trading-day convention.
Fix: Variance scales with time, volatility scales with its square root. Use the number of periods the question states, and 252 for daily data if none is given.
Dividing by n instead of n − 1, or using prices instead of returns.
Students rush and copy the population formula, or count the prices as the observations.
Fix: With n + 1 prices you have n returns. Use n − 1 in the sample variance unless the question tells you to assume a zero mean.
Saying implied volatility is a forecast that is always accurate, or that it equals historical volatility.
Both are called volatility, so students treat them as the same thing.
Fix: Historical is backward-looking from past returns. Implied is the market's forward-looking price of volatility and can include a risk premium. They usually differ.
Drawing the equity skew the wrong way, with implied volatility rising as strike rises.
Students memorize the word smile and picture a U for all assets.
Fix: Equity: implied volatility falls as strike rises, which means a heavier left tail than lognormal. Currency: U-shaped, with both tails heavier.
Believing implied volatility needs an algebraic solution, or that multiple volatilities can match one price.
Students expect to rearrange the BSM formula like a normal equation.
Fix: There is no closed form, so use iteration. Since price increases monotonically in σ, there is exactly one implied volatility per price.
Describing VIX as the implied volatility of one at-the-money S&P 500 option.
Students link implied volatility to a single option inverted through BSM.
Fix: VIX uses a strip of out-of-the-money options across strikes, weighted by 1 ÷ K², and gives an annualized 30-day expectation. It does not need BSM inversion.

Worked examples

Example 1

A stock has five daily log returns: 1%, −1%, 2%, 0% and −2%. Assume the mean daily return is zero and use 252 trading days a year. What is the annualized historical volatility, to one decimal place? (A) 1.4% (B) 22.4% (C) 25.1% (D) 356.4%

Show the solution

The returns in decimals are 0.01, −0.01, 0.02, 0.00 and −0.02. The mean is assumed to be zero.
Sum of squared returns = 0.0001 + 0.0001 + 0.0004 + 0 + 0.0004 = 0.0010.
Because the question tells you to assume a zero mean, divide by n = 5: variance = 0.0010 ÷ 5 = 0.0002.
Daily volatility = √0.0002 = 0.014142, or 1.4142%.
Annualize: 0.014142 × √252 = 0.014142 × 15.8745 = 0.22450, or 22.4%.
Check the distractors: dividing by n − 1 = 4 gives 25.1%. Reporting the daily figure gives 1.4%. Multiplying by 252 instead of √252 gives 356.4%.

Answer: (B) 22.4%

Example 2

A European call is quoted in the market at $4.00. BSM gives a call price of $3.60 with σ = 20% and $4.50 with σ = 25%, all other inputs unchanged. Using linear interpolation, what is the approximate implied volatility? (A) 21.1% (B) 22.2% (C) 23.3% (D) 24.4%

Show the solution

Price rises with σ, so the implied volatility lies between 20% and 25%, because $4.00 lies between $3.60 and $4.50.
Fraction of the way between the two prices = (4.00 − 3.60) ÷ (4.50 − 3.60) = 0.40 ÷ 0.90 = 0.4444.
Interpolate: σ ≈ 20% + 0.4444 × (25% − 20%) = 20% + 2.222% = 22.22%.
Linear interpolation is only an approximation, because price is slightly curved in σ. A calculator or Newton-Raphson would refine it, but the exam answer here is 22.2%.

Answer: (B) 22.2%

Exam tips

Read whether the question says to assume a zero mean. It changes the denominator and the answer, so check the options for the distractor.
For smile questions, name the asset first. Equity means downward skew with a fat left tail, currency means a symmetric smile with both tails fat.
Do not try to solve BSM for σ by algebra. Questions give you bracketing prices or ask conceptual points such as monotonicity and uniqueness.
For VIX, know what it measures: a 30-day expected volatility from out-of-the-money S&P 500 options, quoted as an annualized percentage. The exam tests the concept far more than the arithmetic.
Check units before choosing: daily versus annual, percent versus decimal. A wrong scaling factor is the most common source of a distractor.

Practice questions from The Black-Scholes-Merton Model

Volatility: Historical and Implied in other exams

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

Volatility: Historical and Implied: frequently asked questions

What is the difference between implied and historical volatility?

Historical volatility is calculated from past returns and describes what has already happened. Implied volatility is backed out from current option prices and reflects what the market expects, along with any risk premium. They are usually not equal.

How do you calculate implied volatility from Black-Scholes?

You set the BSM price equal to the observed market price and solve for σ. There is no closed-form solution, so you use a numerical method such as bisection or Newton-Raphson. Because option prices rise with σ, there is only one answer.

What does the volatility smile mean for FRM Part I?

It shows that implied volatility changes with strike, so BSM's constant-volatility assumption does not hold. Equity options show a downward skew, which implies a heavier left tail than lognormal. Currency options show a more symmetric smile, which implies fat tails on both sides.

How is VIX calculated, and do I need to compute it?

VIX is derived from a strip of out-of-the-money S&P 500 call and put prices across many strikes, weighted by 1 ÷ K², for about 30 days to expiry. It is not an inversion of one BSM price. For the exam, understand what it measures and the structure of the formula rather than expecting a long computation.