dsl: Stable lambda distribution

In ecd: Elliptic Lambda Distribution and Option Pricing Model

Stable lambda distribution

Description

Implements the stable lambda distribution (SLD) and the quartic stable lambda distribution (QSLD). Be aware of the performance concerns: (a) The cumulative density function is implemented by direct integration over the density. (b) The quantile function is implemented by root finding on cumulative density function.

Usage

dsl(x, t = 1, nu0 = 0, theta = 1, convo = 1, beta.a = 0, mu = 0, lambda = 4)

dqsl(x, t = 1, nu0 = 0, theta = 1, convo = 1, beta.a = 0, mu = 0)

rqsl(n, t = 1, nu0 = 0, theta = 1, convo = 1, beta.a = 0, mu = 0)

rsl(n, t = 1, nu0 = 0, theta = 1, convo = 1, beta.a = 0, mu = 0, lambda = 4)

pqsl(x, t = 1, nu0 = 0, theta = 1, convo = 1, beta.a = 0, mu = 0)

psl(x, t = 1, nu0 = 0, theta = 1, convo = 1, beta.a = 0, mu = 0, lambda = 4)

qqsl(q, t = 1, nu0 = 0, theta = 1, convo = 1, beta.a = 0, mu = 0)

qsl(q, t = 1, nu0 = 0, theta = 1, convo = 1, beta.a = 0, mu = 0, lambda = 4)

kqsl(t = 1, nu0 = 0, theta = 1, convo = 1, beta.a = 0, mu = 0)

ksl(t = 1, nu0 = 0, theta = 1, convo = 1, beta.a = 0, mu = 0, lambda = 4)

cfqsl(
  s,
  t = 1,
  nu0 = 0,
  theta = 1,
  convo = 1,
  beta.a = 0,
  mu = 0,
  method = "a"
)

cfsl(
  s,
  t = 1,
  nu0 = 0,
  theta = 1,
  convo = 1,
  beta.a = 0,
  mu = 0,
  lambda = 4,
  method = "a"
)

Arguments

x	numeric, vector of responses.
t	numeric, the time parameter, where the variance is t, default is 1.
nu0	numeric, the location parameter, default is 0.
theta	numeric, the scale parameter, default is 1.
convo	numeric, the convolution number, default is 1.
beta.a	numeric, the skewness parameter, default is 0. This number is annualized by sqrt(t).
mu	numeric, the location parameter, default is 0.
lambda	numeric, the shape parameter, default is 4.
n	numeric, number of observations.
q	numeric, vector of quantiles.
s	numeric, vector of responses for characteristic function.
method	character, method of characteristic function (CF) calculation. Default is "a". Method a uses cfstdlap x dstablecnt. Method b uses dstdlap x cfstablecnt. Method c uses direct integration on PDF up to 50 stdev. They should yield the same result.

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 lambda distribution is the stationary distribution for financial asset returns. It is a product of the stable count distribution and the Lihn-Laplace process. The density function is defined as

P_Λ^(m) (x;t,ν_0,θ,β_a,μ) = integrate_ν_0^∞ 1/ν f_L^(m)((x-μ)/ν; t,β_a sqrt(t)) N_α(ν; ν_0,θ) dν

where f_L^{(m)}(.) is the Lihn-Laplace distribution and N_α(.) is the quartic stable count distribution. t is the time or sampling period, α is the stability index which is 2/λ, ν_0 is the floor volatility parameter, θ is the volatility scale parameter, β_a is the annualized asymmetric parameter, μ is the location parameter.
The quartic stable lambda distribution (QSLD) is a specialized distribution with λ=4 aka α=0.5. The PDF integrand has closed form, and all the moments have closed forms. Many financial asset returns can be fitted by QSLD precisely up to 4 standard deviations.

Author(s)

Stephen H-T. Lihn

References

For more detail, see Section 8.2 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.

See Also

dstablecnt for N_α(.), and dstdlap for f_L^{(m)}(.).

Examples

  # generate the quartic pdf for SPX 1-day distribution
  x <- c(-0.1, 0.1, by=0.001)
  pdf <- dqsl(x, t=1/250, nu0=6.92/100, theta=1.17/100, convo=2, beta=-1.31)

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