Adv. Math. Econ. 14, 69–92 (2011)

Stock price process and long memory in trade signs

Koji Kuroda^{1}, Jun-ichi Maskawa^{2} and JoshinMurai ^{4}^{3}

1 Graduate school of Integrated Basic Sciences, Nihon University, Sakura-Josui 3-25-40, Setagaya-ku, Tokyo 156-8550, Japan (e-mail: kuroda@math.chs.nihon-u.ac.jp)

2 Department of Economics, Seijo University 6-1-20 Seijo, Setagaya-ku, Tokyo 157-8511, Japan (e-mail: maskawa@gmail.com)

3 Graduate school of Humanities and Social Sciences, Okayama University, Okayama 700-8530, Japan (e-mail: murai@e.okayama-u.ac.jp)

Received: February 27, 2010 Revised: July 2, 2010

Mathematical Subject Classification (2010): 60K35, 60G22, 60F17, 82B20

Abstract.Empirical study on tick by tick data in stock markets shows us that there exists a long memory in trade signs and signed trade volumes. This means that an order flow is a highly autocorrelated long memory process. We present a mathematical model of trade signs and trade volumes in which traders decompose their orders into small pieces. We prove that fractional Brownian motions are obtained as a scaling limit of the signed volume process induced by the model.

Key words: trade signs, long memory, Hurst index, fractional Brownian motion, Polymer expansion

4Supported by Grant-in-Aid for Scientific Research (C) No. 21510146.