Theoretical characteristic function (CF) of the rapidly decreasing tempered stable distribution.
charRDTS(
t,
alpha = NULL,
delta = NULL,
lambdap = NULL,
lambdam = NULL,
mu = NULL,
theta = NULL
)The CF of the the rapidly decreasing 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.
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.
Rachev, Svetlozar T. & Kim, Young Shin & Bianchi, Michele L. & Fabozzi, Frank J. (2011) 'Financial models with Lévy processes and volatility clustering' tools:::Rd_expr_doi("10.1002/9781118268070")
x <- seq(-5,5,0.25)
y <- charRDTS(x,0.5,1,1,1,0)
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