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FRM Exam Part I · Measuring and Monitoring Volatility

Volatility Forecasting and Mean Reversion for FRM Part I

Updated 11 October 2026 · Fact-checked

Volatility forecasting projects future variance from a model. In GARCH(1,1), the expected variance k days ahead is V_L + (α + β)^k × (σ² − V_L), so it reverts to the long-run variance V_L. EWMA has no mean reversion: the forecast stays flat. Add daily variances to get multi-day variance.

Understand Volatility Forecasting and Mean Reversion

Volatility is not constant. After a shock it is high, and over time it drifts back toward a long-run average. This pull is called mean reversion. A forecast must capture it, or you will overstate risk over long horizons after a calm period or understate it after a spike.

In GARCH(1,1), tomorrow's variance is σ²ₙ = ω + α·u²ₙ₋₁ + β·σ²ₙ₋₁, with α + β < 1. The long-run variance is V_L = ω ÷ (1 − α − β). You can rewrite the model as σ²ₙ − V_L = α(u²ₙ₋₁ − V_L) + β(σ²ₙ₋₁ − V_L). Since the expected value of u² equals σ², the gap between the forecast and V_L shrinks by the factor (α + β) each day.

This gives the k-day-ahead forecast: E[σ²ₙ₊ₖ] = V_L + (α + β)^k × (σ²ₙ − V_L). The persistence α + β sets the speed. Near 1, reversion is slow. Smaller values mean fast reversion. The half-life is the number of days for the gap to halve.

For the total variance over the next T days, you add the expected daily variances. This is not the same as T times today's variance unless variance is flat. The term structure of volatility is the set of annualised volatilities for different horizons. It slopes up when current variance is below V_L and down when it is above.

EWMA is the special case ω = 0 and α + β = 1. There is no long-run level, so the forecast for every future day equals today's variance. Be ready to contrast the two models.

Key formulas to remember

GARCH(1,1) variance
σ²ₙ = ω + α·u²ₙ₋₁ + β·σ²ₙ₋₁
Requires α + β < 1 for stationarity and ω > 0.
Long-run variance
V_L = ω ÷ (1 − α − β)
Long-run volatility = √V_L. Also ω = γ·V_L with γ = 1 − α − β.
k-day-ahead variance forecast
E[σ²ₙ₊ₖ] = V_L + (α + β)^k × (σ²ₙ − V_L)
Here σ²ₙ is the variance for day n, so k = 0 gives σ²ₙ. Check the day indexing in the question.
Persistence
α + β
Closer to 1 means slower mean reversion.
Half-life
h = ln(0.5) ÷ ln(α + β)
Days for the gap to V_L to halve.
Variance over T days
Σ E[σ²ₙ₊ₖ] for k = 0 to T−1
Sum daily variances, then take the square root for T-day volatility.
EWMA forecast
E[σ²ₙ₊ₖ] = σ²ₙ for all k
EWMA has no mean reversion.
Annualisation
σ_annual = σ_daily × √252
Use the number of trading days given in the question.

How to solve Volatility Forecasting and Mean Reversion questions

Use this method for any multi-horizon volatility forecast question.

  1. 1Identify the model: GARCH(1,1) (mean reverting) or EWMA (flat forecast).
  2. 2List ω, α, β and the latest variance. Check whether volatility or variance is given, and whether it is daily or annual.
  3. 3Compute persistence α + β and the long-run variance V_L = ω ÷ (1 − α − β) if not provided.
  4. 4Find the gap: current variance minus V_L.
  5. 5Apply the factor (α + β)^k to the gap and add back V_L to get the k-day-ahead variance.
  6. 6If the question asks about total or average variance over T days, sum the daily forecasts or use the average formula.
  7. 7Convert to volatility with a square root, and annualise if asked.
  8. 8Sanity check: the forecast should lie between the current variance and V_L.

Quickest way: Gap-and-decay shortcut

When to use it: When the question asks for one variance k days ahead, a half-life, or the direction of the term structure.

  1. Compute the gap σ² − V_L. Its sign tells you the direction: positive means variance falls, negative means it rises.
  2. Multiply the gap by (α + β)^k using the calculator's power key.
  3. Add V_L. Eliminate answer options on the wrong side of V_L or beyond the current variance.
  4. For half-life, use ln(0.5) ÷ ln(α + β). Persistence of 0.98 gives about 34 days.
  5. If the model is EWMA, answer immediately: the forecast equals the current variance.

Common mistakes in Volatility Forecasting and Mean Reversion

  • Using k × current variance as the k-day forecast under GARCH.

    The square-root-of-time habit is applied to every model.

    Fix: Under GARCH the daily variance changes with the horizon. Forecast each day, then sum.

  • Applying persistence to volatility instead of variance.

    The question gives volatilities, and students skip the conversion.

    Fix: Square volatilities first. The decay (α + β)^k applies to the variance gap.

  • Forgetting to add V_L back after decaying the gap.

    Students stop once they have (α + β)^k × gap.

    Fix: The forecast is V_L plus the decayed gap. Check that it lies between the current and long-run variance.

  • Claiming EWMA mean reverts.

    EWMA looks like GARCH.

    Fix: EWMA has ω = 0 and α + β = 1, so the forecast is flat.

  • Off-by-one error in k.

    The variance known today is for the next day, so the exponent is unclear.

    Fix: Follow the question's wording. If σ²ₙ is the variance already forecast for day n, then day n+k uses exponent k.

  • Mixing daily and annual units.

    V_L is computed in daily terms but compared with an annual volatility.

    Fix: Convert everything to one unit before computing, and annualise last.

Worked examples

Example 1

A GARCH(1,1) model has ω = 0.000002, α = 0.08, β = 0.90. The current daily variance forecast is 0.00030. Find the expected daily variance 10 days ahead.

Show the solution
  1. Persistence = 0.08 + 0.90 = 0.98.
  2. V_L = 0.000002 ÷ (1 − 0.98) = 0.000002 ÷ 0.02 = 0.0001.
  3. Gap = 0.00030 − 0.0001 = 0.0002.
  4. 0.98^10 = 0.81707.
  5. Decayed gap = 0.0002 × 0.81707 = 0.000163414.
  6. Forecast = 0.0001 + 0.000163414 = 0.000263414.

Answer: About 0.000263, which is a daily volatility of about 1.62%.

Example 2

Using a GARCH(1,1) model with persistence 0.97, how many days does it take for the gap between current variance and long-run variance to halve?

Show the solution
  1. Solve 0.97^h = 0.5.
  2. h = ln(0.5) ÷ ln(0.97).
  3. ln(0.5) = −0.693147 and ln(0.97) = −0.030459.
  4. h = 0.693147 ÷ 0.030459 = 22.76.

Answer: About 22.8 days.

Exam tips

  • Read whether the question gives variance or volatility, and daily or annual. Many wrong options are built from unit errors.
  • Know the contrast: GARCH reverts to V_L, EWMA stays flat. Conceptual questions test this often.
  • Use the direction check. If current variance is above V_L, every forecast must be lower than today and above V_L.
  • Keep five decimals in the power step. Options are often close.
  • Know the half-life formula. Higher persistence means a longer half-life.

Practice questions from Measuring and Monitoring Volatility

Volatility Forecasting and Mean Reversion: frequently asked questions

How do I forecast variance k days ahead with GARCH(1,1)?

Use E[σ²ₙ₊ₖ] = V_L + (α + β)^k × (σ²ₙ − V_L). The gap to the long-run variance shrinks by (α + β) each day.

What is the half-life of mean-reverting volatility?

It is the time for the gap between current variance and long-run variance to halve. It equals ln(0.5) ÷ ln(α + β). Higher persistence gives a longer half-life.

Does EWMA have mean reversion?

No. EWMA is GARCH with ω = 0 and α + β = 1. The variance forecast for every future day equals the current estimate.

How do I get volatility over several days?

Add the expected daily variances over the horizon, then take the square root. Only a constant-variance model lets you simply multiply by the number of days.