On the probability distribution of stock returns in the Mike-Farmer model
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 $α_x$ of the Student distribution can always produce power-law tails in the return distribution $f(r)$ with different tail exponent $α_r$. 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.