TradingCalcGARCH(1,1) Volatility Calculator
RISK/STATS · GARCH VOLATILITY · ADVANCED
GARCH(1,1) volatility model fit by maximum likelihood: estimates volatility persistence and forecasts next-period variance from a return series.
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Fits a constant-mean GARCH(1,1) model by maximum likelihood: estimates how strongly volatility persists period-to-period and forecasts the next period's variance. Needs real sample depth (50+ values) to identify persistence reliably.
The model
A constant-mean GARCH(1,1) model, fit by maximum likelihood on the return series you supply:
- r_t = μ + ε_t, ε_t = σ_t × z_t, z_t ~ N(0,1)
- σ_t² = ω + α × ε_(t-1)² + β × σ_(t-1)²
μ, ω, α, β are estimated jointly by maximizing the Normal log-likelihood over the whole series (a derivative-free Nelder-Mead search from several starting points, not a closed form). α+β (persistence) measures how slowly a volatility shock decays; the model requires α+β < 1 by construction, so volatility always reverts to a finite unconditional level.
Reading the result
- Persistence near 1: a volatility shock (a big move) decays slowly, clustering continues for a while
- Persistence well below 1: volatility reverts to its unconditional level quickly
- Forecast volatility: the model's one-period-ahead estimate, given the series' most recent shock and variance
Where to go next
GARCH gives a forward-looking volatility estimate instead of a flat historical figure. Feed it into a forward-looking risk snapshot with the VaR / CVaR calculator (supply the forecast volatility as the stdev input), see the full trailing risk/return picture with the portfolio tearsheet, or, if your series is infrequently-marked/appraisal-based, check how much smoothing might be understating it in the first place with the return unsmoothing calculator.
Use via API or MCP
This calculation is available as a deterministic API call for bots and AI agents.
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What assumptions and edge cases does this calculator use?
Returns are decimals per period (0.02 is 2%), not prices. It needs at least 50 returns, all finite numbers, and a flat series gives an error. The fit is Gaussian maximum likelihood for GARCH(1,1), and the volatility is per period of your data (no annualization). Not covered: asymmetric models such as GJR or EGARCH, fat-tailed errors, multi-step forecasts, and a guarantee that the optimizer found the global maximum.
What is GARCH(1,1)?
A standard model of volatility clustering: the current period's variance depends on the previous period's squared shock (alpha) and the previous period's variance (beta), plus a baseline level (omega). It's the workhorse model for describing how volatility persists over time.
How are the parameters estimated?
By maximum likelihood: the parameters that make the observed return series most probable under a Normal-innovation GARCH(1,1) model, found via a derivative-free numerical search (Nelder-Mead) from several starting points, since there's no closed-form solution.
What does persistence (alpha+beta) mean?
How slowly a volatility shock decays. Close to 1: a big move keeps elevated volatility around for a while (a fat-tailed, clustered market). Well below 1: volatility reverts to its long-run average quickly.
Is the forecast volatility a prediction of what the market will do?
No. It's a backward-looking statistical estimate: given the model fit to your historical series and its most recent shock, this is what the model implies about next period's variance. It's not a guarantee, and it doesn't account for information the historical series doesn't contain.
How much data do I need?
At least 50 return values, and meaningfully more for a reliable fit - GARCH estimates 4 parameters jointly, and a short series routinely converges to an unstable or degenerate fit.