# Volatility updating with Heston-Nandi model

I'm trying to program a volatility updating rule using iteration. I start with the well-known Heston-Nandi model where the returns dynamics are:

$$\left\{ \begin{array}{rcl} R_{t+1} &=& r + \gamma h_{t+1} + \sqrt{h_{t+1}} z_{t+1}\\ h_{t+1} &=& \beta_0 + \beta_1 h_t + \beta_2\left(z_t - \beta_3 \sqrt{h_t}\right)^2 \end{array} \right.$$

where

• $z_t$ is an IID standard normal random variable
• $h_t$ is time-varying squared volatility
• $\beta_0 > 0,\ \beta_1 \ge 0,\ \beta_2 \ge 0$, and $\gamma > -\frac{1}{2}$.

I want to do is to write the code associate to the volatility updating rule, explained in this algorithm:

1. Define $h_0 = c$ equals the given unconditional variance which is constant,
2. Iteration for $t \in \{1, 2, \cdots, n\}$:

$$\left\{ \begin{array}{rcl} h_{t+1} &=& \beta_0 + \beta_1 h_t + \beta_2\left(z_t - \beta_3 \sqrt{h_t}\right)^2 \\ z_t &=& \frac{[R_t - r - \gamma h_t]}{\sqrt{h_t}} \end{array} \right.$$

to obtain the returns based proxy for spot variances $(h_t^R)_t$. Which yields an updating function that exclusively involves observation:

$$h_{t+1} = \beta_0 + \beta_1 h_t + \frac{\beta_2}{h_t} \left( R_t - r - (\beta_3 + \gamma)h_t \right)^2$$

My program is the following:

library(fGarch)

T=3000
# For the example I simulate a GARCH
#process parameters
eta = 0.2 #eta = 0 is equivalent to Geometric Brownian Motion
mu = 100 #the mean of the process

#GARCH volatility model
specs = garchSpec(model = list(omega = 0.000001, alpha = 0.5, beta = 0.4))
sigma = garchSim(spec = specs, n = T)

P_0 = mu #starting price, known
P = rep(P_0,T)

for(i in 2:T){
P[i] = P[i-1] + eta * (mu - P[i-1]) + sigma[i] * P[i-1]
}

# Set the parameters :
para<-c(0.1,0.2,0.3,0.4,0.5,0.7) # (beta_0,beta_1, beta_2, beta_3, r, gamma)
# Iteration to obtain the volatility associate to the model :

vol = c()
vol[1]=sd(P)
for (i in 2:length(P)){
para_vol <- para[1:6]
vol[i]=para_vol[1]+ (para_vol[2]*vol[i-1])+ (para_vol[3]/vol[i-1])*(P[i-1]-para_vol[5]-(para_vol[4]+para_vol[6])*vol[i-1])
}
vol


This is an example where I simulate a GARCH (as data set). I am trying to extract the volatility associated with the Heston-Nandi model.

I know I'm using a lot of bad things for R, but I could not figure out a better solution. Are there any corrections or suggestions needed to improve this process?