GARCH
Models and forecasts time-varying volatility by capturing the clustering of large price moves that characterize financial markets.
Overview
GARCH (Generalized AutoRegressive Conditional Heteroskedasticity) was developed by Tim Bollerslev in 1986, extending Robert Engle's ARCH model (which earned Engle the 2003 Nobel Prize). GARCH models the time-varying nature of financial market volatility — the well-observed phenomenon that large price moves tend to cluster together and be followed by more large moves, while quiet periods tend to persist. It is the standard academic and industry model for volatility forecasting.
How it looks on a chart
Illustration only — synthetic data generated for visual reference.
You may have noticed that financial markets don't have constant volatility. Some weeks are calm and quiet; others are wild and unpredictable. GARCH captures this "volatility clustering" — the tendency for high-volatility periods and low-volatility periods to each persist for a while. The key insight is that yesterday's market conditions tell you something about how volatile today will be. If the last few days have been very volatile, GARCH will predict higher-than-average volatility today. If the last few days were calm, it predicts lower-than-average volatility. This is useful for trading in several ways. You can adjust your position sizes to risk more when GARCH says volatility is low and less when it says volatility is high (volatility targeting). You can also use GARCH forecasts to price options more accurately or to set dynamic stop-loss levels that adapt to current conditions.
GARCH(1,1): σ²ₜ = ω + α·ε²ₜ₋₁ + β·σ²ₜ₋₁. Here, σ²ₜ is today's conditional variance, ε²ₜ₋₁ is yesterday's squared return (ARCH term — reaction to market shocks), σ²ₜ₋₁ is yesterday's variance forecast (GARCH term — persistence), ω is the long-run average variance, and α + β < 1 ensures mean-reversion to the long-run level. Typical estimated values for daily equity returns: α ≈ 0.05–0.15 (shock impact), β ≈ 0.80–0.90 (persistence). The high β means volatility is very persistent — a volatility shock today still affects the forecast many days ahead. The "half-life" of a volatility shock is log(0.5) / log(α+β). Variants include: EGARCH (allows asymmetric response to positive/negative shocks — the leverage effect), GJR-GARCH (adds a separate term for negative shocks), and TGARCH. The leverage effect — volatility rises more after price drops than after price rises of the same magnitude — is a well-documented empirical regularity in equity markets.
The GARCH framework is the foundation of modern volatility forecasting and options pricing beyond Black-Scholes. Heston's stochastic volatility model (1993) and its calibration to options surfaces is closely related to continuous-time GARCH. For practical options trading, GARCH-based delta hedging can reduce hedging costs by 20–30% compared to constant implied volatility assumptions (Engle & Rosenberg 2000). Realized Volatility (computed from high-frequency intraday data) has been shown to be a more accurate estimator of current volatility than GARCH on daily data (Andersen et al. 2003). The HAR-RV model (Heterogeneous AutoRegressive model of Realized Volatility) uses daily, weekly, and monthly realized volatility as a simple regression for next-day volatility forecasting, outperforming GARCH in many markets. For systematic strategy management, GARCH-based volatility targeting is one of the most robust improvements to any strategy. By scaling position size to achieve a target annualized volatility (e.g., 10%) — dividing the base position by the GARCH forecast — drawdowns are significantly reduced during high-volatility episodes. Research by Moreira and Muir (2017) shows that volatility-managed momentum and value strategies achieve substantially higher Sharpe ratios than unmanaged versions.
Formula
GARCH(1,1): σ²ₜ = ω + α·ε²ₜ₋₁ + β·σ²ₜ₋₁ where ε²ₜ₋₁ is yesterday's squared return, σ²ₜ₋₁ is yesterday's variance Constraint: α + β < 1 (mean-reverting volatility)
- 1.Compute the daily return series: rₜ = log(Pₜ / Pₜ₋₁).
- 2.Initialize with the sample variance; estimate parameters (ω, α, β) via maximum likelihood.
- 3.Each period: σ²ₜ = ω + α × rₜ₋₁² + β × σ²ₜ₋₁.
- 4.Convert conditional variance to annualized volatility: σ_annual = √(σ²ₜ × 252).
- 5.Use σ_annual to size positions (target vol / forecast vol × base position).
Parameters
| Parameter | Default | Range | Description |
|---|---|---|---|
| ARCH Term (alpha) | 0.1 | 0.01–0.3 | Sensitivity to recent return shocks (ARCH effect). |
| GARCH Term (beta) | 0.85 | 0.5–0.99 | Persistence of past volatility forecasts (GARCH effect). |
| Target Volatility | 10 | 5–30 | Annual volatility target for position sizing (%). |
Trading signals
neutral: GARCH volatility forecast rising above 2× long-run average
High volatility regime — reduce position sizes, widen stops.
bullish: GARCH volatility forecast at multi-month low
Low volatility regime — increase position sizes for volatility-targeted strategies.
bearish: GARCH forecast jumps sharply after a large down move
Leverage effect triggered — expect continued elevated volatility.
Limitations
- •GARCH parameters are estimated from historical data and may not reflect current regime dynamics.
- •Model assumes returns are i.i.d. given variance — correlation between assets is ignored in univariate GARCH.
- •Jump processes (large overnight gaps) are not well-captured by GARCH, which models only continuous volatility clustering.
- •Parameter estimation requires at least 500+ observations for stability; short histories give unreliable estimates.
Gilito uses GARCH(1,1) and GJR-GARCH models as core components in its volatility-targeting position sizing system. Every strategy in its library can optionally be run with GARCH-based volatility scaling, and Gilito backtests show which strategies benefit most from volatility management versus fixed fractional sizing.