dstablecnt: Stable Count distribution

In ecd: Elliptic Lambda Distribution and Option Pricing Model

Stable Count distribution

Description

Implements the stable count distribution (based on stabledist package) for stable random walk simulation. Quartic stable distribution is implemented through gamma distribution.

Usage

dstablecnt(x, alpha = NULL, nu0 = 0, theta = 1, lambda = NULL)

pstablecnt(x, alpha = NULL, nu0 = 0, theta = 1, lambda = NULL)

rstablecnt(n, alpha = NULL, nu0 = 0, theta = 1, lambda = NULL)

qstablecnt(q, alpha = NULL, nu0 = 0, theta = 1, lambda = NULL)

cfstablecnt(s, alpha = NULL, nu0 = 0, theta = 1, lambda = NULL)

kstablecnt(alpha = NULL, nu0 = 0, theta = 1, lambda = NULL)

Arguments

x	numeric, vector of responses.
alpha	numeric, the shape parameter, default is NULL. User must provide either alpha or lambda.
nu0	numeric, the location parameter, default is 0.
theta	numeric, the scale parameter, default is 1.
lambda	numeric, alternative shape parameter, default is NULL.
n	numeric, number of observations.
q	numeric, vector of quantiles.
s	numeric, vector of responses for characteristic function.

Value

numeric, standard convention is followed: d* returns the density, p* returns the distribution function, q* returns the quantile function, and r* generates random deviates. The following are our extensions: k* returns the first 4 cumulants, skewness, and kurtosis, cf* returns the characteristic function.

Details

The stable count distribution is the conjugate prior of the stable distribution. The density function is defined as

N_α(ν; ν_0, θ) = α/Γ(1/α) * 1/(ν-ν_0) * L_α(1/(ν-ν_0))

where ν>ν_0. α is the stability index, ν_0 is the location parameter, and θ is the scale parameter.
At α=0.5 aka λ=4, it is called "quartic stable count distribution", which is a gamma distribution with shape of 3/2. It has the closed form of

N_α(ν; ν_0, θ) = 1/(4 sqrt(π) θ^1.5) (ν-ν_0)^0.5 exp(-(ν-ν_0)/(4θ))

Author(s)

Stephen H-T. Lihn

References

For more detail, see Section 2.4 and Section 3.3 of Stephen Lihn (2017). A Theory of Asset Return and Volatility under Stable Law and Stable Lambda Distribution. SSRN: 3046732, https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3046732. This distribution is also documented formally in Wikipedia: https://en.wikipedia.org/wiki/Stable_count_distribution.

Examples

  # generate the pdf of the VIX distribution
  x <- c(0, 100, by=0.1)
  pdf <- dstablecnt(x, nu0=10.4, theta=1.6, lambda=4)
  

ecd documentation built on May 10, 2022, 1:07 a.m.