The Return Distribution of Stocks: A Dynamic Model

An analytical dynamic model for the return distribution of stocks is presented, based on the conservation equation of return-dependent shares of a stock. Two types of a return distribution are established: real and virtual. The real return distribution characterizes the current return distribution of shares of a stock, while the virtual return distribution is derived from traded price data. The main difference between both is that the real return distribution requires the mean holding time of shares. According to the theory, the real return distribution is stationary, featuring a double exponential distribution near mean return with power-law tails. In contrast, the virtual return distribution evolves over time and includes an additional Gaussian component around mean return. Comparison with empirical data validates the model's applicability to stocks satisfying the model's conditions.