Modelling and predicting market risk with Laplace–Gaussian mixture distributions

Abstract

While much of classical statistical analysis is based on Gaussian distributional assumptions, statistical modelling with the Laplace distribution has gained importance in many applied fields. This phenomenon is rooted in the fact that, like the Gaussian, the Laplace distribution has many attractive properties. This paper investigates two methods of combining them and their use in modelling and predicting financial risk. Based on 25 daily stock return series, the empirical results indicate that the new models offer a plausible description of the data. They are also shown to be competitive with, or superior to, use of the hyperbolic distribution, which has gained some popularity in asset–return modelling and, in fact, also nests the Gaussian and Laplace.

Acknowledgements

The research of Haas and Mittnik was supported by the Deutsche Forschungsgemeinschaft (SFB 386). Mittnik conducted part of the research while visiting the Department of Economics at Washington University in St Louis with a grant from the Fulbright Commission. Part of the research of M. Paolella has been carried out within the National Centre of Competence in Research ‘Financial Valuation and Risk Management’ (NCCR FINRISK), which is a research programme supported by the Swiss National Science Foundation. The authors are grateful for the constructive comments from an anonymous referee.

Notes

¹ This has, for example, the consequence that DaimlerChrysler is not included because of the merger of Chrysler and Daimler Benz in 1998.

² Although kurtosis is monotonic in θ _s , the moments are not. For example, the variance is a quadratic function in θ _s , which has its minimum at θ _s  = 2/3, namely E(Z ²) = 2/3.

³ The values were computed via numeric integration. Complicated expressions for the moments do exist however; see e.g. Kuechler et al . (1999, p. 5).