3.1 Historical average position

Fig 2(a) shows the position trajectories for two different banks. The vertical axis denotes the average trader position in these banks. A positive (negative) number represents their long (short) positions. The second top bank accumulates a significant position, while the fourth top bank maintains its position very close to 0.

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Fig 2. Classification of position management strategies.

(a) Average trader positions for the second top (orange) and fourth top (green) banks plotted every 10 minutes. See Fig 1(b) for the definition of the rankings. A positive (negative) number represents a long (short) position in USD/JPY. (b) Schematic of the definition of the historical average position. The historical average position is calculated using the ensemble average of positions observed during past κ transactions undertaken by the considered bank (κ = 3 in the example). Assume that the i-th bank transacted at times t₁, t₂, and t₆, whereas the j-th bank transacted at times t₃, t₄, and t₅. The historical average position of the i-th bank is the average of the positions at t₁, t₂, and t₆, whereas that of the j-th bank is the average of the positions at t₃, t₄, and t₅.

https://doi.org/10.1371/journal.pone.0220645.g002

To characterize the position management strategies, we introduce the historical average position. Over a decade ago, Mizuno et al. [16] found that the diffusion speed of market prices was well-characterized by focusing on the price differences between current and historical average prices. Since the trajectories of these two banks appear to show different diffusion speeds, we also introduce the historical average position to characterize their position management strategies in terms of how banks react to position fluctuations from historical levels. Interpretations on the meaning of the historical average position and the gap between a position and the historical average are presented in the conclusion and discussion section. Note that we calculate the aggregate net position using traders sharing the same bank IDs, and regard this as the bank position for the purpose of obtaining an adequate sample size.

The κ-tick historical average position is calculated as follows. We define a position during this week as (1) where superscript i indicates the i-th bank, Nⁱ(t) is the number of transactions until time t, is the k-th transaction time during the week, and vⁱ(t) is the transaction volume at time t. vⁱ(t) takes a positive (negative) value when a trader has a long (short) contract. As shown in Fig 2(b), the historical average position over κ tick is calculated as (2)

Since a position has to take an integer value, we also round the historical average position to an integer, denoted by . In the following analysis, we set κ = 15. Note that κ values ranging from 10 to 50 provide similar results to those shown in Section 4. However, when κ value becomes too large such as 150, we failed to estimate the position management strategies as shown in S3 Appendix.

It is worth mentioning why we focus on analyzing the response pattern instead of the overall positions of all currency pairs. Ideally, the most accurate way to analyze position management strategies is by tracking the positions of all currencies the banks are holding. However, in reality, it is very difficult to track the positions of all currency pairs across all foreign exchange trading venues (e.x. EBS, Currenex, HotSpot Fxi, FXCM Pro, and FastMatch) to which consistent bank IDs are attached. Since the only dataset we are able to access is the USD/JPY market provided by EBS, we analyze the response pattern to position fluctuations, which can be estimated using data from one currency market. See the conclusion and discussion section for a more detailed explanation of why we expect this method to work.

3.2 Detection of outlier trades

As depicted by the orange line in Fig 2(a), the position trajectory of the second bank includes several sharp falls and rises, which look like outlier trades to the naked eye. To capture these outlier trades statistically, we use the epsilon (ϵ) drawdowns and drawups (EDD) methodology [17–19]. Drawdowns and drawups are traditionally defined as persistent market-price decreases and increases, respectively, over consecutive periods. ϵ is introduced to relax the above definition by allowing small market-price deviations from the current drawdowns or drawups up to a predefined tolerance level ϵ (see [17–19] for details). Therefore, EDD is able to robustly recognize similar local dynamics, and is useful for detecting local similarities in position trajectories (see S4 Appendix for the detailed reason why we employ EDD to detect outlier trades).

First, we calculate the sum of transaction volumes during [t₀, t] (unit: sec): (3) where is the set of transaction times for the i-th bank during [t₀, t]. At each point in time, we need to find the largest position deviation from the previous extremum. (4)

This search process ceases at time t, when the deviation becomes larger than the predefined tolerance ϵ. The time ) determines the end of the drawup (drawdown), and becomes the starting point t₀ for the search for the next drawdown (drawup). The duration of the current drawup or drawdown is defined by . Due to this definition, the EDD recognizes that drawups and drawdowns are alternating. We iterate this process until the end of the week. As an initial condition, we start from a drawdown (drawup) period if where t_S is the beginning of the week.

Since the typical transaction volume differs from bank to bank, it is reasonable to assume that the tolerance ϵ depends on historical fluctuations in the transaction volumes of each bank. Thus we set the tolerance ϵ as ϵ = ϵ₀ σⁱ(t) where ϵ₀ is a constant and is a measure of the volatility of transaction volumes for the i-th bank over the time window w. As discussed in [20], these partitions are dependent on the introduced parameter (ϵ₀, ω). To reduce this dependency, we set ϵ₀ from 0.5 to 5.0 in steps of 0.5 and ω from 60 to 300 minutes in steps of 60 minutes (50 samples in all), and define outlier trades in the next step by considering all of the results obtained by each parameter set. See S5 Appendix for detailed data processing before the application of the EDD method.

Then, we introduce the indicator function to aggregate the end of various drawups and drawdowns. After applying the EDD to the position trajectories, we obtain a set of end dates (unit:sec) and durations (unit:sec) of drawups and drawdowns: (5) (6) (7) (8) where and are the end dates and the duration of the j-th drawup (drawdown), respectively, for the i-th bank obtained by the parameter (ϵ₀, ω). Fig 3(a) shows the example of a series of drawups and drawdowns obtained using parameter sets (ϵ₀, ω) = (3, 120), and (5, 300) for the second top bank. The characteristic of the EDD method is that while the EDD with different parameters principally detects different types of local drawups and drawdowns, it occasionally detects the same end points for the global drawups and drawdowns. See S1 Appendix how the EDD changes the size distributions of transaction volumes.

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Fig 3. Example of the generation process of the outlier signals N^2nd(t) for the second top bank.

(a) The position trajectory of the second top bank from 00:00 to 18:00 on 8 June 2016 is shown by the black points. The top (bottom) translucent red and blue boxes represent the drawups and drawdowns, respectively, obtained by the epsilon drawdown (EDD) method (see text) using parameter (ϵ₀, ω) = (3,120), and (5,300). (b) Aggregation over a set of parameters (ϵ₀, ω) (see text) of the end points of drawups and drawdowns detected by the EDD method for this second top bank, given by Eqs (9), and (11).

https://doi.org/10.1371/journal.pone.0220645.g003

This information is integrated via the indicator function such that: (9) (10) (11) where N_ϵ is the total number of parameters (in this study, N_ϵ = 50), i.e. the cardinal of , where is the set of parameters (ϵ₀, ω), and is the set of analysed banks. The example of this integration process for the second top bank is depicted in Fig 3(b). After the identification of the end dates of drawups and drawdowns obtained by different parameters, we first integrate those end dates over different parameters for a given bank i thus obtaining N_i(t) shown in Fig 3(a), and then integrate them over different banks to create the signal N(t). We depict a slice of the sample trajectory of N(t) on top of the mid-prices from 2:10 to 4:20 on 8 June 2016 (Fig 4(a))). The left and right ordinates represent the market prices and the magnitude of signal N, respectively. A large signal N indicates that several banks concluded their outlier trades at the same time. Seemingly, there is a relationship between mid-price behavior and the magnitude of N, especially when N exhibits strong signals. We analyze this relationship in Section 4.4.

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Fig 4. Statistical properties of signals N(t) recognizing outlier trades, defined by Eq 10.

(a) Example of signals N(t) (green line) on top of the mid-price trajectory (black points). The abscissa represents the time from 2:10 to 4:20 on 8 June 2016. The left and right ordinates represent market prices and the magnitude of signal N, respectively. Positive (negative) signals show the end time of outlier trading for the buy (sell) side, and a large N(t) indicates that several banks concluded their outlier trades simultaneously. (b) Probability distribution function of N. We define extreme outliers as being in the bottom and top 0.05% (total 0.1%), highlighted by the translucent red boxes, and define moderate outliers as being in the top and bottom 2.5%, excluding the extreme outliers (total 4.9%), highlighted by the translucent orange boxes.

https://doi.org/10.1371/journal.pone.0220645.g004

After summing the indicator function over all parameter sets and banks, we obtain a frequency distribution indicating how many times each t is identified as the end of the drawdowns or drawups (Fig 4(b), where N = 0 is excluded). The ends of outlier trades are defined as samples exhibiting a strong signal N. The extreme outliers are defined as the samples in the bottom and top 0.05% (total 0.1%), while the moderate outliers are defined as the samples in the bottom and top 2.5% excluding the extreme outliers (total 4.9%).

Let us now explain the procedure used to identify the beginning of outlier trades. Let us assume that an outlier trade ends at time t. First, we scan for a long trade (a short trade) to find the elements equal to t. Then, we obtain the corresponding duration from and calculate the beginning by subtracting the duration from the end time: , which is the beginning of the j-th drawup (drawdown) for the i-th bank observed for parameter (ϵ₀, ω). Note that there are several beginnings associated with the end of one outlier trade because different parameter sets or different banks can provide different beginnings of drawups and drawdowns. In the following analysis, we regard the outlier trade from beginning to end as one long (short) trade when the signal N at the end of outlier trades has a positive (negative) value even though it includes several opposite type of small contracts.

4.1 Classification of position management strategies

Fig 5(a) shows the probabilities for the second and fourth top banks to have a long contract in their next transaction, conditional on the differences between their current and historical average positions. The red, blue, and grey bars represent positive, negative, and zero differences, respectively, between the current and historical average positions. There is almost an 80% chance that the second top bank will have a long contract in its next transaction when its position leans to the positive, whereas there is less than a 30% chance that the fourth top bank will have a long contract under the same condition. This suggests that position management strategies can be classified into “mean-deviating” and “mean-reverting” strategies.

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Fig 5. Classification into economically-motivated (EM) strategies and arbitrage-motivated (AM) strategies.

(a) Probabilities for the bank to take a long contract in the next time period, conditional on the differences between its current and historical average position (data for the second and fourth top banks). Positive (negative) differences are shown as red (blue) bar plots. When there is no difference, the bar plot is grey. (b) Empirical relationship between the differences in the current and historical average positions and the trading amount vⁱ of the next transaction. We measure the magnitude of the slopes and the range of fitting errors using gnuplot with the Levenberg–Marquardt algorithm [21]. The orange and green lines show the positive or negative slopes, respectively, after taking fitting errors into account. Note that we aggregated samples to ensure the presence of more than 300 samples in each bin. (c,d) Scaling plots of (b). Repulsive (c) and attractive (d) potential functions collapse onto a quadratic master curve. This figure is obtained for κ = 15, which is the number of past transactions over which the averaging over positions is performed (see Eq (2)).

https://doi.org/10.1371/journal.pone.0220645.g005

To verify this statement, Fig 5(b) shows the relationship between the amplitude of the present position unbalance and the next trading volumes vⁱ for the top 30 banks defined in Fig 1(b). We measure the magnitudes of the slopes and the range of fitting error using gnuplot with Levenberg–Marquardt algorithm [21]. See [22] for more information. The orange and green lines represent the existence of two classes of behaviors, characterised respectively by positive and negative slopes, which are clearly identified as such after taking fitting errors into consideration. Three banks were excluded from this figure because their slopes were not significantly different from 0. This figure clearly demonstrates that the relationship suggested in Fig 5(a) for two banks holds also for the top 27 banks; some banks build their positions, whereas other banks maintain their positions close to 0. The banks characterised by a positive slope build up sequentially their positions and we interpret this as an economically-motivated (EM) strategy. The banks characterised by a negative slope reverse their position, without building inventories. We interpret this behavior as an arbitrage-motivated (AM) strategy. We find that 19 (resp. 8) banks follow the EM (resp. AM) strategy and account for 32.3% (resp. 19.8%) of all transactions during the week.

Based on Fig 5(b), the relationship between vⁱ and can be described as potential forces in the following functional form: (12) where bⁱ is constant, and positive or negative bⁱ corresponds to a mean-reverting or mean-deviating strategy, respectively. To verify this statement, we depicted the potential forces estimated by the above relationship and scaled using b_i and κ. Fig 5(c) and 5(d) represent the mean-deviating (c) and mean-reverting (d) strategies characterized by repulsive and attractive potential forces, respectively. To perform a scaling analysis, we used the slopes obtained in Fig 5(b) and shifted the peak of the potentials to the origin in this figure. As can be seen, the potential forces are well-approximated by the quadratic function for the top 27 banks despite the magnitude and signs of the slopes being unique to each bank. This suggests that the position management strategies display similar mathematical characteristics to quadratic potential forces.

We propose to explain the “mean-deviating” strategy (i.e. repulsive potential force) as follows. It is well known that banks split large volumes into small volumes, and subsequently handle these small orders [23]. This strategy further unbalances their positions, which results in the repulsive force. Stop-loss orders are another mechanism that can increase banks’ positions [24]. They are designed to reduce a bank’s potential losses on their positions, and are typically placed in an order book at different prices and volumes and are triggered when market prices reach unfavourable levels relative to their positions. When the stop-loss orders are triggered, the banks will repeatedly trade on the same side to reduce their positions, which also results in a repulsive potential force.

Finally, we study the average limit-order book shapes conditional on the banks following the EM and AM strategies, as well as those of the banks not classified into any of the above strategies. Fig 6 shows the average density of bid and ask limit orders residing at specific distances from the best prices (i.e. depth) given in the abscissa. Banks employing the EM and AM strategies are less likely than other banks to submit limit orders over all depths. This implies that the banks not classified into the above strategies mainly work as market makers (MMs) to provide liquidity to markets. Therefore, we refer to their strategy as a MM strategy.

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Fig 6. Order-book depth distributions for the banks classified according to their used strategy: Economically-motivated (EM), arbitrage-motivated (AM), and market makers (MM) strategies are represented by violet, green, and light-blue lines, respectively.

Depth is the distance from the current best price to the limit-order prices submitted by the banks following each strategy. We use an average for both the bid and ask sides.

https://doi.org/10.1371/journal.pone.0220645.g006

4.2 Prediction of future market prices

In this subsection, we demonstrate that the type of deployed position management strategies and the banks’ positions contain information on future market prices, especially when the overall positions held by the banks following EM or AM strategies become unbalanced.

The left (right) panel of Fig 7 shows the distribution of market mid-price changes fifteen minutes after the observation of the unbalanced position after aggregating over (i) the banks following the EM (AM) strategy and (ii) the past thirty minutes. The red, blue, grey lines, respectively, correspond to the case where the aggregated position is more than γ_EM(AM), less than −γ_EM(AM), and in-between. We can clearly observe skewed distributions, of which the peaks are unique to both the position management strategies and the direction of the position unbalance. The average price changes of the distributions depicted by the red, blue, and grey lines respectively are 4.3 ± 3.6, −18.0 ± 2.8, and 1.3 ± 0.8 tpip for the EM strategy in Fig 7, and −11.5 ± 2.9, 33.0 ± 4.7, and 0.3 ± 0.8 tpip for the AM strategy in Fig 7. The errors represent the standard errors. See S6 Appendix for the robustness of a priori market-price prediction using various parameters (γ_EM, γ_AM) and a different prediction time horizon. The obtained distributions are skewed in the same fashion as shown in Fig 7.

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Fig 7. Conditional distributions of market mid-price changes fifteen minutes after the observation of unbalanced positions in banks following either EM or AM strategies aggregated over the past thirty minutes.

The EM strategy is depicted in the left panel, and the AM strategy is depicted in the right panel. In the prediction analysis, the basic time unit is a minute. When aggregated over the banks following EM or AM strategies respectively is more (less) than the threshold values γ_EM or γ_AM (-γ_EM or -γ_AM), the distributions are depicted by red (blue) lines. When they are in-between, they are depicted by grey lines. The sampling condition used to detect an unbalanced position is (γ_EM, γ_AM) = (8,6). The average price changes in the distributions depicted by red, blue, and grey lines, respectively, are 4.3 ± 3.6, −18.0 ± 2.8, and 1.3 ± 0.8 tpip for the EM strategy, and −11.5 ± 2.9, 33.0 ± 4.7, and 0.3 ± 0.8 tpip for the AM strategy.

https://doi.org/10.1371/journal.pone.0220645.g007

One of the mechanisms generating the skewed distributions is the following. When the aggregated position of the banks following the EM (AM) strategy becomes unbalanced, the banks will simultaneously expand (reduce) their positions by trading in the same (opposite) direction as that of their unbalanced positions. For example, when banks following the EM (AM) strategy overall have long positions, they are likely to simultaneously enter long (short) contracts in their next transaction based on the results in Fig 5(b), 5(c) and 5(d), which drives future market prices upward (downward) and results in skewed distributions, as shown by the red line on the left (right) side of Fig 7. This implies that future market prices are affected by the distribution of the banks’ positions, which is a consequence of historical decision-making processes, and thus such information could be one of the key factors in determining future market prices. See S7 Appendix for an additional test of the future market price prediction using another dataset from 13th to 18th September 2015.

4.3 Average price changes around transactions

In Subsections 4.1 and 4.2, we classified the different position management strategies, and showed that the information on the type of deployed position management strategy and on the positions is useful for an ex-ante market price prediction. In this subsection, we ex-post characterize the position management strategies associated with the market-price behavior around the trading decisions (which is unrelated to ex-ante prediction).

Fig 8 shows the average market prices changes around the transactions as a maker (blue), and as a taker (red) conditional on the bank’s position management strategies. The left, middle, and right panels respectively show the EM, AM, and MM strategies. The price change is defined as (13) where t₀ is the transaction time.

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Fig 8. Average price changes around the transaction timing (Time 0) of the EM, AM, and MM strategies as a taker and a maker.

Error bars represent the standard errors, and 〈⋅〉 represents the ensemble average. Banks employing the EM strategy are likely to transact as a maker during gradual price appreciation, while banks employing the AM strategy are likely to transact as a taker under rapid price depreciation. No strategy characteristic is evident for the MM strategy. In each case, the average price changes after the transactions plateau, which implies that none of the strategies has strategic superiority on average.

https://doi.org/10.1371/journal.pone.0220645.g008

These graphs provide several insights into the position management strategies in terms of the pattern of market-price changes. By considering the slope of the average price changes created by the EM maker strategy, the banks are likely to trade as a maker after a gradual market-price appreciation toward the side the banks are going to hold; they are prone to submitting bid (ask) limit orders when market prices are rising (falling) to follow the current market trends. Given that the EM strategy corresponds to repeated trades on the same side (Fig 5(b)), they seem to focus on reducing a negative market impact on their positions by following market trends and trading as a maker. Since the slope of the average price changes as a taker is much gentler than that as a maker and almost close to zero, it would be difficult to infer taker strategic characteristics associated with market-price behavior.

Looking at the accelerated curve of the average price changes created by the AM taker strategy, we confirm that the banks employing the AM strategy prefer trading as a taker under rapid market-price depreciation toward the side they are going to hold. As Fig 5(b) shows, the AM strategy corresponds to alternate trades on opposite sides, and we can infer the maker strategy as follows. Assume that a bank has a unit of a short contract at a price of 101. When the market price rapidly falls to 100, the bank decides to enter a long contract as a taker to realize immediate gains from the existing short contract. This kind of trading decision process provides a possible explanation for the market-price curve as a taker. The average price changes as a maker’s transactions plateau, and seems to contain almost no information on its strategy.

No peculiar characteristics are evident for the MM strategy in terms of the pattern of market-price changes. Note that the average price changes after the transactions plateau in each case, which implies that, on average, none of the strategies has strategic superiority.

4.4 Market price behavior during outlier trades

Fig 9 shows the market-price profile around the end (a) and beginning (c) of the extreme outlier trades and the end (b) and beginning (d) of the moderate outlier trades. The prices at t = 0 in Fig 9(a) and 9(b) (Fig 9(c) and 9(d)) are the mid-prices observed at the end (beginning) of large volumes. The prices after and before t = 0 are the average price transitions from the market mid-prices observed at t = 0 (i.e. at the end and beginning of outlier trades). We aggregate both maker and taker trades to ensure there are enough samples to calculate the average price changes.

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Fig 9. Average price changes around the end (top panels (a,b)) and the beginning (bottom panels (c,d)) of outlier trades.

The ordinate represents the average price changes at the end (a,b) and the beginning (c,d) of outlier trades, and the abscissa covers three minutes before and after the end (a,b) and the beginning (c,d) of outlier trades, which corresponds to time 0. We used extreme outliers to make Fig (a,c) with sample size of 12 (a) and 102 (c), and moderate outliers to make Fig (b,d) with sample size of 405 (b) and 1827 (d). Error bars represent the standard errors. Note that the ordinate of the extreme outlier case is five times larger than that of the moderate outlier case. These graphs imply the following two points. First, MMs keep jacking up limit-order prices even after the end of the extreme outlier trades (a), whereas they do not behave in this manner after the end of the moderate outlier trades (b). Second, banks prefer to follow (c) (go against (d)) market-price trends when handling extreme (moderate) outlier trades.

https://doi.org/10.1371/journal.pone.0220645.g009

Fig 9(a) and 9(b) shows the response pattern of the MMs to outlier trades. After the end of large volumes at t = 0 in Fig 9(a), the market prices are still increasing without being driven by large volumes. This implies that the MMs recognize the extreme outlier trades, and attempt to jack up limit-order prices in anticipation of the subsequent transactions on the same side. This MM strategy is observed only during extreme outlier trades. As shown in Fig 9(b), the MMs do not behave in the same way during the moderate outlier trades. This implies that the MMs somehow anticipate outlier trades using historical market information and change their strategies to adapt to these outlier events, which is another example of how banks recursively affect future market prices using historical market information. Note that it is an obvious price behavior that market prices increase before t = 0 in Fig 9(a) and 9(b) as a result of outlier trades although we have a small change of trend from t = −1800 to −1000 in Fig 9(b).

Fig 9(c) and (d) imply that, when traders start to handle extreme (moderate) outlier trades, they prefer to follow (go against) market trends. We conjecture that, when handling extreme outlier trades, banks pay attention to reducing their market impact whereas, when handling moderate outlier trades, they pay more attention to price depreciation than to market impact. As we explained above, it is trivial that market mid-prices increase after t = 0 because banks start to handle outlier trades after t = 0, which obviously causes market prices to rise. See S8 Appendix for an analysis of how the limit-order book shape changes around the outlier trades.