On the probability distribution of stock returns in the Mike-Farmer model

TL;DR

This paper investigates the origin of power-law tails in stock return distributions within the Mike-Farmer model, revealing that such tails emerge only when the left part of the order price distribution exhibits power-law behavior.

Contribution

It demonstrates that power-law tails in return distributions are linked to the tail properties of the order price distribution in the MF model, providing insights into their origin.

Findings

Power-law tails in returns occur only when the left part of the order price distribution has a power-law tail.

Return distributions at different timescales are well modeled by Student distributions with tail exponents near the cubic law.

The tail exponent of return distributions increases with the timescale.

Abstract

Recently, Mike and Farmer have constructed a very powerful and realistic behavioral model to mimick the dynamic process of stock price formation based on the empirical regularities of order placement and cancelation in a purely order-driven market, which can successfully reproduce the whole distribution of returns, not only the well-known power-law tails, together with several other important stylized facts. There are three key ingredients in the Mike-Farmer (MF) model: the long memory of order signs characterized by the Hurst index $H_s$, the distribution of relative order prices $x$ in reference to the same best price described by a Student distribution (or Tsallis' $q$-Gaussian), and the dynamics of order cancelation. They showed that different values of the Hurst index $H_s$ and the freedom degree $\alpha_x$ of the Student distribution can always produce power-law tails in the return distribution $f(r)$ with different tail exponent $\alpha_r$. In this paper, we study the origin of the power-law tails of the return distribution $f(r)$ in the MF model, based on extensive simulations with different combinations of the left part $f_L(x)$ for $x<0$ and the right part $f_R(x)$ for $x>0$ of $f(x)$. We find that power-law tails appear only when $f_L(x)$ has a power-law tail, no matter $f_R(x)$ has a power-law tail or not. In addition, we find that the distributions of returns in the MF model at different timescales can be well modeled by the Student distributions, whose tail exponents are close to the well-known cubic law and increase with the timescale.