On Quadratic Forms in Multivariate Generalized Hyperbolic Random Vectors

This article has been accepted for publication in Biometrika Published by Oxford University Press. (https://doi.org/10.1093/biomet/asaa067)

19 Pages Posted: 16 Apr 2019 Last revised: 3 Sep 2020

See all articles by Simon A. Broda

Simon A. Broda

University of Zurich - Department of Banking and Finance

Juan Arismendi-Zambrano

University College Dublin (UCD), College of Business and Law, UCD School of Business, Michael Smurfit Graduate School of Business, Students; University of Reading - ICMA Centre

Date Written: September 10, 2019

Abstract

Countless test statistics can be written as quadratic forms in certain random vectors, or ratios thereof. Consequently, their distribution has received considerable attention in the literature. Except for a few special cases, no closed-form expression for the cdf exists, and one resorts to numerical methods. Traditionally the problem is analyzed under the assumption of joint Gaussianity; the algorithm that is usually employed is that of Imhof (1961). The present manuscript generalizes this result to the case of multivariate generalized hyperbolic random vectors. This flexible distribution nests, among others, the multivariate t, Laplace, and variance gamma distributions. An expression for the first partial moment is also obtained, which plays a vital role in financial risk management. The proof involves a generalization of the classic inversion formula due to Gil-Pelaez (1951). Two numerical applications are considered: first, the finite-sample distribution of the two stage least squares estimator of a structural parameter. Second, the Value at Risk and expected shortfall of a quadratic portfolio with heavy-tailed risk factors. An empirical application is examined, in which a portfolio of Dow Jones Industrial Index stock options is optimized with respect to its expected shortfall. The results demonstrate the benefits of the analytical expression.

Keywords: Characteristic Function, Conditional Value at Risk, Expected Shortfall, Transform Inversion, Two Stage Least Squares

JEL Classification: C02, C34, G32

Suggested Citation

Broda, Simon A. and Arismendi-Zambrano, Juan, On Quadratic Forms in Multivariate Generalized Hyperbolic Random Vectors (September 10, 2019). This article has been accepted for publication in Biometrika Published by Oxford University Press. (https://doi.org/10.1093/biomet/asaa067), Available at SSRN: https://ssrn.com/abstract=3369208 or http://dx.doi.org/10.2139/ssrn.3369208

Simon A. Broda (Contact Author)

University of Zurich - Department of Banking and Finance ( email )

Plattenstr 32
Zurich, 8032
Switzerland

Juan Arismendi-Zambrano

University College Dublin (UCD), College of Business and Law, UCD School of Business, Michael Smurfit Graduate School of Business, Students ( email )

Blackrock, Dublin
Ireland
+353017168077 (Phone)

HOME PAGE: http://https://www.smurfitschool.ie/

University of Reading - ICMA Centre ( email )

Whiteknights Park
P.O. Box 242
Reading, RG6 6BA
United Kingdom

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