pgb: CDF of Generalized Beta
In RND: Risk Neutral Density Extraction Package

Description Usage Arguments Details Value Author(s) References Examples

pgb is the cumulative distribution function (CDF) of a genaralized beta random variable.

1	pgb(x, a, b, v, w)

`x`	value at which the CDF is to be evaluated
`a`	power parameter > 0
`b`	scale paramter > 0
`v`	first beta paramter > 0
`w`	second beta parameter > 0

Let B be a beta random variable with parameters v and w. Then Z = b *(B/(1-B))^(1/a) is a generalized beta random variable with parameters (a,b,v,w).

out

CDF value at x

Kam Hamidieh

R.M. Bookstaber and J.B. McDonald (1987) A general distribution for describing security price returns. Journal of Business, 60, 401-424

X. Liu and M.B. Shackleton and S.J. Taylor and X. Xu (2007) Closed-form transformations from risk-neutral to real-world distributions Journal of Business, 60, 401-424

E. Jondeau and S. Poon and M. Rockinger (2007): Financial Modeling Under Non-Gaussian Distributions Springer-Verlag, London

#
# What does the cdf of a GB look like?
#

a  = 1
b  = 10
v  = 2
w  = 2

x = seq(from = 0, to = 500, by = 0.01)
y = pgb(x = x, a = a, b = b, v = v, w = w)
plot(y ~ x, type = "l")
abline(h=c(0,1), lty=2)