Modeling Stock Returns and Volatility Using Bivariate Gamma Generalized Laplace Law

TL;DR

This paper introduces a bivariate gamma generalized Laplace model for stock returns and volatility, simplifying estimation and revealing unique convergence properties, with practical financial applications.

Contribution

It extends the variance-gamma distribution to a bivariate setting with observed gamma mixing variables, enabling explicit estimators and simplified maximum likelihood estimation.

Findings

Estimators can have nonstandard convergence rates.

The model effectively analyzes stock index returns and volatility.

Maximum likelihood reduces to classical linear regression.

Abstract

We consider a generalization of the variance-gamma (generalized asymmetric Laplace) distribution, defined as a normal mean - variance mixture with a gamma mixing distribution. While this model is typically studied in the univariate setting, we assume that the gamma mixing variable is observed alongside the primary variable, resulting in a bivariate framework. In this setting, maximum likelihood estimation becomes significantly simpler than in the standard univariate case, reducing to a form of classical linear regression. We derive explicit expressions for the resulting estimators. For certain parameter configurations, the estimators exhibit nonstandard convergence rates, exceeding the usual square-root rate. Finally, we illustrate the applicability of this model in financial contexts by analyzing stock index returns and associated volatility for several major indices.