Modeling stock returns’ volatility with GARCH models
In this recipe, we present how to work with an extension of the ARCH model, namely the Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model. GARCH can be considered an ARMA model applied to the variance of a time series—the AR component was already expressed in the ARCH model, while GARCH additionally adds the moving average part.
The equation of the GARCH model can be presented as:
While the interpretation is very similar to the ARCH model presented in the previous recipe, the difference lies in the last equation, where we can observe an additional component. Parameters are constrained to meet the following: 
, and 
.
In the GARCH model, there are additional constraints on coefficients. For example, in the case of a GARCH(1,1) model, 
must be less than 1. Otherwise, the model is unstable.
The two hyperparameters of the GARCH model can be described as:
...