What is the 1-step-ahead forecast from an ARCH(1) model? Can anybody shortly show and explain how to do an 1-step-forward ARCH(1) forecast?
 A: An ARCH(1) model with a constant conditional mean is
\begin{aligned}
y_t &= \mu + u_t, \\
u_t &= \sigma_t \varepsilon_t, \\
\sigma_t^2 &= \omega + \alpha u_{t-1}^2, \\
\varepsilon_t &\sim i.i.d(0,1) \\
\end{aligned}
where $d$ is some distribution, e.g. Normal.
It gives you the conditional distribution of $y_t$ (the current value of $y$) conditioned on $y_{t-1}$ (the previous value of $y$): 
$$
y_t \sim d(\mu,\omega + \alpha u_{t-1}^2).
$$ 
You can move one step ahead in time to get the conditional distribution of $y_{t+1}$ (the next period's value of $y$) based on $y_t$ (the current value of $y$): 
$$
y_{t+1} \sim d(\mu,\omega + \alpha u_t^2).
$$
From this you can tell, for example, that an optimal point forecast of $y_{t+1}$ under square loss is $\hat y_{t+1} = \hat\mu$ where $\hat\mu$ is an estimate of $\mu$. Also, the forecast of the conditional variance $\sigma_{t+1}^2$ is $\hat\sigma_{t+1}^2 = \hat\omega + \hat\alpha\hat\sigma_t^2$ where again $\hat\omega$, $\hat\alpha$ and $\hat\sigma_t^2$ denote estimates of $\omega$, $\alpha$ and $\sigma_t^2$, respectively.

In R, you can use the "rugarch" package for forecasting with an ARCH(1) model.
You (1) specify the model, (2) fit it to the data, and (3) forecast as follows:
library(rugarch)               # load the "rugarch" package
set.seed(1); x = rnorm(1000)   # generate an example time series
spec = ugarchspec(variance.model = list(garchOrder = c(0,1))) # specify the model
fit = ugarchfit(data = x, spec = spec)                        # fit the model
fcst = ugarchforecast(fitORspec = fit, n.ahead = 1)           # generate the forecast
print(fcst)                    # forecast object
print(fcst@forecast$seriesFor) # point forecast
print(fcst@forecast$sigmaFor)  # forecast of the conditional variance

