# 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=sd(P)
for (i in 2:length(P)){
para_vol <- para[1:6]
vol[i]=para_vol+ (para_vol*vol[i-1])+ (para_vol/vol[i-1])*(P[i-1]-para_vol-(para_vol+para_vol)*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?

First, in R we use <- for assignment. It has its advantages.

It is prefered that you don't use T and F as variable names as these are aliases to TRUE and FALSE.

Instead of this P <- rep(P_0, T) you could have P <- vector(mode = "numeric", length = T).

This is just a suggestion: But there is no reason to not name your 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)
para <- c(beta_0 = 0.1,  beta_1 = 0.2, beta_2 = 0.3, beta_4 = 0.4, r = 0.5, gamma = 0.7) # (beta_0,beta_1, beta_2, beta_3, r, gamma)


Then, you may access them like para["gamma"], but not para\$gamma. Also, there is no need for para_vol<-para[1:6] in the loop, but I wager this is remnant from other attempts.

In terms of performance, the pre-allocation I used did a 70 ms to 50 ms. Although, I'm not confident in those numbers, as I did it fairly unscientifically.

Also, the referenced post shows a more direct way of calculating this.

Collapsed version of the code I ended up with:

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)

# library(ggfortify)
# autoplot(sigma)

P_0 <- mu #starting price, known
# P <-  rep(P_0, T)
P <- vector(mode = "numeric", length = T)
P <- P_0

for (i in 2:T) {
P[i] <-  P[i - 1] + eta * (mu - P[i - 1]) + sigma[i] * P[i - 1]
}
# DOESN'T WORK
# P[-T] <-  P[-T] + eta * (mu - P[-T]) + sigma[-1] * P[-T]

# 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 <- vector("numeric", length = length(P))
vol <-  sd(P)
for (i in 2:length(P)) {
# para_vol <- para[1:6]
vol[i] <-
para_vol + (para_vol * vol[i - 1]) + (para_vol /
vol[i - 1]) * (P[i - 1] - para_vol - (para_vol + para_vol) * vol[i - 1])
}
vol

# autoplot(vol)