On the probability distribution of stock returns in the Mike-Farmer model |
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Authors: | G-F Gu and W-X Zhou |
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Institution: | 1.School of Business, East China University of Science and Technology,Shanghai,P.R. China;2.School of Science, East China University of Science and Technology,Shanghai,P.R. China;3.Research Center for Econophysics, East China University of Science and Technology,Shanghai,P.R. China;4.Engineering Research Center of Process Systems Engineering (Ministry of Education), East China University of Science and Technology,Shanghai,P.R. China;5.Research Center on Fictitious Economics & Data Science, Chinese Academy of Sciences,Beijing,P.R. China |
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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 Hs, 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
Hs and the freedom degree αx of the Student distribution
can always produce power-law tails in the return distribution
fr(r) with different tail exponent αr. In this paper, we
study the origin of the power-law tails of the return distribution
fr(r) in the MF model, based on extensive simulations with
different combinations of the left part L(x) for x < 0 and the
right part R(x) for x > 0 of fx(x). We find that power-law
tails appear only when L(x) has a power-law tail, no matter 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. |
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Keywords: | PACS" target="_blank">PACS 89 65 Gh Economics econophysics financial markets business and management 89 75 Da Systems obeying scaling laws 05 40 -a Fluctuation phenomena random processes noise and Brownian motion |
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