Dynamic numerical models of stock market price: from microscopic determinism to macroscopic randomness

A variant of threshold dynamics is introduced to model the behaviors of a large assembly of dealers in a stock market. Although the microscopic evolution dynamics is deterministic the collective behaviors such as market prices show seemingly stochastic fluctuations. The statistical properties of market price change can be well approximated by a simple discrete Langevin-type equation with random amplification. The macroscopic stochastic equation is solved both numerically and analytically showing that the market price change generally follow power-law distributions in the steady state. The reason for the appearance of rapid decay in the distribution tails are discussed.