I was watching this Youtube Video. He used a exchange rates of Euro to Dollar for a few days and apply GARCH(1,1) to get the predicted price. However, I didnt understand variance that he calculates from the table. WHat is the formula?


GARCH model is used to model persistence in volatility. If you square demean exchange rate and calculate autocorrelation you will find significant autocorrelation upto many lags that indicates the clustering of volatility in data.

A simple ARMA(1,0)-GARCH(1,1) model can be written as : $$y_t=\mu + \phi y_{t-1}+e_t $$ $$e_t \sim N(0, \sigma^2_t)$$ $$\sigma_t^2=\gamma + \beta e_{t-1}^2+ \eta\sigma_{t-1}^2$$

After calibrating the above model, you can use the first equation to forecast one day ahead exchange rate(expected exchange rate). For example: $$\mathbb{E}(y_t|\mathscr{F_{t-1}})=\mu_t+\phi y_{t-1}$$

Similarly, using last equation you can get conditional standard deviation. To get unconditional variance take expectation one more time:$$\mathbb{E[\sigma_t^2|\mathscr{F}_{t-1}}]=\gamma + \beta \, \mathbb{E}[e^2_{t-1}]+\eta \,\mathbb{E}[\sigma^2_{t-1}]$$ solving above equation, you will get $$\sigma_t^2=\frac{\gamma}{1-\beta -\eta}$$


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