Quantitative statistical analysis of order-splitting behavior of individual trading accounts in the Japanese stock market over nine years

Econophysics aims to understand the macroscopic behavior of financial markets from the underlying microscopic decision-making dynamics. In particular, the order splitting of large metaorders is one of the most important trading strategies in this literature: while traders have large potential metaorders, they split the large orders into small pieces (called child orders) to minimize market impact. This strategic behavior is believed to be important because it is a promising candidate for the microscopic origin of the long-range correlation (LRC) in the persistent order flow. Indeed, Lillo, Mike, and Farmer (LMF) [Phys. Rev. E 71, 066122 (2005)] introduced a simple microscopic model of the order-splitting traders to predict the asymptotic behavior of the LRC from the microscopic dynamics, even quantitatively. The plausibility of this scenario has been investigated by Tóth et al. [J. Econ. Dyn. Control 51, 218 (2015)] at a qualitative level. However, no solid support has been presented yet on the quantitative prediction by the LMF model in the lack of large microscopic data sets. In this paper, we have provided a quantitative statistical analysis of the order-splitting behavior at the level of each trading account. We analyze a large data set of the Tokyo stock exchange (TSE) market over nine years, including the account data of traders (called virtual servers). The virtual server is a unit of trading accounts in the TSE market, and we can effectively define the trader IDs by an appropriate preprocessing. We apply a strategy clustering to individual traders in terms of market orders to identify the order-splitting traders and the random traders. The length distribution of metaorders is empirically estimated for each stock every year. For most of the stocks, we find that the metaorder length distribution obeys power laws with exponent $\alpha$, such that $P\left(L\right)\propto L^{-\alpha -1}$ with the metaorder length $L$, as theoretically assumed in the LMF model. By analyzing the sign correlation of order flow $C\left(\tau \right)\propto {\tau }^{-\gamma }$, we draw the scatterplot between $\alpha$ and $\gamma$, directly confirming the LMF prediction $\gamma \approx \alpha -1$. Furthermore, we discuss how to estimate the total number of splitting traders only from public data via the autocorrelation function prefactor formula in the LMF model. Our work provides quantitative evidence of the LMF model, strongly supporting the order-splitting hypothesis as the origin of LRC.

Figure 1

Schematic of the order-splitting hypothesis. In this hypothesis, several traders hold large latent orders (called metaorders) and split them into small child orders. The plus (minus) sign “$+$” (“$-$”) represents a buy (sell) market order. By definition, the child orders have the same order sign; thus, the order-sign sequence of the whole market exhibits a long memory due to this order splitting.

Figure 2

Summary statistics on the TSE markets from 2012 to 2020. (a) Daily transaction volume on the arrowhead system $V\left(t\right)$ normalized by the volume in 2012 [i.e., $V\left(2012\right)$]. This figure is based on the public data provided by the TSE about the monthly/yearly transaction-volume statistics. These data show the increase in transaction volumes, particularly in the second section and Mothers markets. (b) Typical daily transaction numbers each year for the stock markets used for this study. As described later, we studied only stocks whose yearly transaction number is over 0.5 million. We calculated each stock's yearly average transaction number and then took its simple average across stocks for this plot.

Figure 3

(a) Schematic example of the fundamental quantities $\mathit{\Gamma }:=({\mathrm{\Omega }}_{\mathrm{TR}},{\left\{\epsilon \left(t\right)\right\}}_t,{\left\{{\epsilon }^{\left(i\right)}\left(t\right)\right\}}_{t,i})$ for the case $N_{\mathrm{TR}}=\left|{\mathrm{\Omega }}_{\mathrm{TR}}\right|=2$. Here, $+$ ($-$) is an abbreviation of $+1$ ($-1$), representing a buy (sell) order. (b) Reduced-sign sequences ${\left\{{\epsilon }_k^{\left(i\right)}\right\}}_k$ are defined by removing zeros from the original order sequences ${\left\{{\epsilon }^{\left(i\right)}\left(t\right)\right\}}_t$ for $i\in {\mathrm{\Omega }}_{\mathrm{TR}}$. Runs ${\left\{L_k^{\left(i\right)}\right\}}_k$ are also defined as the numbers of successively the same signs for the trader $i$.

Figure 4

Schematic of the Lillo-Mike-Farmer (LMF) model proposed in Ref. [19]. At the microscopic dynamics, the total number of traders is $N_{\mathrm{TR}}$ and all the traders are assumed to be splitting traders (STs). STs hold large metaorders and they randomly split them into small orders. Here we assume that the run length $L$ obeys the power-law distribution $P\left(L\right)\propto L^{-\alpha -1}$ with $\alpha >1$. At the macroscopic dynamics, the order-sign sequence of the whole market exhibits the long-range correlation $C\left(\tau \right)\propto {\tau }^{-\gamma }$. Furthermore, the LMF model theoretically predicts $\gamma =\alpha -1$ as Eq. (7), connecting the macroscopic power-law exponent $\gamma$ and the microscopic exponent $\alpha$. The state variables and model parameters are summarized in Table 2.

Figure 5

Verification of the LMF prediction (3) in the previous work [19]. We extracted the data from figure 7 of Ref. [19] by measuring the coordinates of the data points using Adobe Illustrator. Here we additionally plot the regression line (led) with the slope 0.07 (statistically insignificant), far from the theoretical coefficient 1. We note that they measured $\gamma$ based on the DFA (see the related discussion in Sec. 5).

Figure 6

Summary statistics of STs. (a) Empirical distribution of the percentage of STs in each market. Approximately 25% (10–50 %) of traders are classified as STs in each market. (b) Empirical distribution of the market-order contribution percentage by STs. 80% of the total market orders are typically submitted by STs. These two figures suggest that STs dominantly contribute to the market orders, while their number is relatively lower than that of RTs. (c) Yearly plot of average daily transaction numbers by STs, showing the growing presence of STs.

Figure 7

Characters of metaorder-length distribution on STs in each markets. (a) The aggregated metaorder-length CCDF for all STs for Toyota Motor Corporation in 2020 as a typical example. The metaorder-length CCDF obeys the power law such that $P_{>}\left(L\right)\propto L^{-\alpha }$. (b) The empirical PDF of the power-law exponents $\alpha$ in our whole data set. The exponent $\alpha$ was measured by Clauset's algorithm [28, 29] across all the markets. Typically, $\alpha$ distributes within $1<\alpha <2$, consistently with the standard assumption for the LMF model.

Figure 8

Numerical simulations of the LMF model to test consistency and biasedness of the NLLS estimator ${\gamma }_{\mathrm{NLLS}}^{\left(a\right)}$ based on the ACF method. (a) Consistency of the ACF-NLLS estimator ${\gamma }_{\mathrm{NLLS}}^{\left(a\right)}$. The theoretical formula ${\gamma }_{\mathrm{NLLS}}^{\left(a\right)}=\alpha -1$ holds for a sufficiently large sample size $N_{\epsilon }={10}^8$, supporting the consistency of the NLLS estimator ${\gamma }_{\mathrm{NLLS}}^{\left(a\right)}$. The parameter sets $(N_{\mathrm{ST}},\alpha )$ are based on the measurement of our data set. (b) For realistic sample sizes $N_{\epsilon }\lesssim {10}^7$, the ACF-NLLS estimator ${\gamma }_{\mathrm{NLLS}}^{\left(a\right)}$ exhibits biasedness due to the finite sample size. We have simulated the LMF model by assuming that the parameter sets $(N_{\mathrm{ST}},N_{\epsilon },\alpha )$ are identical to those measured in our data set. We find that the theoretical formula ${\gamma }_{\mathrm{NLLS}}^{\left(a\right)}=\alpha -1$ does not hold due to the finite sample size. The deviation from the theoretical line is particularly serious for large $\alpha -1>1$, showing the nonuniform convergence of the NLLS estimator. Based on this empirical finding, we focus on the range $\alpha -1\in (0,1)$ and apply the linear regression (23) to obtain the navy line. (c) Check of the numerically constructed unbiased ACF estimator ${\gamma }_{\mathrm{unbiased}}^{\left(a\right)}$. The unbiased estimator is constructed by ${\gamma }_{\mathrm{unbiased}}^{\left(a\right)}:=({\gamma }_{\mathrm{NLLS}}^{\left(a\right)}-{\beta }_2^{\left(a\right)})/{\beta }_1^{\left(a\right)}$. As expected, the theoretical formula ${\gamma }_{\mathrm{unbiased}}^{\left(a\right)}\approx \alpha -1$ holds even for the realistic sample sizes. (d), (e) Parameter distribution of ${\beta }_1^{\left(a\right)}$ and ${\beta }_2^{\left(a\right)}$, the coefficients of the regression formula (23) in constructing the unbiased estimator ${\gamma }_{\mathrm{unbiased}}^{\left(a\right)}$. The means of ${\beta }_1^{\left(a\right)}$ and ${\beta }_2^{\left(a\right)}$ were given by 0.592 and 0.147, respectively.

Figure 9

Numerical tests for consistency and biasedness of the NLLS estimator ${\gamma }_{\mathrm{NLLS}}^{\left(s\right)}$ based on the PSD method. The test was based on the LMF model. (a) Consistency of ${\gamma }_{\mathrm{NLLS}}^{\left(s\right)}$ for a sufficiently large sample size $N_{\epsilon }={10}^8$. The parameter sets $(N_{\mathrm{ST}},\alpha )$ are based on the measurement of our data set. (b) For realistic sample sizes $N_{\epsilon }\lesssim {10}^7$, the PSD-NLLS estimator ${\gamma }_{\mathrm{NLLS}}^{\left(s\right)}$ exhibits biasedness due to the finite sample size. The LMF model was simulated by assuming that the parameter sets $(N_{\mathrm{ST}},N_{\epsilon },\alpha )$ are identical to those measured in our data set. We applied the linear regression (23) for the range $\alpha -1\in (0,1)$ to obtain the navy line. (c) Confirmation of the numerically constructed unbiased PSD estimator ${\gamma }_{\mathrm{unbiased}}^{\left(s\right)}$. The unbiased estimator is constructed by ${\gamma }_{\mathrm{unbiased}}^{\left(s\right)}:=({\gamma }_{\mathrm{NLLS}}^{\left(s\right)}-{\beta }_2^{\left(s\right)})/{\beta }_1^{\left(s\right)}$. As expected, ${\gamma }_{\mathrm{unbiased}}^{\left(s\right)}\approx \alpha -1$ holds even for the realistic sample sizes. (d), (e) Parameter distribution of ${\beta }_1^{\left(s\right)}$ and ${\beta }_2^{\left(s\right)}$, the coefficients of the regression formula (23) in constructing the unbiased estimator ${\gamma }_{\mathrm{unbiased}}^{\left(s\right)}$. The means of ${\beta }_1^{\left(s\right)}$ and ${\beta }_2^{\left(s\right)}$ were given by 0.753 and 0.083, respectively.

Figure 10

Inconsistency of the DFA estimator ${\gamma }_{\mathrm{DFA}}$ even for the LMF model. We used the same parameters $(N_{\mathrm{ST}},\alpha )$ as those measured in our data set and set $N_{\epsilon }={10}^8$. The theoretical relationship ${\gamma }_{\mathrm{DFA}}=\alpha -1$ does not hold even for large sample size, rejecting the numerical consistency of the DFA estimator ${\gamma }_{\mathrm{DFA}}$ at least under a realistic computational resource.

Figure 11

Direct verification of the LMF prediction (3) based on the ACF method (a)–(c) and the PSD method (d)–(f). (a), (d) Scatterplot between $\alpha$ and ${\gamma }_{\mathrm{unbiased}}$. We set bins along the $\alpha$ axis, and the red line signifies the averages within the bins. (b), (e) Box plot between $\alpha$ and ${\gamma }_{\mathrm{unbiased}}$, where statistical quantities [e.g., the first, second (median), and third quartiles] are calculated within the bins along the $\alpha$ axis. (c), (f) Empirical PDF of the errors $\eta :=\alpha -1-{\gamma }_{\mathrm{unbiased}}$. The average error is very small such that $\langle {\eta }^{\left(a\right)}\rangle =0.03$ for the ACF method and $\langle {\eta }^{\left(s\right)}\rangle =0.003$ for the PSD method, respectively.

Figure 12

Estimation of the total number of STs via the LMF theory for the ACF (a), (b) and PSD (c), (d) methods. (a), (c) Scatterplots between the unbiased LMF estimators ${\left[{log}_{10}{\left(N_{\mathrm{ST}}^{\mathrm{LMF}}\right)}^{1-{\gamma }_{\mathrm{NLLS}}}\right]}_{\mathrm{unbiased}}$ and the actual values of ${log}_{10}{\left(N_{\mathrm{ST}}\right)}^{1-{\gamma }_{\mathrm{NLLS}}}$. (b), (d) Corresponding boxplots. We find that the LMF estimator is strongly correlated with the actual $N_{\mathrm{ST}}$. However, the LMF estimator systematically underestimated the actual $N_{\mathrm{ST}}$, which is consistent with the theoretical inequality (32) for a generalized LMF model in [27].

Figure 13

Temporal market-order activity of Toyota 2020. The yearly average of the temporal market-order ratio $r_{\mathrm{MO}}$ is plotted to show the U-shape profile in its intensity: i.e., the transactions are active near the opening and closing times of the continuous double auction $t=0$, 150, and 300. Note that $r_{\mathrm{MO}}\left(t\right)$ is defined by $r_{\mathrm{MO}}\left(t\right):=N_{\mathrm{MO}}\left(t\right)/N_{\mathrm{MO}}^{\mathrm{tot}}$ with the daily-total number of market orders $N_{\mathrm{MO}}^{\mathrm{tot}}$ and minutely total number of market orders $N_{\mathrm{MO}}\left(t\right)$.

Figure 14

Characters of metaorder-length distribution on RTs. (a) The metaorder-length (run-length) CCDF of all the RTs for Toyota Motor Corporation in 2020. The run-length CCDF obeys the exponential law at least for the body part $L\lesssim 10$, such that $P_{>}\left(L\right)\sim 2^{1-L}$, as theoretically expected. At the same time, we observed a theoretical discrepancy for the tail because the second-kind statistical error was not controlled in our binomial test; a small portion of STs might be included in the RT cluster. (b) The empirical PDF $P\left(L^{*}\right)$ of the decay length $L^{*}$ in our data set, by assuming the exponential metaorder-length CCDF $P_{>}\left(L\right)\approx {\left(L^{*}\right)}^{1-L}$ for the RTs for each data point. The decay length $L^{*}$ is measured by the maximum likelihood estimation (i.e., $L^{*}=\langle L^{\mathrm{RT}}\rangle$). The PDF has a sharp peak around $L^{*}\approx 1.8$, consistent with the theoretical prediction (D1). (c) The aggregated metaorder CCDF only for the active RTs (submitting more than 1000 orders annually), showing the exponential law without fat tails for Toyota 2020. This is an empirically successful symptom of reducing the second kind's error.

Figure 15

Schematic figure of the ACF fitting based on the NLLS estimation for Toyota 2020. We used the orange area $\tau \in [{\tau }_{\mathrm{th}}^{-},{\tau }_{\mathrm{th}}^{+}]$ for the fitting to obtain the power-law guideline with exponent ${\gamma }_{\mathrm{NLLS}}$ (solid black).

Figure 16

Schematic figure of the PSD fitting based on the NLLS estimation for Toyota 2020. We used the orange area $\tau \in [{\omega }_{\mathrm{th}}^{-},{\omega }_{\mathrm{th}}^{+}]$ for the fitting to obtain the power-law guideline with exponent $H_{\mathrm{NLLS}}$ (solid black).

Figure 17

Scatterplot between $\alpha$ and the NLLS estimator ${\gamma }_{\mathrm{NLLS}}$ for our data set based on the ACF method [upper panels, (a), (b)] and the PSD method [lower panels, (c), (d)]. (a), (c) The full scatterplot between $\alpha$ and ${\gamma }_{\mathrm{NLLS}}$. This figure illustrates that the NLLS estimator is actually biased, particularly for $\alpha >2$. (b), (d) The boxplot between $\alpha -1\in (0,1)$ and ${\gamma }_{\mathrm{NLLS}}$. This boxplot shows an approximate linear relation but shows the systematic deviation, as expected by the LMF simulations.

Figure 18

Robustness check of our statistical analysis. Our nine-year data set was split into three data sets from 2012 to 2014, from 2015 to 2017, and from 2018 to 2020. The unbiased estimator ${\gamma }_{\mathrm{unbiased}}$ agrees with the theoretical line $\gamma =\alpha -1$ for the three periods. The left (right) panels are based on the ACF (PSD) methods.

Figure 19

Numerical study of the NLLS estimator ${\left[{log}_{10}{\left(N_{\mathrm{ST}}^{\mathrm{LMF}}\right)}^{1-\gamma }\right]}_{\mathrm{NLLS}}$ based on the ACF (a)–(e) and PSD (f)–(j) methods for the LMF simulations. The superscript $X^{\left(a\right)}$ ($X^{\left(s\right)}$) signifies that the estimator is based on the ACF (PSD) method. (a) Consistency check of the ACF-NLLS estimator for $N_{\epsilon }={10}^8$. The parameters $(N_{\mathrm{ST}},\alpha )$ are the same as in our data set. (b) The ACF-NLLS estimator is biased for $N_{\epsilon }\lesssim {10}^7$. The parameters $(N_{\mathrm{ST}},\alpha ,N_{\epsilon })$ are the same as in our data set. (c) ACF-based unbiased estimator ${\left[{log}_{10}{\left(N_{\mathrm{ST}}^{\mathrm{LMF}}\right)}^{1-\gamma }\right]}_{\mathrm{unbiased}}$ approximately shows the unbiasedness as expected. (d), (e) Histogram of the regression coefficients $({\beta }_3,{\beta }_4)$ in Eq. (H2) for the ACF method. (f)–(j) Corresponding figures for the PSD method.