  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.

Statistical Finance;Physics and Society