On Some Processes and Distributions in a Collective Model of Investors' Behavior

This article considers a model for alternative processes for securities prices and compares this model with actual return data of several securities. The distributions of returns that appear in the model can be Gaussian as well as non-Gaussian; in particular they may have two peaks. We consider a discrete Markov chain model. This model in some aspects is similar to well-known Ising model describing ferromagnetics. Namely we consider a set of N investors, each of whom has either bullish or bearish opinion, denoted by plus or minus respectively. At every time step each of N investors can change his/her sign. The probability of a plus becoming a minus and the probability of a minus becoming a plus depends only on the bullish sentiment described as the number of bullish investors among the total of N investors. The number of bullish investors then forms a Markov chain whose transition matrix is calculated explicitly. The transition matrix of that chain is ergodic and any initial distribution of bullish investors converges to stationary. Stationary distributions of bullish investors in this Markov chain model are similar to continuous distributions of the "theory of social imitation" of Callen and Shapero. Distributions obtained this way can represent 3 types of market behavior: one-peaked distribution that is close to Gaussian, transition market (flattening of the top), and two-peaked distribution.