# Characterizing limit order books in call auctions of a stock market

## TL;DR

This paper studies the structure of limit order books in call auctions and finds that their shapes can be modeled using a hyperbolic tangent function, revealing insights into market states and stock performance.

## Contribution

The study introduces new parameters, 'median spread' and 'width', derived from the hyperbolic tangent fit to analyze call auction order books.

## Key findings

- Limit order book shapes in call auctions are well-fitted by a hyperbolic tangent function.
- The ratio of 'width' to 'median spread' is similar across most stocks, indicating similar execution ratios.
- Deviations in this ratio correlate with stocks of companies making outstanding profits.

## 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.