A Compound Events Model for Security Prices

Abstract

SUMMARY A model for the distribution of security price changes is proposed. The model is similiar to previous analyses of stock market price behavior in that logged price changes are assumed to be independent (random walk models). However, this model differs from earlier work in that the logged price changes are not assumed to follow some stable distribution (which might possibly be normal). Instead, the logged price changes are assumed to follow a distribution that is a Poisson mixture of normal distributions. It is shown that the analytical characteristics of such a distribution agree with what has been found empirically. That is, this distribution is in general skewed, leptokurtic, more peaked at its mean than the distribution of a comparable normal variate, and has greater probability mass in its tails than the distribution of a comparable normal variate. A cumulant matching method is suggested for estimating the four parameters of the distribution. The model is also generalized to the multivariate case in which a group of securities may be studied simultaneously. Parameters are in a manner analogous to that of the univariate case. A brief frequency discussion of the spectrum for the univariate case is also provided. Finally, a univariate empirical study of ten of the Dow Jones Industrial stocks is presented. All parameters are estimated, and graphs are provided to show how actual prices varied from 1926-1960. These graphs are compared with the model trend (linear), with a one standard deviation confidence region. For each stock, two graphs of the cumulative distribution function (c.d.f.) of the price differences are presented. One, the empirical c.d.f. is computed directly from the sample observations. The other, the estimated theoretical c.d.f., is computed by substituting the parameter estimates for the true parameters in the proposed model. Agreement between the two curves is often quite close. The numerical results must be interpreted bearing in mind that the sample sizes may be much too small to provide efficient estimators, and the time periods studied may be much too long to yield small confidence bounds on the price variations that should be expected at reasonable significance levels.