charCTS {TempStable}

Characteristic function of the classical tempered stable (CTS) distribution

Description

Theoretical characteristic function (CF) of the classical tempered stable distribution. See Kuechler & Tappe (2013) for details.

Usage

charCTS(
  t,
  alpha = NULL,
  deltap = NULL,
  deltam = NULL,
  lambdap = NULL,
  lambdam = NULL,
  mu = NULL,
  theta = NULL,
  functionOrigin = "massing"
)

Arguments

t

A vector of real numbers where the CF is evaluated.

alpha

Stability parameter. A real number between 0 and 2.

deltap

Scale parameter for the right tail. A real number > 0.

deltam

Scale parameter for the left tail. A real number > 0.

lambdap

Tempering parameter for the right tail. A real number > 0.

lambdam

Tempering parameter for the left tail. A real number > 0.

mu

A location parameter, any real number.

theta

Parameters stacked as a vector.

functionOrigin

A string. Either "massing", or "kim10".

Details

theta denotes the parameter vector (alpha, deltap, deltam, lambdap, lambdam, mu). Either provide the parameters individually OR provide theta. Characteristic function shown here is from Massing (2023).

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

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

Origin of functions Since the parameterisation can be different for this characteristic function in different approaches, the respective approach can be selected with functionOrigin. For the estimation function TemperedEstim and therefore also the Monte Carlo function TemperedEstim_Simulation and the calculation of the density function dMTS only the approach of Massing (2023) can be selected. If you want to use the approach of Kim et al. (2010) for these functions, you have to clone the package from GitHub and adapt the functions accordingly.

massing

From Massing, T. (2023), 'Parametric Estimation of Tempered Stable Laws'.

kim10

From Kim et al. (2010) 'Tempered stable and tempered infinitely divisible GARCH models'.

Value

The CF of the classical tempered stable distribution.

References

Kim, Y. S.; Rachev, S. T.; Bianchi, M. L. & Fabozzi, F. J.(2010), 'Tempered stable and tempered infinitely divisible GARCH models', doi:10.1016/j.jbankfin.2010.01.015

Kuechler, U. & Tappe, S. (2013), 'Tempered stable distributions and processes' doi:10.1016/j.spa.2013.06.012

Massing, T. (2023), 'Parametric Estimation of Tempered Stable Laws'

Examples

x <- seq(-10,10,0.25)
y <- charCTS(x,1.5,1,1,1,1,0)