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I found big inconsistency in the GARCH models and their underlying assumption of stationarity. GARCH models require that data must be stationary, where stationary means both mean and variance are time invariant. If variance is time invariant i.e. constant then what is logic behind using GARCH models.

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GARCH models are essentially white noise models with some time dependency. The reason GARCH models are used is because they have a lot of nice properties. The main being that the Conditional Volatility is time-dependent. This means that volatility can cluster.

It's true that conditional vol will regress towards "normality" as a random walk process with drift.

The second nice property is the closed-form solution allows you to calculate expected vol with ease.

A third nice property is that the model is simple and very easy to fit to historical data.

As with all models you need to understand it's underlying assumptions so you can assess it's downfalls. In this case your big assumption is the stationary mean and volatility. One of which may not be true and the other nigh impossible to measure.

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