## 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.