Stock prices as branching processes

An extension of the Bienaymé–Galton–Watson branching process is proposed to model the short-term behavior of stock prices. Measured in units of $1/8, prices are integer-valued, yet they have many of the characteristics of the multiplicative random walk: e.g., uncorrelated increments. Unlike the random walk higher moments of returns (price relatives) depend on initial price. Conditional distributions of returns over short periods, such as one day, are thick-tailed, but tail thickness decreases as either initial price or the length of the period increases. As initial price approaches infinity, the normalized return approaches a compound-Poisson process – the ‘compound-events’ model. The model is applied to daily closing prices of a sample of common stocks.