GARCH Process

What is GARCH?

The generalized autoregressive conditional heteroskedasticity (GARCH) process is an econometric model for estimating and forecasting time-varying volatility in financial data. Introduced by Robert F. Engle in 1982 (building on ARCH), GARCH captures volatility clustering — the tendency for high-volatility periods to cluster together — and adapts as market conditions change. It is widely used for risk management, volatility forecasting, option pricing, and portfolio construction.

Key takeaways

• GARCH models volatility as a function of past shocks and past variances, unlike homoskedastic models that assume constant variance.
• They better capture volatility clustering and persistence, making them useful in normal and stressed market conditions.
• Variants (EGARCH, GJR-GARCH, multivariate GARCH) extend the base model to capture leverage effects and cross-asset dynamics.
• Limitations include sensitivity to specification, heavy-tailed returns, and limited ability to predict extreme “black swan” events.

How GARCH works — intuition and basic form

Heteroskedasticity means the variance of errors changes over time rather than remaining constant. GARCH models that changing variance (conditional variance) by making it depend on:
* past squared residuals (recent shocks), and
* past conditional variances (persistence).

A common specification is GARCH(1,1):
sigma_t^2 = ω + α * ε_{t-1}^2 + β * sigma_{t-1}^2

Where:
* sigma_t^2 = conditional variance at time t
* ε_{t-1}^2 = squared error (shock) from time t−1
* ω, α, β = parameters (ω > 0, α ≥ 0, β ≥ 0)

Interpretation:
* α measures the immediate impact of shocks on volatility.
* β measures volatility persistence (how long shocks influence future volatility).
* α + β near 1 indicates high persistence; if >1, variance may be nonstationary.

Typical estimation steps

1. Fit a mean model (e.g., autoregressive model) to capture predictable returns.
2. Compute residuals and examine autocorrelation in squared residuals.
3. Specify and estimate a GARCH(p,q) model; test parameter significance and model diagnostics (e.g., standardized residuals, Ljung–Box on squared residuals).

Comparison to other volatility methods

• Historical volatility (simple standard deviation of past returns): easy to compute but treats past observations equally and ignores evolving dynamics.
• Exponentially weighted moving average (EWMA): gives more weight to recent observations but lacks the explicit autoregressive variance specification of GARCH.
• Homoskedastic OLS: assumes constant variance and misses volatility clustering and persistence.

Practical applications

• Volatility forecasting for risk metrics (Value-at-Risk, expected shortfall).
• Option pricing and implied volatility analysis.
• Portfolio optimization and dynamic asset allocation that incorporate changing risk.
• Stress testing and scenario analysis — helps quantify how volatility can evolve during crises.
• Multivariate GARCH models capture time-varying covariances across assets for portfolio risk aggregation.

Extensions and variants

• EGARCH and GJR-GARCH: allow for asymmetry (leverage effects) where negative shocks raise volatility more than positive shocks.
• Multivariate GARCH (MGARCH): models time-varying covariances among multiple series.
• Component-GARCH, FIGARCH: capture long memory or multiple volatility components.

Limitations and cautions

• Model risk: results depend on specification (order p,q and chosen variant).
• Assumed error distributions (often normal) may understate tail risk; using t-distributions or other heavy-tailed errors helps.
• GARCH models improve volatility modeling but do not eliminate the possibility of extreme, unpredictable events.
• Requires adequate data for stable estimation; parameters can change over regimes.

Conclusion

GARCH provides a practical, widely adopted framework for modeling time-varying volatility in financial markets. By incorporating past shocks and past variances, it captures volatility clustering and persistence that constant-variance models miss. Proper model selection, diagnostics, and awareness of limitations (heavy tails, regime shifts) are essential for reliable forecasting and risk management.