Analysis and modeling of client order flow in limit order markets

Figures & data

Figure 1. Clients submit ‘parent orders’ to execution services and different trading desks. For example, these could be high touch (HT) and low touch (LT) desks. Parent orders are sliced into child orders via ‘Desks’ and ‘Algos’ and routed to different venues for cost-efficient execution. Usually, a routing system (SOR) determines to which venue a child order is sent.

Figure 2. Different views of the same LOB snapshot. Note, only the omniscient view has complete information about the queue priority of each order. (a) anonymized view (b) Broker view (c) Omniscient view.

Figure 3. Exemplary embedding using the first three eigenvectors of the normalized Laplacian matrix using the spectral embedding method from Belkin and Niyogi (2002). The color of the nodes indicates the values of the associated feature. The brighter the points, the higher the value of the corresponding feature. (a) Mean creation time (b) Ratio of specified limit price.

Table 1. Table indicating the ARI between K-means and spectral clustering.

# Clusters	2	3	4	5
ARI	0.7979	0.5107	0.7968	0.6153

Note: The higher the number, the higher the consistency between to partitions; the maximum value ARI can attain is 1, indicating a perfect matching of the clusters.

Table 2. The variance between an observation and its corresponding cluster.

# Clusters	1	2	3	4	5	6	7	8	9
Variances	3.914	3.507	3.210	2.981	2.871	2.738	2.640	2.596	2.536
2nd Diffs.	–	0.108	0.069	0.118	−0.021	0.034	0.053	−0.016	–

Note: The stronger the decrease when K is increased by one cluster, the larger is the marginal effect of the new cluster to separate the data.

Table 3. Exemplary centroids setting K = 4 for one of the 25 most liquid STOXX 600 instruments during December 2019.

	1: Quant	2: Day VWAP	3: Signal	4: Res
Buy ratio	0.63	0.55	0.63	0.65
Cancellation ratio	0.26	0	0.15	0.07
# Trades per month	219.7	1.72	4.08	5.03
Maximum order creation time	16:10:47	08:32:04	15:02:03	14:34:05
Mean order creation time	12:03:50	08:25:34	14:20:00	11:31:56
Mean order size (in ADV)	0.01	0.03	0.06	0.06
Mean momentum (bps)	2.285	−17.87	29.94	38.26
Mean volatility	22.2	23.03	22.42	22
Minimum order creation time	07:58:08	08:19:10	13:34:31	08:40:12
# Active days per month	13.47	2.1	3.71	4.95
St. dev. of order creation time	02:24:49	00:07:12	00:42:16	02:40:46

Notes: Liquidity, in this case, refers to the total number of trades in the data base. The highest (lowest) value of the corresponding features are highlighted in bold (italics).

Table 4. Summary of agent types.

Cluster Name	Cluster Description
Quant	Mainly quantitative traders, many trades, small volumes, orders sent throughout the day, execution with few child orders leading to a large POV, net inventory closer to 0
Day VWAP	Mostly VWAP as execution, few orders, only sent in the morning, almost no cancellation
Signal	Typically trading in the afternoon (especially US market opening), large trades, large amount traded in dark venues, large POV in general
Res	Large orders, medium frequency, sent throughout the day

Figure 4. Embedding of first stage cluster centers from the single time slice clustering process. Colors indicate the first stage clustering result. As discovered in table 3, Res is positioned in-between the other three clusters. (a) Different tickers, same time period. (b) Same ticker, different time periods.

Figure 5. Exemplary centroid features of one ticker for 24 months. The solid lines indicate the actual values over the months, the dashed horizontal lines indicate the average over all months. (a) Number of orders (log scale) (b) Mean submission time (c) Cancellation rate.

Table 5. Log-weighted average entropy and maximum affiliation probability of the agents.

Dimension	Entropy	Maximum Affiliation
Ticker	0.4	0.8
Time	0.5	0.76

Notes: The first row indicates both values for the consistency in ticker dimension. The second row indicates the values in time dimension.

Table 6. Affiliation of clusters to meta clusters for different tickers (time periods in parentheses).

	Meta Quant	Meta Day VWAP	Meta Signal	Meta Res
Quant	22 (23)	0 (0)	0 (0)	3 (1)
Day VWAP	0 (0)	25 (23)	0 (0)	0 (1)
Signal	0 (0)	0 (0)	24 (24)	1 (0)
Res	1 (0)	0 (0)	0 (0)	24 (24)

Figure 6. Distribution of the number of trades and the executed quantity. (a) Number of submitted orders (b) Fraction of executed quantity.

Table 7. Average values of the distributions of clients, trades and executed quantities across different clusters.

	Quant	Day VWAP	Signal	Res
Number of orders	0.69	0.06	0.06	0.20
Executed qty	0.41	0.15	0.11	0.34

Table 8. Aggregated statistics of child orders by clusters.

	C-Quant	C-Day VWAP	C-Signal	C-Res
# Child Orders	0.88	0.02	0.03	0.06
Mean Order Qty.	0.23	0.42	2.74	0.61
Median Order Qty.	0.29	0.28	3.11	0.33
Std Order Qty.	0.52	0.61	2.02	0.85
Mean Exec Qty.	0.12	0.18	3.53	0.17
Median Exec Qty.	0.70	0.75	1.76	0.79
Std Exec Qty.	0.33	0.40	2.95	0.31
DMA Ratio	2.27	0.03	1.63	0.07
Cancellation Ratio	1.07	0.90	1.09	0.94
Median Distance to Best Price	0.52	1.81	0.90	0.77

Notes: This table shows the total number of submitted child orders, the ratio of orders which were sent due to direct market access, the ratio of child orders which were cancelled, the mean, median and standard deviation for both order quantity and the executed quantity of orders. The last row shows the median distance of a limit order from the best opposite price. The number of child orders is normalized by the sum of all clusters; the remainder of the statistics are normalized by the mean of the four clusters.

Table 9. Table indicating the mean cumulative inventory of the clusters.

	C-Quant	C-Day VWAP	C-Signal	C-Res
Average inventory	−0.62	0.12	0.11	0.16

Note: Values are normalized with the absolute inventory of the clusters.

Table 10. Correlation matrix of net inventory between different components.

	Quant	Day VWAP	Signal	Res
Quant	1.00	−0.05	−0.02	−0.04
Day VWAP	−0.05	1.00	0.00	−0.03
Signal	−0.02	0.00	1.00	−0.09
Res	−0.04	−0.03	−0.09	1.00

Table 11. Tables indicating the correlation of the order flow imbalance between different components.

	Quant	Day VWAP	Signal	Res
Quant	1.00	−0.07	0.00	−0.11
Day VWAP	−0.07	1.00	−0.02	−0.04
Signal	0.00	−0.02	1.00	−0.03
Res	−0.11	−0.04	−0.03	1.00

Table 12. Table indicating the average $\mathbb{E}\left(Pn{L}_k^l\right)$ for agent types in basis points (bps).

	Gross return				Excess
	l = 0	l = 1	l = 10	l = 20	l = 1, excess	l = 10, excess	l = 20, excess
Quant	0.38	0.72	3.39	3.76	0.78${}^{\ast }$	3.78	2.83
Day VWAP	−0.25	1.54	−2.04	−3.95	1.42	0.15	1.44
Signal	−0.47${}^{\ast }$	−0.43	3.09	13.17${}^{\ast }$	0.32	2.65	10.63${}^{\ast }$
Res	0.08	0.37	−1.36	−1.51	0.52	−2.24	−2.19

Notes: The suffix _excess indicates market excess returns over the STOXX 600 baseline. A * behind the return in basis points indicates significance using a one-sample t-test and with a confidence level of $95\mathrm{\%}$.

Figure 7. Cumulative PnL $\mathbb{E}\left(Pn{L}_{t,k}^l\right)$ in basis points (bps) of agent types over different time horizons, $l=\{0,1,10,20\}$. (a) l = 0 (b) l = 1 (c) l = 10 (d) l = 20.

Figure 8. Order flow imbalance by cluster, during days of returns with large magnitude.

Table 13. Correlation between OFI and returns of stocks (column logOC) and return of the index (column logOC_index).

Cluster	logOC	logOC_index
Quant	−0.0301	0.0113
Day VWAP	0.1590	0.0229
Signal	−0.1506	0.0499
Res	0.1102	0.1616

Note: $OF{I}^{C_k}$ as in equation (18(18) $OF{I}^{C_k}=\frac{\sum _{p\in {\mathcal{X}}^k}{q}^{exec}}{\sum _{p\in {\mathcal{X}}^k}\left|q^{exec}\right|},$(18) ).

Table 14. Correlation between OFI and returns of stocks (column logOC) and return of the index (column logOC_index).

Cluster	logOC	logOC_index
Quant	0.1368	0.0833
Day VWAP	0.2044	0.1416
Signal	−0.0441	−0.0794
Res	0.2189	0.2390

Note: $OF{I}^{ADV}$ as in equation (19(19) $OF{I}^{ADV}=\frac{\sum _{p\in {\mathcal{X}}^k}{q}^{exec}}{\mathrm{A}\mathrm{D}\mathrm{V}},$(19) ).

Figure 9. Aggregated parent order flow distributions for one stock, over two years, for Quant. (a) Order counts: QUANT order flow. (b) Order size distribution: QUANT order flow (c) Submission times (d) Fraction of buy orders.

Figure 10. Aggregated parent order sizes (on a log scale) for one stock over two years for Day VWAP.

Figure 11. Aggregated parent order flow distributions for one stock over two years for Signal. (a) Order counts (b) Order sizes (c) Submission times.

Table A1. Feature list used for the clustering.

Feature name	Explanation
Buy ratio	Pct. of client orders which is a buy order
Cancellation ratio	Pct. of client orders which are cancelled before full execution
# Trades per month	Number of trades per month.
Inventory	Mean inventory accumulation of a client on a given day
NO Limit price ratio	Percentage of the client's orders for which no limit price has been specified (maximum/minimum price for execution of child orders)
Maximum order creation time	Latest time at which a client submits an order
Mean order creation time	Average time at which a client submits an order
Mean order size (ratio of ADV)	Mean order size measured on the average daily volume of the last 20 days, execution may be less.
Mean momentum (bps)	Mean momentum of entire trading day measured in basis points (bps)
Mean percentage of volume	Mean percentage of traded volume during trade horizon (visible fills + dark fills) / (visible market volume), exceeding 100 is indicator for larger placements in dark venues
Mean volatility	Mean volatility during which a client trades measured on the last 20 days
Minimum order creation time	Earliest time a client creates an order
# Active days per month	Number of days a client trades per month
Mean # orders per active day	Number of orders a client trades if it trades during a day
Standard deviation # orders per active day	Standard deviation of the number of orders of days during which a client places at least one order
Standard deviation of order creation time	Standard deviation of a client's creation time
Standard deviation order size	Standard deviation of a clients order size
Total order size	Cumulative trade size measured on the average daily volume of the last 20 days. Indication of how much a client in average trades at all

Notes: All features are computed on the base of a one-month dataset. For instance, #Trades indicates the number of trades for a particular client per month. The creation time, measured from the time passed since midnight is set to a minimum of 7 as some clients, sending their orders on the evening before disrupt the feature distribution. 7 am in this case involves all orders sent before market opening.

Belkin, M. and Niyogi, P., Laplacian eigenmaps and spectral techniques for embedding and clustering. In Proceedings of the Advances in Neural Information Processing Systems, pp. 585–591, 2002.

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