charMTS {TempStable}

Characteristic function of the modified tempered stable distribution

Description

Theoretical characteristic function (CF) of the modified tempered stable distribution.

Usage

charMTS(
  t,
  alpha = NULL,
  delta = NULL,
  lambdap = NULL,
  lambdam = NULL,
  mu = NULL,
  theta = NULL,
  functionOrigin = "kim08"
)

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.

functionOrigin

A string. Either "kim09", "rachev11" or "kim08". Default is "kim08".

Details

theta denotes the parameter vector (alpha, delta, lambdap, lambdam, mu). Either provide the parameters individually OR provide theta. Characteristic function shown here is from Kim et al. (2008).

\varphi_{MTS}(t;\theta):= E_{\theta}\left[\mathrm{e}^{\mathrm{i}tX}\right]= \exp\left(\mathrm{i}t\mu+G_R\left(t;\alpha,\delta,\lambda_+,\lambda_-\right) +G_R\left(t;\alpha,\delta,\lambda_+,\lambda_-\right)\right),

where

\left. G_R\left(t;\alpha,\delta,\lambda_+,\lambda_-\right)= \frac{\sqrt{\pi}\delta\Gamma(-\frac{\alpha}{2})} {2^{\frac{\alpha+3}{2}}}\left((\lambda_+^{2}+t^{2})^{\frac{\alpha}{2}} -\lambda_+^{\alpha}+(\lambda_-^{2}+t^{2})^{\frac{\alpha}{2}} -\lambda_-^{\alpha} \right)\right.\\

\left. G_I\left(t;\alpha,\delta,\lambda_+,\lambda_-\right)= \frac{\mathrm{i}t\delta\Gamma(\frac{1-\alpha}{2})} {2^{\frac{\alpha+1}{2}}} \left(\lambda_+^{\alpha-1} F\left(1,\frac{1-\alpha}{2};\frac{3}{2};-\frac{t^2}{\lambda_+^2}\right) \right. \right. \\

\left. \left. -\lambda_-^{\alpha-1} F\left(1,\frac{1-\alpha}{2};\frac{3}{2};-\frac{t^2}{\lambda_-^2}\right) \right)\right.

F is the hypergeometric function.

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 Kim et al. (2008) or Rachev et al. (2011) can be selected. If you want to use the approach of Kim et al. (2009) for these functions, you have to clone the package from GitHub and adapt the functions accordingly.

kim09

From Kim et al. (2009) 'The modified tempered stable distribution, GARCH-models and option pricing'. Here alpha is in (-Inf,1) except 0.5.

kim08

From Kim et al. (2008) 'Financial market models with Levy processes and time-varying volatility'. Without further coding, this is the selected function for estimation function from this package.

rachev11

From Rachev et al. (2011) 'Financial Models with Levy Processes and time-varying volatility'. Similar to kim08

Value

The CF of the the modified tempered stable distribution.

References

Kim, Y. S.; Rachev, S. T.; Bianchi, M. L. & Fabozzi, F. J. (2008), 'Financial market models with lévy processes and time-varying volatility' doi:10.1016/j.jbankfin.2007.11.004

Kim, Y. S.; Rachev, S. T.; Bianchi, M. L. & Fabozzi, F. J. (2009), 'A New Tempered Stable Distribution and Its Application to Finance' doi:10.1007/978-3-7908-2050-8_5

Rachev, S. T.; Kim, Y. S.; Bianchi, M. L. & Fabozzi, F. J. (2011), 'Financial models with Lévy processes and volatility clustering' doi:10.1002/9781118268070

Examples

x <- seq(-5,5,0.1)
y <- charMTS(x, 0.5,1,1,1,0)