udghyp function

UNU.RAN object for Generalized Hyperbolic distribution

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.

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

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.

Details

The generalized hyperbolic distribution with parameters $lambda$, $alpha$, $beta$, $delta$, and $mu$

has density

$$f(x) = \kappa \;(\delta^2+(x-\mu)^2)^{1/2 (\lambda-1/2)}\cdot \exp(\beta(x-\mu))\cdot K_{\lambda-1/2}\left(\alpha\sqrt{\delta^2+(x-\mu)^2}\right)f(x) = kappa * (delta^2+(x-mu)^2)^(1/2*(lambda-1/2)) * exp(beta*(x-mu)) * K_{lambda-1/2}(alpha * sqrt(delta^2+(x-mu)^2))$$

where the normalization constant is given by

$$\kappa =\frac{\left(\sqrt{\alpha^2 - \beta^2}/\delta\right)^{\lambda}}{\sqrt{2\pi} \, \alpha^{\lambda-1/2} \,K_{\lambda}\left(\delta \sqrt{\alpha^2-\beta^2}\right)}kappa = (sqrt(alpha^2 - beta^2)/delta)^lambda / (sqrt(2*pi) * alpha^(lambda-1/2) * K_{lambda}(delta*sqrt(alpha^2-beta^2))$$

$K_(lambda)(t)$ 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).

Returns

An object of class "unuran.cont".

See Also

unuran.cont.

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.

Author(s)

Josef Leydold and Wolfgang H"ormann unuran@statmath.wu.ac.at .

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)