Nonlinear dynamics and the distribution of daily stock index returns.

I. Introduction

The independent and identically distributed (i.i.d.) normal assumption is simple and convenient for financial models, but research on stock prices finds the distribution is leptokurtotic, not normal. That is, the empirical distribution of daily price changes has more observations around the mean and in the extreme tails than a normal distribution. In addition, Scheinkman and LeBaron (1989) find evidence of nonlinear dependence in stock returns.

Violating the independence assumption causes ordinary least squares estimates of regression parameters to be inefficient, and hypothesis tests based on i.i.d. normality assumptions to be inconsistent. If the distribution is leptokurtotic, the traditional capital asset pricing model (CAPM) is not valid. Perhaps the most practical reason the observed leptokurtosis should get greater attention is that ignoring it may lead to biased estimates of option premia when using the Black and Scholes (1972) option pricing model. Black and Scholes (1972) find that options based on their model seem to be overpriced for high-variance stocks. Finnerty (1978) confirms that the Black and Scholes model works poorly for options far in and out of money. Jorion (1988) argues the inaccuracy of the Black and Scholes model in currency options may be due to an incorrect assumption about the statistical distribution.

In this study we determine the most descriptive model of daily stock returns by testing a diffusion-jump process, extended versions of GARCH models under normal and student distributions, and a combination of the GARCH and jump processes. The results provide a test of whether the data are consistent with deterministic chaos.

Ball and Torous (1985) argue daily stock returns are characterized by log normally distributed jumps. Akgiray and Booth (1988) suggest a mixed diffusion-jump process for exchange rate changes.(1) The mixture of a Brownian motion and a Poisson process explain exchange rate movements better than the stable, scaled t-distribution and mixture of normal distributions. These mixed distributions can partly explain leptokurtosis, but because they assume that successive observations are independent, the mixed distributions are inconsistent with the empirical work that finds both linear and nonlinear dependence (e.g., Scheinkman and LeBaron (1989)).