A package on the generalized hyperbolic distribution and its special cases

Description

This package provides detailed functionality for working with the univariate and multivariate Generalized Hyperbolic distribution and its special cases (Hyperbolic (hyp), Normal Inverse Gaussian (NIG), Variance Gamma (VG), skewed Student-t and Gaussian distribution). Especially, it contains fitting procedures, an AIC-based model selection routine, and functions for the computation of density, quantile, probability, random variates, expected shortfall and some portfolio optimization and plotting routines as well as the likelihood ratio test. In addition, it contains the Generalized Inverse Gaussian distribution.

Details

Initialize:

Density, distribution function, quantile function and random generation:

Fit to data:

Risk, performance and portfolio optimization:

Utilities:

Plot functions:

Generalized inverse gaussian distribution:

Package vignette:
A document about generalized hyperbolic distributions can be found in the doc folder of this package or on https://cran.r-project.org/package=ghyp.

Existing solutions

There are packages like GeneralizedHyperbolic, HyperbolicDist, SkewHyperbolic, VarianceGamma and fBasics which cover the univariate generalized hyperbolic distribution and/or some of its special cases. However, the univariate case is contained in this package as well because we aim to provide a uniform interface to deal with generalized hyperbolic distribution. Recently an R port of the S-Plus library QRMlib was released. The package QRMlib contains fitting procedures for the multivariate NIG, hyp and skewed Student-t distribution but not for the generalized hyperbolic case. The package fMultivar implements a fitting routine for multivariate skewed Student-t distributions as well.

Object orientation

We follow an object-oriented programming approach in this package and introduce distribution objects. There are mainly four reasons for that:

• Unlike most distributions the GH distribution has quite a few parameters which have to fulfill some consistency requirements. Consistency checks can be performed uniquely when an object is initialized.

• Once initialized the common functions belonging to a distribution can be called conveniently by passing the distribution object. A repeated input of the parameters is avoided.

• Distributions returned from fitting procedures can be directly passed to, e.g., the density function since fitted distribution objects add information to the distribution object and consequently inherit from the class of the distribution object.

• Generic method dispatching can be used to provide a uniform interface to, e.g., plot the probability density of a specific distribution like plot(distribution.object). Additionally, one can take advantage of generic programming since R provides virtual classes and some forms of polymorphism.

Acknowledgement

This package has been partially developed in the framework of the COST-P10 “Physics of Risk” project. Financial support by the Swiss State Secretariat for Education and Research (SBF) is gratefully acknowledged.

Author(s)

David Luethi, Wolfgang Breymann

Institute of Data Analyses and Process Design (https://www.zhaw.ch/en/engineering/institutes-centres/idp/groups/data-analysis-and-statistics/)

Maintainer: Marc Weibel <marc.weibel@quantsulting.ch>

References

Quantitative Risk Management: Concepts, Techniques and Tools by Alexander J. McNeil, Ruediger Frey and Paul Embrechts
Princeton Press, 2005

Intermediate probability: A computational approach by Marc Paolella
Wiley, 2007

S-Plus and R Library for Quantitative Risk Management QRMlib by Alexander J. McNeil (2005) and Scott Ulman (R-port) (2007)

Risk attribution.

Description

Functions to get the contribution of each asset to the portfolio's Expected Shortfall based on multivariate generalized hyperbolic distributions as well as the expected shortfall sensitivity to marginal changes in portfolio allocation.

Usage

ESghyp.attribution(
  alpha,
  object = ghyp(),
  distr = c("return", "loss"),
  weights = NULL,
  ...
)

Arguments

Details

The parameter distr specifies whether the ghyp-object describes a return or a loss-distribution. In case of a return distribution the expected-shortfall on a confidence level \alpha is defined as \hbox{ES}_\alpha := \hbox{E}(X| X \leq F^{-1}_X(\alpha)) while in case of a loss distribution it is defined on a confidence level \alpha as \hbox{ES}_\alpha := \hbox{E}(X | X > F^{-1}_X(\alpha)).

Value

ESghyp.attribution is an object of class ghyp.attribution.

Author(s)

Marc Weibel

See Also

contribution,ghyp.attribution-method, sensitivity,ghyp.attribution-method and weights for Expected Shortfall.

Examples

## Not run: 
data(smi.stocks)
## Fit a NIG model to Novartis, CS and Nestle log-returns
assets.fit <- fit.NIGmv(smi.stocks[, c("Novartis", "CS", "Nestle")], silent = TRUE)
## Define Weights of the Portfolio
weights <- c(0.2, 0.5, 0.3)
## Confidence level for Expected Shortfall
es.levels <- c(0.01)

portfolio.attrib <- ESghyp.attribution(alpha=es.levels, object=assets.fit, weights=weights)

## End(Not run)

Extract parameters of generalized hyperbolic distribution objects

Description

The function coef returns the parameters of a generalized hyperbolic distribution object as a list. The user can choose between the “chi/psi”, the “alpha.bar” and the “alpha/delta” parametrization. The function coefficients is a synonym for coef.

Usage

## S4 method for signature 'ghyp'
coef(object, type = c("chi.psi", "alpha.bar", "alpha.delta"))

## S4 method for signature 'ghyp'
coefficients(object, type = c("chi.psi", "alpha.bar", "alpha.delta"))

Arguments

Details

Internally, the “chi/psi” parametrization is used. However, fitting is only possible in the “alpha.bar” parametrization as it provides the most convenient parameter constraints.

Value

If type is “chi.psi” a list with components:

If type is “alpha.bar” a list with components:

If type is “alpha.delta” a list with components:

Note

A switch from either the “chi/psi” to the “alpha.bar” or from the “alpha/delta” to the “alpha.bar” parametrization is not yet possible.

Author(s)

David Luethi

See Also

ghyp, fit.ghypuv, fit.ghypmv, ghyp.fit.info, transform, [.ghyp

Examples

  ghyp.mv <- ghyp(lambda = 1, alpha.bar = 0.1, mu = rep(0,2), sigma = diag(rep(1,2)),
                  gamma = rep(0,2), data = matrix(rt(1000, df = 4), ncol = 2))
  ## Get parameters
  coef(ghyp.mv, type = "alpha.bar")
  coefficients(ghyp.mv, type = "chi.psi")

  ## Simple modification (do not modify slots directly e.g. object@mu <- 0:1)
  param <- coef(ghyp.mv, type = "alpha.bar")
  param$mu <- 0:1
  do.call("ghyp", param) # returns a new 'ghyp' object

Fitting generalized hyperbolic distributions to multivariate data

Description

Perform a maximum likelihood estimation of the parameters of a multivariate generalized hyperbolic distribution by using an Expectation Maximization (EM) based algorithm.

Usage

fit.ghypmv(data, lambda = 1, alpha.bar = 1, mu = NULL, sigma = NULL,
           gamma = NULL, opt.pars = c(lambda = TRUE, alpha.bar = TRUE, mu = TRUE,
                                      sigma = TRUE, gamma = !symmetric),
           symmetric = FALSE, standardize = FALSE, nit = 2000, reltol = 1e-8,
           abstol = reltol * 10, na.rm = FALSE, silent = FALSE, save.data = TRUE,
           trace = TRUE, ...)

fit.hypmv(data,
          opt.pars = c(alpha.bar = TRUE, mu = TRUE, sigma = TRUE, gamma = !symmetric),
          symmetric = FALSE, ...)

fit.NIGmv(data,
          opt.pars = c(alpha.bar = TRUE, mu = TRUE, sigma = TRUE, gamma = !symmetric),
          symmetric = FALSE, ...)

fit.VGmv(data, lambda = 1,
         opt.pars = c(lambda = TRUE, mu = TRUE, sigma = TRUE, gamma = !symmetric),
         symmetric = FALSE, ...)

fit.tmv(data, nu = 3.5,
        opt.pars = c(lambda = TRUE, mu = TRUE, sigma = TRUE, gamma = !symmetric),
        symmetric = FALSE, ...)

fit.gaussmv(data, na.rm = TRUE, save.data = TRUE)

Arguments

Details

This function uses a modified EM algorithm which is called Multi-Cycle Expectation Conditional Maximization (MCECM) algorithm. This algorithm is sketched in the vignette of this package which can be found in the doc folder. A more detailed description is provided by the book Quantitative Risk Management, Concepts, Techniques and Tools (see “References”).

The general-purpose optimization routine optim is used to maximize the loglikelihood function of the univariate mixing distribution. The default method is that of Nelder and Mead which uses only function values. Parameters of optim can be passed via the ... argument of the fitting routines.

Value

An object of class mle.ghyp.

Note

The variance gamma distribution becomes singular when \mathbf{x} - \mathbf{\mu} = 0. This singularity is catched and the reduced density function is computed. Because the transition is not smooth in the numerical implementation this can rarely result in nonsensical fits.

Providing both arguments, opt.pars and symmetric respectively, can result in a conflict when opt.pars['gamma'] and symmetric are TRUE. In this case symmetric will dominate and opt.pars['gamma'] is set to FALSE.

Author(s)

Wolfgang Breymann, David Luethi

References

Alexander J. McNeil, Ruediger Frey, Paul Embrechts (2005) Quantitative Risk Management, Concepts, Techniques and Tools

ghyp-package vignette in the doc folder or on https://cran.r-project.org/package=ghyp.

S-Plus and R library QRMlib)

See Also

fit.ghypuv, fit.hypuv, fit.NIGuv, fit.VGuv, fit.tuv for univariate fitting routines. ghyp.fit.info for information regarding the fitting procedure.

Examples

  data(smi.stocks)

  fit.ghypmv(data = smi.stocks, opt.pars = c(lambda = FALSE), lambda = 2,
             control = list(rel.tol = 1e-5, abs.tol = 1e-5), reltol = 0.01)

Fitting generalized hyperbolic distributions to univariate data

Description

This function performs a maximum likelihood parameter estimation for univariate generalized hyperbolic distributions.

Usage

fit.ghypuv(data, lambda = 1, alpha.bar = 0.5, mu = median(data),
           sigma = mad(data), gamma = 0,
           opt.pars = c(lambda = TRUE, alpha.bar = TRUE, mu = TRUE,
                        sigma = TRUE, gamma = !symmetric),
           symmetric = FALSE, standardize = FALSE, save.data = TRUE,
           na.rm = TRUE, silent = FALSE, ...)

fit.hypuv(data,
          opt.pars = c(alpha.bar = TRUE, mu = TRUE, sigma = TRUE, gamma = !symmetric),
          symmetric = FALSE, ...)

fit.NIGuv(data,
          opt.pars = c(alpha.bar = TRUE, mu = TRUE, sigma = TRUE, gamma = !symmetric),
          symmetric = FALSE, ...)

fit.VGuv(data, lambda = 1,
         opt.pars = c(lambda = TRUE, mu = TRUE, sigma = TRUE, gamma = !symmetric),
         symmetric = FALSE, ...)

fit.tuv(data, nu = 3.5,
        opt.pars = c(nu = TRUE, mu = TRUE, sigma = TRUE, gamma = !symmetric),
        symmetric = FALSE, ...)

fit.gaussuv(data, na.rm = TRUE, save.data = TRUE)

Arguments

Details

The general-purpose optimization routine optim is used to maximize the loglikelihood function. The default method is that of Nelder and Mead which uses only function values. Parameters of optim can be passed via the ... argument of the fitting routines.

Value

An object of class mle.ghyp.

Note

The variance gamma distribution becomes singular when x - \mu = 0. This singularity is catched and the reduced density function is computed. Because the transition is not smooth in the numerical implementation this can rarely result in nonsensical fits.

Providing both arguments, opt.pars and symmetric respectively, can result in a conflict when opt.pars['gamma'] and symmetric are TRUE. In this case symmetric will dominate and opt.pars['gamma'] is set to FALSE.

Author(s)

Wolfgang Breymann, David Luethi

References

ghyp-package vignette in the doc folder or on https://cran.r-project.org/package=ghyp.

See Also

fit.ghypmv, fit.hypmv, fit.NIGmv, fit.VGmv, fit.tmv for multivariate fitting routines. ghyp.fit.info for information regarding the fitting procedure.

Examples

  data(smi.stocks)

  nig.fit <- fit.NIGuv(smi.stocks[,"SMI"], opt.pars = c(alpha.bar = FALSE),
                       alpha.bar = 1, control = list(abstol = 1e-8))
  nig.fit

  summary(nig.fit)

  hist(nig.fit)

Create generalized hyperbolic distribution objects

Description

Constructor functions for univariate and multivariate generalized hyperbolic distribution objects and their special cases in one of the parametrizations “chi/psi”, “alpha.bar” and “alpha/delta”.

Usage

ghyp(lambda = 0.5, chi = 0.5, psi = 2, mu = 0, sigma = diag(rep(1, length(mu))),
     gamma = rep(0, length(mu)), alpha.bar = NULL, data = NULL)

ghyp.ad(lambda = 0.5, alpha = 1.5, delta = 1, beta = rep(0, length(mu)),
        mu = 0, Delta = diag(rep(1, length(mu))), data = NULL)


hyp(chi = 0.5, psi = 2, mu = 0, sigma = diag(rep(1, length(mu))),
    gamma = rep(0, length(mu)), alpha.bar = NULL, data = NULL)

hyp.ad(alpha = 1.5, delta = 1, beta = rep(0, length(mu)), mu = 0,
       Delta = diag(rep(1, length(mu))), data = NULL)


NIG(chi = 2, psi = 2, mu = 0, sigma = diag(rep(1, length(mu))),
    gamma = rep(0, length(mu)), alpha.bar = NULL, data = NULL)

NIG.ad(alpha = 1.5, delta = 1, beta = rep(0, length(mu)), mu = 0,
       Delta = diag(rep(1, length(mu))), data = NULL)


student.t(nu = 3.5, chi = nu - 2, mu = 0, sigma = diag(rep(1, length(mu))),
          gamma = rep(0, length(mu)), data = NULL)

student.t.ad(lambda = -2, delta = 1, beta = rep(0, length(mu)), mu = 0,
             Delta = diag(rep(1, length(mu))), data = NULL)


VG(lambda = 1, psi = 2*lambda, mu = 0, sigma = diag(rep(1, length(mu))),
   gamma = rep(0, length(mu)), data = NULL)

VG.ad(lambda = 2, alpha = 1.5, beta = rep(0, length(mu)), mu = 0,
      Delta = diag(rep(1, length(mu))), data = NULL)

gauss(mu = 0, sigma = diag(rep(1, length(mu))), data = NULL)

Arguments

Details

These functions serve as constructors for univariate and multivariate objects.

ghyp, hyp and NIG are constructor functions for both the “chi/psi” and the “alpha.bar” parametrization. Whenever alpha.bar is not NULL it is assumed that the “alpha.bar” parametrization is used and the parameters “chi” and “psi” become redundant.

Similarly, the variance gamma (VG) and the Student-t distribution share the same constructor function for both the chi/psi and alpha.bar parametrization. To initialize them in the alpha.bar parametrization simply omit the argument psi and chi, respectively. If psi or chi are submitted, the “chi/psi” parametrization will be used.

ghyp.ad, hyp.ad, NIG.ad, student.t.ad and VG.ad use the “alpha/delta” parametrization.

The following table gives the constructors for each combination of distribution and parametrization.

Have a look on the vignette of this package in the doc folder for further information regarding the parametrization and for the domains of variation of the parameters.

Value

An object of class ghyp.

Note

The Student-t parametrization obtained via the “alpha.bar” parametrization slightly differs from the common Student-t parametrization: The parameter sigma denotes the standard deviation in the univariate case and the variance in the multivariate case. Thus, set \sigma = \sqrt{\nu /(\nu - 2)} in the univariate case to get the same results as with the standard R implementation of the Student-t distribution.

In case of non-finite variance, the “alpha.bar” parametrization does not work because sigma is defined to be the standard deviation. In this case the “chi/psi” parametrization can be used by submitting the parameter chi. To obtain equal results as the standard R implmentation use student.t(nu = nu, chi = nu) (see Examples).

Have a look on the vignette of this package in the doc folder for further information.

Once an object of class ghyp is created the methods Xghyp have to be used even when the distribution is a special case of the GH distribution. E.g. do not use dVG. Use dghyp and submit a variance gamma distribution created with VG().

Author(s)

David Luethi

References

ghyp-package vignette in the doc folder or on https://cran.r-project.org/package=ghyp

See Also

ghyp-class for a summary of generic methods assigned to ghyp objects, coef for switching between different parametrizations, d/p/q/r/ES/gyhp for density, distribution function et cetera, fit.ghypuv and fit.ghypmv for fitting routines.

Examples

  ## alpha.bar parametrization of a univariate GH distribution
  ghyp(lambda=2, alpha.bar=0.1, mu=0, sigma=1, gamma=0)
  ## lambda/chi parametrization of a univariate GH distribution
  ghyp(lambda=2, chi=1, psi=0.5, mu=0, sigma=1, gamma=0)
  ## alpha/delta parametrization of a univariate GH distribution
  ghyp.ad(lambda=2, alpha=0.5, delta=1, mu=0, beta=0)

  ## alpha.bar parametrization of a multivariate GH distribution
  ghyp(lambda=1, alpha.bar=0.1, mu=2:3, sigma=diag(1:2), gamma=0:1)
  ## lambda/chi parametrization of a multivariate GH distribution
  ghyp(lambda=1, chi=1, psi=0.5, mu=2:3, sigma=diag(1:2), gamma=0:1)
  ## alpha/delta parametrization of a multivariate GH distribution
  ghyp.ad(lambda=1, alpha=2.5, delta=1, mu=2:3, Delta=diag(c(1,1)), beta=0:1)

  ## alpha.bar parametrization of a univariate hyperbolic distribution
  hyp(alpha.bar=0.3, mu=1, sigma=0.1, gamma=0)
  ## lambda/chi parametrization of a univariate hyperbolic distribution
  hyp(chi=1, psi=2, mu=1, sigma=0.1, gamma=0)
  ## alpha/delta parametrization of a univariate hyperbolic distribution
  hyp.ad(alpha=0.5, delta=1, mu=0, beta=0)

  ## alpha.bar parametrization of a univariate NIG distribution
  NIG(alpha.bar=0.3, mu=1, sigma=0.1, gamma=0)
  ## lambda/chi parametrization of a univariate NIG distribution
  NIG(chi=1, psi=2, mu=1, sigma=0.1, gamma=0)
  ## alpha/delta parametrization of a univariate NIG distribution
  NIG.ad(alpha=0.5, delta=1, mu=0, beta=0)

  ## alpha.bar parametrization of a univariate VG distribution
  VG(lambda=2, mu=1, sigma=0.1, gamma=0)
  ## alpha/delta parametrization of a univariate VG distribution
  VG.ad(lambda=2, alpha=0.5, mu=0, beta=0)

  ## alpha.bar parametrization of a univariate t distribution
  student.t(nu = 3, mu=1, sigma=0.1, gamma=0)
  ## alpha/delta parametrization of a univariate t distribution
  student.t.ad(lambda=-2, delta=1, mu=0, beta=1)

  ## Obtain equal results as with the R-core parametrization
  ## of the t distribution:
  nu <- 4
  standard.R.chi.psi <- student.t(nu = nu, chi = nu)
  standard.R.alpha.bar <- student.t(nu = nu, sigma = sqrt(nu  /(nu - 2)))

  random.sample <- rnorm(3)
  dt(random.sample, nu)
  dghyp(random.sample, standard.R.chi.psi)   # all implementations yield...
  dghyp(random.sample, standard.R.alpha.bar) # ...the same values

  random.quantiles <- runif(4)
  qt(random.quantiles, nu)
  qghyp(random.quantiles, standard.R.chi.psi)   # all implementations yield...
  qghyp(random.quantiles, standard.R.alpha.bar) # ...the same values

  ## If nu <= 2 the "alpha.bar" parametrization does not exist, but the
  ## "chi/psi" parametrization. The case of a Cauchy distribution:
  nu <- 1
  standard.R.chi.psi <- student.t(nu = nu, chi = nu)

  dt(random.sample, nu)
  dghyp(random.sample, standard.R.chi.psi)   # both give the same result

  pt(random.sample, nu)
  pghyp(random.sample, standard.R.chi.psi) # both give the same result

The Generalized Hyperbolic Distribution

Description

Density, distribution function, quantile function, expected-shortfall and random generation for the univariate and multivariate generalized hyperbolic distribution and its special cases.

Usage

dghyp(x, object = ghyp(), logvalue = FALSE)

pghyp(q, object = ghyp(), n.sim = 10000, subdivisions = 200,
      rel.tol = .Machine$double.eps^0.5, abs.tol = rel.tol,
      lower.tail = TRUE)

qghyp(p, object = ghyp(), method = c("integration", "splines"),
      spline.points = 200, subdivisions = 200,
      root.tol = .Machine$double.eps^0.5,
      rel.tol = root.tol^1.5, abs.tol = rel.tol)

rghyp(n, object = ghyp())

Arguments

Details

qghyp only works for univariate generalized hyperbolic distributions.

pghyp performs a numeric integration of the density in the univariate case. The multivariate cumulative distribution is computed by means of monte carlo simulation.

qghyp computes the quantiles either by using the “integration” method where the root of the distribution function is solved or via “splines” which interpolates the distribution function and solves it with uniroot afterwards. The “integration” method is recommended when only few quantiles are required. If more than approximately 20 quantiles are needed to be calculated the “splines” method becomes faster. The accuracy can be controlled with an adequate setting of the parameters rel.tol, abs.tol, root.tol and spline.points.

rghyp uses the random generator for generalized inverse Gaussian distributed random variates from the Rmetrics package fBasics (cf. rgig).

Value

dghyp gives the density,
pghyp gives the distribution function,
qghyp gives the quantile function,
rghyp generates random deviates.

Note

Objects generated with hyp, NIG, VG and student.t have to use Xghyp as well. E.g. dNIG(0, NIG()) does not work but dghyp(0, NIG()).

When the skewness becomes very large the functions using qghyp may fail. The functions qqghyp, pairs and portfolio.optimize are based on qghyp.

Author(s)

David Luethi

References

ghyp-package vignette in the doc folder or on https://cran.r-project.org/package=ghyp and references therein.

See Also

ghyp-class definition, ghyp constructors, fitting routines fit.ghypuv and fit.ghypmv, risk and performance measurement ESghyp and ghyp.omega, transformation and subsettting of ghyp objects, integrate, spline.

Examples

  ## Univariate generalized hyperbolic distribution
  univariate.ghyp <- ghyp()

  par(mfrow=c(5, 1))

  quantiles <- seq(-4, 4, length = 500)
  plot(quantiles, dghyp(quantiles, univariate.ghyp))
  plot(quantiles, pghyp(quantiles, univariate.ghyp))

  probabilities <- seq(1e-4, 1-1e-4, length = 500)
  plot(probabilities, qghyp(probabilities, univariate.ghyp, method = "splines"))

  hist(rghyp(n=10000,univariate.ghyp),nclass=100)

  ## Mutivariate generalized hyperbolic distribution
  multivariate.ghyp <- ghyp(sigma=var(matrix(rnorm(10),ncol=2)),mu=1:2,gamma=-(2:1))

  par(mfrow=c(2, 1))

  quantiles <- outer(seq(-4, 4, length = 50), c(1, 1))
  plot(quantiles[, 1], dghyp(quantiles, multivariate.ghyp))
  plot(quantiles[, 1], pghyp(quantiles, multivariate.ghyp, n.sim = 1000))

  rghyp(n = 10, multivariate.ghyp)

Get methods for objects inheriting from class ghyp

Description

These functions simply return data stored within generalized hyperbolic distribution objects, i.e. slots of the classes ghyp and mle.ghyp. ghyp.fit.info extracts information about the fitting procedure from objects of class mle.ghyp. ghyp.data returns the data slot of a gyhp object. ghyp.dim returns the dimension of a gyhp object. ghyp.name returns the name of the distribution of a gyhp object.

Usage

ghyp.fit.info(object)

ghyp.data(object)

ghyp.name(object, abbr = FALSE, skew.attr = TRUE)

ghyp.dim(object)

Arguments

Value

ghyp.fit.info returns list with components:

ghyp.data returns NULL if no data is stored within the object, a vector if it is an univariate generalized hyperbolic distribution and matrix if it is an multivariate generalized hyperbolic distribution.

ghyp.name returns the name of the ghyp distribution which can be the name of a special case. Depending on the arguments abbr and skew.attr one of the following is returned.

ghyp.dim returns the dimension of a ghyp object.

Note

ghyp.fit.info requires an object of class mle.ghyp. In the univariate case the parameter variance is returned as well. The parameter variance is defined as the inverse of the negative hesse-matrix computed by optim. Note that this makes sense only in the case that the estimates are asymptotically normal distributed.

The class ghyp contains a data slot. Data can be stored either when an object is initialized or via the fitting routines and the argument save.data.

Author(s)

David Luethi

See Also

coef, mean, vcov, logLik, AIC for other accessor functions, fit.ghypmv, fit.ghypuv, ghyp for constructor functions, optim for possible error messages.

Examples

  ## multivariate generalized hyperbolic distribution
  ghyp.mv <- ghyp(lambda = 1, alpha.bar = 0.1, mu = rep(0, 2), sigma = diag(rep(1, 2)),
                  gamma = rep(0, 2), data = matrix(rt(1000, df = 4), ncol = 2))

  ## Get data
  ghyp.data(ghyp.mv)

  ## Get the dimension
  ghyp.dim(ghyp.mv)

  ## Get the name of the ghyp object
  ghyp.name(ghyp(alpha.bar = 0))
  ghyp.name(ghyp(alpha.bar = 0, lambda = -4), abbr = TRUE)

  ## 'ghyp.fit.info' does only work when the object is of class 'mle.ghyp',
  ## i.e. is created by 'fit.ghypuv' etc.
  mv.fit <- fit.tmv(data = ghyp.data(ghyp.mv), control = list(abs.tol = 1e-3))
  ghyp.fit.info(mv.fit)

Internal ghyp functions

Description

Internal ghyp functions. These functions are not to be called by the user.

Usage

.abar2chipsi(alpha.bar, lambda, eps = .Machine$double.eps)

.besselM3(lambda = 9/2, x = 2, logvalue = FALSE)

.check.data(data, case = c("uv", "mv"), na.rm = TRUE,
            fit = TRUE, dim = NULL)

.check.gig.pars(lambda, chi, psi)

.check.norm.pars(mu, sigma, gamma, dimension)

.check.opt.pars(opt.pars, symmetric)

.fit.ghyp(object, llh = 0, n.iter = 0, converged = FALSE, error.code = 0,
          error.message = "", parameter.variance, fitted.params, aic,
          trace.pars = list())

.ghyp.model(lambda, chi, psi, gamma)

.t.transform(lambda)

.inv.t.transform(lambda.transf)

.integrate.moment.gig(x, moment = 1, ...)

.integrate.moment.ghypuv(x, moment = 1, ...)

.dghypuv(x, lambda = 1, chi = 1, psi = 1, alpha.bar = NULL,
         mu = 1, sigma = 1, gamma = 0, logvalue = FALSE)

.dghypmv(x, lambda, chi, psi, mu, sigma, gamma, logvalue = FALSE)

.mle.default(data, pdf, vars, opt.pars = rep(TRUE, length(vars)),
             transform = NULL, se = FALSE,
             na.rm = FALSE, silent = FALSE, ...)

.p.default(q, pdf, pdf.args, lower, upper, ...)

.q.default(p, pdf, pdf.args, interval, p.lower, ...)

.test.ghyp(object, case = c("ghyp", "univariate", "multivariate"))

.is.gaussian(object)

.is.univariate(object)

.is.symmetric(object)

.is.student.t(object, symmetric = NULL)

.get.stepAIC.ghyp(stepAIC.obj,
                  dist = c("ghyp", "hyp", "NIG", "VG", "t", "gauss"),
                  symmetric = FALSE)

Details

.abar2chipsi
Convert “alpha.bar” to “chi” and “psi” when using the “alpha.bar” parametrization.

.besselM3
Wrapper function for besselK.

.check.data
This function checks data for consistency. Only data objects of typ data.frame, matrix or numeric are accepted.

.check.gig.pars
Some combinations of the GIG parameters are not allowed. This function checks whether this is the case or not.

.check.norm.pars
This function simply checks if the dimensions match.

.check.opt.pars
When calling the fitting routines (fit.ghypuv and fit.ghypmv) a named vector containing the parameters which should not be fitted can be passed. By default all parameters will be fitted.

.fit.ghyp
This function is called by the functions fit.ghypuv and fit.ghypmv to create objects of class mle.ghyp and mle.ghyp.

.ghyp.model
Check if the parameters denote a special case of the generalized hyperbolic distribution.

.t.transfrom
Transformation function used in fit.ghypuv for parameter nu belonging to the Student-t distribution.

.inv.t.transfrom
The inverse of t.transfrom.

.integrate.moment.gig
This function is used when computing the conditional expectation of a generalized inverse gaussian distribution.

.integrate.moment.ghypuv
This function is used when computing the conditional expectation of a univariate generalized hyperbolic distribution.

.dghypuv
This function is used during the fitting procedure. Use dghyp to compute the density of generalized hyperbolic distribution objects.

.dghypmv
This function is used during the fitting procedure. Use dghyp to compute the density of generalized hyperbolic distribution objects.

.mle.default
This function serves as a generic function for maximum likelihood estimation. It is for internal use only. See fit.ghypuv which wraps this function.

.p.default
A generic distribution function integrator given a density function. See pghyp for a wrapper of this function.

.q.default
A generic quantile function calculator given a density function. See qghyp for a wrapper of this function.

.test.ghyp
This function tests whether the object is of class ghyp and sometimes whether it is univariate or multivariate according to the argument case and states a corresponding error if not.

.is.gaussian
Tests whether the object is of a gaussian type.

.is.symmetric
Tests whether the object is symmetric.

.is.student.t
Tests whether the object describes a Student-t distribution.

.is.univariate
Tests whether the object is a univariate ghyp-distribution.

.get.stepAIC.ghyp
Returns a specific model from a list returned by stepAIC.ghyp

Author(s)

Wolfgang Breymann, David Luethi

Classes ghyp and mle.ghyp

Description

The class “ghyp” basically contains the parameters of a generalized hyperbolic distribution. The class “mle.ghyp” inherits from the class “ghyp”. The class “mle.ghyp” adds some additional slots which contain information about the fitting procedure. Namely, these are the number of iterations (n.iter), the log likelihood value (llh), the Akaike Information Criterion (aic), a boolean vector (fitted.params) stating which parameters were fitted, a boolean converged whether the fitting procedure converged or not, an error.code which stores the status of a possible error and the corresponding error.message. In the univariate case the parameter variance is also stored in parameter.variance.

Objects from the Class

Objects should only be created by calls to the constructors ghyp, hyp, NIG, VG, student.t and gauss or by calls to the fitting routines like fit.ghypuv, fit.ghypmv, fit.hypuv, fit.hypmv et cetera.

Slots

Slots of class ghyp:

call:

The function-call of class call.

lambda:

Shape parameter of class numeric.

alpha.bar:

Shape parameter of class numeric.

chi:

Shape parameter of an alternative parametrization. Object of class numeric.

psi:

Shape parameter of an alternative parametrization. Object of class numeric.

mu:

Location parameter of lass numeric.

sigma:

Dispersion parameter of class matrix.

gamma:

Skewness parameter of class numeric.

model:

Model, i.e., (a)symmetric generalized hyperbolic distribution or (a)symmetric special case. Object of class character.

dimension:

Dimension of the generalized hyperbolic distribution. Object of class numeric.

expected.value:

The expected value of a generalized hyperbolic distribution. Object of class numeric.

variance:

The variance of a generalized hyperbolic distribution of class matrix.

data:

The data-slot is of class matrix. When an object of class ghypmv is instantiated the user can decide whether data should be stored within the object or not. This is the default and may be useful when fitting eneralized hyperbolic distributions to data and perform further analysis afterwards.

parametrization:

Parametrization of the generalized hyperbolic distribution of class character. These are currently either “chi.psi”, “alpha.bar” or “alpha.delta”.

Slots added by class mle.ghyp:

n.iter:

The number of iterations of class numeric.

llh:

The log likelihood value of class numeric.

converged:

A boolean whether converged or not. Object of class logical.

error.code:

An error code of class numeric.

error.message:

An error message of class character.

fitted.params:

A boolean vector stating which parameters were fitted of class logical.

aic:

The value of the Akaike Information Criterion of class numeric.

parameter.variance:

The parameter variance is the inverse of the fisher information matrix. This slot is filled only in the case of an univariate fit. This slot is of class matrix.

trace.pars:

Contains the parameter value evolution during the fitting procedure. trace.pars of class list.

Extends

Class “mle.ghyp” extends class "ghyp", directly.

Methods

A “pairs” method (see pairs).
A “hist” method (see hist).
A “plot” method (see plot).
A “lines” method (see lines).
A “coef” method (see coef).
A “mean” method (see mean).
A “vcov” method (see vcov).
A “scale” method (see scale).
A “transform” method (see transform).
A “[.ghyp” method (see [).
A “logLik” method for objects of class “mle.ghyp” (see logLik).
An “AIC” method for objects of class “mle.ghyp” (see AIC).
A “summary” method for objects of class “mle.ghyp” (see summary).

Note

When showing special cases of the generalized hyperbolic distribution the corresponding fixed parameters are not printed.

Author(s)

David Luethi

See Also

optim for an interpretation of error.code, error.message and parameter.variance.
ghyp, hyp, NIG, VG, student.t and gauss for constructors of the class ghyp in the “alpha.bar” and “chi/psi” parametrization. xxx.ad for all the constructors in the “alpha/delta” parametrization. fit.ghypuv, fit.ghypmv et cetera for the fitting routies and constructors of the class mle.ghyp.

Examples

  data(smi.stocks)
  multivariate.fit <- fit.ghypmv(data = smi.stocks,
                                 opt.pars = c(lambda = FALSE, alpha.bar = FALSE),
                                 lambda = 2)
  summary(multivariate.fit)

  vcov(multivariate.fit)
  mean(multivariate.fit)
  logLik(multivariate.fit)
  AIC(multivariate.fit)
  coef(multivariate.fit)

  univariate.fit <- multivariate.fit[1]
  hist(univariate.fit)

  plot(univariate.fit)
  lines(multivariate.fit[2])

Risk and Performance Measures

Description

Functions to compute the risk measure Expected Shortfall and the performance measure Omega based on univariate generalized hyperbolic distributions.

Usage

ESghyp(alpha, object = ghyp(), distr = c("return", "loss"), ...)

ghyp.omega(L, object = ghyp(), ...)

Arguments

Details

The parameter distr specifies whether the ghyp-object describes a return or a loss-distribution. In case of a return distribution the expected-shortfall on a confidence level \alpha is defined as \hbox{ES}_\alpha := \hbox{E}(X | X \leq F^{-1}_X(\alpha)) while in case of a loss distribution it is defined on a confidence level \alpha as \hbox{ES}_\alpha := \hbox{E}(X | X > F^{-1}_X(\alpha)).

Omega is defined as the ratio of a European call-option price divided by a put-option price with strike price L (see References): \Omega(L) := \frac{C(L)}{P(L)}.

Value

ESghyp gives the expected shortfall and
ghyp.omega gives the performance measure Omega.

Author(s)

David Luethi

References

Omega as a Performance Measure by Hossein Kazemi, Thomas Schneeweis and Raj Gupta
University of Massachusetts, 2003

See Also

ghyp-class definition, ghyp constructors, univariate fitting routines, fit.ghypuv, portfolio.optimize for portfolio optimization with respect to alternative risk measures, integrate.

Examples

  data(smi.stocks)

  ## Fit a NIG model to Credit Suisse and Swiss Re log-returns
  cs.fit <- fit.NIGuv(smi.stocks[, "CS"], silent = TRUE)
  swiss.re.fit <- fit.NIGuv(smi.stocks[, "Swiss.Re"], silent = TRUE)

  ## Confidence levels for expected shortfalls
  es.levels <- c(0.001, 0.01, 0.05, 0.1)

  cs.es <- ESghyp(es.levels, cs.fit)
  swiss.re.es <- ESghyp(es.levels, swiss.re.fit)

  ## Threshold levels for Omega
  threshold.levels <- c(0, 0.01, 0.02, 0.05)

  cs.omega <- ghyp.omega(threshold.levels, cs.fit)
  swiss.re.omega <- ghyp.omega(threshold.levels, swiss.re.fit)

  par(mfrow = c(2, 1))

  barplot(rbind(CS = cs.es, Swiss.Re = swiss.re.es), beside = TRUE,
          names.arg = paste(100 * es.levels, "percent"), col = c("gray40", "gray80"),
          ylab = "Expected Shortfalls (return distribution)", xlab = "Level")

  legend("bottomright", legend = c("CS", "Swiss.Re"), fill = c("gray40", "gray80"))

  barplot(rbind(CS = cs.omega, Swiss.Re = swiss.re.omega), beside = TRUE,
          names.arg = threshold.levels, col = c("gray40", "gray80"),
          ylab = "Omega", xlab = "Threshold level")
  legend("topright", legend = c("CS", "Swiss.Re"), fill = c("gray40", "gray80"))

  ## => the higher the performance, the higher the risk (as it should be)

Class ghyp.attribution

Description

The class “ghyp.attribution” contains the Expected Shortfall of the portfolio as well as the contribution of each asset to the total risk and the sensitivity of each Asset. The sensitivity gives an information about the overall risk modification of the portfolio if the weight in a given asset is marginally increased or decreased (1 percent).

The function contribution returns the contribution of the assets to the portfolio expected shortfall.

Usage

contribution(object, ...)

## S4 method for signature 'ghyp.attribution'
contribution(object, percentage = FALSE)

sensitivity(object)

## S4 method for signature 'ghyp.attribution'
sensitivity(object)

## S4 method for signature 'ghyp.attribution'
weights(object)

Arguments

Details

Expected shortfall enjoys homogeneity, sub-additivity, and co-monotonic additivity. Its associated function is continuously differentiable under moderate assumptions on the joint distribution of the assets.

Value

contribution of each asset to portfolio's overall expected shortfall.

sensitivity of each asset to portfolio's overall expected shortfall.

weights of each asset within portfolio.

Slots

ES

Portfolio's expected shortfall (ES) for a given confidence level. Class matrix.

contribution

Contribution of each asset to the overall ES. Class matrix.

sensitivity

Sensitivity of each asset. Class matrix.

weights

Weight of each asset.

Objects from the Class

Objects should only be created by calls to the constructors ESghyp.attribution.

Note

When showing special cases of the generalized hyperbolic distribution the corresponding fixed parameters are not printed.

Author(s)

Marc Weibel

Marc Weibel

See Also

ESghyp.attribution, ghyp.attribution-class to compute the expected shortfall attribution.

Examples

## Not run: 
data(smi.stocks)
multivariate.fit <- fit.ghypmv(data = smi.stocks,
opt.pars = c(lambda = FALSE, alpha.bar = FALSE),
lambda = 2)

portfolio <- ESghyp.attribution(0.01, multivariate.fit)
summary(portfolio)

## End(Not run)

Compute moments of generalized hyperbolic distributions

Description

This function computes moments of arbitrary orders of the univariate generalized hyperbolic distribution. The expectation of f(X - c)^k is calculated. f can be either the absolute value or the identity. c can be either zero or E(X).

Usage

ghyp.moment(object, order = 3:4, absolute = FALSE, central = TRUE, ...)

Arguments

Details

In general ghyp.moment is based on numerical integration. For the special cases of either a “ghyp”, “hyp” or “NIG” distribution analytic expressions (see References) will be taken if non-absolute and non-centered moments of integer order are requested.

Value

A vector containing the moments.

Author(s)

David Luethi

References

Moments of the Generalized Hyperbolic Distribution by David J. Scott, Diethelm Wuertz and Thanh Tam Tran
Working paper, 2008

See Also

mean, vcov, Egig

Examples

  nig.uv <- NIG(alpha.bar = 0.1, mu = 1.1, sigma = 3, gamma = -2)

  # Moments of integer order
  ghyp.moment(nig.uv, order = 1:6)

  # Moments of fractional order
  ghyp.moment(nig.uv, order = 0.2 * 1:20, absolute = TRUE)

The Generalized Inverse Gaussian Distribution

Description

Density, distribution function, quantile function, random generation, expected shortfall and expected value and variance for the generalized inverse gaussian distribution.

Usage

dgig(x, lambda = 1, chi = 1, psi = 1, logvalue = FALSE)

pgig(q, lambda = 1, chi = 1, psi = 1, ...)

qgig(p, lambda = 1, chi = 1, psi = 1, method = c("integration", "splines"),
     spline.points = 200, subdivisions = 200,
     root.tol = .Machine$double.eps^0.5,
     rel.tol = root.tol^1.5, abs.tol = rel.tol, ...)

rgig(n = 10, lambda = 1, chi = 1, psi = 1)

ESgig(alpha, lambda = 1, chi = 1, psi = 1, distr = c("return", "loss"), ...)

Egig(lambda, chi, psi, func = c("x", "logx", "1/x", "var"), check.pars = TRUE)

Arguments

Details

qgig computes the quantiles either by using the “integration” method where the root of the distribution function is solved or via “splines” which interpolates the distribution function and solves it with uniroot afterwards. The “integration” method is recommended when few quantiles are required. If more than approximately 20 quantiles are needed to be calculated the “splines” method becomes faster. The accuracy can be controlled with an adequate setting of the parameters rel.tol, abs.tol, root.tol and spline.points.

rgig relies on the C function with the same name kindly provided by Ester Pantaleo and Robert B. Gramacy.

Egig with func = "log x" uses grad from the R package numDeriv. See the package vignette for details regarding the expectation of GIG random variables.

Value

dgig gives the density,
pgig gives the distribution function,
qgig gives the quantile function,
ESgig gives the expected shortfall,
rgig generates random deviates and
Egig gives the expected value of either x, 1/x, log(x) or the variance if func equals var.

Author(s)

David Luethi and Ester Pantaleo

References

Dagpunar, J.S. (1989). An easily implemented generalised inverse Gaussian generator. Commun. Statist. -Simula., 18, 703–710.

Michael, J. R, Schucany, W. R, Haas, R, W. (1976). Generating random variates using transformations with multiple roots, The American Statistican, 30, 88–90.

See Also

fit.ghypuv, fit.ghypmv, integrate, uniroot, spline

Examples

dgig(1:40, lambda = 10, chi = 1, psi = 1)
qgig(1e-5, lambda = 10, chi = 1, psi = 1)

ESgig(c(0.19,0.3), lambda = 10, chi = 1, psi = 1, distr = "loss")
ESgig(alpha=c(0.19,0.3), lambda = 10, chi = 1, psi = 1, distr = "ret")

Egig(lambda = 10, chi = 1, psi = 1, func = "x")
Egig(lambda = 10, chi = 1, psi = 1, func = "var")
Egig(lambda = 10, chi = 1, psi = 1, func = "1/x")

Histogram for univariate generalized hyperbolic distributions

Description

The function hist computes a histogram of the given data values and the univariate generalized hyperbolic distribution.

Usage

## S4 method for signature 'ghyp'
hist(x, data = ghyp.data(x), gaussian = TRUE,
     log.hist = F, ylim = NULL, ghyp.col = 1, ghyp.lwd = 1,
     ghyp.lty = "solid", col = 1, nclass = 30, plot.legend = TRUE,
     location = if (log.hist) "bottom" else "topright", legend.cex = 1, ...)

Arguments

Value

No value is returned.

Author(s)

David Luethi

See Also

qqghyp, fit.ghypuv, hist, legend, plot, lines.

Examples

  data(smi.stocks)
  univariate.fit <- fit.ghypuv(data = smi.stocks[,"SMI"],
                               opt.pars = c(mu = FALSE, sigma = FALSE),
                               symmetric = TRUE)
  hist(univariate.fit)

Monthly returns of five indices

Description

Monthly returns of indices representing five asset/investment classes Bonds, Stocks, Commodities, Emerging Markets and High Yield Bonds.

Usage

data(indices)

Format

hy.bond

JPMorgan High Yield Bond A (Yahoo symbol “OHYAX”).

emerging.mkt

Morgan Stanley Emerging Markets Fund Inc. (Yahoo symbol “MSF”).

commodity

Dow Jones-AIG Commodity Index (Yahoo symbol “DJI”).

bond

Barclays Global Investors Bond Index (Yahoo symbol “WFBIX”).

stock

Vanguard Total Stock Mkt Idx (Yahoo symbol “VTSMX”).

See Also

smi.stocks

Examples

  data(indices)

  pairs(indices)

Likelihood-ratio test

Description

This function performs a likelihood-ratio test on fitted generalized hyperbolic distribution objects of class mle.ghyp.

Usage

lik.ratio.test(x, x.subclass, conf.level = 0.95)

Arguments

Details

The likelihood-ratio test can be used to check whether a special case of the generalized hyperbolic distribution is the “true” underlying distribution.

The likelihood-ratio is defined as

\Lambda = \frac{sup\{L(\theta | \mathbf{X}) : \theta \in \Theta_0\}} { sup\{L(\theta | \mathbf{X}) : \theta \in \Theta\}}.

Where L denotes the likelihood function with respect to the parameter \theta and data \mathbf{X}, and \Theta_0 is a subset of the parameter space \Theta. The null hypothesis H0 states that \theta \in \Theta_0. Under the null hypothesis and under certain regularity conditions it can be shown that -2 \log(\Lambda) is asymtotically chi-squared distributed with \nu degrees of freedom. \nu is the number of free parameters specified by \Theta minus the number of free parameters specified by \Theta_0.

The null hypothesis is rejected if -2 \log(\Lambda) exceeds the conf.level-quantile of the chi-squared distribution with \nu degrees of freedom.

Value

A list with components:

Author(s)

David Luethi

References

Linear Statistical Inference and Its Applications by C. R. Rao
Wiley, New York, 1973

See Also

fit.ghypuv, logLik, AIC and stepAIC.ghyp.

Examples

  data(smi.stocks)

  sample <- smi.stocks[, "SMI"]

  t.symmetric <- fit.tuv(sample, silent = TRUE, symmetric = TRUE)
  t.asymmetric <- fit.tuv(sample, silent = TRUE)

  # Test symmetric Student-t against asymmetric Student-t in case
  # of SMI log-returns
  lik.ratio.test(t.asymmetric, t.symmetric, conf.level = 0.95)
  # -> keep the null hypothesis

  set.seed(1000)
  sample <- rghyp(1000, student.t(gamma = 0.1))

  t.symmetric <- fit.tuv(sample, silent = TRUE, symmetric = TRUE)
  t.asymmetric <- fit.tuv(sample, silent = TRUE)

  # Test symmetric Student-t against asymmetric Student-t in case of
  # data simulated according to a slightly skewed Student-t distribution
  lik.ratio.test(t.asymmetric, t.symmetric, conf.level = 0.95)
  # -> reject the null hypothesis

  t.symmetric <- fit.tuv(sample, silent = TRUE, symmetric = TRUE)
  ghyp.asymmetric <- fit.ghypuv(sample, silent = TRUE)

  # Test symmetric Student-t against asymmetric generalized
  # hyperbolic using the same data as in the example above
  lik.ratio.test(ghyp.asymmetric, t.symmetric, conf.level = 0.95)
  # -> keep the null hypothesis

Extract Log-Likelihood and Akaike's Information Criterion

Description

The functions logLik and AIC extract the Log-Likelihood and the Akaike's Information Criterion from fitted generalized hyperbolic distribution objects. The Akaike information criterion is calculated according to the formula -2 \cdot \mbox{log-likelihood} + k \cdot n_{par}, where n_{par} represents the number of parameters in the fitted model, and k = 2 for the usual AIC.

Usage

## S4 method for signature 'mle.ghyp'
logLik(object, ...)

## S4 method for signature 'mle.ghyp'
AIC(object, ..., k = 2)

Arguments

Value

Either the Log-Likelihood or the Akaike's Information Criterion.

Note

The Log-Likelihood as well as the Akaike's Information Criterion can be obtained from the function ghyp.fit.info. However, the benefit of logLik and AIC is that these functions allow a call with an arbitrary number of objects and are better known because they are generic.

Author(s)

David Luethi

See Also

fit.ghypuv, fit.ghypmv, lik.ratio.test, ghyp.fit.info, mle.ghyp-class

Examples

  data(smi.stocks)

  ## Multivariate fit
  fit.mv <- fit.hypmv(smi.stocks, nit = 10)
  AIC(fit.mv)
  logLik(fit.mv)

  ## Univariate fit
  fit.uv <- fit.tuv(smi.stocks[, "CS"], control = list(maxit = 10))
  AIC(fit.uv)
  logLik(fit.uv)

  # Both together
  AIC(fit.uv, fit.mv)
  logLik(fit.uv, fit.mv)

Expected value, variance-covariance, skewness and kurtosis of generalized hyperbolic distributions

Description

The function mean returns the expected value. The function vcov returns the variance in the univariate case and the variance-covariance matrix in the multivariate case. The functions ghyp.skewness and ghyp.kurtosis only work for univariate generalized hyperbolic distributions.

Usage

## S4 method for signature 'ghyp'
mean(x)

## S4 method for signature 'ghyp'
vcov(object)

ghyp.skewness(object)

ghyp.kurtosis(object)

Arguments

Details

The functions ghyp.skewness and ghyp.kurtosis are based on the function ghyp.moment. Numerical integration will be used in case a Student.t or variance gamma distribution is submitted.

Value

Either the expected value, variance, skewness or kurtosis.

Author(s)

David Luethi

See Also

ghyp, ghyp-class, Egig to compute the expected value and the variance of the generalized inverse gaussian mixing distribution distributed and its special cases.

Examples

  ## Univariate: Parametric
  vg.dist <- VG(lambda = 1.1, mu = 10, sigma = 10, gamma = 2)
  mean(vg.dist)
  vcov(vg.dist)
  ghyp.skewness(vg.dist)
  ghyp.kurtosis(vg.dist)

  ## Univariate: Empirical
  vg.sim <- rghyp(10000, vg.dist)
  mean(vg.sim)
  var(vg.sim)

  ## Multivariate: Parametric
  vg.dist <- VG(lambda = 0.1, mu = c(55, 33), sigma = diag(c(22, 888)), gamma = 1:2)
  mean(vg.dist)
  vcov(vg.dist)

  ## Multivariate: Empirical
  vg.sim <- rghyp(50000, vg.dist)
  colMeans(vg.sim)
  var(vg.sim)

Pairs plot for multivariate generalized hyperbolic distributions

Description

This function is intended to be used as a graphical diagnostic tool for fitted multivariate generalized hyperbolic distributions. An array of graphics is created and qq-plots are drawn into the diagonal part of the graphics array. The upper part of the graphics matrix shows scatter plots whereas the lower part shows 2-dimensional histogramms.

Usage

## S4 method for signature 'ghyp'
pairs(x, data = ghyp.data(x), main = "'ghyp' pairwise plot",
      nbins = 30, qq = TRUE, gaussian = TRUE,
      hist.col = c("white", topo.colors(100)),
      spline.points = 150, root.tol = .Machine$double.eps^0.5,
      rel.tol = root.tol, abs.tol = root.tol^1.5, ...)

Arguments

Author(s)

David Luethi

See Also

pairs, fit.ghypmv, qqghyp

Examples

  data(smi.stocks)
  fitted.smi.stocks <- fit.NIGmv(data = smi.stocks[1:200, ])
  pairs(fitted.smi.stocks)

Plot univariate generalized hyperbolic densities

Description

These functions plot probability densities of generalized hyperbolic distribution objects.

Usage

## S4 method for signature 'ghyp,missing'
plot(x, range = qghyp(c(0.001, 0.999), x), length = 1000, ...)
## S4 method for signature 'ghyp'
lines(x, range = qghyp(c(0.001, 0.999), x), length = 1000, ...)

Arguments

Details

When the density is very skewed, the computation of the quantile may fail. See qghyp for details.

Author(s)

David Luethi

See Also

hist, qqghyp, pairs, plot, lines.

Examples

  data(smi.stocks)

  smi.fit   <-  fit.tuv(data = smi.stocks[,"SMI"], symmetric = TRUE)
  nestle.fit <- fit.tuv(data = smi.stocks[,"Nestle"], symmetric = TRUE)

  ## Student-t distribution
  plot(smi.fit, type = "l", log = "y")
  lines(nestle.fit, col = "blue")

  ## Empirical
  lines(density(smi.stocks[,"SMI"]), lty = "dashed")
  lines(density(smi.stocks[,"Nestle"]), lty = "dashed", col = "blue")

Plot ES contribution

Description

These functions plot the contribution of each asset to the overall portfolio expected shortfall.

Usage

## S3 method for class 'ghyp.attrib'
plot(
  x,
  metrics = c("contribution", "sensitivity"),
  column.index = NULL,
  percentage = FALSE,
  colorset = NULL,
  horiz = FALSE,
  unstacked = TRUE,
  pie.chart = FALSE,
  sub = NULL,
  ...
)

Arguments

Author(s)

Marc Weibel

See Also

ESghyp.attribution.

Examples

## Not run: 
 data(smi.stocks)
 
 ## Fit a NIG model to Novartis, CS and Nestle log-returns
 assets.fit <- fit.NIGmv(smi.stocks[, c("Novartis", "CS", "Nestle")], silent = TRUE)
 
 ## Define Weights of the Portfolio
 weights <- c(0.2, 0.5, 0.3)
 
 ## Confidence level for Expected Shortfall
 es.levels <- c(0.01)
 portfolio.attrib <- ESghyp.attribution(alpha=es.levels, object=assets.fit, weights=weights)
 
 ## Plot Risk Contribution for each Asset
 plot(portfolio.attrib, metrics='contribution')  

## End(Not run)

Portfolio optimization with respect to alternative risk measures

Description

This function performs a optimization of a portfolio with respect to one of the risk measures “sd”, “value.at.risk” or “expected.shortfall”. The optimization task is either to find the global minimum risk portfolio, the tangency portfolio or the minimum risk portfolio given a target-return.

Usage

portfolio.optimize(object,
                   risk.measure = c("sd", "value.at.risk", "expected.shortfall"),
                   type = c("minimum.risk", "tangency", "target.return"),
                   level = 0.95, distr = c("loss", "return"),
                   target.return = NULL, risk.free = NULL,
                   silent = FALSE, ...)

Arguments

Details

If type is “minimum.risk” the global minimum risk portfolio is returned.

If type is “tangency” the portfolio maximizing the slope of “(expected return - risk free rate) / risk” will be returned.

If type is “target.return” the portfolio with expected return target.return which minimizes the risk will be returned.

Note that in case of an elliptical distribution (symmetric generalized hyperbolic distributions) it does not matter which risk measure is used. That is, minimizing the standard deviation results in a portfolio which also minimizes the value-at-risk et cetera.

Value

A list with components:

Note

In case object denotes a non-elliptical distribution and the risk measure is either “value.at.risk” or “expected.shortfall”, then the type “tangency” optimization problem is not supported.

Constraints like avoiding short-selling are not supported yet.

Author(s)

David Luethi

See Also

transform, fit.ghypmv

Examples

data(indices)

t.object <- fit.tmv(-indices, silent = TRUE)
gauss.object <- fit.gaussmv(-indices)

t.ptf <- portfolio.optimize(t.object,
                            risk.measure = "expected.shortfall",
                            type = "minimum.risk",
                            level = 0.99,
                            distr = "loss",
                            silent = TRUE)

gauss.ptf <- portfolio.optimize(gauss.object,
                                risk.measure = "expected.shortfall",
                                type = "minimum.risk",
                                level = 0.99,
                                distr = "loss")

par(mfrow = c(1, 3))

plot(c(t.ptf$risk, gauss.ptf$risk),
     c(-mean(t.ptf$portfolio.dist), -mean(gauss.ptf$portfolio.dist)),
     xlim = c(0, 0.035), ylim = c(0, 0.004),
     col = c("black", "red"), lwd = 4,
     xlab = "99 percent expected shortfall",
     ylab = "Expected portfolio return",
     main = "Global minimum risk portfolios")

legend("bottomleft", legend = c("Asymmetric t", "Gaussian"),
       col = c("black", "red"), lty = 1)

plot(t.ptf$portfolio.dist, type = "l",
     xlab = "log-loss ((-1) * log-return)", ylab = "Density")
lines(gauss.ptf$portfolio.dist, col = "red")

weights <- cbind(Asymmetric.t = t.ptf$opt.weights,
                 Gaussian = gauss.ptf$opt.weights)

barplot(weights, beside = TRUE, ylab = "Weights")

Quantile-Quantile Plot

Description

This function is intended to be used as a graphical diagnostic tool for fitted univariate generalized hyperbolic distributions. Optionally a qq-plot of the normal distribution can be added.

Usage

qqghyp(object, data = ghyp.data(object), gaussian = TRUE, line = TRUE,
       main = "Generalized Hyperbolic Q-Q Plot",
       xlab = "Theoretical Quantiles", ylab = "Sample Quantiles",
       ghyp.pch = 1, gauss.pch = 6, ghyp.lty = "solid",
       gauss.lty = "dashed", ghyp.col = "black", gauss.col = "black",
       plot.legend = TRUE, location = "topleft", legend.cex = 0.8,
       spline.points = 150, root.tol = .Machine$double.eps^0.5,
       rel.tol = root.tol, abs.tol = root.tol^1.5, add = FALSE, ...)

Arguments

Author(s)

David Luethi

See Also

hist, fit.ghypuv, qghyp, plot, lines

Examples

  data(smi.stocks)

  smi <- fit.ghypuv(data = smi.stocks[, "Swiss.Re"])

  qqghyp(smi, spline.points = 100)

  qqghyp(fit.tuv(smi.stocks[, "Swiss.Re"], symmetric = TRUE),
         add = TRUE, ghyp.col = "red", line = FALSE)

Scaling and Centering of ghyp Objects

Description

scale centers and/or scales a generalized hyperbolic distribution to zero expectation and/or unit variance.

Usage

## S4 method for signature 'ghyp'
scale(x, center = TRUE, scale = TRUE)

Arguments

Value

An object of class ghyp.

Author(s)

David Luethi

See Also

transform, mean, vcov.

Examples

  data(indices)

  t.fit <- fit.tmv(indices)
  gauss.fit <- fit.gaussmv(indices)

  ## Compare the fitted Student-t and Gaussian density.
  par(mfrow = c(1, 2))

  ## Once on the real scale...
  plot(t.fit[1], type = "l")
  lines(gauss.fit[1], col = "red")

  ## ...and once scaled to expectation = 0, variance = 1
  plot(scale(t.fit)[1], type = "l")
  lines(scale(gauss.fit)[1], col = "red")

Daily returns of five swiss blue chips and the SMI

Description

Daily returns from January 2000 to January 2007 of five swiss blue chips and the Swiss Market Index (SMI).

Usage

data(smi.stocks)

Format

SMI

Swiss Market Index.

Novartis

Novartis pharma.

CS

Credit Suisse.

Nestle

Nestle.

Swisscom

Swiss telecom company.

Swiss.Re

Swiss reinsurer.

See Also

indices

Examples

  data(smi.stocks)

  pairs(smi.stocks)

Perform a model selection based on the AIC

Description

This function performs a model selection in the scope of the generalized hyperbolic distribution class based on the Akaike information criterion. stepAIC.ghyp can be used for the univariate as well as for the multivariate case.

Usage

stepAIC.ghyp(data, dist = c("ghyp", "hyp", "NIG", "VG", "t", "gauss"),
             symmetric = NULL, ...)

Arguments

Value

A list with components:

Author(s)

David Luethi

See Also

lik.ratio.test, fit.ghypuv and fit.ghypmv.

Examples

  data(indices)

  # Multivariate case:
  aic.mv <- stepAIC.ghyp(indices, dist = c("ghyp", "hyp", "t", "gauss"),
                         symmetric = NULL, control = list(maxit = 500),
                         silent = TRUE, nit = 500)

  summary(aic.mv$best.model)

  # Univariate case:
  aic.uv <- stepAIC.ghyp(indices[, "stock"], dist = c("ghyp", "NIG", "VG", "gauss"),
                         symmetric = TRUE, control = list(maxit = 500), silent = TRUE)


  # Test whether the ghyp-model provides a significant improvement with
  # respect to the VG-model:
  lik.ratio.test(aic.uv$all.models[[1]], aic.uv$all.models[[3]])

mle.ghyp summary

Description

Produces a formatted output of a fitted generalized hyperbolic distribution.

Usage

## S4 method for signature 'mle.ghyp'
summary(object)

Arguments

Value

Nothing is returned.

Author(s)

David Luethi

See Also

Fitting functions fit.ghypuv and fit.ghypmv, coef, mean, vcov and ghyp.fit.info for accessor functions for mle.ghyp objects.

Examples

  data(smi.stocks)
  mle.ghyp.object <- fit.NIGmv(smi.stocks[, c("Nestle", "Swiss.Re", "Novartis")])
  summary(mle.ghyp.object)

Linear transformation and extraction of generalized hyperbolic distributions

Description

The transform function can be used to linearly transform generalized hyperbolic distribution objects (see Details). The extraction operator [ extracts some margins of a multivariate generalized hyperbolic distribution object.

Usage

## S4 method for signature 'ghyp'
transform(`_data`, summand, multiplier)

## S3 method for class 'ghyp'
x[i = c(1, 2)]

Arguments

Details

If X \sim GH, transform gives the distribution object of “multiplier * X + summand”, where X is the argument named _data.

If the object is of class mle.ghyp, iformation concerning the fitting procedure (cf. ghyp.fit.info) will be lost as the return value is an object of class ghyp.

Value

An object of class ghyp.

Author(s)

David Luethi

See Also

scale, ghyp, fit.ghypuv and fit.ghypmv for constructors of ghyp objects.

Examples

  ## Mutivariate generalized hyperbolic distribution
  multivariate.ghyp <- ghyp(sigma=var(matrix(rnorm(9),ncol=3)), mu=1:3, gamma=-2:0)

  ## Dimension reduces to 2
  transform(multivariate.ghyp, multiplier=matrix(1:6,nrow=2), summand=10:11)

  ## Dimension reduces to 1
  transform(multivariate.ghyp, multiplier=1:3)

  ## Simple transformation
  transform(multivariate.ghyp, summand=100:102)

  ## Extract some dimension
  multivariate.ghyp[1]
  multivariate.ghyp[c(1, 3)]