FRM Exam Part I · Measuring and Monitoring Volatility

Implied vs Historical Volatility for FRM Part I

Updated 11 October 2026 · Fact-checked

Historical volatility is the standard deviation of past returns, annualized. Implied volatility is the volatility that, put into an option pricing model such as Black-Scholes-Merton, returns the observed market option price. Historical looks backward at what happened. Implied reflects the market's forward-looking view and risk premium.

Understand Implied vs Historical Volatility

Volatility measures how much an asset's returns vary. You cannot observe it directly. You must estimate it, and there are two main ways.

Historical volatility uses past prices. You compute daily returns, take their sample standard deviation, then scale to a year. With daily data and 252 trading days, annual volatility = daily volatility × √252. It is easy to compute, but it assumes the past describes the future. It also depends on the window length and the weighting you choose.

Implied volatility works backward from an option price. The Black-Scholes-Merton (BSM) model takes the stock price, strike, maturity, risk-free rate, dividends and volatility, and gives an option price. All inputs except volatility are observable. So you take the market price and solve for the volatility that makes the model price match. There is no closed-form inverse, so you use a numerical search such as Newton-Raphson or bisection. Vega, the price sensitivity to volatility, drives the search. The BSM price is strictly increasing in volatility, so a market price that lies within the no-arbitrage bounds maps to a unique implied volatility. A higher option price therefore means a higher implied volatility, all else equal.

The two measures differ for several reasons. Implied volatility is forward-looking and covers the option's remaining life. Historical volatility covers the past window. Implied volatility tends to exceed subsequent realized volatility on average, especially for equity index options, because of a volatility risk premium: option sellers demand compensation for bearing volatility and jump risk, and investors pay for protection. This is an empirical tendency, not a rule for every asset or period. Implied volatility also varies by strike (the smile or skew), which shows that BSM's constant-volatility, lognormal assumptions do not hold in practice.

The VIX is a well-known implied volatility index. It is built from prices of out-of-the-money S&P 500 calls and puts with near-term maturities, and it measures the market's expected 30-day volatility, quoted in annualized percentage points. It is model-free: it does not need BSM inversion. It is often called a fear gauge because it tends to spike when markets fall.

Key formulas to remember

Annualizing historical volatility: σ_annual = σ_daily × √252; Use the number of periods per year in your data (252 daily, 52 weekly, 12 monthly). Volatility scales with the square root of time, not time.
Sample historical volatility: σ = √[ Σ (rᵢ − r̄)² ÷ (n − 1) ]; rᵢ are returns, often ln(Sᵢ ÷ Sᵢ₋₁). Many exam questions use the zero-mean shortcut σ² ≈ Σ rᵢ² ÷ n; follow the question's instruction.
Log return: rᵢ = ln(Sᵢ ÷ Sᵢ₋₁); Continuously compounded return used with BSM.
Implied volatility definition: Find σ such that BSM(S, K, T, r, q, σ) = market option price; Solved numerically. The BSM price of a European call or put is strictly increasing in σ, so a market price within the no-arbitrage bounds gives a unique implied volatility.
Newton-Raphson update: σ_new = σ_old − (Model price − Market price) ÷ Vega; Vega is positive, so a model price below market means you raise σ.
Scaling volatility over time: σ_T = σ_annual × √T; For a 1-month horizon, T = 1/12. VIX is quoted annualized; divide by √12 for the one-month standard deviation.

How to solve Implied vs Historical Volatility questions

Use this method for any question on implied versus historical volatility.

1Identify which measure the question asks about: estimated from past returns (historical) or backed out of option prices (implied).
2For historical: list the returns, compute the mean if needed, then the sample variance and standard deviation. Check whether returns are simple or log, and whether the zero-mean shortcut is allowed.
3Annualize by multiplying by the square root of periods per year. Do not multiply by the periods themselves.
4For implied: note the market price and compare it with the BSM price at a trial volatility. If the model price is too low, raise volatility. If too high, lower it.
5Use vega to refine the estimate if the question gives it. The change in σ ≈ price gap ÷ vega.
6For comparison questions, explain the gap: forward-looking versus backward-looking, volatility risk premium, skew, and model assumptions.
7Check units: volatility as a percentage per year, time in years, and VIX as annualized percentage points.

Quickest way: Vega shortcut and square-root scaling

When to use it: When the exam gives a price gap and vega, or asks you to convert between annual and short-horizon volatility.

Price gap ÷ vega gives the approximate change in volatility. Check that vega's units match (per 1.00 or per 1%).
For horizon conversion, divide annual volatility by √(periods per year): √252 for days, √52 for weeks, √12 for months.
For conceptual choices, remember: implied is forward-looking and includes a risk premium; historical is backward-looking and depends on the window.
Eliminate any option that says implied volatility is directly observable or that it always equals realized volatility.

Common mistakes in Implied vs Historical Volatility

Annualizing by multiplying daily volatility by 252 instead of √252.
Returns add over time, so students assume volatility does too.
Fix: Variance grows linearly with time, so volatility grows with the square root of time. Use √252.
Treating implied volatility as a forecast that is always accurate.
It is forward-looking, so it seems like the best predictor.
Fix: Implied volatility contains a risk premium and is model-dependent. It tends to exceed later realized volatility on average, especially for equity index options, but not for every asset or period.
Using n instead of n − 1 in sample standard deviation without checking the question.
Population and sample formulas get mixed up.
Fix: Use n − 1 for a sample unless the question says to assume a zero mean and use n.
Adjusting implied volatility in the wrong direction when matching a market price.
Students forget that option prices increase with volatility.
Fix: Model price below market means raise σ. Model price above market means lower σ.
Saying the VIX measures realized volatility or is an average of historical returns.
The word volatility suggests a historical statistic.
Fix: The VIX is derived from S&P 500 option prices and measures expected 30-day volatility, annualized.
Assuming one implied volatility applies to all strikes.
BSM uses a single constant σ.
Fix: Implied volatility differs by strike and maturity (smile or skew). This shows BSM's assumptions are imperfect.

Worked examples

Example 1

A stock has daily log returns over five days of 1%, −2%, 0%, 3% and −1%. Using the sample standard deviation (n − 1), estimate the annualized historical volatility with 252 trading days. Give the answer to the nearest 0.1%.

Show the solution

Mean = (1 − 2 + 0 + 3 − 1) ÷ 5 = 1 ÷ 5 = 0.2%.
Deviations: 0.8, −2.2, −0.2, 2.8, −1.2.
Squared deviations: 0.64, 4.84, 0.04, 7.84, 1.44. Sum = 14.80.
Sample variance = 14.80 ÷ 4 = 3.70, so daily σ = √3.70 = 1.9235%.
Annualize: 1.9235% × √252 = 1.9235 × 15.8745 = 30.53%.

Answer: About 30.5% annualized.

Example 2

A European call has a market price of 5.40. At a trial volatility of 20%, the BSM price is 4.90 and vega is 0.25 per 1 percentage point of volatility. Using one Newton-Raphson step, what is the new implied volatility estimate?

Show the solution

Price gap = market − model = 5.40 − 4.90 = 0.50.
The model price is below market, so volatility must rise.
Vega is 0.25 per 1 percentage point, so change = 0.50 ÷ 0.25 = 2 percentage points.
New estimate = 20% + 2% = 22%.

Answer: Approximately 22%.

Exam tips

Be ready for conceptual questions on why implied volatility tends to exceed realized volatility, especially for equity index options: the answer is the volatility risk premium plus demand for protection.
Always check the annualization factor in the question. Quick errors here cost easy marks.
For VIX questions, remember it is derived from index options, expected 30-day horizon, quoted annualized.
For smile or skew questions, link them to fat tails and BSM's constant-volatility assumption.
A financial calculator's standard deviation function (sample, Sx) saves time on historical volatility. Check you read Sx and not σx.

Practice questions from Measuring and Monitoring Volatility

Implied vs Historical Volatility in other exams

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

Implied vs Historical Volatility: frequently asked questions

What is the difference between implied and historical volatility?

Historical volatility is calculated from past returns and describes what already happened. Implied volatility is backed out of current option prices and reflects the market's forward-looking expectation plus a risk premium.

How is implied volatility calculated from Black-Scholes?

You set the BSM price equal to the observed market price and solve for volatility, holding the other inputs fixed. There is no closed-form solution, so you use a numerical method such as Newton-Raphson or bisection.

Why does implied volatility usually exceed realized volatility?

Option sellers demand a volatility risk premium for bearing the risk of large moves and jumps. Investors also pay up for downside protection, which pushes option prices and implied volatility higher.

What does the VIX measure?

The VIX measures the market's expectation of S&P 500 volatility over the next 30 days, calculated from a range of index option prices and quoted as an annualized percentage.