charKRTS {TempStable}

Characteristic function of the Kim-Rachev tempered stable distribution

Description

Theoretical characteristic function (CF) of the Kim-Rachev tempered stable distribution.

Usage

charKRTS(
  t,
  alpha = NULL,
  kp = NULL,
  km = NULL,
  rp = NULL,
  rm = NULL,
  pp = NULL,
  pm = 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 1.

kp, km, rp, rm

Parameter of KR-distribution. A real number >0.

pp, pm

Parameter of KR-distribution. A real number >-alpha.

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_{KRTS}(t;\theta):= E_{\theta}\left[\mathrm{e}^{\mathrm{i}tX}\right]= \exp\left(\mathrm{i}t\mu-\mathrm{i}t\Gamma(1-\alpha) \left(\frac{k_+r_+}{p_++1}-\frac{k_-r_-}{p_-+1}\right) \right.\\

\left. +k_+H(\mathrm{i}t;\alpha,r_+,p_+)+k_ -H(-\mathrm{i}t;\alpha,r_-,p_-)\right),

where

\left. H\left(x;\alpha,r,p\right)= \frac{\Gamma(-\alpha)}{p}\left(F\left(p,-\alpha;1+p;rx\right)-1\right)\right.

F denotes the hypergeometric Function.

Value

The CF of the the Kim-Rachev 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 <- charKRTS(x,0.5,1,1,1,1,1,1,0)