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

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.

• You may look at this: stackoverflow.com/questions/9969962/… – Neeraj Mar 4 '16 at 17:50
• Thanks @Neeraj, so it seems I am doing the very same thing. Then the error must be in the way I calculate the weights for the assets. – Masher Mar 4 '16 at 18:22
• Can you please elaborate what are residuals from GARCH model and what are Gaussian residuals? @Masher If you have done something please write it here? Your Edit 2 is incomprehensible. – Neeraj Mar 4 '16 at 18:33
• @Neeraj I deleted edit2 as it is incorrect and I don't want someone to get the wrong idea from it. As seen in the post you have referred me to, the correct approach to simulating the series does not involve any scaling. So basically the whole question is answered by your comment :) – Masher Mar 5 '16 at 16:45

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)