| charRDTS {TempStable} | R Documentation |
Characteristic function of the rapidly decreasing tempered stable (RDTS) distribution
Description
Theoretical characteristic function (CF) of the rapidly decreasing tempered stable distribution.
Usage
charRDTS(
t,
alpha = NULL,
delta = NULL,
lambdap = NULL,
lambdam = NULL,
mu = NULL,
theta = NULL
)
Arguments
t |
A vector of real numbers where the CF is evaluated. |
alpha |
Stability parameter. A real number between 0 and 2. |
delta |
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
The CF of the RDTS distribution is given by (Rachev et al. (2011)):
\varphi_{RDTS}(t;\theta):=
E_{\theta}\left[\mathrm{e}^{\mathrm{i}tX}\right]=
\exp\left(\mathrm{i}t\mu+\delta(G(\mathrm{i}t;\alpha,\lambda_+)
+G(-\mathrm{i}t;\alpha,\lambda_-))\right),
where
G\left(x;\alpha,r,\lambda\right)=
2^{-\frac{\alpha}{2}-1}\lambda^\alpha\Gamma\left(-\frac{\alpha}{2}\right)
\left(M\left(-\frac{\alpha}{2},\frac{1}{2};\frac{x^2}{2\lambda^2}\right)
-1\right)\\
+2^{-\frac{\alpha}{2}-\frac{1}{2}}\lambda^{\alpha-1}x
\Gamma\left(\frac{1-\alpha}{2}\right)
\left(M\left(\frac{1-\alpha}{2},\frac{3}{2};\frac{x^2}{2\lambda^2}\right)
-1\right).
M stands for the confluent hypergeometric function.
Value
The CF of the the rapidly decreasing tempered stable distribution.
References
Rachev, Svetlozar T. & Kim, Young Shin & Bianchi, Michele L. & Fabozzi, Frank J. (2011) 'Financial models with Lévy processes and volatility clustering' doi:10.1002/9781118268070
Examples
x <- seq(-5,5,0.25)
y <- charRDTS(x,0.5,1,1,1,0)