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FRM Exam Part I · Measuring and Monitoring Volatility

GARCH(1,1) Model: Formula, Persistence and Forecasting

Updated 11 October 2026 · Fact-checked

GARCH(1,1) updates variance as σ²ₙ = ω + α·uₙ₋₁² + β·σ²ₙ₋₁. Persistence is α + β, and it must be below 1. Long-run variance is ω ÷ (1 − α − β). Forecasts of future variance revert toward that long-run level at a rate set by α + β.

Understand GARCH(1,1) Model

Volatility is not constant. Large moves tend to follow large moves, and calm periods follow calm periods. This is volatility clustering. GARCH(1,1) is a model built to capture it.

The model says today's variance forecast has three parts: a constant ω, yesterday's squared return uₙ₋₁² (the latest shock) weighted by α, and yesterday's variance estimate σ²ₙ₋₁ weighted by β. The "(1,1)" means one lag of the squared return and one lag of the variance.

The sum α + β is the persistence. It tells you how slowly a shock fades. A value near 1, such as 0.98, means shocks last a long time. For the variance process to be stable you need α + β < 1. If α + β ≥ 1 there is no finite long-run variance.

Because the constant ω is positive, the model pulls variance back to a long-run variance V_L = ω ÷ (1 − α − β). This is mean reversion. When current variance is above V_L, forecasts fall toward it. When it is below V_L, forecasts rise toward it.

EWMA is a special case with ω = 0, α = 1 − λ and β = λ. Then α + β = 1, there is no long-run variance, and forecasts do not mean-revert. That is the main GARCH vs EWMA difference the exam tests.

Key formulas to remember

GARCH(1,1) variance equation
σ²ₙ = ω + α·uₙ₋₁² + β·σ²ₙ₋₁
uₙ₋₁ is the return from the previous period. σ²ₙ₋₁ is the previous variance estimate. Parameters ω, α, β are positive.
Persistence
α + β
Must be less than 1 for a stable model. Closer to 1 means shocks decay more slowly.
Long-run variance
V_L = ω ÷ (1 − α − β)
Defined only when α + β < 1. Long-run volatility is √V_L.
Alternative parametrisation
ω = γ·V_L, with γ = 1 − α − β
Gives σ²ₙ = γ·V_L + α·uₙ₋₁² + β·σ²ₙ₋₁. The weights γ, α, β sum to 1.
Expected future variance
E[σ²ₙ₊ₜ] = V_L + (α + β)ᵗ × (σ²ₙ − V_L)
Gap to V_L shrinks by the factor (α + β) each period.
Half-life of a variance shock
t = ln(0.5) ÷ ln(α + β)
Periods for the gap to V_L to halve.
EWMA as a special case
σ²ₙ = λ·σ²ₙ₋₁ + (1 − λ)·uₙ₋₁²
GARCH with ω = 0, α = 1 − λ, β = λ. Persistence equals 1, so no mean reversion.

How to solve GARCH(1,1) Model questions

Use this order for any GARCH(1,1) question. Work in variance, and take the square root only at the end.

  1. 1Write the equation and identify ω, α, β from the question. If given γ and V_L, convert using ω = γ·V_L.
  2. 2Check units. Convert percentages to decimals, so 2% becomes 0.02 and its square is 0.0004. Keep returns and variances on the same period basis (usually daily).
  3. 3Compute persistence α + β and confirm it is below 1.
  4. 4If asked for long-run variance, compute ω ÷ (1 − α − β). Take the square root for long-run volatility.
  5. 5If asked for the next variance, substitute the latest return squared and the latest variance. Do not use the latest volatility without squaring it.
  6. 6If asked for a forecast t periods ahead, use V_L + (α + β)ᵗ × (σ²ₙ − V_L). Make sure σ²ₙ is the right starting point.
  7. 7Take the square root to convert variance to volatility. To annualise, multiply daily volatility by √252 (or the convention the question gives).
  8. 8Sanity check: the forecast must lie between current variance and V_L.

Quickest way: Shortcut: gap-to-long-run method

When to use it: Multi-period forecast questions and questions that give V_L or persistence directly.

  1. Compute V_L = ω ÷ (1 − α − β) first. Most answers hinge on it.
  2. Find the gap: current variance minus V_L.
  3. Multiply the gap by (α + β)ᵗ.
  4. Add V_L back to get the forecast variance, then take the square root.
  5. Eliminate options quickly: the forecast must sit between current variance and V_L, and must be closer to V_L as t grows.

Common mistakes in GARCH(1,1) Model

  • Using ω ÷ (α + β) or ω ÷ (1 − α) for long-run variance.

    The formula is memorised loosely.

    Fix: Always divide ω by (1 − α − β). It equals ω divided by γ, the weight on the long-run variance.

  • Plugging return or volatility into the equation instead of squared return and variance.

    Questions often quote volatility, such as 1.2% daily, and students use it directly.

    Fix: Square the return and square the volatility before substituting. Convert 1.2% to 0.012, then square to 0.000144.

  • Forgetting to convert percentages to decimals.

    The numbers look cleaner as 2 than 0.02.

    Fix: Convert to decimals first. A squared 2% return is 0.0004, not 4, and ω is usually a very small number.

  • Saying GARCH and EWMA both mean-revert.

    Both use a weighted average of past variance and squared returns.

    Fix: EWMA has no ω, so α + β = 1 and no long-run variance. Only GARCH with α + β < 1 mean-reverts.

  • Using (α + β) as the decay factor on volatility instead of variance.

    Students forecast volatility directly.

    Fix: The decay applies to the gap in variance. Forecast variance, then take the square root.

  • Treating a persistence of 0.98 as a 98% chance of anything, or ignoring that α + β ≥ 1 breaks the model.

    Persistence is read like a probability.

    Fix: Read it as the fraction of the gap to V_L that remains each period. If α + β ≥ 1, no finite long-run variance exists.

Worked examples

Example 1

A daily GARCH(1,1) model has ω = 0.000002, α = 0.08 and β = 0.90. Yesterday's return was −2% and yesterday's volatility estimate was 1.2%. Find the persistence, the long-run daily volatility, and today's variance and volatility estimate.

Show the solution
  1. Persistence = α + β = 0.08 + 0.90 = 0.98, which is below 1.
  2. Long-run variance V_L = ω ÷ (1 − α − β) = 0.000002 ÷ 0.02 = 0.0001.
  3. Long-run daily volatility = √0.0001 = 0.01, or 1%.
  4. Squared return: uₙ₋₁² = (−0.02)² = 0.0004. Previous variance: σ²ₙ₋₁ = 0.012² = 0.000144.
  5. σ²ₙ = 0.000002 + 0.08 × 0.0004 + 0.90 × 0.000144 = 0.000002 + 0.000032 + 0.0001296 = 0.0001636.
  6. Volatility = √0.0001636 ≈ 0.01279, or about 1.279% per day.

Answer: Persistence is 0.98, long-run daily volatility is 1%, and today's estimate is variance 0.0001636, or about 1.28% daily volatility.

Example 2

Using the same model (ω = 0.000002, α = 0.08, β = 0.90), the current daily variance estimate is σ²ₙ = 0.0001636. Forecast the expected daily variance and volatility 10 days ahead, and the half-life of a variance shock.

Show the solution
  1. V_L = 0.0001 and α + β = 0.98, from the model.
  2. Gap now = σ²ₙ − V_L = 0.0001636 − 0.0001 = 0.0000636.
  3. Decay factor = 0.98¹⁰ ≈ 0.8171.
  4. Remaining gap = 0.0000636 × 0.8171 ≈ 0.00005197.
  5. Forecast variance = 0.0001 + 0.00005197 = 0.00015197.
  6. Forecast volatility = √0.00015197 ≈ 0.01233, or about 1.233% per day.
  7. Half-life = ln(0.5) ÷ ln(0.98) = −0.6931 ÷ −0.0202 ≈ 34.3 days.

Answer: Expected variance 10 days ahead is about 0.000152, or about 1.23% daily volatility. The half-life is about 34 days.

Exam tips

  • Memorise V_L = ω ÷ (1 − α − β). Many questions only ask for this, or give V_L and ask you to find ω.
  • Check whether the question gives returns or volatility, and square accordingly. Mistakes here are the easiest marks to lose.
  • For conceptual questions, link persistence to the speed of mean reversion. Higher α + β means slower reversion and a longer half-life.
  • If a question says ω = 0 or α + β = 1, think EWMA: no long-run variance and a flat variance forecast.
  • A financial calculator helps with powers: enter 0.98, press the power key, then 10. Use the memory key to store V_L.

Practice questions from Measuring and Monitoring Volatility

GARCH(1,1) Model: frequently asked questions

What is the GARCH(1,1) formula for FRM Part I?

σ²ₙ = ω + α·uₙ₋₁² + β·σ²ₙ₋₁. Here uₙ₋₁ is the previous period's return and σ²ₙ₋₁ is the previous variance estimate. The model needs α + β < 1 for a finite long-run variance.

How do I calculate long-run variance from omega, alpha and beta?

Divide ω by (1 − α − β). For example, ω = 0.000002 with α + β = 0.98 gives 0.000002 ÷ 0.02 = 0.0001. Take the square root for long-run volatility, here 1% per day.

What does persistence α + β tell you?

It shows how slowly shocks to variance fade. Each period, the gap between current variance and long-run variance is multiplied by α + β. A value close to 1 means a slow return to the long-run level.

What is the difference between GARCH and EWMA?

EWMA is GARCH with ω = 0, so α + β = 1 and there is no long-run variance or mean reversion. GARCH(1,1) with α + β < 1 pulls variance back to a long-run level. That makes GARCH better suited to multi-period forecasts.