HestonNandiGarchFit: Heston-Nandi Garch(1,1) Modelling
In fOptions: Rmetrics - Pricing and Evaluating Basic Options

Description Usage Arguments Details Value Author(s) References Examples

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 Heston-Nandi Garch(1,1) process,
`hngarchFit`	MLE for a Heston Nandi Garch(1,1) model,
`hngarchStats`	True moments of the log-Return distribution,
`print.hngarch`	Print method,
`summary.hngarch`	Diagnostic summary.

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, ...)

`description`	a brief description of the porject of type character.
`innov`	[hngarchSim] - is a univariate time series or vector of innovations to produce the series. If not provided, `innov` will be generated using the random number generator specified by `rand.gen`. Missing values are not allowed. By default the normal random number generator will be used.
`model`	a list of GARCH model parameters with the following entries: `lambda`, `omega`, the constant coefficient of the variance equation, `alpha` the autoregressive coefficient, `beta` the variance coefficient, `gamma` the asymmetry coefficient, and `rf`, the risk free rate, numeric values.
`n`	[hngarchSim] - is the length of the series to be simulated. The default value is 1000.
`n.start`	[hngarchSim] - gives the number of start-up values to be discarded. The default value is 100.
`object`	[summary] - a fitted HN-GARCH(1,1) time series object of class `"hngarch"` as returned from the function `hngarchFit`.
`rand.gen`	[hngarchSim] - is the function which is called to generate the innovations. Usually, `rand.gen` will be a random number generator. Additional arguments required by the random number generator `rand.gen`, usually the location, scale and/or shape parameter of the underlying distribution function, have to be passed through the `dots` argument.
`start.innov`	[hngarchSim] - is a univariate time series or vector of innovations to be used as start up values. Missing values are not allowed.
`symmetric`	[hngarchFit] - a logical, if TRUE a symmetric model is estimated, otherwise the parameters are estimated for an asymmetric HN Garch(1,1) model.
`title`	a character string which allows for a project title.
`trace`	[hngarchFit] - a logical value. Should the optimizarion be traced? If `trace=FALSE`, no tracing is done of the iteration path.
`x`	[hngarchFit] - an univariate vector or time series. [print] - a fitted HN-GARCH(1,1) time series object of class `"hngarch"` as returned from the function `hngarchFit`.
`...`	additional arguments to be passed.

Path Simulation:

The function hngarchSim simulates a Heston-Nandi 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 log-likelihood approach the parameters either for a symmetric or an asymmetric Heston-Nandi 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 Heston-Nandi 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.

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.

Diethelm Wuertz for the Rmetrics R-port.

Heston S.L., Nandi S. (1997); A Closed-Form GARCH Option Pricing Model, Federal Reserve Bank of Atlanta.

## hngarchSim -
   # Simulate a Heston Nandi Garch(1,1) Process:
   # Symmetric Model - Parameters:
   model = list(lambda = 4, omega = 8e-5, alpha = 6e-5, 
     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 - 
   # HN-GARCH log likelihood Parameter Estimation:
   # To speed up, we start with the simulated model ...
   mle = hngarchFit(model = model, x = ts, symmetric = TRUE)
   mle
   
## summary.hngarch - 
   # HN-GARCH Diagnostic Analysis:
   par(mfrow = c(3, 1), cex = 0.75)
   summary(mle)    

## hngarchStats - 
   # HN-GARCH Moments:
   hngarchStats(mle$model)

Loading required package: timeDate
Loading required package: timeSeries
Loading required package: fBasics


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Analysing Markets and calculating Basic Statistics
Copyright (C) 2005-2014 Rmetrics Association Zurich
Educational Software for Financial Engineering and Computational Science
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Rmetrics Package fOptions
Pricing and Evaluating Basic Options
Copyright (C) 2005-2014 Rmetrics Association Zurich
Educational Software for Financial Engineering and Computational Science
Rmetrics is free software and comes with ABSOLUTELY NO WARRANTY.
https://www.rmetrics.org --- Mail to: info@rmetrics.org
Warning message:
In nlm(.llhHNGarch, par.start, trace = trace, symmetric = symmetric,  :
  NA/Inf replaced by maximum positive value

Title:
 Heston-Nandi Garch Parameter Estimation 

Call:
 hngarchFit(x = ts, model = model, symmetric = TRUE)

Parameters:
lambda   omega   alpha    beta   gamma      rf  
 4e+00   8e-05   6e-05   7e-01   0e+00   0e+00  

Coefficients: lambda, omega, alpha, beta, gamma
   lambda      omega      alpha       beta      gamma  
3.308e+00  3.402e-05  7.460e-05  8.001e-01  0.000e+00  

Log-Likelihood:
 2436.63 

Persistence and Variance:
 0.800118 
 0.0005434072 

Description:
 Wed Jun 14 20:11:40 2017 by user: anon 


Title:
Heston-Nandi Garch Parameter Estimation 

Call:
hngarchFit(x = ts, model = model, symmetric = TRUE)

Parameters:
lambda   omega   alpha    beta   gamma      rf  
 4e+00   8e-05   6e-05   7e-01   0e+00   0e+00  

Coefficients: lambda, omega, alpha, beta, gamma
   lambda      omega      alpha       beta      gamma  
3.308e+00  3.402e-05  7.460e-05  8.001e-01  0.000e+00  

Log-Likelihood:
2436.63 

Persistence and Variance:
0.800118 
0.0005434072 
$mean
[1] 0.001797332

$variance
[1] 0.0005437456

$skewness
[1] 0.02422895

$kurtosis
[1] 3.315409

$persistence
[1] 0.800118

$leverage
[1] 0

$meansigma2
[1] 0.0005434072

$meansigma4
[1] 3.262268e-07

$meansigma6
[1] 2.177046e-10

$meansigma8
[1] 1.621986e-13