charGTS {TempStable}

Characteristic function of the generalized classical tempered stable (GTS) distribution.

Description

Theoretical characteristic function (CF) of the generalized classical tempered stable distribution. See Rachev et al. (2011) for details. The GTS is a more generalized version of the CTS charCTS, as alpha = alphap = alpham for CTS. The characteristic function is given - with a small adjustment - by Rachev et al. (2011):

Usage

charGTS(
  t,
  alphap = NULL,
  alpham = NULL,
  deltap = NULL,
  deltam = NULL,
  lambdap = NULL,
  lambdam = NULL,
  mu = NULL,
  theta = NULL
)

Arguments

t

A vector of real numbers where the CF is evaluated.

alphap, alpham

Stability parameter. A real number between 0 and 2.

deltap, deltam

Scale parameter. A real number > 0.

lambdap, lambdam

Tempering parameter. A real number > 0.

mu

A location parameter, any real number.

theta

Parameters stacked as a vector.

Details

theta denotes the parameter vector (alphap, alpham, deltap, deltam, lambdap, lambdam, mu). Either provide the parameters individually OR provide theta. Characteristic function shown here is from Rachev et al. (2011).

\varphi_{GTS}(t;\theta):= E_{\theta}\left[\mathrm{e}^{\mathrm{i}tX}\right]= \exp\left(\mathrm{i}t\mu-\mathrm{i}t\Gamma(1-\alpha_+) \left(\delta_+\lambda_+^{\alpha_+-1}\right)\right.\\

\left. +\mathrm{i}t\Gamma(1-\alpha_-) \left(\delta_-\lambda_-^{\alpha_--1}\right)\right.\\

\left.+\delta_+\Gamma(-\alpha_+) \left(\left(\lambda_+-\mathrm{i}t\right)^{\alpha_+} -\lambda_+^{\alpha_+}\right) \right.\\

\left.+\delta_-\Gamma(-\alpha_-) \left(\left(\lambda_-+\mathrm{i}t\right)^{\alpha_-} -\lambda_-^{\alpha_-}\right)\right)

Value

The CF of the the generalized classical tempered stable distribution.

References

Rachev, S. T.; Kim, Y. S.; Bianchi, M. L. & Fabozzi, F. J. (2011), 'Financial models with Lévy processes and volatility clustering' doi:10.1002/9781118268070

Examples

x <- seq(-5,5,0.25)
y <- charGTS(x,0.3,0.2,1,1,1,1,0)