Financial volatility is not a constant background parameter. It clusters, decays, spikes and changes its memory across regimes. ARCH and GARCH models turn that behaviour into a conditional variance process that can be forecast, monitored and challenged.

01 / Conditional variance

The basic GARCH(1,1) structure can be written as r_t = mu + epsilon_t, epsilon_t = sigma_t z_t, and sigma_t^2 = omega + alpha epsilon_{t-1}^2 + beta sigma_{t-1}^2. The model separates return direction from volatility state: the shock is random, but the scale of that shock is time-varying.

02 / Shock memory

The alpha term reacts to fresh surprises. The beta term carries yesterday's variance into today. When alpha + beta is close to one, volatility is persistent: shocks decay slowly and the model keeps risk elevated. That persistence is often more informative than the point forecast itself.

03 / Where the simple model breaks

Plain GARCH is symmetric. It does not naturally know that equity markets often react more violently to negative returns than to positive returns. Extensions such as EGARCH or GJR-GARCH add asymmetry, while heavy-tailed innovations handle the fact that standardized residuals are rarely Gaussian in stressed markets.

04 / Risk use case

In a portfolio setting, the model is useful when it feeds VaR, Expected Shortfall, scenario scaling or position sizing. The raw forecast matters less than the decomposition: new shock, persistent variance, residual distribution and confidence in the calibration window.

05 / Interface implication

A strong analytics screen should show volatility forecast, realized volatility, standardized residuals, persistence, exceedance count and regime breaks together. The goal is to see not only that volatility is rising, but whether the model still understands why.

Volatility is not noise around the model. In markets, volatility is often the model.

Reference note: Nobel Prize note on Engle and ARCH/GARCH.