https://orcid.org/0009-0003-4322-7678
Figures
Abstract
Statistical and dynamical characteristics of stock markets have been extensively studied, providing a solid basis for econophysics and its application as “stylized facts”. However, most of those studies are for markets under the continuous auction, i.e., trades are executed sequentially. There has been less research on another major type of auction, call auctions, where orders are accumulated and those are executed at once in the final moment. This study focuses on the structure of the limit order books of stocks under call auctions. Using the data of all stocks listed in the Tokyo Stock Exchange, we find that the shapes of the limit order books in call auctions are well fitted by a simple functional form of a hyperbolic tangent. From the fitting, we define the “median spread” and the “width” of limit orders. The ratios of the “width” to the “median spread” of most stocks are found to be similar, indicating that the execution ratios (the trading volume relative to the total number of orders) are nearly equal among them. Furthermore, the deviation in this ratio from the majority is found to be a good indicator to find the stocks of the companies making outstanding profit. Our results demonstrate that those parameters of the structure of the limit order book well characterize the states of the market under call auctions.
Citation: Nagumo S, Shimada T (2025) Characterizing limit order books in call auctions of a stock market. PLoS One 20(7): e0327430. https://doi.org/10.1371/journal.pone.0327430
Editor: Junhuan Zhang, Beihang University, CHINA
Received: October 7, 2024; Accepted: June 14, 2025; Published: July 7, 2025
Copyright: © 2025 Nagumo, Shimada. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: The data that support the findings of this study are available from Tokyo Stock Exchange, Inc. but restrictions apply to the availability of these data, which were used under the non-disclosure agreement for the study, and so are not publicly available. However, in some cases, the data might be shared by Japan Exchange Group, Inc. (the parent company of Tokyo Stock Exchange, Inc.) under the non-disclosure agreement. [Contact for inquiries regarding the data] Japan Exchange Group, Inc. E-mail: jpx-research@jpx.co.jp
Funding: The author(s) received no specific funding for this work.
Competing interests: The authors have declared that no competing interests exist.
Introduction
A limit order book is a list of orders to sell or buy at a certain price (limit orders). The execution prices and the trading volumes are determined by the structure of the limit order book and the inflow of the market orders (orders which do not specify the price). Therefore, understanding the distribution of the limit order book and the dynamics of that is essential to predicting the dynamics of price, trading volume, volatility, etc. of that stock [1–6].
The classical indicators to characterize the shape of the limit order books are “spread”, “depth”, and “resiliency” [7]. The spread refers to the distance between sell orders and buy orders. The most commonly used one is the Best Bid-Offer spread (BBO spread), which is the distance between the best ask (the lowest sell orders) and the best bid (the highest buy orders). It is known that stocks of the company with lower profit tend to have wider BBO spreads. One possible explanation for this is that traders demand a high return for taking the risk, and hence place high sell orders and low buy orders [8]. Moreover, it is known that orders tend to be submitted inside the best ask and the best bid when the BBO spread is larger than its time-series median, while orders tend to be submitted outside the best ask and the best bid when the BBO spread is smaller than its time-series median [9]. The depth characterizes the volume of orders near the price of the best ask and the best bid. Especially, order volume at the price of the best ask and the best bid is commonly used. The resiliency refers to the speed with which the market price (especially, the BBO spread) returns to its original value after the execution. Another characteristic of the shape of the limit order books is “slope”, which is the increment of the order volume to the increment of the price on the limit order books. It is known that the slope of the limit order book is negatively correlated with the trading volume [10].
However, these studies focus on the continuous auction, where orders are placed and executed sequentially. There has been little study on the call auction, where orders are accumulated and those are executed all at once at the end of the auction. The share of the call auction is at around 30% in the Tokyo Stock Exchange [11, 12] in the European exchanges. Moreover, the share of the call auction in the European exchanges is increasing [13, 14].
In this study, we investigate the shape of the limit order books on the call auction, and see the differences among the stocks with different attributes. Moreover, we clarify the relation between the shape and the trading volume.
Results
Fitting function of limit order book
We use the data of the limit order books at the moment of the execution (9:00 A.M.) for all the stocks listed on the Tokyo Stock Exchange in 2022. In order to analyze the shape of the limit order book for each stock, we aggregate the limit order books of all the days in the data for each stock. Because the price levels and the order volumes vary day to day, we standardize them for aggregation. The details of the data preparation are described in Method.
For the aggregated order distribution, we denote the number density of sell (Ask) limit orders at price x as , and the number density of buy (Bid) limit orders at price x as
. We denote the cumulative number of sell and buy limit orders as
and
, respectively. Also, the number of sell market orders and the buy market orders are denoted by
and
, respectively.
The execution price X and the trading volume V are determined by the relation
As shown in Fig 1, the execution price X is in the range between the first quartile of sell limit orders, a1 and the first quartile of buy limit orders, b1. Therefore, in order to keep the information to characterize the price and the volume, we fit in the range
, and fit
in the range
by functions introduced below, where a0 is the price of the best ask and b0 is the price of the best bid. Fig 2 shows a typical shape of a limit order book. We treat the sell market orders as the sell limit orders placed at a0, and treat the buy market orders as the buy limit orders placed at b0. The red solid line represents the sell limit orders and market orders
, and the blue dashed line represents the buy limit orders and market orders
. It is found that the cumulative orders
and
typically start from a slow rise and then steepens at a certain point like the sigmoid function, so we adopt the hyperbolic tangent function as the fitting function for its mathematical convenience. Namely, we fit
and
as
where and
are the total numbers of sell and buy limit orders.
and
are the fitting parameters which correspond to the typical widths of the sell and buy orders.
and
represent the medians of the sell and buy fitting functions. The example of the fit for the limit order book is shown in Fig 2 by the black lines.
Almost all execution prices are within the first quartiles.
We treat sell market orders as the sell limit orders at the best ask a0, so the number of sell orders placed at a0 includes sell market orders , which appears Likewise, the cumulative total number of buy limit orders and buy market orders
starts from the best bid b0, and the number of buy orders placed at b0 includes buy market orders
.
is fitted by
in the range
, and
is fitted by
in the range
, where
and
are the fitting functions in Eq 2.
Asymmetry in the width of the sell and buy limit order books
Using the fitting functions in Eq 2, we observe the relation between the typical widths of the sell and buy orders, and
. Fig 3 shows the scatter plot of stocks in the
-
plane with its density contours. This indicates that
and
are correlated, with the clear asymmetry that
tends to be larger than
. In the continuous auction, the asymmetry in the sell-side and the buy-side has been reported for the number of orders [15]. For the distributions of incoming orders, there seems to be no difference between the sell-side and the buy-side when the sell (buy) order prices are normalized by the best ask (bid) [16]. The asymmetry in the shape of the limit order book we have found here is different from the above reported ones, which may be specific to the call auction.
Classification of stocks
We now compare the distance between the sell orders and the buy orders to those widths, to capture the basic information of the limit order book of each stock. We use to characterize the width of the orders, and
to characterize the distance between the sell orders and the buy orders.
Fig 4a shows the distribution of stocks in the and
plain. This distribution can be classified into three clusters. The first cluster (Cluster 1) is characterized by the small width and the small spread (
). The second cluster (Cluster 2) is the group of stocks which are distributed around a line:
. The other stocks are distributed below this line, forming the third cluster (Cluster 3).
The density contours are plotted by the solid lines. The dashed line corresponds to the equal execution ratio (
) line, and the dash-dotted curve
is for the eye guide.
We find that those clusters are strongly correlated with the types of stocks. While most stocks in Cluster 2 and Cluster 3 are individual stocks, most stocks in Cluster 1 are Exchange-Traded Funds (ETFs) and Exchange-Traded Notes (ETNs) (Fig 4b). ETFs and ETNs are investment funds linked to an index composed of different stocks and other assets, and therefore the fluctuations of the prices tend to be small, merely because of the law of large numbers. An interesting observation is that both Cluster 1 and Cluster 2 are distributed along the same line: . This constant slope of the distribution corresponds to the constant execution ratio. As is shown in the Method, the execution ratio
for given fitting functions is determined only by the slope
as
The lines for the constant execution ratio are shown in Fig 5. The slope corresponds to the execution ratio
. The fact that Cluster 1 and Cluster 2 are distributed along the single line suggests that self-organizing mechanism may work among traders. That is, traders have an empirical knowledge of the normal shape of the limit order book, i.e. the ratio of the spread to the width. Based on this knowledge, they place orders, which result in the normal shape of the limit order book. As for Cluster 3, they are distributed along a curve:
, which is below the line with slope
. For these stocks, the theoretical execution ratios
are higher than
. The origin of this non-linearity of the curve is not clear yet. It turns out that most of the individual stocks in Cluster 3 are separated from Cluster 2 by whether the company’s net profit in that year was more than 10 billion yen or not (Fig 4c, 4d).
The slope 3.0 corresponds to the theoretical execution ratio . Lines with gentler slopes correspond to higher execution ratios.
Generality of the results
It should be noted that the results we have shown are not dependent on the particular choice of the fitting function to the hyperbolic tangent. For example, analysis using a piecewise linear function for fitting gives the same results in the relation between the width and spreads, the clustering structure, etc., while the goodness of fitting is worse than the hyperbolic tangent (see Supporting Information S1 Fig for detail).
Also, we can confirm the generality of our results among different periods under different market environment. To show this, we take the data set of year 2019, which is before the COVID-19 pandemic event and hence the social and economic situations were evidently very much different from that of year 2022. The difference between these periods is also apparent in quantitative terms: The annual average of the Nikkei Stock Average, which indicates the market level, was 21,697 yen in 2019, while it was 27,257 yen in 2022. The annual average of Nikkei Stock Average Volatility Index, a widely recognized indicator of market turbulence, was 17.1 points and 22.5 points in 2019 and 2022 respectively. In spite of such clear difference, we can find the consistent results in the data set of 2019 (see Supporting Information S2 Fig for detail).
Summary and discussion
Intensive studies have been made on the limit order books under the continuous auction. On the other hand, there has been little research on the limit order books in the call auction. In this study, we focused on the shape of the limit order books in the call auction at the moment of its execution. The empirical findings we have discovered are following.
First, we fitted the sell (A) and buy (B) limit order books of each stock under call auction by the hyperbolic tangent function to obtain the typical widths and the medians
of them. We find that the widths on the sell-side and on the buy-side are correlated and asymmetric,
. This suggests that the valuations of sell traders are more diverse than that of buy traders. A possible explanation is that each sell trader has a different initial buy price which serves as a reference point for placing a sell order, therefore that leads to the wider variety in sell order prices. The asymmetric functionality form in the expected utility of traders to the change, which is established in the prospect theory of behavioral economics [17], may also contribute to the asymmetry. Because such an asymmetry in the sell and the buy sides has not been reported from the continuous auction, the difference between the call auction and the continuous auction and its relation to the distribution of reference prices and to the behavioral economics should be further investigated.
Next, we examined the relation between the average width and the median spread
. From that, we found that stocks are roughly classified into three clusters. Cluster 1 is the group of stocks whose width and spread are small (
). Cluster 2 is the group of stocks which are distributed around the line:
. Cluster 3 is the group of stocks which are distributed below Cluster 2, forming a branch around a sub-linear curve:
. Most stocks in Cluster 1 are ETFs and ETNs. Most stocks in Cluster 2 and Cluster 3 are individual stocks. Cluster 2 is formed mainly by the stocks of the companies whose net profits are less than 10 billion yen, and the stocks in Cluster 3 tends to be the ones with net profits larger than 10 billion yen. Moreover, we showed that the theoretical execution ratio
is determined solely by the slope
. The slope for Cluster 1 and Cluster 2,
, corresponds to the execution ratio
. The smaller slope for Cluster 3 corresponds to higher execution ratio:
.
These empirical findings are found to be robust against the choice of the fitting function among other simple functions (e.g. piecewise linear functions) and against the different observation periods (e.g. 2019 data). It remains to be verified whether such consistency can be found in other markets such as foreign exchanges.
The similarity in the execution ratio of the majority of the stocks (Cluster 2) suggests the existence of an auto-regulation mechanism. A possible scenario is described as the following. When the spread is large relative to the width, traders will place lower sell orders or higher buy orders because this may allow their orders to be executed prior to others, with keeping enough gain, i.e. the difference between the actual execution price and the trader’s valuation price to that stock. On the other hand, when the spread is small relative to the width, traders will place higher sell orders or lower buy orders to increase their gain with expecting to keep considerable possibility of the execution. In the former case, the execution ratio increases, and in the latter case, the execution ratio decreases. Therefore, as a result of this mechanism, the execution ratio tends to be auto-regulated around a constant value. For continuous auctions, this type of self-regulating mechanism has been reported, e.g. the BBO spread tends to recover to its original length [9]. To our knowledge, for the call auctions, this study is the first case to report the characteristics which suggest self-regulatory mechanism. Detailed confirmation and analysis of self-regulating dynamics in call auction, and its comparison to the continuous auction, need more investigation. In terms of theoretical studies, such mechanism is absent in the conventional models of limit order book structure, such as the Maslov model, which have successfully explained essential characteristics of the real market dynamics such as the price fluctuations [18, 19]. Since the trader’s character to seek higher execution ratio and gain should be also relevant in the continuous auction, introducing such mechanism into the continuous auction models is an important direction for future study. The relation between trading volume and gain has been also studied by a simpler mathematical model [20, 21], which is again without any auto-regulatory mechanism. Applying such mechanism into this framework will improve the understanding of market, such as the merit of the increase of the number of participants, clarifying the quantity which is subject to the maximization in the trader’s strategy, and so on.
The finding that Cluster 3 which mainly consists of stocks with profits more than 10 billion yen is distributed below the majority (Cluster 2) could be interpreted in terms of traders’ risk appetite, as explained in the previous study [8]. That is, for such favored stocks, traders’ appetite is mostly on trading that stocks (i.e. the higher execution probability) and not so much on the gain. However, we do not have good understanding for why Cluster 3 appears as a distinct branch apart from Cluster 2, rather than as a continuously distributed tail. Also, the origin of the sub-linear shape of Cluster 3 remains unknown.
Clusters 1, 2, and 3 have been discovered on the plain of width and median spread
. However, if we substitute the median spread by conventional BBO spread
, such clustering structure disappears (see Supporting Information S3 Fig). This illustrates the importance of taking the shape of the limit order book apart from best ask (bid) to better capture the state of the stock market.
Our finding on the shape of limit order books could be applied to the study on market impact. It is well known that the market impact in the continuous auction is proportional to the square root of the order volume. This fact has been explained by the latent order book model, which is based on the linearity of the limit order book near the execution price [22]. From the curvature of the shape of limit order books, which we have found in this study, the market impact in the call auction can be evaluated more precisely.
As another practical application, we can consider a daily monitoring of limit order books simply by the ratio of to
in reference to, for example, 3. It is because, while a large portion of all stocks are distributed around the line
as shown in Fig 4a, only 0.8% of the individual stocks whose net profit is more than 10 billion yen have the ratio
more than 3 (Fig 4d). This feature is consistently found in the 2019 data: while most of the entire stocks are found around the line
, only
of the individual stocks whose net profit is more than 10 billion yen are found in the regime of
. Therefore, the drastic change of the ratio in time can be used as an indicator of the anomaly which is possibly happening on that stock. This knowledge helps stock exchanges to detect erroneous orders or illegal activities such as insider trading. For traders, unusual position of
and
may indicate unexpected changes in the company performance such as the profit, and contribute to their investment decisions.
Method
Data set
We use the data of limit order books at the time of the call auction at 9:00 A.M. for 244 business days from January 4 to December 30, 2022 for all the listed stocks on the Tokyo Stock Exchange (4,024 stocks per day on average). In order to examine the shape of the limit order book for each stock, we aggregate the limit orders of all the days in the data for each stock. 74 stocks were excluded from the data because they had no sell or buy orders throughout the year, and therefore we cannot define at least one of those fitting parameters, ,
,
, or
.
Data preparation
The order volumes vary day to day. Sometimes artificial events such as stock split are conducted by the Tokyo Stock Exchange. Therefore, when we aggregate the limit order books, the weight of the order volume for each day is set to be constant.
Also, the stock price levels varies from day to day, and according to these differences, the position of the limit order book varies from day to day. Therefore, we define a reference price for each day for each stock, and calculate order prices divided by the reference price. We aggregate the limit order books based on the relative price. This concept of the relative price is used in a previous study [16], for instance. As for the definition of the reference price, the execution price of the call auction is one candidate. However, the execution price cannot be defined when the sell and buy limit order books are not overlapped. Next idea of the definition of the reference price shall be the middle price of the best ask and the best bid. However, for many cases of call auctions, the best ask and the best bid become the lower and the upper limit price, which are artificially set by the Tokyo Stock Exchange. Therefore, the middle price of the best ask and the best bid does not reflect the stock price level of the day. Thus, we define the reference price by the average of the first quartile of the sell limit order book and the first quartile of the buy limit order book.
Theoretical execution price and execution ratio
As Eq 1, the execution is conducted at the point where the cumulative numbers of sell and buy orders cross. By using the fitting functions and
obtained in Eq 2, the execution price
and the trading volume
shall satisfy the following relation
Under the approximations that ,
, and
, we obtain the execution price
and the execution ratio
as follows.
Supporting information
S1 Fig. Fitting orders by the piecewise linear functions
https://doi.org/10.1371/journal.pone.0327430.s001
(PDF)
Disclaimer
The views expressed in this paper are those of the authors and do not represent the views of the organizations to which they belong.
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