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):
charGTS(
t,
alphap = NULL,
alpham = NULL,
deltap = NULL,
deltam = NULL,
lambdap = NULL,
lambdam = NULL,
mu = NULL,
theta = NULL
)The CF of the the generalized classical tempered stable distribution.
A vector of real numbers where the CF is evaluated.
Stability parameter. A real number between 0 and 2.
Scale parameter. A real number > 0.
Tempering parameter. A real number > 0.
A location parameter, any real number.
Parameters stacked as a vector.
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).
$$\varphi_{GTS}(t;\theta):=
E_{\theta}\left[\mathrm{e}^{\mathrm{i}tX}\right]=
\exp\left(\mathrm{i}t\mu-\mathrm{i}t\Gamma(1-\alpha_+)
\left(\delta_+\lambda_+^{\alpha_+-1}\right)\right.\\$$
$$\left. +\mathrm{i}t\Gamma(1-\alpha_-)
\left(\delta_-\lambda_-^{\alpha_--1}\right)\right.\\$$
$$\left.+\delta_+\Gamma(-\alpha_+)
\left(\left(\lambda_+-\mathrm{i}t\right)^{\alpha_+}
-\lambda_+^{\alpha_+}\right) \right.\\$$
$$\left.+\delta_-\Gamma(-\alpha_-)
\left(\left(\lambda_-+\mathrm{i}t\right)^{\alpha_-}
-\lambda_-^{\alpha_-}\right)\right)$$
Rachev, S. T.; Kim, Y. S.; Bianchi, M. L. & Fabozzi, F. J. (2011), 'Financial models with Lévy processes and volatility clustering' tools:::Rd_expr_doi("10.1002/9781118268070")
x <- seq(-5,5,0.25)
y <- charGTS(x,0.3,0.2,1,1,1,1,0)
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