HestonNandi Garch(1,1) Modelling
Description
A collection and description of functions to model
the GARCH(1,1) price paths which underly Heston and
Nandi's option pricing model.
The functions are:
hngarchSim  Simulates a HestonNandi Garch(1,1) process, 
hngarchFit  MLE for a Heston Nandi Garch(1,1) model, 
hngarchStats  True moments of the logReturn distribution, 
print.hngarch  Print method, 
summary.hngarch  Diagnostic summary. 
Usage
1 2 3 4 5 6 7 8 9 10  hngarchSim(model, n, innov, n.start, start.innov, rand.gen, ...)
hngarchFit(x, model = list(lambda = 0.5, omega = var(x), alpha =
0.1 * var(x), beta = 0.1, gamma = 0, rf = 0), symmetric = TRUE,
trace = FALSE, title = NULL, description = NULL, ...)
hngarchStats(model)
## S3 method for class 'hngarch'
print(x, ...)
## S3 method for class 'hngarch'
summary(object, ...)

Arguments
description 
a brief description of the porject of type character. 
innov 
[hngarchSim]  
model 
a list of GARCH model parameters with the following entries:

n 
[hngarchSim]  
n.start 
[hngarchSim]  
object 
[summary]  
rand.gen 
[hngarchSim]  
start.innov 
[hngarchSim]  
symmetric 
[hngarchFit]  
title 
a character string which allows for a project title. 
trace 
[hngarchFit]  
x 
[hngarchFit]  
... 
additional arguments to be passed. 
Details
Path Simulation:
The function hngarchSim
simulates a HestonNandi Garch(1,1)
process with structure parameters specified through the list
model(lambda, omega, alpha, beta, gamma, rf)
.
Parameter Estimation:
The function hngarchFit
estimates by the maximum loglikelihood
approach the parameters either for a symmetric or an asymmetric
HestonNandi Garch(1,1) model from the log returns x
of a
financial time series. For optimization R's optim
function is
used. Additional optimization parameters may be passed throught the
...
argument.
Diagnostic Analysis:
The function summary.hngarch
performs a diagnostic analysis
and recalculates conditional variances and innovations from the time
series.
Calculation of Moments:
The function hngarchStats
calculates the first four true
moments of the unconditional log return distribution for a stationary
HestonNandi Garch(1,1) process with standard normally distributed
innovations. In addition the persistence and the expectation values
of sigma to the power 2, 4, 6, and 8 can be accessed.
Value
hngarchSim
returns numeric vector with the simulated time
series points neglecting those from the first start.innov
innovations.
hngarchFit
returns list with two entries: The estimated model parmeters
model
, where model
is a list of the parameters
itself, and llh
the value of the log likelihood.
hngarchStats
returns a list with the following components:
persistence
, the value of the persistence,
meansigma2
, meansigma4
, meansigma6
, meansigma8
,
the expectation value of sigma to the power of 2, 4, 6, and 8,
mean
, variance
, skewness
, kurtosis
,
the mean, variance, skewness and kurtosis of the log returns.
summary.hngarch
returns list with the following elements: h
,
a numeric vector with the conditional variances, z
, a numeric
vector with the innovations.
Author(s)
Diethelm Wuertz for the Rmetrics Rport.
References
Heston S.L., Nandi S. (1997); A ClosedForm GARCH Option Pricing Model, Federal Reserve Bank of Atlanta.
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24  ## hngarchSim 
# Simulate a Heston Nandi Garch(1,1) Process:
# Symmetric Model  Parameters:
model = list(lambda = 4, omega = 8e5, alpha = 6e5,
beta = 0.7, gamma = 0, rf = 0)
ts = hngarchSim(model = model, n = 500, n.start = 100)
par(mfrow = c(2, 1), cex = 0.75)
ts.plot(ts, col = "steelblue", main = "HN Garch Symmetric Model")
grid()
## hngarchFit 
# HNGARCH log likelihood Parameter Estimation:
# To speed up, we start with the simulated model ...
mle = hngarchFit(model = model, x = ts, symmetric = TRUE)
mle
## summary.hngarch 
# HNGARCH Diagnostic Analysis:
par(mfrow = c(3, 1), cex = 0.75)
summary(mle)
## hngarchStats 
# HNGARCH Moments:
hngarchStats(mle$model)
