**1. Introduction**

To gain a deeper understanding of the mechanisms of financial markets, it is necessary to clarify the order strategies of individual market participants. In financial markets, recent developments in information technology have made it possible to track the transactions of individual market participants in detail. These technological advances have led to the analysis of the trading strategies of individual market participants and how these strategies affect financial markets. For example, Odean [1], and Grinblatt and Keloharju [2], reported the relationship between historical returns and market participants' decisions to buy and sell stocks. The position managemen<sup>t</sup> strategies of individual market participants were analyzed based on the data, which confirmed that these strategies actually affected market prices in the near future [3]. Individual strategies for placing buy and sell orders in response to market price changes were approximated using a simple mathematical model, and the basic statistical properties of financial Brownian motion were theoretically derived based on the kinetic theory in a manner parallel to traditional statistical physics [4–6].

In particular, high-frequency traders (HFTs) have recently attracted attention. HFTs are algorithmic traders who can react to market changes in milliseconds and place or cancel, buy and sell orders at high frequencies [7]. Because of the development of information technology, they have a large presence in financial markets around the world. In fact, HFTs accounted for 68.3% of the total trading volume in the stock market [8]. Furthermore, HFTs currently account for the majority of orders shown in the order book [9–11]. The availability of high-frequency trading data has triggered the academic study of HFTs [12,13]. Previous

**Citation:** Watari, H.; Takayasu, H.; Takayasu, M. Analysis of Individual High-Frequency Traders' Buy–Sell Order Strategy Based on Multivariate Hawkes Process. *Entropy* **2022**, *24*, 214. https://doi.org/10.3390/e24020214

Academic Editors: Ryszard Kutner, Christophe Schinckus and H. Eugene Stanley

Received: 7 January 2022 Accepted: 28 January 2022 Published: 29 January 2022

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studies have generally agreed that HFTs make market spreads smaller and enhance market liquidity [8,14,15]. As an indicator for predicting the short-term volatility of market prices from the order book information, the volume-synchronized probability of informed trading (VPIN) has been proposed and actively studied [16–20]. In addition, informed trading using the advantage of information such as public news and confidential information has been studied using high-frequency data [21–25]. We believe that it is crucial to gain a deep understanding of their trading behavior in current financial markets, where HFTs provide most of the liquidity.

In this study, we used a multivariate Hawkes process to investigate the processes used by individual HFTs for generating sell and buy limit orders in the USD/JPY forex market, and clarified when each HFT placed buy–sell limit orders. The Hawkes process is a type of non-homogeneous Poisson process proposed by Hawkes [26]. As will be explained later, it is characterized by an intensity function which determines the probability of the occurrence of an event in a point process. It utilizes an excitation term that is affected by past events, and can describe a point process associated with past events. Similar ideas have been independently introduced for financial markets to explain the strong correlation to past events, such as the "autoregressive conditional duration model" [27] and "self-modulation processes" [28]. The Hawkes process is a useful model for interpreting financial phenomena, in which many factors interact to produce complex aspects. In this paper, we show that it is also useful for interpreting the behavior of HFTs. Specifically, we introduced a multivariate Hawkes process in which the process of generating HFTs' buy–sell limit orders is mutually excited by a total of eight events, such as the creation of limit orders, the cancellation of limit orders, and execution in the order book, showing that the order behaviors of many HFTs can be modeled by the Hawkes process.

Hawkes processes [29] have various applications in the financial field, such as those related to volatility clustering [30], market activity and risk [31–33], and market impact [34]. In particular, the Hawkes process has been actively employed as an approach to the dynamic description of order books, where a set of order types is specified and a multivariate Hawkes process is fitted to their timestamps [35–41]. However, there has been no study that used a multivariate Hawkes process to investigate the order generation processes of individual HFTs. In today's financial markets, where the majority of order books are made up of HFTs' orders, our empirical results provide new information from a more microscopic perspective. We believe that this study shed light on how HFTs provide liquidity to the market.

The remainder of this paper is organized as follows. Section 2 explains the datasets and describes the HFTs that were analyzed in this research. Section 3 introduces the multivariate Hawkes process and describes the method used for parameter estimation. In Section 4, a clustering analysis of 134 HFTs is introduced to categorize their strategies based on the estimated Hawkes' parameters. In Section 5, we discuss our results.

### **2. Data**

First, we provide a basic description of the order data for the USD/JPY forex market (EBS market), along with individual trader IDs (see Section 2.1). We then define the HFTs in this market (see Section 2.2) and show some examples to explain how their buy–sell limit order generation is linked to changes in the order book (see Section 2.3).

#### *2.1. EBS Market Data Description*

In this study, we used high-frequency data for the USD/JPY forex market provided by the EBS. EBS is an interbank forex market and one of the largest financial platforms in the world. Because it is an interbank market, most market participants are professional traders from banks, hedge funds, and other financial institutions, and our forex dataset contains their trading data. Our dataset contains information from five days (from 21:00 GMT on 5 June 2016 to 21:00 GMT on 10 June 2016), with a total of approximately 2.8 million orders and a transaction volume of USD 68 billion corresponding to this period. Table 1

shows an example of the raw data we used. The data for each of the 2.8 million orders contained not only the order type, price, volume, and timestamp (in milliseconds), but also an anonymized trader ID that could identify who submitted the order. Using these trader IDs, we could track individual traders' full orders in milliseconds. In addition, the minimum price unit that a trader could submit was JPY 0.005, and the minimum transmission volume was USD 1 million.


**Table 1.** Examples of raw data. Each order datum is tied to an anonymized trader ID.

The EBS market is open 24 h a day from Monday morning to Friday at midnight, and trading is conducted via a double auction system in the order book. Figure 1 shows a schematic of the trading in the order book, where the horizontal axis is the price and the vertical axis is the volume. There are six order types for trading: buy/sell limit orders, buy/sell cancel orders, and buy/sell market orders. Limit orders are submitted at the trader's desired price and remain in the order book until traded or cancelled. Cancel orders are submitted by a trader to cancel a limit order that they previously submitted. Market orders are submitted at the current best limit price. Transactions that are executed by buy market orders are called hit sell transactions, and transactions executed by sell market orders are called hit buy transactions (see Figure 1). If the best price worsens (e.g., the best sell limit price becomes higher) before the market receives the market order at the best price, the market order is automatically invalidated. In fact, this study found that 79.5% of the market orders were invalidated without being executed.

**Figure 1.** Schematic of trading in order book. In the EBS market, even a sell (buy) limit order becomes a hit buy (sell) if a buy (sell) limit order at the same price is already in the order book.

Figure 2a shows the average trading price per 10 min window over the 5 days we analyzed. During this period, there are no market crashes or spikes. Figure 2b shows the number of each type of order per day, which looks stable.

**Figure 2.** (**a**) Average trading price per 10 min window; (**b**) daily number of orders for the 6 types of orders in unit of 105.

#### *2.2. Definition of HFTs*

"HFTs" is a general term for traders who place and cancel orders at high speed and high frequency according to an algorithm, but there are various definitions. In this study, we define an HFT as a trader who places both buy and sell limit orders and presents an average of 500 or more limit orders per day following the previous report written by a researcher from EBS [9]. Based on this definition, the number of HFTs was 134 out of the 1031 traders included in this 5-day data set. These 134 HFTs accounted for 89.6% of the market's total number of limit orders.

Figure 3a shows the histogram of the minimum time interval between orders for each HFT. There is no description in the data to identify whether the ID is a human or a computer; however, Figure 3a shows that most of the intervals are within 0.1 s, which are difficult for a human to execute.

In Figure 3b, we plot the number of HFTs and non-HFTs participating in the market every hour, indicating that the number of HFTs is relatively stable compared to non-HFTs. Figure 3c shows the percentage of limit orders placed by HFTs every hour, demonstrating that the majority of the limit orders are provided by the HFTs.

**Figure 3.** (**a**) Histogram of the minimum time intervals between orders for 134 HFTs individually; (**b**) hourly changes in the number of HFTs and non-HFTs participating in trading; and (**c**) hourly change of the percentage of limit orders provided by HFTs.

#### *2.3. Basic Properties of HFTs*

In this study, we focused on the limit order generation process of HFTs, which accounted for the majority of limit orders in the order book. Naturally, the order strategies (i.e., the processes used to submit limit orders) of HFTs differed from every algorithm. However, it is natural for them to see the quotes in the order book when submitting their

limit orders. Figure 4a,b plot the numbers of buy–sell limit orders per 10 min window for three HFTs, respectively, and Figure 4c plots the numbers for six types of orders in the order book. From Figure 4, we can observe that the numbers of buy–sell limit orders from the three HFTs increased or decreased simultaneously and tended to be in sync. More interestingly, the numbers of these HFTs' buy–sell limit orders tend to be in sync with the numbers for each type of order in the order book where all market participants' orders are submitted. Since the above synchronization phenomenon was confirmed for many HFTs, we believe that many HFTs react instantaneously to some changes in the market when submitting limit orders.

**Figure 4.** Numbers of (**a**) sell limit orders and (**b**) buy limit orders per 10 min window for three HFTs (green: HFT with 4th highest order frequency; purple: HFT with 7th highest order frequency; yellow: HFT with 10th highest order frequency). (**c**) Numbers for six types of orders per 10 min window in the order book (red: sell limit order; blue: buy limit order; red dotted line: sell cancel; blue dotted line: buy cancel; orange: hit sell; sky blue: hit buy). The vertical axis of each figure shows the number of each type of order per 10 min window, and the horizontal axis shows the time from 0:00 to 18:00 on 6 June 2016.
