Stochastic Lotka-Volterra Systems of Competing Auto-Catalytic Agents Lead Generically to Truncated Pareto Power Wealth Distribution, Truncated Levy Distribution of Market Returns, Clustered Volatility, Booms and Craches

TL;DR

This paper models stock market dynamics using stochastic Lotka-Volterra systems, showing how individual auto-catalytic wealth growth and competition lead to Pareto wealth distribution, Levy-distributed returns, and market phenomena like volatility, booms, and crashes.

Contribution

It introduces a microscopic agent-based model linking wealth dynamics to market statistics, revealing how auto-catalysis and competition produce realistic financial distributions and behaviors.

Findings

Wealth distribution follows a truncated Pareto power law.

Market returns exhibit truncated Levy distribution with intermittency.

System dynamics resemble noisy low-dimensional chaos under certain conditions.

Abstract

We give a microscopic representation of the stock-market in which the microscopic agents are the individual traders and their capital. Their basic dynamics consists in the auto-catalysis of the individual capital and in the global competition/cooperation between the agents mediated by the total wealth invested in the stock (which we identify with the stock-index). We show that such systems lead generically to (truncated) Pareto power-law distribution of the individual wealth. This, in turn, leads to intermittent market (short time) returns parametrized by a (truncated) Levy distribution. We relate the truncation in the Levy distribution to the (truncation in the Pareto Power Law i.e. to the) fact that at each moment no trader can own more than the current total wealth invested in the stock. In the cases where the system is dominated by the largest traders, the dynamics looks similar to noisy low-dimensional chaos. By introducing traders memory and/or feedback between individual and collective wealth fluctuations (the later identified with the stock returns), one obtains clustered "volatility" as well as market booms and crashes. The basic feedback loop consists in: - computing the market price of the stock as the sum of the individual wealths invested in the stock by the traders and - determining the time variation of the individual trader wealth as his/her previous wealth multiplied by the stock return (i.e. the variation of the stock price).