2.1 Short review of our analysis

To identify the trading strategies, we focused on the response pattern of limit orders and market orders to historical trends. We first introduced the coarse-grained tick intervals to calculate market price changes and the time lags to measure trends. To find the optimal parameters for both the tick intervals and the time lags, we then employed the multi-linear regression analysis for limit orders (see 2.2) and the multi-logistic analysis for market orders (see 2.4), respectively. In the following subsections, we described the detailed methodology of the identification of the optimal parameters.

2.2 Limit-order analysis

We first quantified the timescale of trend-following behavior of each trader by studying the correlation between historical price trends and future limit-order price changes by traders. Let us look at the two sample trajectories of limit orders issued by two different traders, which illustrate the variety of the limit-order response speed to the change of transaction prices (Fig 2(A)). To quantify such heterogeneity in response timescales, we introduced a coarse-grained tick interval to calculate the market price changes and the maximum time lag up to which a trader refers in his/her memory in Fig 2(B). For example, let us compare the interpretation based on the blue and red lines in Fig 2(B). The blue line is based on 3 maximum time lag with 1 tick coarse-graining and indicates downward trends. On the other hand, the red line is based on 3 maximum time lag with 4 tick coarse-graining and indicates upward trends though the given transaction time series is the same. It is therefore necessary to determine (i) the timescale for coarse-graining and (ii) the maximum time lag for each trader.

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Fig 2. Determining timescales of the response to trends, and validation test results.

A, Sample trajectories of limit orders issued by two FTs during six minutes: the lifetimes of ask and bid orders (red and blue lines, respectively), and a trajectory of transaction prices (black line). B, Schematic of the difference interpretation of trends from a single price trajectory. If a trader sees short-term price changes, the prices are in a down-trend (the blue curved arrow), whereas if a trader sees longer-term price changes, prices are in an up-trend (the red curved arrow). Different timescales lead to the different interpretations to historical trends. C, Relationship between the limit-order price change and trends. The historical periods over which to take an average are 1 tick (orange), 3 ticks (light-blue), and 8 ticks (violet). The hyperbolic tangent relation between them, empirically shown in early works [10, 11] focusing on the last single tick price change, also establishes price changes over several ticks. D, Three sample-normalized weights of regressors obtained by Eq (2) (left side) and the weights of 161 FTs after scaling (right side). As the three sample weights are well-approximated by exponential functions, we scale all FTs’ weights by scaling function d_i exp(−k/τ_i). The right-hand-side graph shows about 85% of target traders (161 of 191 traders) determine trends by the exponential moving average. The inset plots the scaled weights on a log scale. Although there are deviations around the distribution tail, the overall trend is well captured by the exponential function.

https://doi.org/10.1371/journal.pone.0208332.g002

In this paper, we determined such strategy parameters by maximizing the correlation between the historical market price changes and the future limit-order price changes of the trader. The historical price change calculated according to j_i tick coarse-graining and k-th time lag is (1) and the limit-order price change for i-th trader is given by Δz_i(t) ≡ z_i(t + 1) − z_i(t). Here is the mid-price of the best ask price and the best bid price . P(t) is the transaction price at time t. When there is no bid (ask) quote, () is substituted by the last bid (ask) quote price, and extreme limit-order price changes more than 100 tpip are excluded from the following analysis. We showed the average changes of Δz_i(t) conditional on the historical price changes in Fig 2(C). In this figure, we employed Δz_i(t) of the 6th trader (orange), the 65th trader (light-blue), and the 180th trader (violet), respectively. Correspondingly, is calculated on the basis of the 1 tick coarse-graining and 5 maximum time lags (orange), the 3 tick coarse-graining and 10 maximum time lags (light-blue), and the 8 tick coarse-graining and 8 maximum time lags (violet). We found that these three examples can be well-approximated by the hyperbolic tangent curves denoted by the black line. It is worth noting that this relationship is a straightforward generalization of the formula found in Refs. [10, 11]. In their paper, they found the special case of this relationship by fixing j_i = 1 such that where are constants.

Given this result, the general relationship between Δz_i(t) and can be formulated by (2) (3) where K_i is the number of time lags, w_i(k) is the coefficient of regressors at time k, ϵ_i(t) is the white noise, and α_i, σ are constants.

On the basis of this relation, we retroactively incremented the number of time lags under multiple time-coarse-graining and optimize the parameter set to maximize the correlation between the historical market price change and the future quoted price of a trader. Parameters α_i and w_i(k) are estimated through the multi-linear regression analysis, and the trend-following strength c_i and the maximum time lag K_i are determined by another process explained in 2.3. The multi-linear regression analysis was performed only when Δz_i(t) ≠ 0. We reduced this non-linear equation to a linear equation using the inverse function of hyperbolic tangent and then performed a multi-linear regression analysis.

We next found that coefficients w_i(k) approximately decays exponentially, whereby the characteristic timescale of trend-following can be defined by the decay timescale in Fig 2(D). After determining the maximum time lag K_i and the time-coarse-graining such that the adjusted coefficient of determination () takes a maximum, we show three examples of coefficients of the regressors with the approximate exponential functions (Fig 2(D)). is the coefficient of determination, and N_i is the number of samples. We found the typical feature of the coefficients for the three traders (the 6th, the 65th, and the 180th) is their exponential form, (4) with various time constants τ_i and heights d_i, at least for the body part of the coefficient. This finding is established not only for these three traders, but also for 161 traders as illustrated by the scaling w_i/d_i (Fig 2(D)), where the body of the distributions approximately collapse onto the exponential master curve despite several deviations especially around its tail. We note that we could not identify the function form for the tail in the absence of sufficient number of data points. Indeed, the typical maximum time lag is five and is not sufficient to conclude whether the true tail obeys other functions such as a power-law tail or not. Fortunately, however, the body part of the weight function is the most important to measure trends and thus we employed the exponential fitting function for simplicity in this paper.

This result shows the direct evidence that the EMA is a typical metrics to measure market price trends [12]. We excluded from these plots data of traders for which the sum of the squared errors (SSE) of the prediction normalized by the d_i exceeds the 0.1 as the fitting threshold. Eleven traders were excluded from the EMA trend-followers and classified among the non-EMA trend-following cluster. We define the reference time for the ith trader as a product of the optimal tick interval j_i with the estimated time constant τ_i. If there is only one data point satisfying the significant level for the correlation, we skip the fitting procedure and assume τ_i = 1.

2.3 Determination of the trend-following strength c_i and the maximum time lag K_i in Eqs (2) and (3)

We explain the way to determine both c_i and K_i introduced in Eqs (2) and (3). We performed the following iteration method with a given coarse-graining time interval j ranging from 1 to 20 ticks.

1. P1. Set c_i = 5 and as initial values. Here is a candidate of K_i.
2. P2. Calculate .
3. P3. Perform the multi-linear regression analysis for the timeseries of as a regressand, and those of as regressors.
4. P4. If the p-value of the coefficient obtained above is lower than the threshold, we consider the obtained coefficient as statistically significant. We then increment and iterate this process until . See Ref. [13] for the calculation of the p-value. In this analysis, we employ 0.001 for the threshold of the p-value, which is also employed in Ref. [14].
5. P5. Once the last p-value of the coefficient is higher than the threshold, we stop this iteration process and set .
6. P6. Calculate the relationship between the average limit-order price changes and trends on the basis of the obtained w_i and K_i, and then calculate a new c_i by fitting the hyperbolic tangent function as well as .
7. P7. When the relative difference between the current and previous c_i is larger than 1%, we repeat back to the process P2. Otherwise, the iteration process is terminated for the given j. We repeat this fitting of c_i 100 times for convergence. Note that, in this study, three traders were classified to the non EMA trend-following cluster since their c_i did not converge for all coarse-graining time intervals.

2.4 Market-order analysis

We describe how to determine the time interval referred to by traders in making a decision to issue market orders. Sample data points for which market orders are issued by two real traders were plotted (Fig 3(A)), and traders also seemingly have different responses to price trends due to the similar reason explained in limit-order analysis. Note that traders are allowed to attach the acceptable transaction price to market orders. If the current best price is worse than that price, a market order fails. To analyze how traders respond to trends, we used logistic regression in the parallel method to analyze limit orders.

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Fig 3. Determining timescales of the response to trends, and validation test results.

A, Sample trajectories of market orders by two FTs during six minutes. Upward (downward) triangles are sell (buy) orders. The colour of triangles represents the status of market orders: orange (gray) signifying filled (missed). The black line is the trajectory of transaction prices. B, Weights w_i in Eq (3) of 131 traders obtained from a logistic analysis Eq (5) are plotted at j_ik. To obtain the optimal w_i, we follow the same procedure with the limit-order analysis except for using the SSE, not . C, The horizontal and vertical axis mean the historical trends and the probabilities controlling the direction of market orders, respectively. The black is the standard logistic function. On top of that, we depict the magnitude of the historical trends for two traders as cross-marks, which is obtained by the Eq (5). The market orders issued to the buy (sell) side are depicted by the cross-marks at 1 (0).

https://doi.org/10.1371/journal.pone.0208332.g003

To determine the time-coarse-graining, we use a similar analytic method to the aforementioned limit order analysis, based on the multi-logistic regression instead of the multi-linear regression: (5) where q_i(t + 1) and 1 − q_i(t + 1) show the probabilities for buying orders and for selling orders when issuing market orders, and β_i is a constant parameter; the notation for other parameters is the same as in the limit-order analysis. We set the threshold of the p-value at 0.05 for this market order analysis, which is smaller than that for the limit order analysis since the number of limit orders is approximately ten times larger than that of market orders (Fig 1(B)). Note that despite the weaker threshold employed in this section, this criteria is generally accepted in the field of statistics [13].

After determining both the time-coarse-graining and the maximum time lag for each trader, we plotted the coefficients obtained by the multi-logistic regression for 131 traders (Fig 3(B)). Most of the coefficients are positive, but a few are negative. We classify their strategies based on the sign of .

We next show the fitting result based on our logistic regression method. The horizontal and vertical axis of Fig 3(C) respectively indicate the historical trends and the probabilities controlling the direction of market orders (i.e. buy or sell). The black line in this figure is the standard logistic function. In addition, we marked the magnitude of historical trends as cross-marks for two traders when market orders are issued, which are calculated according to Eq (5). The market orders issued to the buy (sell) side are depicted by the cross-marks at 1 (0). Given the vertical axis showing the probabilities controlling the direction of market orders, the top (bottom) graph shows a trader weakly (strongly) motivated by historical trends.

3.1 Clustering of limit-order strategies

To understand financial markets as a market ecology, we are interested in the typical differences of limit-order strategies, rather than the detailed differences of them in this paper. We thus cluster the limit-order strategies by the similarity of trend-following timescales, and then track the differences of the limit-order activities back to the differences of their limit-order book shapes, which has been a topic of study of late [10, 11, 15–19].

Fig 4(A) shows the distribution of the reference times. Using the k-means method, we classified the reference times into three clusters: the short-time (typically 4 ticks; 30 sec), intermediate-time (typically 20 ticks; 2.5 min), and long-time (typically 40 ticks; 5 min) clusters. To determine the cluster size, we employ the silhouette method [20] and compared clusters ranging from size 2 to 5. We conclude that three clusters form an optimal size in terms of both the silhouette coefficient and the thickness of clusters. Note that all FTs were classified as either EMA trend-followers or non-EMA trend-followers.

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Fig 4. Analysis of limit-order strategies.

A, Distribution of the trend-following reference time of 161 FTs. There are three typical clusters ranging from 1 tick to 10 ticks (short-time cluster, marked in orange), from 11 ticks to 23 ticks (intermediate-time cluster, in light-blue), and from 24 ticks to 50 ticks (long-time cluster, in violet), all which are obtained using the k-means method. They typically correspond to half, three, and six minutes given the average transaction interval is 9 seconds in this week. Two samples around 60 ticks were excluded as exceptions. B, The average number of limit orders (red) and transactions as limit orders (blue) for each cluster. The gradations in the plot bars presents a heat map of the ascending number of limit orders and that of transactions as limit orders by a trader in each cluster. The short-time (long-time) trend-followers submit the most (least) frequently, whereas the number of transactions for intermediate-time trend-followers is least despite a relatively large number of submissions. C, Probability density functions of the limit-order distributions (PDFs) conditional on the limit-order strategies. The peak of PDF of intermediate-time trend-followers lies far behind the best prices compared with other trend-followers, which reduce the transaction frequencies of intermediate-time trend-followers. D, Time-series of the ratio for the number of limit orders in the order book issued by each cluster. Each bar represents the hourly average ratio. JST, LST, and EST signify Japan, London, and Eastern Standard Times, respectively. The clock in the figure starts from 9:00 am to 6:00 pm for each standard time. Dark-gray bars represent the fraction of limit orders issued by LFTs.

https://doi.org/10.1371/journal.pone.0208332.g004

What does this timescale difference imply? To answer this question, we studied the average number of limit-order submissions and that of transactions as limit orders for each cluster (Fig 4(B)). Although the number of submissions has a trivial correlation in that short-time (long-time) trend-followers submit the most (least) frequently, the number of transactions has a nontrivial correlation; the number of transactions for intermediate-time trend-followers is least despite a relatively large number of submissions. To investigate this nontrivial correlation, we studied the limit order book shape for each cluster, representing the typical depth of order placements (Fig 4(C)). These order-book profiles provide clear answers to the nontrivial behaviour. The short-time and long-time trend-followers maintain their orders near the best prices, leading to frequent transactions. The non-EMA trend-followers also transact frequently because they leave their orders without price modifications. However, the intermediate-time trend-followers maintain their orders relatively far from the best prices compared with other trend-followers and therefore are less likely to transact.

We remark on the intraday pattern of limit-order strategies. Fig 4(D) is the hourly limit-order component ratio in the order book. In Tokyo, trend-following of short duration is the dominant strategy during the daytime, whereas in New York it is of intermediate duration. Given the order-book shape in Fig 4(C), Tokyo (New York) traders are bullish (bearish) on transactions at current best prices in the daytime.

3.2 Clustering of market-order strategies

We report the detail properties of market-order strategies. Fig 5(A) is the distribution of market-order strategies of FTs, which is quantified by : positive (negative) implies that the ith trader is a trend-follower (contrarian), who issues buy orders during positive (negative) trends, and sell orders during negative (positive) trends. In our market-order analysis, we found several FTs were contrarians but most were trend-followers. Note that traders showing no significant correlation with trends were classified within the random cluster.

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Fig 5. Analysis of market-order strategies.

A, Distribution of quantifying the average strength of trend-following for market orders. The original samples w_i are shown in Fig 4. A positive (negative) denotes a ith trader is a trend-follower (contrarian), and represented by a green (pink) plot bar. B, Average number of market orders (red) and that of transactions as market orders (blue) for each cluster. The gradation in plot bars presents a heat map of the ascending number of market orders and that of transactions as market orders by a trader in each cluster. Contrarians are active despite their small size. C,D, Failure probabilities of market orders in transactions (C) and the probabilities in which market orders are issued at prices better than the current best prices (D). The green (gray) bars and circles represent the strategic properties of trend-followers (random traders). The median failure probability of trend-followers (random traders) is 83% (24%), and the probability of trend-followers issuing market orders at better prices than the current best price is 70% (15%). Trend-follower may be attempting to obtain better prices than current best prices by submitting market orders in advance. E, Time-series of the ratio for the number of market orders issued by each cluster. Each bar represents the hourly average ratio. The dark-gray bars signify the fraction of market orders issued by LFTs.

https://doi.org/10.1371/journal.pone.0208332.g005

To extract features of the strategies, we studied the number of market orders and that of transactions as market orders for each cluster (Fig 5(B)). We found that the contrarians are overwhelmingly active despite their small size. Indeed, the first and second most frequent traders were contrarians in our dataset. Notably, a previous study [14] reports the existence of contrarians at the trader group level.

Another feature is the difference in the degree of contributions to transactions. Given the large number of market orders trend-followers issue, the transaction count is relatively small compared with that for random traders. To clarify this imbalance, we defined the failure probability as the fraction of failed market orders to total market orders (see S1 Appendix). Fig 5(C) shows that the typical failure probability of the trend-followers (84%) is almost four times higher than that of the random traders (23%). Why did trend-followers submit such “meaningless” market orders? One of our conjectures is that trend-followers may aim the latency during price-matching processes (we have another conjecture based on pinging strategies [14, 21–24] which is illustrated in S2 Appendix). Given this latency, a good strategy may be to hit in advance better prices than the current best prices following their trend prediction. Indeed, as illustrated in Fig 4(D), most of the market orders by trend-followers were issued at better prices (70%) than the current best prices compared with random traders (15%), a practice commonly observed during the week (Fig 4(E)). Note that the individual Tokyo traders in the daytime behave as contrarians, which is consistent with a previous work [25] indicating that contrarian behaviour is the favoured and profitable strategy in Japan.

3.3 Strategy-matrix analysis

Finally, as a strategy-analysis summary including LFTs, Fig 6(A) shows an ecological matrix (4 × 5 elements with two empty elements) illustrating the number of traders, submissions, and transactions for the combination strategies of both order types. This figure shows the following two characteristics: one is the immense contribution to submissions and transactions by the short-time trend-followers for limit orders and the trend-followers for market orders (i.e., the clusters surrounded by the chain line). They occupy 44% (40%) of submissions as limit (market) orders and 28% (35%) of transactions as limit (market) orders, though their population is relatively small (80 traders, 7.8% of all population). The other characteristic is the tendency that there are many traders who submit mainly either limit or market orders (i.e., the cluster surrounded by the dot line; 189 traders, 19% of all population). This characteristic implies that they might be specialized in either limit or market order strategy.

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Fig 6. Illustrating the overall strategic properties as an ecological matrix.

A, The frequencies of order submissions and that of transactions issued by traders employing strategies using various combinations of limit and market orders. The box size represents the number of traders. The blank elements indicates the absence of traders adopting corresponding combination strategies. We confirm (i) the immense contributions to submissions and transactions by the short-time cluster for limit order strategies and the trend-follower cluster for market-order strategies surrounded by a chain line, and (ii) a large population of traders tend to mainly submit either limit orders or market orders given the size of boxes surrounded by a dotted line.

https://doi.org/10.1371/journal.pone.0208332.g006

We further investigate the relationship between the limit-order and market-order strategies with (i) trader’s role for liquidity and (ii) their trading performances in Fig 7(A). We show pie charts quantifying the overall balance between liquidity providers and consumers. Each component is highlighted to illustrate trading performances as measured by the Sharpe ratio (see S3 Appendix). As one may notice, there exists the strong correlation between the Sharpe ratios and liquidity consumption probabilities (0.54 as measured by the Spearman rank correlation (Fig 7(B))). This correlation suggests the traders consuming (providing) the liquidity are likely to exhibit good (bad) trading performances as they take on risk for (not for) their sake. This result is consistent with the analysis concerning the inventory risk for liquidity providers to the decline in asset prices [26].

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Fig 7. Illustrating the trading performances as an ecological matrix.

A The pie charts quantify the overall balance between the liquidity provision and consumption of the cluster. Here the liquidity provision (consumption) is measured as the total volumes transacted as makers (takers). Each cluster is classified as either a liquidity provider or consumer through a statistical test on the significance of the imbalance between liquidity provision (P) and consumption (C). N/A signifies that the imbalance is not statistically significant. In addition, clusters are colour coded (red, yellow, or blue) to mark their trading performances as measured by the Sharpe ratio. The breakdown of trading performances and trading profits of clusters the high performance clusters coded by a brown and green line are further investigated in Fig 8. B, Scattering plot between the liquidity consumption ratios and the Sharpe ratios with Spearman’s rank correlation coefficient. This positive correlation implies that more frequently traders transact as takers, better performances traders are likely to exhibit.

https://doi.org/10.1371/journal.pone.0208332.g007

It would be interesting to explore why the two opposite types of clusters surrounded by a brown line (typically high frequency traders (HFTs)) and a green line (LFTs) in Fig 7 exhibit high trading performances. We therefore provide the breakdown properties of these two clusters as a case study. After aggregating traders at the bank level, we plotted the distributions of trading profits calculated every 20 minutes, the total trading profits in this week, and the Sharpe ratios (Fig 8(A), 8(B) and 8(C), respectively). From these figures, the maximum profit (trading efficiency) of HFTs is smaller (larger) than that of LFTs, indicating HFTs (LFTs) make small (large) profits using small (large) inventory at stake. Given the previous study highlighting that HFTs are highly profitable by taking advantage of response speed, this result indicates counterintuitively that strategies of HFTs and LFTs seem equilibrium-balanced by optimization according to different metrics at least in our dataset.

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Fig 8. Illustrating the profit distributions and the trading performances of HFTs and LFTs.

A,B,C Trading profits and trading performances of banks, the traders in the high-performance cluster of which are aggregated. We exclude from the aggregation traders with transaction counts below 100 in the week. Specifically, (A)–(C) plots the trading profit distributions per trader calculated every 20 minutes during a week, the cumulative profit distributions at the end of the week, and the distribution of the trading performances measured by the Sharpe ratio.

https://doi.org/10.1371/journal.pone.0208332.g008