Financial Brownian Particle in the Layered Order-Book Fluid and Fluctuation-Dissipation Relations

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Financial Brownian Particle in the Layered Order-Book Fluid and Fluctuation-Dissipation Relations

Yoshihiro Yura, Hideki Takayasu, Didier Sornette, and Misako Takayasu

Phys. Rev. Lett. 112, 098703 – Published 7 March 2014

Abstract

We introduce a novel description of the dynamics of the order book of financial markets as that of an effective colloidal Brownian particle embedded in fluid particles. The analysis of comprehensive market data enables us to identify all motions of the fluid particles. Correlations between the motions of the Brownian particle and its surrounding fluid particles reflect specific layering interactions; in the inner layer the correlation is strong and with short memory, while in the outer layer it is weaker and with long memory. By interpreting and estimating the contribution from the outer layer as a drag resistance, we demonstrate the validity of the fluctuation-dissipation relation in this nonmaterial Brownian motion process.

Received 12 April 2013

DOI:https://doi.org/10.1103/PhysRevLett.112.098703

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Authors & Affiliations

Yoshihiro Yura^1,4, Hideki Takayasu^2,3, Didier Sornette⁴, and Misako Takayasu^1,*

¹Department of Computational Intelligence and Systems Science, Interdisciplinary Graduate School of Science and Engineering, Tokyo Institute of Technology 4259 Nagatsuta-cho, Yokohama 226-8502, Japan
²Sony Computer Science Laboratories, 3-14-13, Higashi-Gotanda, Shinagawa-ku, Tokyo 141-0022, Japan
³Meiji Institute for Advanced Study of Mathematical Sciences, Meiji University, 4-21-1 Nakano, Nakano-ku, Tokyo 164-8525, Japan
⁴Department of Management, Technology, and Economics, ETH Zurich, Scheuchzerstrasse 7, 8092 Zurich, Switzerland

^*takayasu@dis.titech.ac.jp

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Issue

Vol. 112, Iss. 9 — 7 March 2014

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Images

Figure 1
Schematic representation of the FBP model. (a) An order-book configuration of buy orders (blue in outer layer and green in inner layer) and sell orders (red in outer layer and orange in inner layer) on the price axis. (b) Corresponding configuration of outer-layer particles (blue and red disks), inner-layer particles (green and orange disks), the colloidal Brownian particle’s interaction range (yellow-green torus), and the core (yellow). (c) After $Δ t$ , the configuration of surrounding particles changes.
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Figure 2
Statistical properties supporting the physical Brownian analogy in Fig. 1. (a) Correlation function between velocity, $v (t)$ , and the change in particle number at depth $γ$ for buy orders (green and blue triangles) and sell orders (orange and red circles) where $γ_{c} = 18$ . ( $b_{1}$ ) Scatter plot of $v (t)$ as a function of $F_{i} (t)$ for $Δ t = 10^{2}$ . ( $b_{2}$ ) Scatter plot of $v (t)$ as a function of the change in the particle number in the outer layer for buy orders [ $F_{o}^{-} (t)$ ] and for sell orders [ $F_{o}^{+} (t)$ ] with $Δ t = 10^{2}$ (blue line and red line, respectively). Dots and bars represent the mean values and the standard deviations. (c) Correlation coefficients between $v (t)$ and $F_{i} (t)$ (pink dots), and between $v (t)$ and the “order flow” (black dots), defined by the number of buy orders minus that of sell orders. (d) Time-shifted correlation functions between $v (t)$ and ${F_{i} (t + Δ t), F_{o}^{+} (t + Δ t), F_{o}^{-} (t + Δ t)}$ (pink line, blue line, and red line, respectively). The inserted figure shows a log-log plot of these correlation functions as a function of $Δ t$ and the dotted line shows a power law, $1 / Δ t$ .
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Figure 3
Schematic diagram of the space-time configuration of particles. (a) Creation and annihilation of fluid particles are shown together with the trajectory of the colloid motion (black line) and the inner layer (the region between black dotted lines). The black disks represent $a_{o i}$ particles. (b) Share percentages of the different particle types.
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Figure 4
Statistical properties of $a_{o i}$ particles (see text for definition): (a) Correlation functions of ${v (t), g_{o i} (t + Δ t)}$ (upper red) and ${v (t), g_{i i} (t + Δ t)}$ (lower blue). (b) Cumulative lifetime distribution of $a_{i i}$ (left blue) and $a_{o i}$ (right red), with the power law $τ^{- 0.5}$ shown as a guideline (red dotted line). (c) Power spectra of $v (t) Δ t$ (top green), $v_{I} (t) Δ t$ (middle red), and ratio of $| v_{I}^{2} (ω) | / | v^{2} (ω) |$ (bottom gray), with the blue dotted guideline $a / (b + ω^{2})$ with $(a, b) = (0.00725, 0.0764)$ . The power spectra are averaged over 200 samples of time series, of size $2^{8}$ ticks. The right vertical axis is for $v (t) Δ t$ and $v_{I} (t) Δ t$ , and the left vertical axis is for the ratio. (d) Time series of $x (t)$ (U.S. dollars—Japanese yen exchange rates for the week of March 14, 2011). (e) Time series of the ratio $a_{o i} / a_{i}$ , measured for the time interval $[t - 1000, t]$ , and the averaged value for the whole period (the blue dotted line).
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