GARCH(1,1) Volatility Forecasting Lab

1. Volatility Clustering & ARCH Foundations

Mandelbrot (1963) observed that in financial asset prices, "large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes." Robert Engle (1982) formalized this via the Autoregressive Conditional Heteroskedasticity (ARCH) model, establishing that while unconditional return variance may be stationary, conditional variance \(\sigma_t^2 = \text{Var}(r_t \mid \mathcal{F}_{t-1})\) evolves endogenously over time.

2. The Bollerslev (1986) GARCH(1,1) Specification

Tim Bollerslev generalized the ARCH model into GARCH(1,1) by incorporating autoregressive lagged conditional variance terms, drastically reducing the parameter count needed to fit persistent volatility: \[ r_t = \mu + \epsilon_t, \quad \epsilon_t = \sigma_t z_t, \quad z_t \overset{\text{i.i.d.}}{\sim} \mathcal{N}(0, 1) \] \[ \sigma_t^2 = \omega + \alpha \epsilon_{t-1}^2 + \beta \sigma_{t-1}^2 \] where \(\omega > 0\), \(\alpha \ge 0\), and \(\beta \ge 0\). The persistence is \(P = \alpha + \beta\). If \(P < 1\), the unconditional long-run variance exists and equals: \[ V_L = \mathbb{E}[\sigma_t^2] = \frac{\omega}{1 - \alpha - \beta} \] The half-life of a volatility shock—the number of trading sessions required for variance deviation to decay by 50%—is: \[ t_{1/2} = \frac{\ln(0.5)}{\ln(\alpha + \beta)} \]

3. Asymmetric Leverage & The GJR-GARCH Formulation

Standard GARCH treats positive return shocks (rallies) and negative return shocks (crashes) symmetrically, because \(\epsilon_{t-1}^2 = (-\epsilon_{t-1})^2\). However, Fischer Black (1976) identified the asymmetric leverage effect: debt-to-equity ratios increase when share prices drop, amplifying equity risk. Glosten, Jagannathan, and Runkle (1993) introduced the GJR-GARCH model: \[ \sigma_t^2 = \omega + \bigl(\alpha + \gamma \mathbb{I}_{\{\epsilon_{t-1} < 0\}}\bigr) \epsilon_{t-1}^2 + \beta \sigma_{t-1}^2 \] where \(\mathbb{I}\) is an indicator function equal to 1 if \(\epsilon_{t-1} < 0\) and 0 otherwise. The parameter \(\gamma > 0\) governs the asymmetric penalty of market crashes. For GJR-GARCH, persistence is \(P = \alpha + \beta + \frac{1}{2}\gamma\).

4. Multi-Step Forward Term Structure of Volatility

Using conditional expectations, the \(k\)-step-ahead variance forecast is: \[ \mathbb{E}_t\bigl[\sigma_{t+k}^2\bigr] = V_L + (\alpha + \beta)^k \bigl(\sigma_t^2 - V_L\bigr) \] This mathematical structure explains why implied volatility curves (such as the VIX term structure) flatten and invert during crises: high spot volatility \(\sigma_t^2 \gg V_L\) decays geometrically over the forecast horizon toward long-run equilibrium \(V_L\).