  There has been little research on limit order books in call auctions of stock
markets, where trades are executed at a single point in time, although there
has been extensive research on continuous auctions, where trades are executed
sequentially. This study focuses on the limit order books in the call auctions,
with the aim of identifying their shapes and clarifying the relation between
these shapes and the trading volume. Using data for all stocks listed on the
Tokyo Stock Exchange, we found that the shape of the limit order books in call
auctions are well fitted by the hyperbolic tangent function. We defined the
median spread and the width of limit orders from the fitting parameters. When
the median spread and the width were plotted for all stocks, we observed
interesting distributions depending on the attributes of the stocks and the
company performances. Furthermore, we showed that the trading volume can be
represented by the median spread and the width. The main contribution of this
research is to demonstrate that median spread and the width are useful
indicators for characterizing the states of the limit order books in the call
auctions.
