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Multivariate generalized hyperbolic laws for modeling financial log‐returns: Empirical and theoretical considerations

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  • Stergios B. Fotopoulos
  • Alex Paparas
  • Venkata K. Jandhyala

Abstract

The aim of this study is to consider the multivariate generalized hyperbolic (MGH) distribution for modeling financial log‐returns. Beginning with the multivariate geometric subordinated Brownian motion for asset prices, we first demonstrate that the mean‐variance mixing model of the multivariate normal law is natural for log‐returns of financial assets. This multivariate mean‐variance mixing model forms the basis for deriving the MGH family as a class of distributions for modeling the behavior of log‐returns. While theory suggests MGH to be an appropriate family, empirical considerations must also support such a proposition. This article reviews various empirical criteria in support of the MGH family. From a theoretical perspective, we present an alternative form of the density for the MGH family. This alternative density for the MGH family is more convenient for deriving certain limiting results. Numerical study on the distributional behavior of six stocks from the US market forms the foundation of investigating the suitability of the MGH family and some of its well‐known subfamilies. Along the way, we implement the multicycle expectation conditional maximization algorithm for estimating the parameters of the MGH family. Then adapting a recently developed algorithmic procedure, we develop a self‐contained methodology for carrying out goodness‐of‐fit tests for the MGH distribution. The numerical study on the six stocks confirms the suitability of the MGH distribution for different time scales of the log‐returns such as daily, monthly, and 6‐monthly data.

Suggested Citation

  • Stergios B. Fotopoulos & Alex Paparas & Venkata K. Jandhyala, 2020. "Multivariate generalized hyperbolic laws for modeling financial log‐returns: Empirical and theoretical considerations," Applied Stochastic Models in Business and Industry, John Wiley & Sons, vol. 36(5), pages 757-775, September.
    • Handle: RePEc:wly:apsmbi:v:36:y:2020:i:5:p:757-775
    DOI: 10.1002/asmb.2525
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