Alternatives to the normal model of stock returns: Gaussian mixture, generalised logF and generalised hyperbolic models

Simple parametric models of the marginal distribution of stock returns are an essential building block in many areas of applied finance. Even though it is well known that the normal distribution fails to represent most of the “stylised” facts characterising return distributions, it still dominates much of the applied work in finance. Using monthly S&P 500 stock index returns (1871–2005) as well as daily returns (2001–2005), we investigate the viability of three alternative parametric families to represent both the stylised and empirical facts: the generalised hyperbolic distribution, the generalised logF distribution, and finite mixtures of Gaussians. For monthly return data, all three alternatives give reasonable fits for all sub-periods. However, the generalised hyperbolic distribution fails to describe some features of the marginal distributions in some sub-periods. The daily return data are much more symmetric and expose another problem for all three distributions: the parameters describing the behaviour of the tails also influence the scale so that simpler alternatives or restricted parameterisations are called for.

[1]  Underlying dynamics of typical fluctuations of an emerging market price index: The Heston model from minutes to months , 2005, physics/0506101.

[2]  Javier Estrada,et al.  Empirical distributions of stock returns: European securities markets, 1990-95 , 1997 .

[3]  Michael Sørensen,et al.  Stock returns and hyperbolic distributions , 1999 .

[4]  Samuel Kotz,et al.  The Laplace Distribution and Generalizations: A Revisit with Applications to Communications, Economics, Engineering, and Finance , 2001 .

[5]  R. Cont.  Empirical properties of asset returns: stylized facts and statistical issues , 2001 .

[6]  O. Barndorff-Nielsen.  Exponentially decreasing distributions for the logarithm of particle size , 1977, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[7]  Christian Bauer,et al.  Value at risk using hyperbolic distributions , 2000 .

[8]  E. Eberlein,et al.  Hyperbolic distributions in finance , 1995 .

[9]  Sangit Chatterjee,et al.  On Measuring Skewness and Elongation in Common Stock Return Distributions: The Case of the Market Index , 1988 .

[10]  M. Abramowitz,et al.  Handbook of Mathematical Functions With Formulas, Graphs and Mathematical Tables (National Bureau of Standards Applied Mathematics Series No. 55) , 1965 .

[11]  Victor M. Yakovenko,et al.  Exponential distribution of financial returns at mesoscopic time lags: a new stylized fact , 2004 .

[12]  Olivier V. Pictet,et al.  From the bird's eye to the microscope: A survey of new stylized facts of the intra-daily foreign exchange markets , 1997, Finance Stochastics.

[13]  J. Wellner,et al.  Empirical Processes with Applications to Statistics , 2009 .

[14]  D. Rubin,et al.  Maximum likelihood from incomplete data via the EM - algorithm plus discussions on the paper , 1977 .

[15]  Hisanao Takahashi.  Ehrenfest Model with Large Jumps in Finance , 2003, cond-mat/0311594.

[16]  Adrian Pagan,et al.  Estimating the Density Tail Index for Financial Time Series , 1997, Review of Economics and Statistics.

[17]  V. Yakovenko,et al.  Probability distribution of returns in the Heston model with stochastic volatility , 2002, cond-mat/0203046.

[18]  Leland Wilkinson,et al.  An Analytic Approximation to the Distribution of Lilliefors's Test Statistic for Normality , 1986 .

[19]  B. Everitt,et al.  Finite Mixture Distributions , 1981 .

[20]  Lawrence B. Levy,et al.  The log F: A Distribution for All Seasons , 2002, Comput. Stat..

[21]  Amado Peiró.  The distribution of stock returns: international evidence , 1994 .

[22]  A. Buckley,et al.  An overview of returns in Europe , 1999 .

[23]  G. McLachlan,et al.  The EM algorithm and extensions , 1996 .

[24]  S. Heston.  A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options , 1993 .

[25]  F. Longin.  The Asymptotic Distribution of Extreme Stock Market Returns , 1996 .

[26]  Milton Abramowitz,et al.  Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables , 1964 .

[27]  Application of the heston and hull–white models to german dax data , 2004 .

[28]  Stanley J. Kon.  Models of Stock Returns—A Comparison , 1984 .