Simulating returns from ARMA(1,0)-GARCH(1,1) model

Question

I want to obtain a simulation of one-step ahead forecasts of stock returns process governed by ARMA(1,0)-GARCH(1,1) process. The returns are of form:

$x_t = \mu + \delta x_{t-1} + \sigma_t z_t$

From my GARCH model I can forecast the conditional mean $\mu + \delta x_{t-1}$ and the conditional standard deviation $\sigma_t$. Let's assume that the distribution of $z_t$ is Gaussian.

So now I am wondering how to obtain the simulation of the stock returns using the above-described approach. My initial solution would be to simulate a number of random variables from the Gaussian distribution $N(0,1)$ and then create my one-step ahead forecast simulations as:

conditional mean (from time $t+1$) + $N(0,1)$ random variable * conditional standard deviation (from time $t+1$)

edit: what is in case of the Gaussian distribution equivalent to: $x_{t+1} \sim N((\mu + \delta x_{t}) ,\sigma_{t+1})$

Is this approach for simulating one-step ahead forecasts of stock returns appropriate? I need those simulations to create asset allocation strategies.

Answer 1

This question has already been answered on Stack Overflow. As it is important to Quant Finance, so I have added R code here. Others users may add code of other programming software to simulate ARMA(1,0)-GARCH(1,1) model.

sim.GARCH <- function(
  horizon=5, N=1e4, 
  h0 = 2e-4, 
  mu = 0, omega=0,
  alpha1 = 0.027,
  beta1  = 0.963
){
  ret <- zt <- et <- ht <- matrix(NA, nc=horizon, nr=N)
  ht[,1] <- h0
  for(j in 1:horizon){
    zt[,j]   <- rnorm(N,0,1)
    et[,j]   <- zt[,j]*sqrt(ht[,j])
    ret[,j]  <- mu + et[,j]
    if( j < horizon )
      ht[,j+1] <- omega+ alpha1*et[,j]^2 + beta1*ht[,j]
  }
  apply(ret, 1, sum)
}
x <- sim.GARCH(N=1e5)