# Heavy tailed distributions in closing auctions

M. Derksen1,2, B. Kleijn2 and R. de Vilder1,2
1 Deep Blue Capital N.V., Amsterdam
2 Korteweg-de Vries Institute for Mathematics, University of Amsterdam

###### Abstract

We study the tails of closing auction return distributions for a sample of
liquid European stocks. We use the stochastic call auction model of Derksen et
al. (2020a), to derive a relation between tail exponents of limit order
placement distributions and tail exponents of the resulting closing auction
return distribution and we verify this relation empirically. Counter-
intuitively, large closing price fluctuations are typically not caused by
large market orders, instead tails become heavier when market orders are
removed. The model explains this by the observation that limit orders are
submitted so as to counter existing market order imbalance.
Key Words: Closing auction; Closing prices; Stochastic models; Price
formation; Heavy tails;

##  1 Introduction

During the trading day, most securities change hands in continuous double
auctions, in which buy and sell orders are immediately matched if possible.
However, to determine opening and closing prices, call auctions are often
conducted. In a call auction, orders are aggregated for an interval of time,
after which all possible transactions are conducted against a single clearing
price that maximizes trading volume. In this paper we study the tails of
closing auction return distributions.

Nowadays it is widely recognized that distributions of (stock) price changes
exhibit heavy tails: extreme price changes (of e.g. more than three standard
deviations) are much more likely than in a Gaussian model or other models with
exponentially decaying tails. This issue was first adressed by Mandelbrot
(1963) in his analysis of cotton prices, where he proposed Lévy stable
distributions to model price fluctuations. It is generally assumed that the
tails follow a power law asymptotically. That is, the distribution of a return
$X$ over some time interval satisfies111Here, $\sim$ denotes _asymptotic
equivalence_ , defined as $f\sim
g\Leftrightarrow\lim_{x\to\infty}\frac{f(x)}{g(x)}=1$.,

$\displaystyle{\mathbb{P}}(X>x)\sim Cx^{-a},\text{ as }x\to\infty,$ (1)

where $C>0$ is a constant (sometimes also replaced by a slowly varying factor
$L(x)$) and $a>0$ is the _tail exponent_ , determining how heavy the tail is .
In early work (Fama, 1965), the exponent $a$ was believed to be below 2 for
stock prices (in line with the stable distributions of Mandelbrot (1963)).
However, subsequent analyses have shown that the exponent is more likely to be
around 3 on intraday time scales (see e.g. Gopikrishnan et al. (1998, 1999);
Gu et al. (2008); Pagan (1996); Plerou and Stanley (2008), among many others).
Although it is generally accepted to model the tails as power laws, the exact
functional form is also subject of debate. For example, Malevergne et al.
(2005) conclude that the tails decay slower than stretched exponential
distributions, but somewhat faster than power laws. In this paper, we do not
aim to answer this question, but use power laws because they describe the
tails in enough detail for our analysis. Theoretically, the functional form in
equation (1) is justified by extreme value theory, in the Fréchet (heavy
tailed) case (see e.g. Embrechts et al. (2003)).

Although most part of the relevant literature focuses on description of the
tails of stock price return distributions, some effort has gone towards
explanations of this tail behaviour. Gabaix et al. (2003, 2006) argue that
large price fluctuations are due to large orders submitted by large market
participants. However, Farmer et al. (2004) and Weber and Rosenow (2006) study
the issue on the microscopic level and find that large returns are not due to
large transactions, but instead are caused by big gaps in the order book, i.e.
fluctuations in liquidity. Mike and Farmer (2008) propose a simulation based
model for continuous trading, which suggests heavy tails in return
distributions are caused by market microstructure effects, such as heavy tails
in limit order placement and long memory in order flow. More theoretically,
Bak et al. (1997) and Cont and Bouchaud (2000) propose models linking heavy
tails to herd behaviour.

###  1.1 Main results

In this paper, we use the model of Derksen et al. (2020a) to study the
distribution of returns in the closing auction. In the model, limit orders are
submitted to the auction randomly, with a limit price that is sampled from an
_order placement distribution_ $F_{A}$ (for sell orders) or $F_{B}$ (for buy
orders). We study the closing auctions of liquid European stocks listed on
Euronext exchanges and find that both return distributions and order placement
distributions exhibit heavy tails, with different tail exponents. Zovko and
Farmer (2002) conclude _‘It seems that the power law for price fluctuations
should be related to that of relative limit prices, but the precise nature and
the cause of this relationship is not clear.’_ Here, we solve this problem in
the context of the closing auction: we provide analytical relations between
the tails of order placement distributions and the tails of the closing price
return distribution. In a version of the model without market orders, the
tails of the closing price distribution behave as the product of the tails of
the order placement distributions $F_{A}$ and $F_{B}$. When we incorporate
market orders, this relation changes, depending on a proportionality relation
between market order and limit order imbalances. We empirically verify the
relations between tail exponents of order placement and auction return
distributions predicted by the model.

In theory, large market orders are a possible cause of large price
fluctuations. We show however that this is typically not the case in closing
auctions, which is our second important result. Somewhat counter-intuitively,
the empirical study shows that closing auction return distributions would have
_heavier_ tails if market orders are removed, suggesting that market orders
have a stabilizing effect on price formation in closing auctions.
Theoretically, we show (for the right tail) that this (initially perhaps
somewhat puzzling) empirical fact can only arise whenever

$\displaystyle 0<\frac{M_{B}-M_{A}}{N_{A}-N_{B}}\leq\frac{a_{A}}{a_{B}},$ (2)

under the assumption that $F_{B}$ and $F_{A}$ have heavy right tails with tail
exponents $a_{B}$ and $a_{A}$ satisfying $a_{B}>a_{A}>0$. Here, $N_{A}$ is the
sell limit order volume, $N_{B}$ the buy limit order volume and $M_{A}$ and
$M_{B}$ denote the sell and buy market order volume. This equation poses two
conditions that should be fulfilled to make it theoretically possible that
tails of closing auction return distributions are heavier without market
orders. First, limit order imbalance and market order imbalance should be of
opposite signs (when $M_{B}>M_{A}$, it should hold that $N_{A}>N_{B}$ and vice
versa) and limit order imbalance $N_{A}-N_{B}$ should be larger in absolute
value than market order imbalance $M_{B}-M_{A}$, meaning that limit orders
overcompensate for market order imbalance. Second, $a_{B}$ should not be too
large, i.e. the right tail of the buy limit order placement distribution needs
to be sufficiently heavy. We show that equation (2) is indeed satisfied on
average empirically, which is explained by the chronology of the closing
auction: most of the market orders are submitted in the first seconds,
revealing early in the auction the market order imbalance. This leads to
strategic behaviour in which limit orders are placed _against_ the direction
of the market order imbalance: when there are more buy than sell market
orders, one can submit a (possibly large) sell order without adversely
impacting the price. Our results suggest that large closing price fluctuations
are not caused by large market orders (at least, not directly), but by
placement of limit orders, in accordance with the intraday results of Farmer
et al. (2004) and Weber and Rosenow (2006). Also, our results suggest that
heavy tails are market microstructure effects and that the tail exponents vary
between different stocks and different market mechanisms, in line with the
view of Mike and Farmer (2008).

The remainder of this paper is structured as follows. In section 2 the model
is described and theoretical results are derived. Then in section 3 the
empirical results are presented and the relations that are predicted by the
model are verified. Concluding remarks are made in section 4 and proofs of the
mathematical theory are collected in the appendix.

##  2 Theoretical results

In this section we recall the auction model of Derksen et al. (2020a) and
derive analytical expressions for the tail behaviour of the return
distribution, given the tails of order placement distributions.

###  2.1 A stochastic model of the call auction

In the standard call auction, orders are aggregated over an interval of time
and then matched to transact at a clearing price that maximizes the total
transacted volume. Suppose $N_{A}$ sell limit orders and $N_{B}$ buy limit
orders are submitted to the auction (all orders have unit size). We assume
market participants on both sides of the market formulate their orders
independently, according to certain order placement distributions $F_{A}$ and
$F_{B}$. Here, $F_{A}$ denotes the distribution of sell orders and $F_{B}$ the
distribution of buy orders. That is, we model the sell order prices
$(A_{1},\dots,A_{N_{A}})$ as an i.i.d. sample from $F_{A}$ and the buy order
prices $(B_{1},\dots,B_{N_{B}})$ as an i.i.d. sample from $F_{B}$222Of course
these assumptions are not all realistic. In reality, orders have different
sizes and market participants may react to each other’s orders. Despite these
simplifying assumptions, the model provides a reliable stochastic description
of auction price formation (see Derksen et al. (2020a))..

For convenience we consider the log return axis instead of the real price
axis. We assume there is some _reference price_ $x_{0}$ (for example the last
traded price before the auction starts or a volume weighted averaged version
thereof) and all prices are expressed as log returns relative to this
reference price. So $F_{A}$ and $F_{B}$ are distributions on
$(-\infty,\infty)$ and $F_{A}(x)$ or $F_{B}(x)$ denotes the probability that a
sell or buy order price is below $x_{0}e^{x}$. Given $(N_{A},N_{B})$, we
denote by $\mathbb{F}_{A}$ and $\mathbb{F}_{B}$ the empirical distribution
functions corresponding to the samples $(A_{1},\dots A_{N_{A}})$ and
$(B_{1},\dots,B_{N_{B}})$, meaning

$\displaystyle\mathbb{F}_{A}(x)=\frac{1}{N_{A}}\sum_{i=1}^{N_{A}}\mathbf{1}_{\\{A_{i}\leq
x\\}},$
$\displaystyle\quad\mathbb{F}_{B}(x)=\frac{1}{N_{B}}\sum_{i=1}^{N_{B}}\mathbf{1}_{\\{B_{i}\leq
x\\}}$

Furthermore, we define the (monotone increasing) _supply curve_ ,

$\mathbb{D}_{A}(x)=N_{A}\mathbb{F}_{A}(x)$

and the (monotone decreasing) _demand curve_ ,

$\mathbb{D}_{B}(x)=N_{B}(1-\mathbb{F}_{B}(x)).$

The supply curve denotes for every $x\in\mathbb{R}$ the number of sell orders
below $x_{0}e^{x}$, the demand curve gives for every $x\in\mathbb{R}$ the
number of buy orders above $x_{0}e^{x}$. Given all buy and sell orders, the
clearing price is the price that maximizes the transactable volume in the
auction, which is the price where supply and demand curves cross. That is, the
_clearing price_ $X$ is defined as the solution to the market clearing
equation,

$\displaystyle\mathbb{D}_{A}(X)=\mathbb{D}_{B}(X).$ (3)

This definition of $X$ may give rise to problems with uniqueness and existence
of solutions to equation (3), as illustrated in figure 1. To solve these
issues, consider the following definition.

###### Definition 2.1

For given supply curve $\mathbb{D}_{A}$ and demand curve $\mathbb{D}_{B}$, the
_lower clearing price_ is defined by

$\displaystyle\underline{X}$
$\displaystyle=\inf\\{x\in{\mathbb{R}}:\mathbb{D}_{A}(x)\geq\mathbb{D}_{B}(x))\\}$
(4)

and the _upper clearing price_ is defined by

$\displaystyle\overline{X}$
$\displaystyle=\sup\\{x\in{\mathbb{R}}:\mathbb{D}_{A}(x)\leq\mathbb{D}_{B}(x)\\}$
$\displaystyle=\inf\\{x\in{\mathbb{R}}:\mathbb{D}_{A}(x)>\mathbb{D}_{B}(x))\\}.$
(5)

The interval $[\underline{X},\overline{X})$ is the interval of all possible
clearing prices.

da_db_simple2(0.42) [t]40,106;$\mathbb{D}_{A}(\cdot)$
[t]40,100.5;$\mathbb{D}_{B}(\cdot)$ [t]70,7;$\underline{X}$
[t]83,7;$\overline{X}$ da_db_cross2(0.42) [t]134,66;$\mathbb{D}_{A}(\cdot)$
[t]134,60.5;$\mathbb{D}_{B}(\cdot)$ [t]71,7;$\underline{X}=\overline{X}$

Figure 1:  Two examples of the supply curve $\mathbb{D}_{A}(\cdot)$ (the
increasing (red) step function) and the demand curve $\mathbb{D}_{B}(\cdot)$
(the decreasing (blue) step function). Left panel: a situation in which there
is no unique point of intersection, but an interval
$[\underline{X},\overline{X})$ of possible clearing prices. Right panel: a
situation in which there is a unique intersection point
$\underline{X}=\overline{X}$.

###### Remark 2.2

Euronext’s closing auction rules say that when there are more possible
clearing prices, the price closest to the last traded price is taken
(Euronext, 2019, Rule 4401/3). This means that when there is a large positive
return, the closing price is equal to the lower clearing price
$\underline{X}$. So in order to study the right tail of the closing price
return distribution, we should study $\underline{X}$. The same reasoning
implies that for the left tail we should consider $\overline{X}$. Note that
the model is symmetric when the roles of $\overline{X}$ and $\underline{X}$
and the sides of the market are interchanged. That is, the left tail of the
distribution of $\overline{X}$ behaves the same as the right tail of the
distribution of $\underline{X}$, when $F_{A}$ and $F_{B}$ and $N_{A}$ and
$N_{B}$ are interchanged. So without loss of generality we focus on the right
tail of $\underline{X}$.

The distribution of the lower clearing price, conditional on $(N_{A},N_{B})$,
has an analytically tractable distribution function, given in the following
theorem (see Derksen et al. (2020a), theorem 2.3).

######  Theorem 2.3 (Lower clearing price distribution)

The distribution of the lower clearing price $\underline{X}$, conditional on
$(N_{A},N_{B})$, is given by its survival function,

$\displaystyle{\mathbb{P}}$ $\displaystyle(\underline{X}>M|N_{A},N_{B})$
$\displaystyle=\sum_{k=0}^{N_{A}}\sum_{l=k+1}^{N_{B}}{N_{A}\choose
k}{N_{B}\choose
l}(1-F_{A}(M))^{N_{A}-k}F_{A}(M)^{k}(1-F_{B}(M))^{l}F_{B}(M)^{N_{B}-l}.$ (6)

In the situation described above, only limit orders are submitted to the
auction. However, market participants also have the possibility to submit
market orders. We define the (possibly stochastic) _market order imbalance_ by
$\Delta=M_{B}-M_{A}$, where $M_{B}$ is the number of buy market orders and
$M_{A}$ is the number of sell market orders. Note that market orders only play
a role through $\Delta$, as matching market orders are executed against each
other without affecting the price formation process. When market order
imbalance $\Delta$ is taken into account, the market clearing equation (3)
becomes

$\mathbb{D}_{A}(X)=\mathbb{D}_{B}(X)+\Delta$

and the definitions of $\underline{X}$ and $\overline{X}$ change accordingly.
A positive (negative) value of $\Delta$ means there is more buy (sell) market
order volume than sell (buy) market order volume, possibly pushing the price
up (down). The market order imbalance alters the clearing price distribution
as in the following proposition (a special case of proposition 2.8 in Derksen
et al. (2020a)).

######  Proposition 2.4 (Lower clearing price distribution in case of market
order imbalance)

When market order imbalance $\Delta$ plays a role, the lower clearing price
distribution as computed in theorem 2.3 modifies into

$\displaystyle{\mathbb{P}}$
$\displaystyle(\underline{X}>M|N_{A},N_{B},\Delta)$
$\displaystyle=\sum_{k=0}^{N_{A}}\sum_{l=\max(k-\Delta+1,0)}^{N_{B}}{N_{A}\choose
k}{N_{B}\choose
l}(1-F_{A}(M))^{N_{A}-k}F_{A}(M)^{k}(1-F_{B}(M))^{l}F_{B}(M)^{N_{B}-l}.$

###  2.2 Limit order auctions

Next we concentrate on the right tail of the lower clearing price return
distribution, as a function of the tails of the order placement distributions
$F_{A}$ and $F_{B}$, initially without market orders. We make the following
assumption on the tails of $F_{A}$ and $F_{B}$.

###### Assumption 1

Assume $F_{A}$ has a heavier right tail than $F_{B}$. That is, there exists
functions $T_{A},T_{B}$ such that

$\displaystyle 1-F_{A}(M)\sim T_{A}(M),\quad 1-F_{B}(M)\sim
T_{B}(M),\quad\text{ as }M\to\infty$

and

$\displaystyle\lim_{M\to\infty}\frac{T_{B}(M)}{T_{A}(M)}=0.$

This assumption is intuitively reasonable and empirically verified in section
3.1. Furthermore, we will assume that $(N_{A},N_{B})$ follows a distribution
$P_{N_{A},N_{B}}$ on

$\mathcal{N}=\\{1,\dots,N\\}\times\\{1,\dots,N\\},$

for some $N\in{\mathbb{N}}$, with probability mass function $p_{N_{A},N_{B}}$
assigning positive probability to any point in $\mathcal{N}$ (we exclude the
possibilities that $N_{A}=0$ or $N_{B}=0$, which describe failing auctions in
which clearing prices do not exist).

In the following proposition we first derive an expression for the right tail
of the lower clearing price distribution, conditional on $(N_{A},N_{B})$.
Finding an expression for the tail of the clearing price distribution amounts
to finding the slowest decaying term in the double sum of theorem 2.3. This is
made formal in the following proposition, the proof of which is found in the
appendix.

###### Proposition 2.5

Under assumption 1, we have

$\displaystyle{\mathbb{P}}(\underline{X}>M|N_{A},N_{B})\sim
N_{B}T_{B}(M)T_{A}(M)^{N_{A}},\text{ as }M\to\infty.$ (7)

###### Remark 2.6

The proof of proposition 2.5 reveals the event that corresponds to the slowest
decaying term in the double sum of theorem 2.3, namely $l=1,k=0$,
corresponding to the event that $\mathbb{D}_{A}(M)=0,~{}\mathbb{D}_{B}(M)=1$,
meaning all sell orders, but only one buy order, are above $M$. This is
interpreted as an auction in which there is little consensus between both
sides of the market (buy and sell orders do not overlap), but there is a very
aggressive buyer willing to pay a high price.

When the conditional result of proposition 2.5 is summed with respect to the
distribution of $(N_{A},N_{B})$, the unconditional tail of $\underline{X}$ is
discovered again by selecting the slowest decaying term. This leads to the
main result of this subsection, a relation between the tail of the closing
price return distribution and the tail of the order placement distributions in
a setting without market orders (its proof is again postponed to the
appendix).

######  Theorem 2.7 (Right tail of the lower clearing price distribution)

Under assumption 1 we have

${\mathbb{P}}(\underline{X}>M)\sim CT_{A}(M)T_{B}(M),\text{ as }M\to\infty,$

where
$C=\sum_{n=1}^{N}np_{N_{A},N_{B}}(1,n)=\mathbb{E}[N_{B}\mathbf{1}_{\\{N_{A}=1\\}}]>0$.

The constant $C$ indicates that the slowest decaying term in the sum
corresponds to the event that $N_{A}=1$: large positive returns are possible
if there are only few sell orders.

###  2.3 Market orders

In this subsection we incorporate market orders in the derivation of
subsection 2.2. First consider the following assumption for the market order
imbalance $\Delta$.

###### Assumption 2

We assume that $\Delta\in(-N_{B},N_{A})$ with probability one.

This assumption is necessary, because otherwise the clearing prices attain the
values $\pm\infty$ with non-zero probability. Under this assumption, the right
tail of the conditional lower clearing price distribution is given by the next
proposition (the proof is again postponed to the appendix and
$x_{+}=\max(x,0)$ and $x_{-}=\max(-x,0)$ denote the positive and negative part
of $x\in{\mathbb{R}}$).

###### Proposition 2.8

Under assumptions 1 and 2, we have

$\displaystyle{\mathbb{P}}(\underline{X}>M|N_{A},N_{B},\Delta)\sim
K(N_{A},N_{B},\Delta-1)T_{B}(M)^{(\Delta-1)_{-}}T_{A}(M)^{N_{A}-(\Delta-1)_{+}},$
(8)

as $M\to\infty$, where

$K(N_{A},N_{B},\Delta)=\begin{cases}{N_{A}\choose\Delta}&\text{ if
}\Delta>0\\\ {N_{B}\choose-\Delta}&\text{ if }\Delta\leq 0\end{cases}.$

This proposition shows that market orders potentially influence the tails
heavily: if $\Delta$ is positive and large (close to $N_{A}$) the influence of
the faster decaying term $T_{B}(M)$ is erased and only the slower decaying
term $T_{A}(M)$ is left, possibly leading to very heavy tails. On the other
hand, if $\Delta$ is negative, the influence of the faster decaying term
$T_{B}$ grows, leading to less heavy tails. However, which combinations are
possible depends on the joint distribution of $(N_{A},N_{B},\Delta)$. Until
now, the tails $T_{A}$ and $T_{B}$ were unspecified and few assumptions were
made on the distribution of $(N_{A},N_{B})$. To work towards an empirically
testable theory, we will make the following assumptions on the distribution of
$(N_{A},N_{B},\Delta)$ and the tails of $F_{A},F_{B}$. Empirically, these
assumptions are verified in section 3.

###### Assumption 3

Assume $(N_{A},N_{B},\Delta)$ follows a distribution $P$ on
$\\{1,\dots,N\\}\times\\{1,\dots,N\\}\times\\{-N,\dots,N\\}$, with probability
mass function denoted by $p$, for some $N\in{\mathbb{N}}$. Furthermore, assume
that market order imbalance $M_{B}-M_{A}$ is proportional to limit order
imbalance $N_{A}-N_{B}$ (in the opposed direction), that is,

$\displaystyle\Delta=M_{B}-M_{A}=c(N_{A}-N_{B}),$ (9)

almost surely for some $c\in(0,1)$ and $P(\Delta=0)=0$ (as the case $\Delta=0$
is already considered in subsection 2.2). Finally, assume that all possible
combinations have positive probability, i.e.

$p(n,m,d)>0,~{}\text{for all
}n,m\in\\{1,\dots,N\\},d\in\pm\\{1,\dots,N\\}\text{ such that }d=c(n-m).$

Equation (9) states that limit order imbalance points in the opposed direction
of market order imbalance, which resembles that limit order submitters adjust
their orders to the market order imbalance. This relation ensures assumption 2
holds and is empirically verified in section 3.3.

###### Assumption 4

Assume $F_{A},F_{B}$ both have power law tails, that is,

$\displaystyle 1-F_{A}(M)\sim T_{A}=M^{-a_{A}},1-F_{B}(M)\sim
T_{B}(M)=M^{-a_{B}},\text{ as }M\to\infty,$

for tail exponents $a_{B}>a_{A}>0$.

Under assumptions 3 and 4, the following theorem (which is proved in the
appendix) describes the tail behaviour of the clearing price distribution in
terms of the parameters $c$ (controlling the relation between market and limit
order imbalance) and $a_{A}$ and $a_{B}$ (controlling the heaviness of the
tails of the buy and sell limit order placement distribution).

######  Theorem 2.9 (Right tail of the lower clearing price distribution with
market orders)

Under assumptions 3 and 4, there exists a constant $\underline{C}>0$, such
that

${\mathbb{P}}(\underline{X}>M)\sim\underline{C}M^{-\underline{a}},\text{ as
}M\to\infty,$

where

$\displaystyle\underline{a}=\min\left(\frac{(c+1)a_{A}}{c},a_{A}+2a_{B}\right).$
(10)

Note that without market order imbalance $\Delta$ we have by theorem 2.7
$\underline{a}=a_{A}+a_{B}$. This theorem makes testable predictions about the
relation between the tails of the closing price return distribution, the tails
of the limit order placement distributions and the limit and market order
imbalance. In the next section we will investigate this relation empirically.

##  3 Empirical results

In this section we investigate empirically the relation between the tails of
the closing auction return distributions and the tails of the limit order
placement distributions. In order to do so, we obtain detailed order-by-order
data over 2018 and 2019, for 100 liquid European stocks (with market
capitalization above EUR 1 bn) listed on Euronext exchanges in Amsterdam,
Paris, Brussels or Lisbon.

Estimating the tails of a distribution comes with a couple of problems. First,
the power law of equation (1) is not assumed to hold for all values of $x$,
but only for the tail. This necessarily involves a starting point
$x_{\text{min}}$ such that the power law holds for all $x>x_{\text{min}}$ (see
Newman (2005) for a discussion). Unfortunately, the eventual estimate for the
tail exponent will depend on this cut-off point: if $x_{\min}$ is taken too
small, the bulk instead of the tail will determine the estimates. The cut-off
is often made through visual inspection of a double logarithmic plot. Then the
second problem arises, because the cut-off eliminates most of the available
data, leaving only a small fraction of the data available for estimation.
Finally, models are often designed to describe only ‘generic’ situations well
and are not intended to explain extreme events. It is a noteworthy advantage
of the call auction model of section 2 that it is suitable to model both the
bulk of the data (as in Derksen et al. (2020a)) and extreme events, as in the
current paper.

Concerning the amount of data relevant for the tails, in every closing auction
a large amount of orders is submitted, so the tails of order placement
distributions can be studied per stock. Unfortunately, this is not possible
for the closing auction return distribution: per stock, we have only around
500 trading days (two years of around 250 trading days per stock) and thus
only that many closing auction returns, which is far insufficient to examine
the tails. For example, if we take the 0.05-quantile for the cut-off point
$x_{\min}$, only about 25 data points reside in the tail, which is too few for
meaningful statistical analysis. So to investigate the tails of the closing
auction return distribution, we merge together the closing auction returns of
all stocks in the sample.

In the entire section, the reference price $x_{0}$ will be the _volume
weighted average price_ over the last five minutes of continuous trading.
Closing auction returns will be measured in log returns with respect to
$x_{0}$. Following Bouchaud et al. (2002) and Zovko and Farmer (2002), limit
order prices are measured in the number of ticks a limit order is placed away
from the reference price $x_{0}$.

###  3.1 Tails of order placement distributions

The mechanism of the call auction makes it possible to study _both_ tails of
_both_ order placement distributions. In figure 2, both tails of the sell
limit order distribution $F_{A}$ and the buy limit order distribution $F_{B}$
are shown in log-log plots, for four stocks that are representative for the
sample.

fa_right_4s(0.4) [t]90,5;$x$ [t]3,60;  CDF [t]60,33;ASML right tail $F_{A}$
[t]56,28;SAINT GOBAIN [t]45,23;SIGNIFY [t]46,18;UBISOFT

(a) Right tail of $F_{A}$.

fb_right_4s(0.4) [t]90,5;$x$ [t]3,60;  CDF [t]60,33;ASML right tail $F_{B}$
[t]56,28;SAINT GOBAIN [t]45,23;SIGNIFY [t]45,18;UBISOFT

(b) Right tail of $F_{B}$.

fa_left_4s(0.4) [t]90,5;$x$ [t]3,60;  CDF [t]59,33;ASML left tail $F_{A}$
[t]56,28;SAINT GOBAIN [t]45,23;SIGNIFY [t]46,18;UBISOFT

(c) Left tail of $F_{A}$.

fb_left_4s(0.4) [t]90,5;$x$ [t]3,60;  CDF [t]59,33;ASML left tail $F_{B}$
[t]56,28;SAINT GOBAIN [t]45,23;SIGNIFY [t]45,18;UBISOFT

(d) Left tail of $F_{B}$.

Figure 2:  Log-log plots of the tails of the order placement distributions for
4 selected stocks (ASML Holding NV, Compagnie de Saint Gobain SA, Signify NV,
Ubisoft Entertainment SA ). The $x$-axes show the number of ticks above (for
the right tail) or below (for the left tail) the reference price $x_{0}$.

Let us first focus on the right tails, i.e. the upper panels (a) and (b) of
figure 2. The plots of the right tails of $F_{A}$ show apparent power law
behaviour in the range between 10 and 1000 ticks above the reference price.
After circa 1000 ticks the tails decay faster for a while, but starting around
5000 ticks a new part of the distribution seems to start. The plot is cut-off
at 10 000 ticks, but some even reach until 100 000 ticks. These extremes do
not contribute to price formation in the auction at all. We focus on the
interval of the price axis where price formation occurs: the intersection of
the supports of $F_{A}$ and $F_{B}$. For the right tail that means $F_{B}$
provides the effective upper bound (note that the closing price can never take
a value above the highest buy order). The support of $F_{B}$ ranges until
around 1000-2000 ticks above the reference price so that is the region we use
in our analysis, roughly in line with the intraday results from Zovko and
Farmer (2002)333The sell orders (far) above this region can be thought of as
coming from another distribution describing patient sellers not relevant to
the auction result. To sketch how irrelevant those orders are: the tick size
of a stock is normally between 1 and 5 basis points. Assuming a tick size of
2.5 basis points, 2000 ticks correspond to a return of 50%, while a closing
auction return in the order of 1% is already high.. Power law behaviour is
less clear for $F_{B}$, but in the range of 100 until 1000 ticks power law
behaviour can be recognized for the liquid stocks ASML and Saint Gobain. For
the less liquid stocks Signify and Ubisoft it stops earlier around 500 ticks,
but this can also be due to smaller volumes of available data. The lower
panels (c) and (d) of figure 2 show the left tails of the order placement
distributions. These are very similar to the right tails, when the roles of
$F_{A}$ and $F_{B}$ are switched. Also, on the left side there is a real cut-
off point, corresponding to price 0, which is found somewhere between 2000 and
10 000 ticks.

both_right_ASML(0.4) [t]90,5;$x$ [t]3,60;  CDF [t]58,33;ASML right tail
$F_{A}$ [t]67,28;Linear fit, slope =-1.07 [t]58,23;ASML right tail $F_{B}$
[t]67,18;Linear fit, slope =-2.37 both_right_GOBAIN(0.4) [t]90,5;$x$ [t]3,60;
CDF [t]64,33;GOBAIN right tail $F_{A}$ [t]67,28;Linear fit, slope =-0.40
[t]64,23;GOBAIN right tail $F_{B}$ [t]67,18;Linear fit, slope =-2.01
both_right_UBISOFT(0.4) [t]90,5;$x$ [t]3,60;  CDF [t]64,33;UBISOFT right tail
$F_{A}$ [t]67,28;Linear fit, slope =-0.58 [t]64,23;UBISOFT right tail $F_{B}$
[t]67,18;Linear fit, slope =-3.63 both_right_SIGNIFY(0.4) [t]90,5;$x$ [t]3,60;
CDF [t]64,33;SIGNIFY right tail $F_{A}$ [t]67,28;Linear fit, slope =-0.89
[t]64,23;SIGNIFY right tail $F_{B}$ [t]67,18;Linear fit, slope =-3.69

Figure 3:  Log-log plots of the right tails of the order placement
distributions for 4 stocks (ASML Holding NV, Compagnie de Saint Gobain SA,
Signify NV, Ubisoft Entertainment SA). The $x$-axes show the number of tick
sizes above the reference price $x_{0}$. Linear fits are also plotted, fitted
on the 0.05-quantile of $F_{B}$ until the 0.001-quantile of $F_{B}$, to
estimate $a_{B}$ and $a_{A}$

In figure 3 we zoom in on the right tails of $F_{B}$ and $F_{A}$ until around
1000 tick sizes above the reference price and provide linear fits as
estimators for the values of $a_{A}$ and $a_{B}$ (the tail exponents of
$F_{A}$ and $F_{B}$ as in assumption 4). We perform linear least square fits
on the log-log plots of the tails of $F_{B}$, starting at its 0.05-quantile.
Visual inspection shows that in the extreme tails, available data points are
too sparse to form a coherent picture. So we stop the fit at the
0.001-quantile of $F_{B}$, which seems reasonable when inspecting the plots
and we make fits for $F_{A}$ on the same interval444These choices are somewhat
arbitrary, but cut-off choices need to be made in any practical tail analysis
(see Newman (2005)) and moreover, results do not change substantially when we
extend the fit to e.g. the 0.0001-quantile, or e.g. start the fit at the
0.01-quantile.. For example for ASML, we obtain $a_{A}\approx
1.07,~{}a_{B}\approx 2.37$, fitted on the interval of 168 until 862 tick
sizes. For all four stocks, $F_{A}$ shows a straight, slowly decaying line,
resembling a power law with exponents around or even below 1. Furthermore, the
tails of $F_{B}$ decay faster than the tails of $F_{A}$, with exponents
between 2 and 4 (more results are discussed in section 3.4).

###  3.2 Tails of closing auction return distributions

For every stock $i$ and day $1\leq d\leq n$ we have a closing auction return
$X_{i,d}$, defined as

$X_{i,d}=\log(C^{i,d})-\log(x_{0}^{i,d}),$

where $C^{i,d}$ is the closing price of stock $i$ on day $d$ and $x_{0}^{i,d}$
is the reference price of stock $i$ on day $d$. Following e.g. Gopikrishnan et
al. (1998) we standardize the returns per stock. That is, we divide for every
stock $i$ the return sample $\\{X_{i,d}:1\leq d\leq n\\}$ by its standard
deviation and obtain a sample of standardized returns of size $n\approx 500$.
These samples are all merged together into one large sample to study the tails
of the closing auction return distributions. In figure 4 the right and left
tails of the return distribution are shown in log-log plots, showing clear
power law behaviour from 2 up to 10 standard deviations for both tails. Linear
least square fits are also shown (starting the fit at 2 standard deviations),
giving tail exponents $a=5.28$ for the left tail and $a=4.74$ for the right
tail.

right_left_tails_100s_long(0.55) [t]90,5;$x$ [t]7,90;  CDF [t]106,153;Right
tail [t]123,148;Linear fit, slope$=-4.74$ [t]104.5,143;Left tail
[t]123,138;Linear fit, slope$=-5.28$

Figure 4:  Log-log plot of the tails of the closing auction return
distribution for all 100 stocks in our sample. Blue dots show the right tail,
that is $P(X>x)$, red squares the left tail, that is $P(X<-x)$, the $x$-axis
is in standardized returns. Linear fits are also plotted, giving a tail
exponent of $a=4.74$ for the right tail and $a=5.28$ for the left tail.

This suggests closing auction returns are less heavy tailed than intraday
returns over short time intervals, for which a tail exponent $a\approx 3$ is
widely supported in the literature (see e.g. Gopikrishnan et al. (1998)). This
difference might be explained in qualitative terms by the large transacted
volumes in the closing auctions. It is known that tails of return
distributions become thinner when longer time intervals are considered, an
effect that is known as _aggregational Gaussianity_ (the empirical fact that
return distributions converge to normal distributions when the interval length
increases, see e.g. Cont (2001)). This is theoretically supported by the call
auction model: the clearing price distribution approaches a normal
distribution, when the number of orders tends to infinity (see Derksen et al.
(2020a), theorem 3.1). Moreover, the empirical effect is known to be stronger
if time intervals are measured in trade time (Chakraborti et al., 2011). In
Europe nowadays around 30% of the daily volume is transacted in the closing
auction, which makes the duration of the closing auction in trade time similar
to approximately half a day of continuous trading555The fraction of daily
transacted volume that is transacted in closing auctions has increased greatly
over the past years, especially since the introduction of MiFID II, see
Derksen et al. (2020b)..

###  3.3 The effect of market orders

Before we study the influence of market orders on the tail behaviour of
closing auction return distributions, we first investigate the relation
between the market order imbalance and the limit order imbalance. In figure 5
the market order imbalance $\Delta=M_{B}-M_{A}$ in every closing auction is
plotted against the limit order imbalance $N_{A}-N_{B}$ in that closing
auction, for the four stocks that were also studied in section 3.1. The figure
shows that the proportionality relation between $\Delta$ and $N_{A}-N_{B}$
introduced in equation (9) holds approximately, with values of $c$ in the
range 0.2-0.4, estimated using linear least square regression. This means that
_limit order imbalance is generally in the opposite direction of market order
imbalance._ An explanation for this lies in the chronology of the closing
auction. We observe in auction data that the vast majority of market orders is
submitted in the first seconds of the closing auction, revealing the market
order imbalance early in the auction (during the accumulation phase of the
auction, information on the imbalance and an indicative price is released, so
it is possible to act on this information). Subsequently, limit orders are
placed against the direction of the market order imbalance, reflecting
strategic behaviour: when there is a large positive market order imbalance
(more buy market orders than sell market orders), one can submit a (possibly
large) sell order without adversely affecting the price.

cvalue_ASML(0.4) [t]0,70;  $\Delta$ [t]90,7;$N_{A}-N_{B}$ [t]54,104;ASML,
$c$=0.329 cvalue_GOBAIN(0.4) [t]0,70;  $\Delta$ [t]90,7;$N_{A}-N_{B}$
[t]58,104;GOBAIN, $c$=0.236 cvalue_UBISOFT(0.4) [t]0,70;  $\Delta$
[t]90,7;$N_{A}-N_{B}$ [t]112,104;UBISOFT, $c$=0.204 cvalue_SIGNIFY(0.4)
[t]0,70;  $\Delta$ [t]90,7;$N_{A}-N_{B}$ [t]58,104;SIGNIFY, $c$=0.252

Figure 5:  The difference $N_{A}-N_{B}$ plotted against the market order
imbalance $\Delta$, showing limit order imbalance goes against the direction
of market order imbalance. The dashed red line is the result of linear least
square regression, to estimate the value of $c$ in equation (9), which is the
slope of the dashed red line (outliers of more than four standard deviations
away from the mean are removed).

Next, we will investigate the effect of market orders on the tail exponents.
Consider figure 6, where two auction results are shown. Supply and demand
curves are represented by the solid lines and the point of intersection is the
closing price, indicated by the black star. When market orders are removed,
translated supply and demand curves (plotted by the dashed lines) lead to an
alternative closing price, represented by the black square. The upper panel
shows a situation in which a large positive closing auction return is caused
by a high market order imbalance. When the market order imbalance would be
removed, the closing price would be much lower (black square). The lower panel
shows a very different situation: a small positive closing auction return, but
a strongly negative market order imbalance. If in this case the market order
imbalance would be removed, the closing auction return would get much higher
(black square).

atypical_auction_ASML_2018-03-16(0.4) [t]180,10;Price [t]15,130;  Volume
[t]259,119;$\mathbb{D}_{A}(x)+M_{A}$ [t]259,113;$\mathbb{D}_{B}(x)+M_{B}$
[t]249,108;$\mathbb{D}_{A}(x)$ [t]249,103;$\mathbb{D}_{B}(x)$
[t]265,97;Reference price $x_{0}$ [t]257,92;Closing price
[t]273,87;Alternative closing price

(a) Closing auction ASML, 2018-03-16.

typical_auction_KPN_2018-02-07(0.4) [t]180,10;Price [t]15,130;  Volume
[t]259,119;$\mathbb{D}_{A}(x)+M_{A}$ [t]259,113;$\mathbb{D}_{B}(x)+M_{B}$
[t]249,108;$\mathbb{D}_{A}(x)$ [t]249,103;$\mathbb{D}_{B}(x)$
[t]265,97;Reference price $x_{0}$ [t]257,92;Closing price
[t]273,87;Alternative closing price

(b) Closing auction KPN, 2018-02-07.

Figure 6:  Two closing auction results. Solid lines are the supply (red) and
demand (blue) curves of the particular closing auction, including market
orders (for convenience sell (buy) market orders are placed just below (above)
the lowest sell (highest buy) limit order). Dashed lines show the supply and
demand curves without market orders. The black dot denotes the reference price
$x_{0}$, the black star denotes the closing price and the black square denotes
the alternative price when only limit orders are considered.

The two scenarios presented in figure 6 raise the question which is more
common: are large closing auction returns caused by large market order
imbalances or is this potential effect cancelled by limit order imbalance and
are limit orders usually the driver of large returns? To answer this question,
we also investigate the tails of the return distribution of the _alternative
closing price_ , defined as the intersection point of the supply and demand
curves when the market orders are removed (black squares in figure 6). So, for
every stock $i$ and day $d$ we have an alternative closing auction return
$\tilde{X}_{i,d}$, defined as

$\tilde{X}_{i,d}=\log(\tilde{C}^{i,d})-\log(x_{0}^{i,d}),$

where $\tilde{C}_{i,d}$ is the alternative closing price of stock $i$ on day
$d$. We again standardize these returns per stock, giving for every stock
around 500 alternative closing auction returns, which are merged to study the
tails. In figure 7 the tails of the alternative closing price return
distribution are shown, together with the tails of the real closing price
return distribution from figure 4. The figure shows that the tails become
_heavier_ when market orders are removed. For the right tail we document a
tail exponent $a=3.75$ without market orders, compared to $a=4.74$ with market
orders. For the left tail, the tail exponent becomes $a=3.9$ when market
orders are removed, compared to the value $a=5.28$ when market orders are
included.

right_tails_mos_100s(0.4) [t]90,5;$x$ [t]117,153;Without market orders
[t]118,148;Linear fit, slope$=-3.75$ [t]112,143;With market orders
[t]118,138;Linear fit, slope$=-4.74$

(a) Right tail return distribution, $P(X>x)$

left_tails_mos_100s(0.4) [t]90,5;$x$ [t]117,153;Without market orders
[t]118,148;Linear fit, slope$=-3.9$ [t]112,143;With market orders
[t]118,138;Linear fit, slope$=-5.28$

(b) Left tail return distribution, $P(X<-x)$

Figure 7:  Log-log plot of the tails of the closing auction return
distribution for all 100 stocks in the sample. Blue dots show the tails for
the real closing auction return distribution, red dots the tails for the
alternative closing auction returns that emerge when market orders are
removed.

It is thus concluded that large closing price fluctuations are in general not
caused by a large market order imbalance (at least, not directly). The
explanation for this counter-intuitive result lies in the chronology of the
auction and the placement of limit orders: when the market order imbalance is
positive (negative), there are more sell (buy) limit orders submitted (cf.
figure 5). Theorems 2.7 and 2.9 give the model’s view on the matter and state
that without market orders the tail exponent is equal to $a_{A}+a_{B}$ and
with market orders it is equal to $\min(\frac{(c+1)a_{A}}{c},a_{A}+2a_{B})$.
This means that tails get heavier without market orders, whenever

$\displaystyle c\leq\frac{a_{A}}{a_{B}}.$ (11)

This equation in fact resembles two conditions that should be fulfilled to
make it possible that tails are heavier without market orders (see also
equation (2)). First, $c$ should be small and positive, reflecting that the
abovementioned strategic behaviour is strong: when there is a large market
order imbalance, in general the limit order difference overcompensates for
this. Second, $a_{B}$ should not be too large compared to $a_{A}$. This is a
condition on the right tail of the buy limit order distribution. Without
market orders, the highest buy limit order serves as an upper bound for the
closing price. So to obtain heavier tails without market orders, the right
tail of $F_{B}$ should be sufficiently heavy (small $a_{B}$). It turns out
that condition (11) is indeed satisfied for most of the stocks: for example,
for ASML we obtained estimators $a_{B}\approx 2.37,a_{A}\approx 1.07$,
$c\approx 0.329$ (cf. figures 3 and 5), satisfying the condition in equation
(11). Indeed, theorems 2.7 and 2.9 imply that the tail exponent for closing
auction returns of ASML is $a_{A}+a_{B}=3.44$ without market orders and
$\frac{(c+1)a_{A}}{c}=4.32$ with market orders. In the next subsection we will
verify the theoretical results on the whole sample consisting of 100 stocks.

###  3.4 Model-predicted and realized tail exponents compared

In this subsection the relations predicted by the model are tested over the
whole sample of 100 stocks. For every stock we estimate the tail exponents of
the order placement distributions ($a_{A}$ and $a_{B}$) and the value of the
parameter $c$ (as in equation (9)). The results are shown in table 2 (for 50
stocks with the lower market capitalizations) and table 3 (for 50 stocks with
the higher market capitalizations). To estimate the parameter $c$, we use
linear least squares regression and to estimate the values of $a_{A}$ and
$a_{B}$ we use the method described in section 3.1: for every stock, we make
linear least square fits on double logarithmic plots as in figure 3, on the
interval between the 0.05- and 0.001-quantiles of $F_{B}$. The absolute values
of the resulting slopes are the estimators for $a_{A}$ and $a_{B}$. For
example, for ASML we obtain in this way estimators $a_{B}\approx
2.37,~{}a_{A}\approx 1.07$ and for Ubisoft we find $a_{B}\approx
3.63,~{}a_{A}\approx 0.58$, cf. figure 3. In tables 2 and 3 the results are
shown for all stocks in the sample, the columns $a_{B}(r)$ and $a_{A}(r)$ give
the estimated tail exponents for the right tails of $F_{B}$ and $F_{A}$. For
the left tails, the same method applies when the roles of $F_{B}$ and $F_{A}$
are interchanged. On the left side, $F_{B}$ has a heavier tail and $F_{A}$
provides the effective lower bound.

both_left_ASML(0.4) [t]90,5;$x$ [t]3,60;  CDF [t]58,33;ASML left tail $F_{A}$
[t]67,28;Linear fit, slope =-2.50 [t]58,23;ASML left tail $F_{B}$
[t]67,18;Linear fit, slope =-1.17 both_left_UBISOFT(0.4) [t]90,5;$x$ [t]3,60;
CDF [t]64,33;UBISOFT left tail $F_{A}$ [t]67,28;Linear fit, slope =-2.81
[t]64,23;UBISOFT left tail $F_{B}$ [t]67,18;Linear fit, slope =-0.87

Figure 8:  Log-log plots of the left tails of the order placement
distributions for ASML Holding NV and Ubisoft Entertainment SA. The $x$-axis
shows the number of tick sizes above the reference price $x_{0}$. Linear fits
are also plotted, fitted on the 0.05-quantile of $F_{A}$ until the
0.001-quantile of $F_{A}$, to estimate $a_{B}$ and $a_{A}$

smallcaps_right_long(0.4) [t]90,5;$x$ [t]5,90;  CDF [t]117,153;Without market
orders [t]119,148;Linear fit, slope$=-4.10$ [t]112,143;With market orders
[t]119,138;Linear fit, slope$=-4.95$

(a) Small caps, right tail return distribution.

smallcaps_left_long(0.4) [t]90,5;$x$ [t]5,90;  CDF [t]117,153;Without market
orders [t]119,148;Linear fit, slope$=-4.11$ [t]112,143;With market orders
[t]119,138;Linear fit, slope$=-5.26$

(b) Small caps, left tail return distribution.

largecaps_right_long(0.4) [t]90,5;$x$ [t]5,90;  CDF [t]117,153;Without market
orders [t]119,148;Linear fit, slope$=-3.48$ [t]112,143;With market orders
[t]119,138;Linear fit, slope$=-4.60$

(c) Large caps, right tail return distribution.

largecaps_left_long(0.4) [t]90,5;$x$ [t]5,90;  CDF [t]117,153;Without market
orders [t]119,148;Linear fit, slope$=-3.75$ [t]112,143;With market orders
[t]119,138;Linear fit, slope$=-5.39$

(d) Large caps, left tail return distribution.

Figure 9:  Log-log plots of the tails of the closing auction return
distributions for the 50 small cap stocks of table 2 (upper panel) and 50
large cap stocks of table 3 (lower panel). Blue dots show the tails for the
real closing auction return distribution, red dots the tails for the
alternative closing auction returns that emerge when market orders are
removed.

In figure 8 the left tails of the order placement distributions are shown for
ASML and Ubisoft, as well as the linear least square fits, showing that for
the left tails $a_{A}\approx 2.50,~{}a_{B}\approx 1.17$ for ASML and
$a_{A}\approx 2.81,~{}a_{B}\approx 0.87$ for Ubisoft. In tables 2 and 3 the
columns $a_{B}(l)$ and $a_{A}(l)$ give the estimated tail exponents for the
left tails of $F_{B}$ and $F_{A}$. Estimates for $a_{A},~{}a_{B}$ and $c$ give
rise to an estimate for the tail exponent $\underline{a}$ for the return
distribution of that particular stock. With market orders
$\underline{a}=\min(\frac{(c+1)a_{A}}{c},a_{A}+2a_{B})$ (cf. theorem 2.9) and
without market orders $\underline{a}=a_{A}+a_{B}$ (cf. theorem 2.7)666Note
that for the left tails the roles of $a_{A}$ and $a_{B}$ need to be
interchanged (see also remark 2.2).. Ideally, we would test these predictions
against the realized tail exponents of the return distribution _for every
stock_. However, as noted in the beginning of this section, this is not
possible, because we only have around 500 closing auction returns per stock.
Instead, we can verify the predictions over groups of stocks, by comparing
estimated tail coefficients with the model’s average predicted values.

First, consider the whole sample of 100 stocks. In figure 7 it was shown that
the right tail of the closing price return distribution has an estimated tail
exponent of $a=4.74$, which changes to $a=3.75$ if market orders are removed.
If we take the average of the model’s predictions over all 100 stocks, we find
an average predicted tail exponent of 4.89 with market orders (column
‘$\underline{a}(r)$ MO’ in tables 2 and 3) and 3.89 without market orders
(column ‘$\underline{a}(r)$ no MO’ in tables 2 and 3). Furthermore, figure 7
shows that the left tail of the closing price return distribution has an
estimated tail exponent of $a=5.28$, which changes to $a=3.90$ if the market
orders are removed. For the left tail, the average predicted tail exponent
over all 100 stocks equals 5.01 with market orders (column ‘$\underline{a}(l)$
MO’ in tables 2 and 3) and 3.72 without market orders (column
‘$\underline{a}(l)$ no MO’ in tables 2 and 3). The predicted tail exponents
vary a lot between the different stocks, suggesting that the heaviness of the
tails depends on the stock. To additionally test if these per stock
predictions give information about the real tail exponents, we split our
sample into 50 stocks with the lower market caps (those in table 2) and 50
stocks with the higher market caps (table 3). In that way, the groups are kept
large enough to examine the tails of the closing auction return distributions.

In figure 9 the tails of the closing auction return distribution for the 50
small caps and the 50 large caps are shown in double logarithmic plots, again
with and without market orders (similar to figure 7). The linear fits to the
double logarithmic plots are the realized tail exponents for the both groups,
which can again be compared to the average predicted values in tables 2 and 3.
The results are summarized in table 1, showing first of all that the model’s
predicted exponents are quite close to the realized exponents. Given that
estimation of tails (and tail exponents in particular) is generally thought of
as a difficult statistical problem, the congruence is quite remarkable.
Second, based on the modelling assumption in equation (9), the model predicts
correctly that the tails get heavier if market orders are removed, and by how
much. The theoretical predictions are especially accurate for the case without
market orders, which is not surprising: theorem 2.7 holds very generally and
follows directly from the mechanics of the closing auction. For the case with
market orders, more assumptions were made (see assumption 3). Most
importantly, we assumed equation (9) holds true, which of course in reality
holds only approximately (see also figure 5). When looking at tables 2 and 3,
the predictions for the case with market orders vary strongly between the
stocks. We do not claim that the most extreme values that are predicted are
close to reality, but we have shown that, on average, model predicted and
realized tail exponents match remarkably well.

| Left tails |  | Right tails
---|---|---|---
| MO | No MO |  | MO | No MO
| Predicted | Realized | Predicted | Realized |  | Predicted | Realized | Predicted | Realized
All stocks | 5.01 | 5.28 | 3.72 | 3.90 |  | 4.89 | 4.74 | 3.89 | 3.75
Small caps | 5.76 | 5.26 | 4.14 | 4.11 |  | 5.19 | 4.95 | 4.17 | 4.10
Large caps | 4.25 | 5.39 | 3.29 | 3.75 |  | 4.59 | 4.60 | 3.61 | 3.48
Table 1:  Average predicted tail exponents compared to realized tail
exponents. Predicted exponents are averages over tables 2 and 3, realized
exponents are the results of the linear fits in figures 7 (all stocks) and 9
(small and large caps), for the cases with (MO) and without (No MO) market
orders.

Stock | Exch. | Mcap |  $a_{A}$(l) |  $a_{B}$(l) |  $a_{A}$(r) |  $a_{B}$(r) | $c$ |  $\underline{a}$(l) no MO |  $\underline{a}$(l) MO |  $\underline{a}$(r) no MO |  $\underline{a}$(r) MO
---|---|---|---|---|---|---|---|---|---|---|---
ASM INTL | AMS | 6.7 | 3.312 | 1.154 | 0.717 | 3.377 | 0.127 |  4.466 |  7.778 |  4.094 |  6.357
AALBERTS | AMS | 3.8 | 3.178 | 0.763 | 1.207 | 3.301 | 0.174 |  3.941 |  5.135 |  4.508 |  7.808
WDP REIT | BRU | 5.2 | 3.142 | 3.279 | 1.739 | 3.032 | 0.113 |  6.421 |  9.562 |  4.771 |  7.804
REXEL | PAR | 3.2 | 2.105 | 1.782 | 1.465 | 2.626 | 0.184 |  3.887 |  5.992 |  4.091 |  6.718
EURONEXT | PAR | 6.8 | 3.319 | 1.943 | 1.918 | 3.229 | 0.164 |  5.262 |  8.582 |  5.146 |  8.375
IMCD GROUP | AMS | 6.0 | 3.328 | 0.972 | 1.859 | 3.016 | 0.158 |  4.300 |  7.118 |  4.875 |  7.892
SIGNIFY | AMS | 4.6 | 3.485 | 1.181 | 0.888 | 3.690 | 0.252 |  4.666 |  5.866 |  4.579 |  4.413
ALTEN | PAR | 2.8 | 3.443 | 1.476 | 1.017 | 2.420 | 0.137 |  4.919 |  8.362 |  3.437 |  5.857
BIC | PAR | 1.9 | 3.025 | 1.644 | 0.916 | 3.820 | 0.324 |  4.669 |  6.720 |  4.736 |  3.746
EUTELSAT COM | PAR | 1.9 | 2.969 | 0.993 | 0.677 | 3.752 | 0.201 |  3.962 |  5.972 |  4.429 |  4.068
INGENICO GROUP | PAR | 8.5 | 1.534 | 0.970 | 0.740 | 2.996 | 0.129 |  2.504 |  4.038 |  3.735 |  6.461
EURAZEO | PAR | 3.3 | 4.512 | 1.953 | 1.536 | 3.928 | 0.134 |  6.464 |  10.976 |  5.464 |  9.393
AEGON | AMS | 5.0 | 1.552 | 0.801 | 0.421 | 2.561 | 0.15 |  2.353 |  3.905 |  2.982 |  3.220
KPN KON | AMS | 10.0 | 2.363 | 1.011 | 0.584 | 3.942 | 0.207 |  3.373 |  5.736 |  4.526 |  3.413
RANDSTAD | AMS | 8.4 | 3.071 | 1.130 | 0.939 | 3.603 | 0.291 |  4.201 |  5.016 |  4.541 |  4.166
KLEPIERRE REIT | PAR | 3.5 | 3.242 | 1.016 | 0.503 | 3.317 | 0.371 |  4.259 |  3.754 |  3.820 |  1.858
SUEZ | PAR | 9.9 | 1.416 | 0.866 | 0.724 | 2.588 | 0.250 |  2.281 |  3.697 |  3.312 |  3.613
GALP ENERGIA | LIS | 6.8 | 4.467 | 2.173 | 1.438 | 3.977 | 0.484 |  6.640 |  6.663 |  5.415 |  4.410
ARKEMA | PAR | 7.2 | 4.292 | 1.231 | 1.354 | 4.441 | 0.278 |  5.522 |  5.659 |  5.795 |  6.225
COVIVIO | PAR | 5.2 | 3.400 | 3.485 | 2.230 | 3.279 | 0.304 |  6.884 |  10.284 |  5.509 |  8.788
ICADE REIT | PAR | 3.4 | 4.179 | 1.649 | 1.371 | 3.499 | 0.231 |  5.828 |  8.798 |  4.870 |  7.314
IPSEN | PAR | 6.5 | 2.515 | 1.463 | 0.844 | 2.594 | 0.200 |  3.978 |  6.493 |  3.439 |  5.072
ORPEA | PAR | 5.9 | 1.605 | 0.786 | 0.987 | 1.683 | 0.126 |  2.391 |  3.997 |  2.671 |  4.354
SCOR | PAR | 4.5 | 3.363 | 1.475 | 0.941 | 3.944 | 0.378 |  4.837 |  5.372 |  4.885 |  3.427
GETLINK | PAR | 6.4 | 1.910 | 1.341 | 1.166 | 3.294 | 0.148 |  3.251 |  5.161 |  4.460 |  7.753
J.MARTINS SGPS | LIS | 9.2 | 4.043 | 1.022 | 1.027 | 3.621 | 0.276 |  5.065 |  4.730 |  4.648 |  4.752
DASSAULT AVIAT | PAR | 6.3 | 4.021 | 1.155 | 0.764 | 4.088 | 0.282 |  5.176 |  5.247 |  4.852 |  3.471
EDENRED | PAR | 10.1 | 3.832 | 2.070 | 2.352 | 3.763 | 0.282 |  5.902 |  9.419 |  6.115 |  9.879
PUBLICIS GROUPE | PAR | 7.6 | 2.600 | 1.251 | 0.704 | 3.221 | 0.390 |  3.851 |  4.462 |  3.925 |  2.509
ATOS | PAR | 7.5 | 2.323 | 0.834 | 0.499 | 2.449 | 0.267 |  3.157 |  3.959 |  2.948 |  2.368
JCDECAUX | PAR | 2.9 | 3.682 | 1.578 | 1.177 | 3.845 | 0.229 |  5.260 |  8.482 |  5.022 |  6.323
EIFFAGE | PAR | 6.9 | 4.086 | 1.324 | 1.785 | 4.083 | 0.248 |  5.410 |  6.656 |  5.868 |  8.973
GECINA | PAR | 7.8 | 2.988 | 2.394 | 2.004 | 2.928 | 0.321 |  5.382 |  8.370 |  4.932 |  7.860
NATIXIS | PAR | 6.5 | 0.763 | 0.513 | 0.524 | 1.672 | 0.155 |  1.276 |  2.040 |  2.196 |  3.868
SES FDR | PAR | 3.0 | 3.138 | 0.649 | 0.905 | 3.218 | 0.186 |  3.786 |  4.132 |  4.123 |  5.768
SEB | PAR | 7.6 | 3.998 | 1.284 | 0.998 | 3.807 | 0.215 |  5.282 |  7.246 |  4.805 |  5.629
UBISOFT | PAR | 10.2 | 2.814 | 0.870 | 0.578 | 3.635 | 0.204 |  3.684 |  5.136 |  4.213 |  3.409
ALSTOM | PAR | 9.4 | 1.988 | 0.715 | 0.866 | 2.955 | 0.279 |  2.703 |  3.280 |  3.821 |  3.975
TECHNIPFMC | PAR | 2.6 | 1.113 | 0.729 | 0.552 | 2.478 | 0.217 |  1.842 |  2.955 |  3.029 |  3.097
ACCOR | PAR | 5.9 | 2.384 | 0.677 | 0.645 | 3.079 | 0.251 |  3.061 |  3.374 |  3.724 |  3.215
VEOLIA | PAR | 9.8 | 1.476 | 0.838 | 0.484 | 2.598 | 0.267 |  2.314 |  3.791 |  3.083 |  2.296
COLRUYT | BRU | 7.3 | 3.340 | 1.373 | 1.245 | 3.390 | 0.318 |  4.713 |  5.689 |  4.635 |  5.159
AGEAS | BRU | 6.7 | 2.408 | 1.115 | 1.254 | 2.735 | 0.321 |  3.524 |  4.585 |  3.989 |  5.155
SOLVAY | BRU | 8.0 | 1.866 | 0.856 | 0.440 | 1.241 | 0.193 |  2.722 |  4.589 |  1.681 |  2.725
UMICORE | BRU | 8.9 | 2.428 | 0.834 | 0.486 | 2.003 | 0.314 |  3.262 |  3.490 |  2.489 |  2.033
PROXIMUS | BRU | 5.2 | 2.618 | 0.845 | 0.666 | 2.820 | 0.271 |  3.463 |  3.964 |  3.486 |  3.126
ABN AMRO BANK | AMS | 6.9 | 1.989 | 1.216 | 0.627 | 3.004 | 0.195 |  3.205 |  5.194 |  3.631 |  3.838
CNP ASSURANCES | PAR | 7.3 | 3.899 | 1.615 | 2.046 | 2.756 | 0.381 |  5.514 |  5.855 |  4.802 |  7.418
UNIBAIL RODAMCO | AMS | 5.7 | 1.983 | 1.368 | 1.052 | 1.947 | 0.390 |  3.351 |  4.876 |  2.999 |  3.750
SODEXO | PAR | 8.8 | 3.559 | 1.852 | 1.696 | 3.668 | 0.245 |  5.411 |  8.970 |  5.364 |  8.629
Average | - | 6.4 | 2.84 | 1.30 | 1.06 | 3.12 | 0.24 |  4.14 |  5.76 |  4.17 |  5.19
Table 2:  Table of results, for the 50 stocks in our sample with the lower market cap. The column Exch. displays the exchange the stock is traded on (Amsterdam, Paris, Brussels or Lisbon) and the column Mcap shows the market capitalization of the stock in billions of euros (in October 2020). Then, $a_{A}$ and $a_{B}$ are the estimated tail exponents of sell and buy limit order distributions, for the left (l) and right (r) tail. $c$ is the estimator for the constant in equation (9) and $\underline{a}=a_{A}+a_{B}$ without market orders (no MO), and $\underline{a}=\min(\frac{(c+1)a_{A}}{c},a_{A}+2a_{B})$ with market orders (MO), both displayed for left (l) and right (r) tails. Stock | Exch. | Mcap |  $a_{A}$(l) |  $a_{B}$(l) |  $a_{A}$(r) |  $a_{B}$(r) | $c$ |  $\underline{a}$(l) no MO |  $\underline{a}$(l) MO |  $\underline{a}$(r) no MO |  $\underline{a}$(r) MO
---|---|---|---|---|---|---|---|---|---|---|---
AMUNDI | PAR | 12.3 | 2.066 | 1.544 | 0.816 | 2.496 | 0.110 |  3.610 |  5.676 |  3.311 |  5.807
BIOMERIEUX ORD | PAR | 16.4 | 2.815 | 1.595 | 2.294 | 3.312 | 0.247 |  4.410 |  7.225 |  5.606 |  8.917
NN GROUP | AMS | 10.9 | 3.187 | 1.051 | 1.010 | 3.797 | 0.376 |  4.238 |  3.847 |  4.807 |  3.698
SARTORIUS | PAR | 28.9 | 2.928 | 1.069 | 1.449 | 2.474 | 0.280 |  3.997 |  4.892 |  3.923 |  6.397
WORLDLINE | PAR | 13.2 | 2.386 | 1.001 | 1.131 | 2.902 | 0.241 |  3.386 |  5.146 |  4.034 |  5.819
EDP | LIS | 18.0 | 3.236 | 0.710 | 1.043 | 3.683 | 0.341 |  3.946 |  2.793 |  4.727 |  4.106
TELEPERFORMANCE | PAR | 16.1 | 3.869 | 1.862 | 1.256 | 4.010 | 0.134 |  5.731 |  9.601 |  5.266 |  9.276
BOUYGUES | PAR | 11.7 | 2.302 | 1.124 | 0.953 | 2.452 | 0.331 |  3.425 |  4.518 |  3.405 |  3.833
AHOLD DEL | AMS | 26.6 | 1.334 | 0.727 | 0.875 | 2.234 | 0.326 |  2.061 |  2.960 |  3.108 |  3.562
AKZO NOBEL | AMS | 17.7 | 2.678 | 1.921 | 1.190 | 2.669 | 0.292 |  4.599 |  7.276 |  3.859 |  5.268
ASML HOLDING | AMS | 138.3 | 2.497 | 1.165 | 2.369 | 1.073 | 0.329 |  3.662 |  4.704 |  3.442 |  4.334
DSM KON | AMS | 24.9 | 3.283 | 1.333 | 1.158 | 3.381 | 0.312 |  4.616 |  5.601 |  4.539 |  4.866
HEINEKEN | AMS | 45.4 | 3.917 | 1.101 | 1.146 | 3.833 | 0.270 |  5.018 |  5.177 |  4.979 |  5.391
ING GROEP | AMS | 24.7 | 1.344 | 1.067 | 0.419 | 2.333 | 0.139 |  2.411 |  3.755 |  2.752 |  3.428
PHILIPS KON | AMS | 37.5 | 3.129 | 0.498 | 0.884 | 3.566 | 0.318 |  3.626 |  2.063 |  4.450 |  3.666
UNILEVER | AMS | 138.7 | 2.929 | 0.619 | 1.417 | 2.879 | 0.322 |  3.548 |  2.543 |  4.297 |  5.818
WOLTERS KLUWER | AMS | 19.3 | 3.473 | 1.417 | 1.823 | 2.997 | 0.244 |  4.891 |  7.235 |  4.820 |  7.817
DANONE | PAR | 34.6 | 2.195 | 1.112 | 0.761 | 2.312 | 0.404 |  3.307 |  3.862 |  3.073 |  2.642
BNP PARIBAS ACT.A | PAR | 40.2 | 1.326 | 0.771 | 0.636 | 2.133 | 0.223 |  2.098 |  3.424 |  2.769 |  3.496
AXA | PAR | 36.1 | 0.740 | 0.698 | 0.548 | 1.878 | 0.321 |  1.438 |  2.178 |  2.426 |  2.253
SOCIETE GENERALE | PAR | 10.2 | 0.815 | 0.923 | 0.383 | 2.051 | 0.160 |  1.738 |  2.553 |  2.434 |  2.777
L’OREAL | PAR | 163.0 | 3.201 | 0.685 | 1.209 | 2.552 | 0.270 |  3.886 |  3.222 |  3.760 |  5.683
SANOFI | PAR | 108.1 | 1.245 | 0.741 | 0.790 | 2.080 | 0.411 |  1.986 |  2.544 |  2.869 |  2.712
SAINT GOBAIN | PAR | 20.0 | 1.242 | 0.936 | 0.402 | 2.014 | 0.237 |  2.178 |  3.420 |  2.416 |  2.099
LEGRAND | PAR | 18.6 | 4.179 | 1.304 | 1.790 | 3.851 | 0.328 |  5.482 |  5.283 |  5.641 |  7.254
TOTAL | PAR | 74.8 | 0.592 | 0.632 | 0.549 | 0.534 | 0.308 |  1.224 |  1.816 |  1.082 |  1.616
HEINEKEN HOLDING | AMS | 20.2 | 3.198 | 1.244 | 2.004 | 3.506 | 0.333 |  4.442 |  4.982 |  5.509 |  8.025
ESSILORLUXOTTICA | PAR | 50.8 | 2.433 | 0.555 | 1.040 | 2.691 | 0.347 |  2.988 |  2.155 |  3.731 |  4.037
AB INBEV | BRU | 93.5 | 1.800 | 0.809 | 0.499 | 2.441 | 0.210 |  2.610 |  4.410 |  2.940 |  2.879
DASSAULT SYSTEM | PAR | 41.4 | 3.381 | 1.439 | 1.144 | 2.779 | 0.341 |  4.819 |  5.653 |  3.923 |  4.495
CHRISTIAN DIOR SE | PAR | 74.0 | 3.482 | 1.585 | 1.860 | 4.092 | 0.095 |  5.068 |  8.550 |  5.952 |  10.044
ARCELORMITTAL | AMS | 13.4 | 0.450 | 0.610 | 0.566 | 2.015 | 0.138 |  1.060 |  1.511 |  2.580 |  4.595
SAFRAN | PAR | 38.7 | 2.944 | 1.467 | 1.154 | 2.380 | 0.386 |  4.411 |  5.267 |  3.534 |  4.141
ENGIE | PAR | 28.3 | 0.485 | 0.652 | 0.269 | 1.846 | 0.138 |  1.136 |  1.621 |  2.114 |  2.220
EDF | PAR | 31.9 | 0.789 | 0.872 | 0.906 | 2.429 | 0.234 |  1.662 |  2.451 |  3.335 |  4.780
CREDIT AGRICOLE | PAR | 21.2 | 0.642 | 0.780 | 0.670 | 1.729 | 0.193 |  1.422 |  2.065 |  2.399 |  4.128
CAPGEMINI | PAR | 18.5 | 2.622 | 0.754 | 0.747 | 2.987 | 0.256 |  3.377 |  3.701 |  3.735 |  3.667
AIRBUS | PAR | 50.4 | 2.921 | 1.232 | 1.320 | 3.268 | 0.116 |  4.153 |  7.075 |  4.588 |  7.856
ORANGE | PAR | 25.3 | 0.480 | 0.925 | 0.515 | 2.162 | 0.250 |  1.405 |  1.885 |  2.676 |  2.577
THALES | PAR | 13.9 | 2.656 | 1.241 | 0.865 | 2.351 | 0.317 |  3.897 |  5.161 |  3.216 |  3.598
MICHELIN | PAR | 16.6 | 2.360 | 0.740 | 0.416 | 2.724 | 0.320 |  3.100 |  3.050 |  3.139 |  1.713
KERING | PAR | 73.7 | 2.572 | 0.864 | 0.684 | 2.463 | 0.332 |  3.436 |  3.466 |  3.146 |  2.742
PERNOD RICARD | PAR | 37.1 | 2.740 | 1.090 | 1.153 | 2.256 | 0.297 |  3.829 |  4.761 |  3.409 |  5.036
SCHNEIDER ELECTRIC | PAR | 58.2 | 2.644 | 1.534 | 1.058 | 2.594 | 0.326 |  4.178 |  6.240 |  3.652 |  4.304
PEUGEOT | PAR | 14.2 | 1.725 | 0.888 | 0.805 | 2.248 | 0.128 |  2.613 |  4.339 |  3.053 |  5.301
ROYAL DUTCH SHELL | AMS | 83.0 | 0.677 | 0.529 | 0.350 | 0.753 | 0.137 |  1.206 |  1.883 |  1.103 |  1.856
GBL | BRU | 11.9 | 2.952 | 1.323 | 1.095 | 2.457 | 0.194 |  4.275 |  7.227 |  3.552 |  6.009
KBC | BRU | 18.5 | 2.124 | 0.702 | 0.716 | 2.881 | 0.217 |  2.826 |  3.944 |  3.597 |  4.018
UCB | BRU | 17.9 | 2.607 | 0.760 | 1.136 | 3.077 | 0.228 |  3.367 |  4.101 |  4.213 |  6.126
VIVENDI | PAR | 27.9 | 1.968 | 0.936 | 0.733 | 2.806 | 0.318 |  2.904 |  3.874 |  3.539 |  3.033
Average | - | 39.7 | 2.27 | 1.02 | 1.00 | 2.61 | 0.26 |  3.29 |  4.25 |  3.61 |  4.59
Table 3:  Table of results, for the 50 stocks in our sample with the higher
market cap. The column Exch. displays the exchange the stock is traded on
(Amsterdam, Paris, Brussels or Lisbon) and the column Mcap shows the market
capitalization of the stock in billions of euros (in October 2020). Then,
$a_{A}$ and $a_{B}$ are the estimated tail exponents of sell and buy limit
order distributions, for the left (l) and right (r) tail. $c$ is the estimator
for the constant in equation (9) and $\underline{a}=a_{A}+a_{B}$ without
market orders (no MO), and
$\underline{a}=\min(\frac{(c+1)a_{A}}{c},a_{A}+2a_{B})$ with market orders
(MO), both displayed for left (l) and right (r) tails.

##  4 Conclusions

In this paper we study the tails of closing auction return distributions, both
from a theoretical and empirical point of view, focusing on large closing
price fluctuations. Using the stochastic call auction model of Derksen et al.
(2020a), we relate tail exponents of order placement distributions and tail
exponents of the return distribution. Empirical analysis supports the model’s
predictions. In theory, large market orders could be a cause of large closing
price fluctuations, but this potential effect is cancelled by limit orders
that are submitted against the direction of the market order imbalance.
Instead, limit order placement appears to be the primary cause of observed
heavy tails in closing auction return distributions.

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##  Appendix A Proofs

See 2.5

Proof The expression for the conditional distribution of $\underline{X}$ in
equation (2.3), implies

$\displaystyle\lim_{M\to\infty}$
$\displaystyle\frac{{\mathbb{P}}(\underline{X}>M|N_{A},N_{B})}{T_{B}(M)T_{A}(M)^{N_{A}}}$
$\displaystyle=\sum_{k=0}^{N_{A}}\sum_{l=k+1}^{N_{B}}\lim_{M\to\infty}{N_{A}\choose
k}{N_{B}\choose
l}\left(\frac{1-F_{A}(M)}{T_{A}(M)}\right)^{N_{A}}(1-F_{A}(M))^{-k}$
$\displaystyle\quad\times\left(\frac{1-F_{B}(M)}{T_{B}(M)}\right)(1-F_{B}(M))^{l-1}$
$\displaystyle=\sum_{k=0}^{N_{A}}\sum_{l=k+1}^{N_{B}}\lim_{M\to\infty}{N_{A}\choose
k}{N_{B}\choose
l}\left(\frac{1-F_{B}(M)}{1-F_{A}(M)}\right)^{k}(1-F_{B}(M))^{l-1-k}$
$\displaystyle=\sum_{k=0}^{N_{A}}\sum_{l=k+1}^{N_{B}}\lim_{M\to\infty}{N_{A}\choose
k}{N_{B}\choose
l}\left(\frac{T_{B}(M)}{T_{A}(M)}\right)^{k}(1-F_{B}(M))^{l-1-k}$
$\displaystyle=N_{B},$

where we exchange limit and sum by dominated convergence, and the last line
follows because all terms are 0, except when $l=1,k=0$. $\Box$

See 2.7

Proof The result of proposition 2.5 implies

$\displaystyle\lim_{M\to\infty}\frac{{\mathbb{P}}(\underline{X}>M)}{CT_{A}(M)T_{B}(M)}$
$\displaystyle=\lim_{M\to\infty}\frac{\mathbb{E}N_{B}T_{B}(M)T_{A}(M)^{N_{A}}}{CT_{A}(M)T_{B}(M)}$
$\displaystyle=\lim_{M\to\infty}\frac{\sum_{i=1}^{N}\sum_{j=1}^{N}p_{N_{A},N_{B}}(i,j)jT_{B}(M)T_{A}(M)^{i}}{CT_{A}(M)T_{B}(M)}$
$\displaystyle=\sum_{i=1}^{N}\sum_{j=1}^{N}\frac{jp_{N_{A},N_{B}}(i,j)}{C}\lim_{M\to\infty}T_{A}(M)^{i-1}$
$\displaystyle=\frac{\sum_{j=1}^{N}jp_{N_{A},N_{B}}(1,j)}{C}=1,$

where sum and limit are exchanged by dominated convergence and the last line
follows by the fact that all terms in the sum are 0, except for $i=1$. $\Box$

See 2.8

Proof Suppose first that $\Delta-1>0$. Then

$\displaystyle\lim_{M\to\infty}$
$\displaystyle\frac{{\mathbb{P}}(\underline{X}>M|N_{A},N_{B},\Delta)}{T_{A}(M)^{N_{A}-(\Delta-1)}}$
$\displaystyle=\sum_{k=0}^{N_{A}}\sum_{l=\max(k-\Delta+1,0)}^{N_{B}}\lim_{M\to\infty}{N_{A}\choose
k}{N_{B}\choose l}\left(\frac{1-F_{A}(M)}{T_{A}(M)}\right)^{N_{A}-(\Delta-1)}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\times(1-F_{A}(M))^{\Delta-1-k}(1-F_{B}(M))^{l}$
$\displaystyle=\sum_{k=0}^{N_{A}}\sum_{l=\max(k-\Delta+1,0)}^{N_{B}}\lim_{M\to\infty}{N_{A}\choose
k}{N_{B}\choose l}\frac{T_{B}(M)^{l}}{T_{A}(M)^{k-\Delta+1}}$
$\displaystyle={N_{A}\choose\Delta-1},$

where the last line follows because all terms are 0, except when
$k=\Delta-1,l=0$.
Now let $\Delta-1<0$, then

$\displaystyle\lim_{M\to\infty}$
$\displaystyle\frac{{\mathbb{P}}(\underline{X}>M|N_{A},N_{B},\Delta)}{T_{A}(M)^{N_{A}}T_{B}(M)^{1-\Delta}}$
$\displaystyle=\sum_{k=0}^{N_{A}}\sum_{l=k+1-\Delta}^{N_{B}}\lim_{M\to\infty}{N_{A}\choose
k}{N_{B}\choose l}\left(\frac{1-F_{A}(M)}{T_{A}(M)}\right)^{N_{A}-k}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\times
T_{A}(M)^{-k}\left(\frac{1-F_{B}(M)}{T_{B}(M)}\right)^{l}T_{B}(M)^{l+\Delta-1}$
$\displaystyle=\sum_{k=0}^{N_{A}}\sum_{l=k+1-\Delta}^{N_{B}}\lim_{M\to\infty}{N_{A}\choose
k}{N_{B}\choose l}\frac{T_{B}(M)^{l+\Delta-1}}{T_{A}(M)^{k}}$
$\displaystyle={N_{B}\choose 1-\Delta},$

where the last line follows because all terms are 0, except when
$k=0,l=1-\Delta$. $\Box$

See 2.9

Proof Under assumptions 3 and 4, proposition 2.8 transforms into,

$\displaystyle{\mathbb{P}}(\underline{X}>M)\sim\sum_{n=1}^{N}\sum_{m=1}^{N}\sum_{d=-N}^{N}K(n,m,d-1)M^{-(a_{A}(n-d+1)+(a_{B}-a_{A})\max(-d+1,0))}p(n,m,d),$

as $M\to\infty$. Here, we used that $\max(-x,0)-\max(x,0)=-x$, for all
$x\in{\mathbb{R}}$. By noting that $K(n,m,d)$ is bounded from above and below
(by ${N\choose N/2}$ and 1), we see that the statement of the theorem holds
true, for

$\underline{a}=\min_{n,d:~{}p(n,d)>0}(a_{A}(n-d+1)-(d-1)(a_{B}-a_{A})\mathbf{1}_{\\{d<1\\}}),$

where the minimum is taken over all $n,d$ such that
$p(n,d)=\sum_{m}p(n,m,d)>0$. Now note that the function
$\underline{F}(n,d):=a_{A}(n-d+1)-(d-1)(a_{B}-a_{A})\mathbf{1}_{\\{d<1\\}}$ is
increasing in $n$, for every $d$. So the minimum is attained for the lowest
$n$ with positive probability. Recall that we assumed $\Delta=c(N_{A}-N_{B})$
and $N_{B}\in\\{1,\dots,N\\}$, so $p(n,d)=0$ for $n<\frac{d}{c}+1$, so the
lowest $n$ with positive probability is $\hat{n}(d)=\max(\frac{d}{c}+1,1)$,
for given $d$. Inserting into $\underline{F}$ leads to

$\underline{F}(\hat{n}(d),d)=\begin{cases}a_{A}-(d-1)a_{B}&\text{ if
}d\leq-1\\\ a_{A}((\frac{1}{c}-1)d+2)&\text{ if }d\geq 1\end{cases},$

which is minimal for $d=\pm 1$, proving that

$\overline{a}=\min\left(a_{A}\left(\frac{1}{c}+1\right),a_{A}+2a_{B}\right)=\min\left(\frac{(c+1)a_{A}}{c},a_{A}+2a_{B}\right).$

$\Box$