Stock Price Processes with Infinite Source Poisson Agents

TL;DR

This paper models stock prices using a Poisson agent framework, demonstrating how different trading intensities lead to fractional Brownian motion or stable Levy motion as limits.

Contribution

It introduces a unified stochastic process model for stock prices that captures different market behaviors through scaling limits and Poisson agent interactions.

Findings

Scaling in the model is equivalent to time scaling.

High-frequency trading leads to fractional Brownian motion.

Lower trading frequency results in stable Levy motion.

Abstract

We construct a general stochastic process and prove weak convergence results. It is scaled in space and through the parameters of its distribution. We show that our simplified scaling is equivalent to time scaling used frequently. The process is constructed as an integral with respect to a Poisson random measure which governs several parameters of trading agents in the context of stock prices. When the trading occurs more frequently and in smaller quantities, the limit is a fractional Brownian motion. In contrast, a stable Levy motion is obtained if the rate of trading decreases while its effect rate increases.