QuadraticFormsMGHyp.jl

Introduction

This package implements the algorithms from our paper On Quadratic Forms in Multivariate Generalized Hyperbolic Random Vectors, which deals with tail probabilities and partial moments of quadratic forms. The algorithms generalize those of Imhof (Biometrika, 1961) and Broda (Mathematical Finance, 2012).

Consider the random variable

\[L\equiv a_0+\mathbf{a}^{\mathrm{\scriptscriptstyle T}}X+X^{\mathrm{\scriptscriptstyle T}}\mathbf{A}X,\]

a quadratic plus a linear form in the random vector $X\sim \mathrm{MGHyp}(\boldsymbol{\mu},\mathbf{C},\boldsymbol{\gamma},\lambda,\chi,\psi)$; i.e., $X$ has a $d$-variate generalized hyperbolic distribution with stochastic representation

\[X=\boldsymbol{\mu}+Y \boldsymbol{\gamma} +\surd{Y}\mathbf{C}Z,\]

where $Z$ has a $d$-variate standard Normal distribution, $\boldsymbol{\mu}$ and $\boldsymbol{\gamma}$ are constant $d$-vectors, $\mathbf{C}$ is a $d\times d$ matrix, and $Y$ has a univariate generalized inverse Gaussian distribution with density

\[f_{GIG}(y;\lambda,\chi,\psi)\equiv\frac{y^{\lambda-1}}{k_\lambda(\chi,\psi)}\exp\left\{-\frac{1}{2}\left(\chi y^{-1}+\psi y\right)\right\},\]

where

\[k_\lambda(\chi,\psi)\equiv\begin{cases}\frac{\psi}{2}^{-\lambda}\Gamma(\lambda),\text{ if }\chi=0\\ \frac{\chi}{2}^{\lambda}\Gamma(-\lambda),\text{ if }\psi=0\\ 2\left(\frac{\chi}{\psi}\right)^{\lambda/2}K_\lambda(\sqrt{\chi\psi}), \text{ if }\chi\neq0 \text{ and }\psi\neq0,\end{cases}\]

and $K_\lambda(z)$ is the modified Bessel function of the second kind of order $\nu$.

The generalized hyperbolic distribution contains as special cases, among others, the Variance-Gamma ($\lambda>0$), Student's $t$ ($\lambda=-\nu/2, \chi=\nu, \psi=0$), Normal Inverse Gaussian ($\lambda=-1/2$), and Hyperbolic ($\lambda=1$) distributions.

Installation

QuadraticFormsMGHyp is a registered Julia package. It can be installed with using Pkg; Pkg.add("QuadraticFormsMGHyp"). Figure 1 in the paper, reproduced in the logo above, can be generated with using Pkg; Pkg.test("QuadraticFormsMGHyp").

Usage

The package exports a single function, qfmgh. Its signature is

qfmgh(x, a0, a, A, C, mu, gam, lam, chi, psi; do_spa=false, order=2)

The first argument, x, can be passed as either a scalar or a vector. The latter is more efficient than the more Julian way of calling qfmgh.(xvec, ...), because certain parts of the computation, such as calculating the eigenvalues of $A$, can be hoisted out of the loop over xvec.

The keyword argument do_spa controls whether an exact result or a saddlepoint approximation is computed. The order of the latter is controlled with the second keyword argument, order, which can be either 1 or 2.

The function returns a tuple containing (vectors of) the tail probability $\mathbb{P}[L>x]$, and the tail conditional mean $\mathbb{E}[L\mid L>x ]$.

Docstrings

qfmgh(x::Union{AbstractVector{<:Real}, Real}, a0, a, A, C, mu, gam, lam, chi, psi; do_spa=false, order=2)

`do_spa`: whether to return the exact result or a saddlepoint approximation
`order`: order of the saddlepoint approximation