11.2 Volatility Modelling & Forecasting using GARCH

• The Generalised Autoregressive Conditional Heteroskedasticity (GARCH) models (, ; ), most popular time series models used for forecasting conditional volatiltiy.

• These models are conditional heteroskedastic as they take into account the conditional variance in a time series. GARCH models are one of the most widely used models for forecasting financial risk measures like VaR and Conditional VaR in financial risk modelling and management.

• The GARCH models are a generalised version of ARCH models. A standard ARCH(p) process with p lag terms designed to capture volatility clustering can be written as follows

$$$\sigma_{t}^{2}=\omega+\sum_{i=1}^{p}\alpha_{i}Y_{t-i}^{2} \tag{11.3}$$$

where the return on day t, is $$Y_{t}=\sigma_{t}Z_{t}$$ and $$Z_{t}\sim i.i.d(0,1)$$, i.e., the innovation in returns are driven by random shocks

• The GARCH(p,q) model include lagged volatility in an ARCH(p) model to incorporate the impact of historical returns which can be written as follows

$$$\sigma_{t}^{2}=\omega+\sum_{i=1}^{p}\alpha_{i}Y_{t-i}^{2}+\sum_{j=1}^{q}\beta_{j}\sigma_{t-j}^{2} \tag{11.4}$$$

• GARCH(1,1) which employs only one lag per order, is the most common version used in empirical research and analysis.

References

Bollerslev, Tim. 1986. “Generalized Autoregressive Conditional Heteroskedasticity.” Journal of Econometrics 31 (3): 307–27.
Engle, Robert. 2001. “GARCH 101: The Use of ARCH/GARCH Models in Applied Econometrics.” Journal of Economic Perspectives, 157–68.
Engle, Robert F. 1982. “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation.” Econometrica: Journal of the Econometric Society, 987–1007.