Finance / Bollerslev GARCH(1,1) conditional variance recursion

GARCH(1,1) Calculator — Volatility Forecasting

Forecast volatility with the GARCH(1,1) recursion: long-run variance, persistence, half-life, multi-step forecasts, and conditional return densities.

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ZorvLabs · Volatility forecasting

GARCH Volatility Forecast Calculator

Univariate GARCH(1,1) conditional variance recursion with exact multi-step forecasts, log-likelihood diagnostics, fan bands, and 2D, 3D, and combined views.

Stability

stationary

Persistence φ = α + β = 0.98000

One-step variance

0.00024114

σ²_{t+1} exact recursion

Long-run variance

0.00100000

ω / (1 − α − β)

Horizon variance

0.00057761

σ² at h = 30

Half-life

34.3096

ln(0.5)/ln(φ)

View mode

3D surface

Display scale

Mean return μ

GARCH constant ω

ARCH coefficient α

GARCH coefficient β

Initial variance σ²_0

Forecast horizon h

Scenario length

Fan quantile z-score

Density window (σ)

Surface α min

Surface α max

Surface β min

Surface β max

Surface X resolution

Surface Y resolution

Persistence φ

0.980000

α + β

Stationarity

Yes

φ < 1 required

Scenario logL

557.3242

Gaussian filter diagnostic

One-step σ

0.015529

√σ²_{t+1}

ARCH/GARCH recursion: σ²_t = ω + αε²_{t-1} + βσ²_{t-1}.

Forecast: E_t[σ²_{t+h}] = σ̄² + (α+β)^{h-1}(σ²_{t+1} − σ̄²).

2D filtered path

Deterministic scenario shocks, conditional variance, and realized squared innovations

2D forecast term structure

Exact multi-step conditional variance forecast

2D return fan chart

Gaussian conditional return intervals from the forecast variance path

2D density slice

Exact conditional density of returns at the selected horizon

3D exact surface

Forecast variance over ARCH/GARCH coefficients at fixed horizon

Key formulas

Canonical ARCH / GARCH recursion and forecast algebra

Engle (1982), ARCH(q): ε_t = σ_t z_t, σ_t² = ω + Σ α_i ε²_{t-i} with E[z_t]=0 and Var[z_t]=1.

Bollerslev (1986), GARCH(p,q): σ_t² = ω + Σ α_i ε²_{t-i} + Σ β_j σ²_{t-j}.

GARCH(1,1) forecast (φ = α+β < 1): E_t[σ²_{t+h}] = σ̄² + φ^(h−1)(σ²_{t+1} − σ̄²), σ̄² = ω/(1−φ).

Gaussian log-likelihood: ℓ = −½ Σ [ log(2π) + log(σ_t²) + ε_t²/σ_t² ].

This tool follows the standard GARCH(1,1) specification and the exact multi-step variance forecast formula.

finance methodology

A clear path from inputs to results.

For risk managers and quant analysts who need tomorrow’s volatility, not last month’s average. The solver implements the GARCH(1,1) conditional variance recursion of Bollerslev (1986): volatility reacts to the latest squared shock and decays back toward a long-run level at a rate set by the persistence. It reports one-step and multi-step variance forecasts, the long-run variance, the shock half-life, and the conditional return density at your chosen horizon.

Step-by-step methodWorked exampleAssumptions explained

How to use

Set the GARCH parameters: the constant omega and the ARCH and GARCH coefficients alpha and beta; persistence is alpha + beta.
Set the current conditional variance sigma-squared-0, which seeds the recursion, along with the mean return mu.
Choose the forecast horizon h (1 to 120 steps); the one-step, long-run, and horizon variance cards update immediately.
Check the diagnostics: persistence with its stationarity flag, the long-run variance omega/(1 - alpha - beta), and the half-life ln(0.5)/ln(persistence).
Review the filtered scenario path (conditional variance against realized squared shocks), the forecast term structure, the fan bands, and the return density at h.
Use the 3D surface to sweep the forecast variance across alpha-beta combinations at the fixed horizon.

The formula

\sigma_t^2 = \omega + \alpha\,\varepsilon_{t-1}^2 + \beta\,\sigma_{t-1}^2, \qquad \mathbb{E}_t\left[\sigma_{t+h}^2\right] = \bar{\sigma}^2 + (\alpha + \beta)^{h-1}\left(\sigma_{t+1}^2 - \bar{\sigma}^2\right), \qquad \bar{\sigma}^2 = \frac{\omega}{1-\alpha-\beta}

Reference: Engle (1982) for ARCH; Bollerslev (1986), "Generalized Autoregressive Conditional Heteroskedasticity," Journal of Econometrics 31, for GARCH; forecast algebra as in standard time-series texts (e.g. Tsay, Analysis of Financial Time Series).

Worked example

Daily GARCH(1,1) with 98% persistence, forecast 30 steps out

Inputs

omega = 0.00002, alpha = 0.08, beta = 0.90, initial variance 0.00035, mean return 0.0004, horizon h = 30, default 180-step scenario.

Calculation path

1
Persistence is alpha + beta = 0.98 < 1, so the process is covariance-stationary with long-run variance omega/(1 - alpha - beta) = 0.00002/0.02 = 0.001, a 3.16% long-run daily volatility, and a shock half-life of ln(0.5)/ln(0.98) = 34.3 steps. The default scenario leaves the recursion at variance 0.000240 with a last innovation of 0.0081, so the one-step forecast is 0.00002 + 0.08 x 0.0000664 + 0.90 x 0.0002398 = 0.0002411. The 30-step forecast is the long-run variance plus 0.98^29 = 0.5566 of the remaining gap: 0.001 + 0.5566 x (0.0002411 - 0.001) = 0.0005776, and iterating the recursion step by step reproduces that value exactly.

Result

Persistence 0.98; long-run variance 0.001 (3.16% daily volatility); half-life 34.3 steps; one-step forecast 0.0002411 (1.55% daily volatility); 30-step forecast 0.0005776 (2.40% daily volatility), still below the long-run level as the current calm mean-reverts upward.

Interpreting the result

Alpha measures how hard volatility reacts to the latest return, beta how much it remembers, and their sum how fast shocks die out: at 0.98, about 55.7% of a shock’s excess over the long run is still present after 30 steps. The forecast term structure therefore reverts geometrically to omega/(1 - alpha - beta); short-horizon risk is dominated by the latest shock, while long-horizon risk converges to the unconditional level. Watch the stationarity flag: if alpha + beta reaches 1 there is no finite long-run variance, and the forecast follows the raw recursion without a reversion anchor.

Assumptions & limitations

GARCH(1,1) with Gaussian innovations; richer specifications such as GJR, EGARCH, or Student-t tails are not modeled.
Parameters are supplied by you, not estimated: the tool filters and forecasts with the values on the sliders, so results are only as good as those inputs.
The illustrative path uses deterministic shocks so volatility clustering is visible and every run is reproducible; it is a display device, not market data.
One period is one step, typically one trading day, and forecasts assume the recursion continues unchanged.
Mean reversion requires alpha + beta < 1; at or above 1 the long-run variance is infinite and only the exact finite-horizon recursion is meaningful.

Frequently asked questions

What do alpha and beta tell me?

Alpha is the news coefficient: how much the latest squared return feeds into current variance. Beta is the memory: how much of yesterday’s variance carries over. Their sum is persistence, which sets the half-life of a shock at ln(0.5)/ln(alpha + beta); typical daily equity values put that half-life in the tens of steps.

Does this tool estimate the parameters from data?

No. You provide omega, alpha, and beta, for example from a maximum-likelihood fit in an econometrics package, and the tool filters the variance path, forecasts, and computes diagnostics with them. That makes it a forecasting and teaching instrument; the estimation step lives elsewhere.

What is the half-life of a volatility shock?

It is the number of steps until a shock’s excess over the long-run variance halves: ln(0.5)/ln(alpha + beta). At the default persistence of 0.98 that is 34.3 steps; at 0.95 it is 13.5. It is the most intuitive single summary of how long turbulent regimes last under the model.

Why does alpha + beta have to stay below 1?

Below 1 the process is covariance-stationary and shocks decay geometrically toward the finite long-run variance omega/(1 - alpha - beta). At exactly 1 (the IGARCH case) there is no finite unconditional variance and forecasts wander without a reversion anchor; above 1 the recursion is explosive. The solver flags both regimes and uses the exact finite-horizon recursion when persistence is at or above 1.

How far ahead is a GARCH forecast meaningful?

The point forecast formula is exact for any horizon, but its information content decays: at the default 0.98 persistence about 56% of a shock’s excess survives 30 steps and the half-life is 34 steps. Month-ahead forecasts are informative; year-ahead forecasts are mostly the long-run variance plus noise.

Why does the scenario use deterministic shocks?

So that the filtered variance path and its log-likelihood are exactly reproducible and comparable across parameter changes, without attaching the tool to any particular market dataset. The recursion, forecasts, and densities behave identically whether innovations come from that scenario or from real returns you analyzed elsewhere.

Methodology last reviewed 2026-10-06. Results are informational and not financial or engineering advice — see our disclaimer.