Runuran (version 0.40)

udghyp: UNU.RAN object for Generalized Hyperbolic distribution

Description

Create UNU.RAN object for a Generalized Hyperbolic distribution with shape parameter lambda, shape parameter alpha, asymmetry (shape) parameter beta, scale parameter delta, and location parameter mu.

[Distribution] -- Generalized Hyperbolic.

Usage

udghyp(lambda, alpha, beta, delta, mu, lb=-Inf, ub=Inf)

Value

An object of class "unuran.cont".

Arguments

lambda

shape parameter.

alpha

shape parameter (must be strictly larger than absolute value of beta).

beta

asymmetry (shape) parameter.

delta

scale parameter (must be strictly positive).

mu

location parameter.

lb

lower bound of (truncated) distribution.

ub

upper bound of (truncated) distribution.

Author

Josef Leydold and Wolfgang H\"ormann .

Details

The generalized hyperbolic distribution with parameters \(\lambda\), \(\alpha\), \(\beta\), \(\delta\), and \(\mu\) has density $$ f(x) = \kappa \; (\delta^2+(x-\mu)^2)^{1/2 (\lambda-1/2)} \cdot \exp(\beta(x-\mu)) \cdot K_{\lambda-1/2}\left(\alpha\sqrt{\delta^2+(x-\mu)^2}\right) $$ where the normalization constant is given by $$ \kappa = \frac{\left(\sqrt{\alpha^2 - \beta^2}/\delta\right)^{\lambda}}{ \sqrt{2\pi} \, \alpha^{\lambda-1/2} \, K_{\lambda}\left(\delta \sqrt{\alpha^2-\beta^2}\right)} $$

\(K_{\lambda}(t)\) is the modified Bessel function of the third kind with index \(\lambda\).

Notice that \(\alpha>|\beta|\) and \(\delta>0\).

The domain of the distribution can be truncated to the interval (lb,ub).

References

Barndorff-Nielsen, O., Blaesild, P., 1983. Hyperbolic distributions. In: Johnson, N. L., Kotz, S., Read, C. B. (Eds.), Encyclopedia of Statistical Sciences. Vol. 3. Wiley, New York, p. 700--707.

Prause, K., 1997. Modelling financial data using generalized hyperbolic distributions. FDM preprint 48, University of Freiburg.

Prause, K., 1999. The generalized hyperbolic model: Estimation, financial derivatives, and risk measures. Ph.D. thesis, University of Freiburg.

See Also

unuran.cont.

Examples

## Create distribution object for generalized hyperbolic distribution
distr <- udghyp(lambda=-1.0024, alpha=39.6, beta=4.14, delta=0.0118, mu=-0.000158)
## Generate generator object; use method PINV (inversion)
gen <- pinvd.new(distr)
## Draw a sample of size 100
x <- ur(gen,100)