# Clearing time randomization and transaction fees for auction market design

Thibaut Mastrolia111UC Berkeley, Department of Industrial Engineering and
Operations Research<EMAIL_ADDRESS>and Tianrui Xu222UC
Berkeley, Department of Mathematics<EMAIL_ADDRESS>
###### Abstract

Flaws of a continuous limit order book mechanism raise the question of whether
a continuous trading session and a periodic auction session would bring better
efficiency. This paper wants to go further in designing a periodic auction
when both a continuous market and a periodic auction market are available to
traders. In a periodic auction, we discover that a strategic trader could take
advantage of the accumulated information available along the auction duration
by arriving at the latest moment before the auction closes, increasing the
price impact on the market. Such price impact moves the clearing price away
from the efficient price and may disturb the efficiency of a periodic auction
market. We thus propose and quantify the effect of two remedies to mitigate
these flaws: randomizing the auction’s closing time and optimally designing a
transaction fees policy. Our results show that these policies encourage a
strategic trader to send their orders earlier to enhance the efficiency of the
auction market, illustrated by data extracted from Alphabet and Apple stocks.

Keywords: Microstructure, auction market design, market making, optimal
stopping

##  1 Introduction

###  1.1 Periodic auction and continuous limit order book

Continuous limit order book (CLOB for short or continuous double auction) and
periodic auction (also called batch auction or call auction) are the two most
commonly used electronic trading systems around the world. For example, New
York Stock Exchange, NASDAQ (U.S.), London Stock Exchange all use the
continuous limit order book system during normal trading hours and switch to a
periodic auction system for the opening and closing auctions to determine the
open price and the closing price of each trading day. CBOE Europe have both
continuous limit order book and periodic auction open during the normal
trading hours. CowSwap a cryptocurrency exchange uses periodic auction to
settle orders.

The trading mechanisms of the two systems are as follows. A continuous limit
order book market executes incoming orders continuously, i.e., when there is a
matching order between a market order and a limit order. Every matched order
trades at a price that depends on its requested price and the price of the
limit order it is matched to. In comparison, a periodic auction market
executes incoming orders as a batch and applies a uniform price to all
executed orders in this batch after a specific time horizon. To be more
specific, a market order would initiate an auction that would be open for a
specific time interval until a terminal time, which is called the clearing
time. During the auction, the exchange receives orders from market
participants. Market participants give the exchange a proposed price at which
they are willing to buy or sell the asset and a specific volume. When the
auction closes, the exchange determines a clearing price by setting it to
maximize the number of fulfilled orders (or to minimize the imbalance). Every
order may be executed at the clearing price instead of their proposed price.
Due to this rule, limit buying orders with a proposed price below the clearing
price and limit selling orders with a proposed price above the clearing price
would not be executed. Despite the difference between the two trading systems,
a continuous limit order book can be considered a periodic auction whose
duration equals $0$ second, see for example [Jusselin et al. (2021)].

###  1.2 Comparison and main flaws of limit order book

The literature to promote general market quality, discover better trading
mechanisms, or improve market competition has been studied since the 60s; see
[Garbade and Silber (1979)]. The continuous trading system has the advantage
of providing ”immediate execution”. No one likes to wait; in [Kalay et al.
(2002)], it is shown empirically that people prefer to trade in a continuous
market instead of an auction market. However, such immediacy also creates a
problem, especially after the emergence of high-frequency traders. [Budish et
al. (2015)], [Wah and Wellman (2013)], and [Farmer and Skouras (2012)]
question the efficiency of limit order mechanism rather than periodic auction.
They study the efficiency of periodic auctions to monitor high-frequency
trading advantages and increase market efficiency. [Budish et al. (2015)]
compares two highly correlated stocks from real data and finds that the
continuous trading system creates arbitrage opportunities in small time
intervals. These arbitrage opportunities could be caught by high-frequency
traders and thus incite competition in speed rather than price. High-frequency
trading has brought down the execution time from several seconds at the start
of the 2000s to microseconds nowadays. [Wah and Wellman (2013)] reaches a
similar conclusion as they use simulations to show that high-frequency traders
are latency arbitrageurs and widen the bid-ask spread. [Farmer and Skouras
(2012)] discusses the negative impacts of high-frequency trading and proposes
using periodic auctions (they also propose pro-rata rules with continuous
market and randomized auction duration with periodic auctions).

Following these works, more studies focus on the advantages and disadvantages
of continuous limit order books and periodic auctions. [Aquilina et al.
(2022)] use exchange message data to quantify the speed competition in [Budish
et al. (2015)]. However, restricting the competition in speed is only one of
the characteristics of the periodic auction system compared to the continuous
system. Recall that the other characteristic of a periodic auction differing
from the CLOB is that the clearing price is set by combining the opinions of a
batch of orders instead of just two. Such characteristics view market supply
and demand more comprehensively and thus could improve the price discovery
process. [Jusselin et al. (2021)] shows that optimally setting a clearing rule
(price discovery and auction duration) for the periodic auction system enables
the clearing price of most assets to be closer to the efficient prices
compared to the continuous limit order book system. However, the continuous
system could sometimes be optimal regarding the above mentioned price
discovery process. [Graf et al. (2024)] shows by using real data that if we
replace the continuous German Electricity Market with a frequent batch
auction, there will be less traded volume but better price discovery (price is
less noisy and closer to the fundamental value) and less liquidity cost
measured in round-trip (CRT) cost. There are, of course, different opinions.
In [Zhang and Ibikunle (2023)], they show empirically that sub-second frequent
batch auction leads to a decline in adverse selection cost but an increase in
relative spread and a decrease in information efficiency measured by
”autocorrelation of midpoint returns”. One thing to note about all these works
is that researchers use different assumptions, models, and measures to reach
their conclusions, so seemingly contrasting conclusions do not necessarily
imply a contradiction.

Echoing the conclusion of [Jusselin et al. (2021)], “One size does not fit
all”. Neither periodic auction nor continuous limit order book is the best by
all measures and neither would benefit all affiliated groups. The interest of
this paper is not to compare periodic auction and CLOB, but to study the
possibility of a co-existence of the two systems. [Derchu et al. (2020)]
proposes an Ad Hoc Electronic Auction Design (AHEAD) which allows traders to
switch between continuous trading sessions and periodic trading sessions. They
show that this design enables a less volatile clearing price and traders
especially the smaller players benefit from this design compared to a
continuous system or a periodic auction system.

In addition to the CLOB and the periodic auction market, there have been
focuses and advances in other trading mechanisms. Dark pools differ from CLOB
and auction in that orders are not displayed to the public; see [Ye (2011)],
[Zhu (2014)], [Ye (2024)], [Baldacci et al. (2023)] for studies on whether a
dark pool harm or help with market efficiency. See [Melton (2017)] for a
latency floor design on CLOB to limit high frequency trading. Note also that
cryptocurrency trading markets have interesting mechanism designs as well; see
[Canidiom and Fritsch (2024)] for combining batch auction with automated
market maker.

We also want to mention additionnal works as for example [Du and Zhu (2017)],
[Brinkman and Wellman (2017)], [Fricke and Gerig (2018)], and [Jusselin et al.
(2021)]. Each of these articles study the design of periodic auction to answer
the following question: What is the optimal auction duration? [Wah et al.
(2016)] uses simulation to show how fast and slow traders would choose between
continuous market and periodic auction if the two markets run together; the
model in this work does not consider strategic timing though. [Duffie and Zhu
(2017)] proposes to add a size discovery market (”workup”) along with a batch
auction market to increase allocation efficiency. A size discovery market
allows traders to exchange inventory at a fixed-price so traders need not to
worry about their price impact. The main differences between their model and
ours are that they assume a strategic player in a batch auction submits a
demand function instead of an order price or quantity and they focus on
balancing the inventory level of each strategic player as they assume that a
equal distribution of inventory among traders is the most desired. Despite the
difference, their proposed design is worth to consider. Relating to [Duffie
and Zhu (2017)]’s concern, [Goldberg and Tenorio (1997)] uses a Nash
equilibrium model to show that strategic players in an auction lower demand
and supply to avoid moving the clearing price away from their interests and
such behavior could lead to a loss of trading volume in the market.

###  1.3 Optimal policies to cure auction’s inefficiencies and related works

In this paper, we are interested in furthering the design of AHEAD, the
concurrence of a continuous market and a periodic auction market, and we focus
specifically on the periodic auction part. We want to see whether a strategic
trader could take advantage of the current setting when an auction is open.
The main question raising our motivation is the following.

Would a strategic player disturbs market efficiency by strategically picking
their arrival time?

If so, how could we improve the auction design to bring back efficiency?

The efficiency measure we use is a price discovery measure, the difference
between the actual clearing price of an asset and its theoretical efficient
price. To get a larger picture, “market efficiency” usually refers to either
“external efficiency” or “internal efficiency” (see [West (1975)]). A market
is externally efficient if its prices reflect all available information. Fama
develops this definition in [Fama (1970)] (also see [Malkiel (2003)]).
Relevant criteria to measure the market quality include auto-correlation of
return, delay measure, and Sharpe ratio (see [Griffin et al. (2010)] and [Liu
and Chen (2020)])…. Such efficiency is called external because it depends not
on trading systems but on the outside world, such as how information is spread
among traders. Internal efficiency refers to whether a market could enable
traders to trade at prices close enough to their desired prices. The
difference could be generated by the distance between the executed price and
the true price (also called pricing error, price discovery measure, or price
formation measure) and by transaction costs such as liquidation and bid-ask
spread. We adopt the price discovery measure because it suits our purpose the
most and because most relevant work uses such measure (see all relevant work
cited in the introduction and see [Madhavan (1992)]).

The main results of our paper are as following. By studying the behavior of a
strategic trader in a periodic auction, we emphasize that the trader benefit
from arriving only at the end of an auction and right before the auction
closes. Such strategic choice of arrival time enables the trader to take
advantage of all known information to submit a strategic price and have the
greatest price impact on the market. Not only is such strategic timing unfair
to other market participants, but also the strategic pricing could drive the
clearing price away from the efficient price of the underlying asset if the
strategic trader has an incorrect guess about the efficient price. It can be
seen as a disturbance to market efficiency.

We propose two regulatory policies to respond to this mechanical problem
inherent in periodic auction markets. First, we introduce a randomization
design of the auction’s closing time. Second, we introduce a transaction fee
indexed on the arrival time of a strategic trader in the auction, i.e., the
later a trader arrives, the higher they pay. We prove that the randomization
and the transaction fee design address the problem efficiently and bring
better quality to the market. Note that the randomization of the closing time
is mentioned in [Farmer and Skouras (2012)] and [Wah et al. (2016)]. However,
none of these works study the possible effects of such design and use
quantification methods to address it. In reality, randomization of closing
time is implemented, so it is possible to conduct an empirical analysis of
such a design. London Stock Exchange adds a 30-second random period to the
opening auction and closing auction. Cboe randomizes the whole duration of its
periodic auction, i.e., an auction might close in 0 to 100 milliseconds.
However, we caution that any conclusions on empirical analysis of real-world
auction design might not apply to auction design because real-world auctions
have specific settings, including auction duration and priority rules.
Excluding or changing any of the settings could lead to a different
conclusion.

Additionally, we would like to remind that our model of a periodic auction
adopts the most basic set of rules. Periodic auction markets in reality could
be much more complicated. Apart from Cboe’s priority rule we mentioned above,
the opening and closing auction of NYSE and Nasdaq each add their own matching
and execution rules. These rules could possibly limit many advantages of a
periodic auction market; see [Jegadeesh and Wu (2022)] for an empirical
analysis of NYSE and Nasdaq’s closing auctions.

The structure of this study is the following. In Section 2, we introduce the
auction mechanism and the modeling without fees or randomization. Section 2
presents the main charcateristics of the market, the mathematical model and
the information set available for the strategic trader. Section 2.1 set the
clearing price rule to trade the considered asset ensuring the largest number
of matching orders (see Proposition 2.1). Section 2.3 define the optimization
problem of the strategic trader without regulation (randomization of the
clearing time and transaction fees) together with its impact of the market
quality. Section 2.4 introduce the data set together with the calibration of
the relevant parameters for the study. Section 2.5 presents the solution of
the problem when the strategic trader has a full information on the traded
asset illustrated with numerical results. Section 2.6 studies the case where
the strategic trader is imperfectly informed about the efficient price of the
asset. Section 3 turns to the impact of a clearing time randomization and
transaction fees policy on the strategic trader behavior, the price impact and
the market quality. The bilevel optimization is first introduced, following by
the result considering only a randomization of the clearing time without fees
then adding the fees. We consider two different problems from the exchange
perspective to improve the market quality: either to reduce the price impact
including the fees paid by the trader (Section 3.3) or to reduce the distance
between the clearing price and the efficient price while benefiting from the
fees structure (Section 3.4). Section 4 concludes the study and provides
future perspective.

##  2 Auctions market modeling with transaction fees and randomization

###  2.1 The market characteristics

We consider an auction to trade a risky asset starting at time $0$ with
duration $T>0$. We denote by $P^{cl}_{T}$ the clearing price of the auction
determined by the exchange to maximize the number of trades at the clearing
time $T$. During the auction’s duration, limit orders arrived such that each
limit order $i$ is characterized by a limit price $P_{i}$ at which a trader is
willing to buy or sell the asset and a volume determined by a supply function
$Q_{i}=K(P^{cl}_{T}-P_{i})$. The parameter $K>0$ is the slope of the supply
function assumed to be fixed for each limit order. We assume that
$(P_{i})_{i\geq 1}$ is a family of independent normally distributed random
variables with mean $\mu^{mm}$ and standard deviation $\sigma^{mm}$. Note that
if $Q_{i}\leq 0$, the order is a buying order; if $Q_{i}\geq 0$, the order is
a selling order. We assume that the efficient price of the risky asset denoted
by $P^{*}$ is a normal random variable with mean $\mu^{*}$ and standard
deviation $\sigma^{*}$. We assume $\mu^{mm}=\mu^{*}$ and
$\sigma^{mm}=\sigma^{*}=\sigma$, for some $\sigma>0$. We model the arrival of
these limit orders by a Poisson process $M$ with intensity
$\lambda_{t}=\lambda\times t$ where $\lambda$ is a positive constant. In other
words, $N_{t}:=M_{t}+1$ denotes the number of market makers active in the
auction up to time $t$.333Here, ”+1” practically means that there is at least
one trader in the auction market for it to be open and theoretically to avoid
division by zero. This assumption is consistent with the existence of
liquidity in a CLOB transfered to an auction at for example the end of the
day, see [Jusselin et al. (2021)]. We model this process as limit orders pre-
existing in a CLOB, set by market makers and sent continuously in the
coexistence system of CLOB and auction market. These limit orders have been
set in the CLOB and are transferred to the auction for execution as block
trades.

We define a family of $\sigma$-algebras
$\mathcal{F}=\\{\mathcal{F}_{t}\\}_{0\leq t\leq T}$ generated by the available
information up to time $t$ and composed with the number of limit orders
arrived in the auction and their limit prices $P_{i}$ that is
$\mathcal{F}_{t}:=\sigma\\{N_{t},(P_{i})_{i=1}^{N_{t}}\\}$.

We consider a strategic seller joining the auction at a deterministic time
$\tau$ between 0 and $T$ aiming at optimally liquidating her position in the
risky asset.444We could similarly assume that the strategic trader is a buyer
but only consider the seller case motivated by optimal liquidation problem for
the sake of simplicity. The strategic seller controls the exact time $\tau$
she arrives in the auction together with the price $P$ at which she is willing
to sell the asset, and the direction of the trade (which is selling). In this
case, the volume sends by this strategic seller is $Q=K(P^{cl}_{T}-P)$. We
assume that the price $P=P_{\tau}^{\mu}$, sent at time $\tau$ in the auction,
is a normal distribution with mean $\mu$ where $\mu$ is a random variable
controlled by the seller and variance $\sigma^{2}$ fixed measurable with
respect to the information available for the trader. Note that while the
variance for the market makers and the strategic seller are the same for the
sake of simplicity, the strategic trader knows up to time $t$ the number of
arrivals $N_{t}$ and the prices of these orders $\\{P_{i}\\}_{i=1}^{N_{t}}$.
In particular, the strategic trader may be imperfectly informed about the
efficient price of the asset since $\mu$ is determined through $\mathcal{F}$
which does not take into account the efficient price. The seller uses the
information available at time $t$ to determine $\mu_{t}$, so $\mu$ could be
viewed as a function $\mu_{t}=\mu(t,N_{t},\\{P_{i}\\}_{i=1}^{N_{t}})$. By
denoting $P^{\mu}_{t}$ the price the seller proposes entering in the auction
at time $t$, we assume that $P_{t}$ is a normal distribution
$\mathcal{N}(\mu(t,N_{t},\\{P_{i}\\}_{i=1}^{N_{t}}),\sigma^{2})$ where the
function $\mu$ is controlled by the strategic seller when she enters at time
$t$ and sees $N_{t}$ arrivals with associated price
$\\{P_{i}\\}_{i=1}^{N_{t}}$.

Since the strategic seller controls the direction she trades, her order would
not be executed if $P^{cl}_{T}<P^{\mu}_{t}$. However, since all other market
makers do not control the directions of their trades, their orders will be
executed for sure. The quantity the strategic seller trades is thus given by
$K(P^{cl}_{T}-P^{\mu}_{t})$ if $P^{cl}_{T}>P^{\mu}_{t}$, and $0$ otherwise.

The strategic trader does not necessarily know exactly the mean $\mu^{*}$ of
the efficient price and the mean $\mu^{mm}$ of other transferred limit orders.
We denote by $\mu_{g}^{*}$ and $\mu_{g}^{mm}$ the estimations the strategic
trader of $\mu^{*}$ and $\mu^{mm}$ respectively. We denote by $\mathbb{E}_{g}$
the expectation when the mean of $P^{*}$ and $P_{i}$ corresponds to these
estimations and we denote by $\mathbb{E}$ the expectation in the case
$\mu_{g}=\mu^{*}$ and $\mu_{g}^{mm}=\mu^{mm}$.

###  2.2 Clearing Price rule

The clearing price of an auction is set to maximize trading volumes. When
there is no strategic trader, corresponding to the case when no traders
control the direction of the proposed prices, this clearing price is the
equilibrium between supply and demand, see [Du and Zhu (2017)] or [Jusselin et
al. (2021), Section 2.3]. That is

$P^{cl}_{T}=\frac{\sum_{i=1}^{N_{T}}P_{i}}{N_{T}},$ (2.1)

which sets the clearing price to eliminate any imbalance between buy and sell
orders.

Assume now that the strategic seller sends a price $P^{\mu}_{t}$ at time $t$
to trade the asset at the clearing time $T$ and will active only if the
clearing price is above $P^{\mu}_{t}$, we have the following result.

######  Proposition 2.1.

The clearing price of the auction is determined by

$P^{cl}_{T}=\left\\{\begin{aligned}
\frac{\sum_{i=1}^{N_{T}}P_{i}+P^{\mu}_{t}}{N_{T}+1}&\qquad\text{ if
}\frac{\sum_{i=1}^{N_{T}}P_{i}}{N_{T}}>P^{\mu}_{t}\\\ \\\
\frac{\sum_{i=1}^{N_{T}}P_{i}}{N_{T}}&\qquad\text{
otherwise.}\end{aligned}\right.$ (2.2)

###### Proof.

The governing rule of setting a clearing price is that the price maximizes the
traded volume.

Assume first that $\frac{\sum_{i=1}^{N_{T}}P_{i}}{N_{T}}>P^{\mu}_{t}$ then
$\frac{\sum_{i=1}^{N_{T}}P_{i}+P_{t}}{N_{T}+1}>\frac{P_{t}N_{T}+P^{\mu}_{t}}{N_{T}+1}=P^{\mu}_{t}.$
If we set $P^{cl}_{T}=\frac{\sum_{i=1}^{N_{T}}P_{i}+P^{\mu}_{t}}{N_{T}+1}$,
$P^{\mu}_{t}$ would be executed as a selling order. Then it follows from (2.1)
that $P^{cl}_{T}$ is set as above.

Assume now $\frac{\sum_{i=1}^{N_{T}}P_{i}}{N_{T}}\leq P^{\mu}_{t}$, there are
two possible cases, either $P^{cl}_{T}<P^{\mu}_{t}$ or $P^{cl}_{T}\geq
P^{\mu}_{t}$. If $P^{cl}_{T}<P^{\mu}_{t}$, then the strategic seller’s order
would not be executed anyway, so
$P^{cl}_{T}=\frac{\sum_{i=1}^{N_{T}}P_{i}}{N_{T}}$ follows from (2.1). If
$P^{cl}_{T}\geq P^{\mu}_{t}$, number of executed buying orders:

$\displaystyle N^{buy}$
$\displaystyle=\min\big{\\{}K\sum_{i:P_{i}>P^{cl}_{T}}(P_{i}-P^{cl}_{T}),\quad
K\sum_{i:P_{i}<P^{cl}_{T}}(P^{cl}_{T}-P_{i})+K(P^{cl}_{T}-P^{\mu}_{t})\big{\\}}$
$\displaystyle=K\sum_{i:P_{i}>P^{cl}_{T}}(P_{i}-P^{cl}_{T})\text{\quad because
\quad}K\sum_{i=1}^{N_{T}}(P_{i}-P^{cl}_{T})\leq
KN_{T}(P^{\mu}_{t}-P^{cl}_{T})\leq 0\leq K(P^{cl}_{T}-P^{\mu}_{t})$

However, $N^{buy}\leq
K\sum_{i:P_{i}>\frac{\sum_{i=1}^{N_{T}}P_{i}}{N_{T}}}(P_{i}-\frac{\sum_{i=1}^{N_{T}}P_{i}}{N_{T}})$.
This implies that traded volumes would be larger if the clearing price is set
as $\frac{\sum_{i=1}^{N_{T}}P_{i}}{N_{T}}$. This violates the clearing price
setting rule when $P^{cl}_{T}>P^{\mu}_{t}$. Thus, $P^{cl}_{T}\leq P^{\mu}_{t}$
and so $P^{cl}_{T}=\frac{\sum_{i=1}^{N_{T}}P_{i}}{N_{T}}$.

∎

The clearing price (2.2) could also be written as:

$P^{cl}_{T}=\frac{\sum_{i=1}^{N_{T}}P_{i}+\mathbf{1}_{\frac{\sum_{i=1}^{N_{T}}P_{i}}{N_{T}}>P^{\mu}_{t}}P^{\mu}_{t}}{(N_{T})+\mathbf{1}_{\frac{\sum_{i=1}^{N_{T}}P_{i}}{N_{T}}>P^{\mu}_{t}}}.$

The clearing price $P^{cl}_{T}$ depends on the strategic trader’s input
$P^{\mu}_{t}$. In the following, we would write $P_{T}^{cl}$ instead of
$P_{T}^{cl}(P^{\mu}_{t})$ for convenience; however, $P_{T}^{cl}$ is a function
of $P^{\mu}_{t}$.

###  2.3 Strategic Trader’s optimization and market quality

####  2.3.1 Strategic trader optimization

The strategic seller sends at time $\tau$ a volume
$K(P_{T}^{cl}-P^{\mu}_{\tau})$ in the auction to sell the asset under the
condition $P^{cl}_{T}>P^{\mu}_{\tau}$. If executed, the value of the strategic
seller’s portfolio at time $T$ is compared with the efficient price. The
seller’s payoff at the clearing time is thus
$K(P^{cl}_{T}-P^{\mu}_{\tau})(P^{cl}_{T}-P^{*})$ if $P^{cl}_{T}>P_{\tau}$, and
$0$ otherwise.

For any $(t,n,(p_{i})_{i=1}^{n})\in[0,T]\times\mathbb{N}\times\mathbb{R}^{n}$,
we define

$\hat{\mu}(t,n,(p_{i})_{i=1}^{n}):=\arg\max_{\mu}\mathbb{E}_{g}[\mathbf{1}_{p_{\mu}\leq
P_{T}^{cl}}K(P^{cl}_{T}-p_{\mu})(P^{cl}_{T}-P^{*})|N_{t}=n,(P_{i})_{i=1}^{N_{t}}=(p_{i})_{i=1}^{n}],$
(2.3)

where $p_{\mu}$ refers to a normal random variable
$\mathcal{N}(\mu,\sigma^{2})$.

######  Remark 2.2.

In the proof of Theorem 2 and remark 2.4 below, we will show the uniqueness of
$\hat{\mu}$. However, if equation (2.3) outputs more than one $\arg\max$
value, we define $\hat{\mu}$ as the minimum of these outputs.

We set

$\displaystyle\hat{\mu}_{\tau}:=\hat{\mu}(\tau,N_{\tau},(P_{i})_{i=1}^{N_{\tau}}),$
$\displaystyle\hat{P}_{\tau}=P_{\tau}^{\hat{\mu}}\sim\mathcal{N}(\hat{\mu}_{\tau},\sigma^{2}).$

Since the strategic seller wants to maximize payoff, the seller’s problem is
written as:

$\displaystyle\begin{split}V^{\circ}=\sup_{\tau}\;V^{\circ}(\tau),\end{split}$
(2.4)

with

$V^{\circ}(\tau)=\mathbb{E}_{g}\Big{[}\mathbf{1}_{\hat{P}_{\tau}\leq
P_{T}^{cl}}\big{\\{}K(P^{cl}_{T}-\hat{P}_{\tau})(P^{cl}_{T}-P^{*})\big{\\}}\Big{]}.$

Intuitively, the optimal arrival of the trader should be always $T$ as the
trader could use more information to make decisions. The following theorem
proves that the optimal time to arrive is indeed $\tau=T$; the later the
trader joins the auction the better the payoff.

######  Theorem 1.

The strategic trader benefits from arriving as late as possible in an auction.
In other words, for any $0\leq s\leq t\leq T$ we have $V^{\circ}(s)\leq
V^{\circ}(t)$.

###### Proof.

We have

$\displaystyle V^{\circ}(s)$
$\displaystyle=\mathbb{E}_{g}\Big{[}\mathbf{1}_{\hat{P}_{s}\leq
P_{T}^{cl}}K(P^{cl}_{T}-\hat{P}_{s})(P^{cl}_{T}-P^{*})\Big{]}$
$\displaystyle=\mathbb{E}_{g}\Big{[}\mathbb{E}_{g}\big{[}\mathbf{1}_{\hat{P}_{s}\leq
P_{T}^{cl}}K(P^{cl}_{T}-\hat{P}_{s})(P^{cl}_{T}-P^{*})|\mathcal{F}_{s}\big{]}\Big{]}$
$\displaystyle=\mathbb{E}_{g}\Bigl{\\{}\mathbb{E}_{g}\Big{[}\mathbb{E}_{g}[\mathbf{1}_{\hat{P}_{s}\leq
P_{T}^{cl}}K(P^{cl}_{T}-\hat{P}_{s})(P^{cl}_{T}-P^{*})|\mathcal{F}_{t}]\Big{|}\mathcal{F}_{s}\Big{]}\Bigl{\\}}$

where the last equality is based on $\mathcal{F}_{s}\subset\mathcal{F}_{t}$.

Similarly,

$\displaystyle V^{\circ}(t)$
$\displaystyle=\mathbb{E}_{g}\Bigl{\\{}\mathbb{E}_{g}\Big{[}\mathbb{E}_{g}[\mathbf{1}_{\hat{P}_{t}\leq
P_{T}^{cl}}K(P^{cl}_{T}-\hat{P}_{t})(P^{cl}_{T}-P^{*})|\mathcal{F}_{t}]\Big{|}\mathcal{F}_{s}\Big{]}\Bigl{\\}}.$

As $\hat{\mu}_{s}$ is $\mathcal{F}_{s}$-measurable and
$\mathcal{F}_{s}\subset\mathcal{F}_{t}$, $\hat{\mu}_{s}$ is
$\mathcal{F}_{t}$-measurable. Then by the definition of $\hat{\mu}$ in (2.3),

$\mathbb{E}_{g}\big{[}\mathbf{1}_{\hat{P}_{t}\leq
P_{T}^{cl}}K(P^{cl}_{T}-\hat{P}_{t})(P^{cl}_{T}-P^{*})|\mathcal{F}_{t}\big{]}\geq\mathbb{E}_{g}\big{[}\mathbf{1}_{\hat{P}_{s}\leq
P_{T}^{cl}}K(P^{cl}_{T}-\hat{P}_{s})(P^{cl}_{T}-P^{*})|\mathcal{F}_{t}\big{]},\;\mathbb{P}-a.s..$

Then $V^{\circ}(t)\geq V^{\circ}(s)$.

∎

####  2.3.2 Market quality and exchange’s viewpoint

While the strategic trader wants to maximize her payoffs, the exchange would
benefit from an arrival of the strategic trader which minimizes the spread
between the clearing price $P^{cl}_{T}$ and the efficient price $P^{*}$. We
denote by $P^{cl,\tau}_{T}$ the clearing price given by (2.2) for
$P=\hat{P}_{\tau}$, that is the clearing price if the strategic seller arrives
at time $\tau$. We have

$P^{cl,\tau}_{T}=\left\\{\begin{aligned}
\frac{\sum_{i=1}^{N_{T}}P_{i}+\hat{P}_{\tau}}{N_{T}+1}&\qquad\text{ if
}\frac{\sum_{i=1}^{N_{T}}P_{i}}{N_{T}}>\hat{P}_{\tau}\\\ \\\
\frac{\sum_{i=1}^{N_{T}}P_{i}}{N_{T}}&\qquad\text{
otherwise.}\end{aligned}\right.$

We introduce two different measures (disutility functions) of market quality:

$\displaystyle(MQ)(\tau):$
$\displaystyle=\mathbb{E}[|P^{cl,\tau}_{T}-P^{*}|^{2}],$
$\displaystyle(MQ)^{\rho}(\tau):$
$\displaystyle=\mathbb{E}[\exp{(\rho|P^{cl,\tau}_{T}-P^{*}|)}],$

where $\rho>0$ is the risk aversion of the exchange with respect to the
spread. As a benchmark and first-best case scenario, we assume that the
exchange controls the arrival of the strategic seller. The exchange aims at
solving

$\min_{\tau_{reg}}MQ(\tau_{reg})=\min_{\tau_{reg}}\mathbb{E}[|P^{cl,\tau_{reg}}_{T}-P^{*}|^{2}],$
(2.5)

or

$\min_{\tau_{reg}}MQ^{\rho}(\tau_{reg})=\min_{\tau_{reg}}\mathbb{E}[\exp{(\rho|P^{cl,\tau_{reg}}_{T}-P^{*}|)}].$
(2.6)

###  2.4 Data and numerical analysis

We now investigate the optimal arrival of the strategic seller solving (2.4)
together with the optimal deviation form the efficient price $\hat{\mu}$
proposed in the auction and the market quality given by (2.5) or (2.6). We
refer to Appendix A for the details of the computations to perform the
numerical study studied in this section.

We set $T=10$ and discretize the time span by assuming that traders only join
the markets at time $\tau\in\\{1,2,...,9,10\\}$. We use trading data extracted
from YahooFinance for Apple and Alphabet (Google) stock on period Oct-2-2023
to Dec-29-2023 to calibrate the parameters $\mu^{mm}$ and $\sigma$. We
consider three months’ data to avoid being affected by any single period’s
abnormal behavior.

We set the mean $\mu^{mm}$ of $P^{*}$ and $\\{P_{j}\\}_{j}$ to be the average
day price of the three month period considered. For each trading day, we
compute each stock’s day price by (Open Price + Close Price + High Price + Low
Price)/4. We get Apple’s $\mu^{mm}=184.39$ and Alphabet’s $\mu^{mm}=134.24$.

As for the standard deviation $\sigma$ of $P^{*}$ and so $P_{t}$ and
$\\{P_{j}\\}_{j}$, we use the formula

$\sigma^{2}=\frac{1}{N}\sum_{k=1}^{N}(S_{k}-S_{k-1})^{2},$

where $S_{k}$ and $S_{k-1}$ are the price of the stock day $k$ and day $k-1$
respectively, $N$ denotes the total number of days of the period. We got from
the data set Apple’s $\sigma=1.76$ and Alphabet’s $\sigma=2.11$.

When the strategic seller is imperfectly informed about the drift of efficient
price and the drift of the price proposed by the other traders, the estimation
of $\mu^{*}$ and $\mu^{mm}$ by the strategic seller, we work under two
scenarios. We first consider the case where the strategic seller under-
estimates these parameters and named this case Case $(-)$, that is
$\mu_{g}=\mu^{*}-\sigma$ and $\mu_{g}^{mm}=\mu^{mm}-\sigma$. Symmetrically, we
consider the case where the strategic seller over-estimates $\mu^{*}$ and
$\mu^{mm}$ and named this case Case $(+)$, that is $\mu_{g}=\mu^{*}+\sigma$
and $\mu_{g}^{mm}=\mu^{mm}+\sigma$.

We further set a bound for the strategic trader’s $\hat{\mu}$ in the bounded
set $[\mu_{g}^{*}-4\sigma,\mu_{g}^{*}+4\sigma]$.555Note that this assumption
is not unrealistic in view of the existing literature since most of market
making mathematical models require to chose a spread in a bounded interval.
Moreover, in remark 2.4, we see that $\hat{\mu}$ could go to infinity for some
cases, so it is necessary to bound $\hat{\mu}$ when solving numerically.

We assume that the slope of the supply function is $K=10$ and take $\lambda=1$
in consideration of the computation cost.

###  2.5 Strategic trader with full information: efficient but unfair market

In a perfect world setting, $\mu^{*}_{g}=\mu^{*}$ and $\mu^{mm}_{g}=\mu^{mm}$,
i.e. the strategic trader either guesses correctly or has insider information.

####  2.5.1 The stock exchange prefers a strategic trader to join the market

In this subsection, we emphasize the benefit for the exchange to attract a
strategic trader in the market. We modify the assumption in section 2.1 by
assuming $\mathcal{F}_{T}=\sigma\\{N_{T},(P_{i})_{i=1}^{N_{T}},P^{*}\\}$. Note
that such modification does not change the result of Theorem 1 which says that
a strategic trader always join at $\tau=T$. We define $P_{T}^{cl,\emptyset}$
to be the clearing price in an auction market when there is no strategic
trader. According to (2.2), we have

$P_{T}^{cl,\emptyset}=\frac{\sum_{i=1}^{N_{T}}P_{i}}{N_{T}}.$

Hence, the market quality when no strategic trader arrives in the auction is
given by

$MQ^{\emptyset}=\mathbb{E}[|P^{cl,\emptyset}_{T}-P^{*}|^{2}].$

######  Remark 2.3.

Without a strategic trader, the exchange would prefer more limit orders in the
auction. This is because
$MQ^{\emptyset}=\sigma^{2}(1+\frac{1-e^{-T\lambda}}{T\lambda})$ which
decreases monotonically with $\lambda$.

######  Theorem 2.

Assume that the strategic trader is either a seller or a buyer, i.e., the
trader’s objective is defined as

$\displaystyle\sup_{\tau}\mathbb{E}[K(P^{cl}_{T}-\overline{\mu}_{\tau})(P^{cl}_{T}-P^{*})],$
(2.7)

where $\overline{\mu}_{t}$ is the optimizer for
$\operatorname*{ess\,sup}_{\mu}\mathbb{E}[K(P^{cl}_{T}-\mu)(P^{cl}_{T}-P^{*})|\mathcal{F}_{t}]$.
Then $MQ(\hat{\tau})=\frac{1}{4}MQ^{\emptyset}<MQ^{\emptyset}$ where
$\hat{\tau}$ is the optimal arrival time of the strategic trader optimizing
(2.7). In other words, the exchange prefers the arrival of a strategic trader
in the auction to improve the market quality.

###### Proof.

Since $\hat{\tau}=T$, the strategic trader’s problem is to find the
$\mathcal{F}_{T}-$measurable optimizer $\bar{\mu}$ for

$\displaystyle\begin{split}&\operatorname*{ess\,sup}_{\mu}\mathbb{E}\big{[}\big{\\{}K(P^{cl}_{T}-\mu)(P^{cl}_{T}-P^{*})\big{\\}}\big{|}{\mathcal{F}_{T}}\big{]}\\\
=&\operatorname*{ess\,sup}_{\mu}\mathbb{E}\big{[}\big{\\{}K(\frac{\sum_{i=1}^{N_{T}}P_{i}+\mu}{N_{T}+1}-\mu)(\frac{\sum_{i=1}^{N_{T}}P_{i}+\mu}{N_{T}+1}-P^{*})\big{\\}}\big{|}{\mathcal{F}_{T}}\big{]}\\\
=&\operatorname*{ess\,sup}_{\mu}\mathbb{E}\big{[}\big{\\{}\frac{-N_{T}}{(N_{T}+1)^{2}}\mu^{2}+\mu(\frac{(1-N_{T})\sum_{i=1}^{N_{T}}P_{i}}{(N_{T}+1)^{2}}+\frac{N_{T}}{N_{T}+1}P^{*})+\frac{\sum_{i=1}^{N_{T}}P_{i}}{N_{T}+1}(\frac{\sum_{i=1}^{N_{T}}P_{i}}{N_{T}+1}-P^{*})\big{\\}}\big{|}{\mathcal{F}_{T}}\big{]}.\end{split}$
(2.8)

The function $f$ defined by

$\displaystyle f(x)$
$\displaystyle=\mathbb{E}\left[\frac{-N_{T}}{(N_{T}+1)^{2}}\big{|}\mathcal{F}_{T}\right]x^{2}+\mathbb{E}\left[\frac{(1-N_{T})\sum_{i=1}^{N_{T}}P_{i}}{(N_{T}+1)^{2}}+\frac{N_{T}}{N_{T}+1}P^{*}\big{|}\mathcal{F}_{T}\right]x+\mathbb{E}\left[\frac{\sum_{i=1}^{N_{T}}P_{i}}{N_{T}+1}(\frac{\sum_{i=1}^{N_{T}}P_{i}}{N_{T}+1}-P^{*})\big{|}\mathcal{F}_{T}\right]$
$\displaystyle=\left[\frac{-N_{T}}{(N_{T}+1)^{2}}\right]x^{2}+\left[\frac{(1-N_{T})\sum_{i=1}^{N_{T}}P_{i}}{(N_{T}+1)^{2}}+\frac{N_{T}}{N_{T}+1}P^{*}\right]x+\left[\frac{\sum_{i=1}^{N_{T}}P_{i}}{N_{T}+1}(\frac{\sum_{i=1}^{N_{T}}P_{i}}{N_{T}+1}-\mu^{*})\right]$

is maximized at

$x=-\frac{\frac{(N_{T}(N_{T}+1)P^{*}-(N_{T}-1)\sum_{i=1}^{N_{T}}P_{i})}{(N_{T}+1)^{2}}}{2\frac{-N_{T}}{(N_{T}+1)^{2}}}.$

By the symmetry of the parabola $f$, the optimizer is

$\bar{\mu}=-\frac{\frac{(N_{T}(N_{T}+1)P^{*}-(N_{T}-1)\sum_{i=1}^{N_{T}}P_{i})}{(N_{T}+1)^{2}}}{2\frac{-N_{T}}{(N_{T}+1)^{2}}}.$

If the strategic trader sends price at $\bar{\mu}$, then

$P_{T}^{cl}(\bar{\mu})-P^{*}=\frac{1}{2}(\frac{\sum_{i=1}^{N_{T}}P_{i}}{N_{T}}-P^{*})=\frac{1}{2}(P_{T}^{cl,\emptyset}-P^{*}).$

Consequently,

$\mathbb{E}[|P_{T}^{cl}(\bar{\mu})-P^{*}|^{2}]=\frac{1}{4}\mathbb{E}[|P_{T}^{cl,\emptyset}-P^{*}|^{2}]=\frac{1}{4}MQ^{\emptyset}.$

∎

######  Remark 2.4.

In this remark, we illustrate where $\hat{\mu}$ is achieved. Since
$\hat{\tau}=T$, the strategic seller’s problem in (2.4) is to find the
optimizer $\hat{\mu}$ for

$\operatorname*{ess\,sup}_{\mu}\mathbb{E}\big{[}\mathbf{1}_{P^{\mu}_{T}\leq
P_{T}^{cl}}\big{\\{}K(P^{cl}_{T}-P^{\mu}_{T})(P^{cl}_{T}-P^{*})\big{\\}}\big{|}\mathcal{F}_{T}\big{]}.$

The difference between the above problem and problem (2.8) is the presence of
the indicator function $\mathbf{1}_{P^{\mu}_{T}\leq P_{T}^{cl}}$ and the
trader sends $P^{\mu}_{T}$ centering around $\mu$ instead of sending $\mu$. If
remove the indicator function from the above problem, we have
$\hat{\mu}=\bar{\mu}$ by the symmetry of the probability distribution of a
normal random variable. (1) Suppose for the chosen event
$\omega\in\mathcal{F}_{T}$, $(P_{T}^{cl,\emptyset}-P^{*})\geq 0$. By Theorem
2, $\hat{\mu}(\omega)\geq\bar{\mu}(\omega)$ due to the presence of the
indicator function $\mathbf{1}_{P^{\mu}_{T}\leq P_{T}^{cl}}$ which shifts
$\hat{\mu}(\omega)$ upward (see appendix B for more details). (2) Suppose for
the chosen event $\omega\in\mathcal{F}_{T}$, $(P_{T}^{cl,\emptyset}-P^{*})<0$.
For any given $\mu$, when $P^{\mu}_{T}>P_{T}^{cl}$,
$\mathbf{1}_{P^{\mu}_{T}\leq
P_{T}^{cl}}K(P^{cl}_{T}-P^{\mu}_{T})(P^{cl}_{T}-P^{*})$ = 0 and when
$P^{\mu}_{T}\leq P_{T}^{cl}$, $\mathbf{1}_{P^{\mu}_{T}\leq
P_{T}^{cl}}K(P^{cl}_{T}-P^{\mu}_{T})(P^{cl}_{T}-P^{*})=K(P^{cl}_{T}-P^{\mu}_{T})(P^{cl}_{T}-P^{*})\leq
K(P_{T}^{cl}-P^{\mu}_{T})(P_{T}^{cl,\emptyset}-P^{*})<0$. Thus, to maximize
the objective (2.4), the strategic trader would pick $\hat{\mu}(\omega)$ as
large as possible, $\hat{\mu}(\omega)=\infty$.

####  2.5.2 Numerical results and economical insights

The numerical analysis is presented in Table 1. We find the optimal
$\hat{\tau}_{reg}=10$ by solving (2.5) or (2.6), which implies that the
regulator benefits from an arrival of a strategic trader with full information
at the end of an auction. Figure 1 shows that the absolute value of the slope
of $MQ^{\rho}$ increases when $\rho$ gets larger. This implies that the
exchange would prefer a late-arriving trader in an auction even more if the
exchange is highly risk averse (higher $\rho$) on market quality spread. In
addition, we find the optimal $\hat{\tau}=10$ for the arrival of the strategic
seller solving (2.4) which confirms Theorem 1 that traders prefer to arrive at
the end.

The result $\hat{\tau}_{reg}=\hat{\tau}=10$ confirms the “Law of One Price”
which says that arbitragers brings price convergence to the efficient
price.666Suppose the efficient price is $5 per share and the market price is
$4 share, then arbitragers would buy at $4 to sell later at $5. As they buy,
they would push the market price upward until it reaches $5 at which
arbitragers stop buying thus stop moving the market price. From this law, the
exchange would prefer the strategic trader to arrive at a time that maximizes
their arbitrage impact which is $\tau=10$ when the trader knows everything
about the market. In Table 1, two columns show that the strategic trader’s
arbitrage impact increases as $\tau$ increases: column ‘Strategic Seller’
shows that the seller’s expected payoff increases with the arrival time and
column ‘Price Impact’ shows that the trader moves the clearing price more away
from other transferred limit orders’ aggregated opinion the later the trader
joins the auction.777Price Impact
:$=\mathbb{E}[|P_{T}^{cl,\emptyset}-P_{T}^{cl}|^{2}]$ From the last column, we
could see that the strategic trader exerts price impacting power by proposing
higher prices as $\tau$ increases. Note that the proposed price is always
higher than the true mean of the efficient price of underlying assets,
implying that the seller wants to sell high to gain profits; in other words,
the later the trader arrives, the larger the spread they proposes.

The agreement between $\hat{\tau}_{reg}=\hat{\tau}$ seems satisfying but
indulging the strategic trader to join only at $\tau=10$ has several concerns.
The strategic trader obviously uses time as a lever to gain advantage of
earlier joined traders. Recall that our model of an auction market is a zero-
sum game, so the greater gain of the trader, the larger loss of the other
traders. This information unfairness could result in general traders losing
interest in the auction market and turning entirely to the continuous market.
In addition, the price impact of the strategic trader is a concern. From the
table, we see the trader has the largest price manipulation power at
$\tau=10$. This does not produce a problem in a perfect world with full
information but is unrealistic and leads to flaws in a less perfect world as
we discuss in the next section.

Table 1: Results of Apple and Alphabet Arrival Time of | Apple |
---|---|---
Strategic Seller | Strategic Seller (2.4) | $MQ$ | $MQ^{0.1}$ | Price Impact | $\mathbb{E}[\hat{\mu}]$ |
t=1 | 0.5135 | 3.2592 | 1.1509 | 0.0561 | 185.2150 |
t=2 | 0.7489 | 3.2516 | 1.1509 | 0.0721 | 185.2238 |
t=3 | 0.9871 | 3.2435 | 1.1506 | 0.0874 | 185.2996 |
t=4 | 1.2273 | 3.2354 | 1.1503 | 0.1022 | 185.3871 |
t=5 | 1.4691 | 3.2274 | 1.1501 | 0.1166 | 185.4779 |
t=6 | 1.7123 | 3.2195 | 1.1499 | 0.1307 | 185.5669 |
t=7 | 1.9566 | 3.2117 | 1.1496 | 0.1444 | 185.6473 |
t=8 | 2.2014 | 3.2035 | 1.1493 | 0.1586 | 185.7147 |
t=9 | 2.4451 | 3.1948 | 1.1487 | 0.1731 | 185.7359 |
t=10 | 2.6843 | 3.1832 | 1.1471 | 0.1869 | 185.5862 |
Arrival Time of | Alphabet |
---|---|---
Strategic Seller | Strategic Seller (2.4) | $MQ$ | $MQ^{0.1}$ | Price Impact | $\mathbb{E}[\hat{\mu}]$ |
t=1 | 0.7381 | 4.6863 | 1.188 | 0.0879 | 135.6279 |
t=2 | 1.0765 | 4.6755 | 1.1879 | 0.1134 | 135.7143 |
t=3 | 1.4188 | 4.6641 | 1.1876 | 0.1371 | 135.829 |
t=4 | 1.764 | 4.6525 | 1.1872 | 0.1602 | 135.9517 |
t=5 | 2.1116 | 4.6412 | 1.1869 | 0.1819 | 136.0635 |
t=6 | 2.4612 | 4.6299 | 1.1866 | 0.2035 | 136.1738 |
t=7 | 2.8123 | 4.6186 | 1.1863 | 0.225 | 136.2715 |
t=8 | 3.164 | 4.607 | 1.186 | 0.2467 | 136.3597 |
t=9 | 3.5143 | 4.5945 | 1.1853 | 0.2696 | 136.4248 |
t=10 | 3.8581 | 4.5779 | 1.1836 | 0.2907 | 136.3411 |

Figure 1:  Apple’s (left) and Google’s (right) $MQ^{\rho}$. From the bottom
curve to the top curve, $\rho$ fixed at $0.1,0.2,0.5,0.8,1,1.2$ respectively.
For each curve, it presents $MQ^{\rho}(t)$, with $t=1,2,...,10$.

###  2.6 Imperfect information and inefficiency of auctions

In a more realistic framework, the strategic trader has a misconception of
$\mu^{*}$ and $\mu^{mm}$. In the previous case, the trader uses price
impacting power to drag clearing price towards the efficient price, but when
the trader has a different target than the efficient price, the story changes.

We test two cases: case $(-)$ when $\mu^{*}_{g}=\mu^{*}-\sigma$ and
$\mu^{mm}_{g}=\mu^{mm}-\sigma$ and case $(+)$ when
$\mu^{*}_{g}=\mu^{*}+\sigma$ and $\mu^{mm}_{g}=\mu^{mm}+\sigma$.888Note that
Case $(-)$ is more likely to happen than Case $(+)$ since we are studying a
seller not a buyer.

The numerical analysis is presented in Table 2.999Alphabet’s graphs carry the
same spirit as Apple’s graphs. For the sake of simplicity, we only analyze the
data from Apple in this part. In the case $(-)$ we find the optimal
$\hat{\tau}_{reg}=1$ by solving (2.5) or (2.6), which implies that the
regulator wants the strategic trader to arrive at the beginning of an auction.
Under this case, we see that the optimal arrival time of a strategic trader
disturbs market efficiency as $\hat{\tau}=10$. In addition, we observe that
the market quality becomes worse as $\tau$ increases, while the strategic
seller’s payoff increases with $\tau$. Thus the exchange would prefer the
strategic trader to arrive as early as possible to minimize the spread between
the efficient price and the clearing price, while the trader prefers to arrive
as late as possible to maximize expected payoff.

In the case $(+)$, we find the optimal $\hat{\tau}_{reg}=1$ when using measure
$MQ^{\rho}$ with large $\rho$, such $\rho=1,\rho=1.2$ and
$\hat{\tau}_{reg}=10$ when using measure $MQ$ or $MQ^{\rho}$ with small
$\rho$. It is important to note that the difference between $MQ(t)$ or
$MQ^{\rho}(t)$ across different $t$ is not significant, see Figure 2. A
possible explanation is that the trader tends to propose a high price due to
the misconception $\mu^{*}_{g}=\mu^{*}+\sigma$, which results in many of the
high-price orders not getting executed due to the fact that $P_{T}^{cl}<P$. As
the trader’s price manipulation power gets restricted in this way, the market
quality measure varies little across different $\tau$ (the arrival time of the
strategic trader). Following the same logic, the reason why we see a divided
result (i.e., what is $\hat{\tau}_{reg}$) between large $\rho$ and small
$\rho$ is that when $\rho$ grows the exchange becomes extremely sensitive to
any possible difference between the clearing price and efficient price. Thus
small chance events get exaggerated; for example, the event when $P_{T}^{cl,}$
is high above $P^{*}$ and the strategic trader manages to propose a high $P$
such that $P<P_{T}^{cl}$. The strategic trader would be able to impact the
price and catch this opportunity more if the trader arrives at $\tau=10$
instead of $\tau=1$, resulting $MQ^{\rho}(\tau=10)>MQ^{\rho}(\tau=1)$. In
addition, note that $MQ^{\rho}(\tau=4)>MQ^{\rho}(\tau=10)$ even though the
trader has larger price impact at $\tau=10$. Recall Section 2.5.1, the
exchange prefers a trader to join an auction market. The trader has more
information to enable a successful execution (i.e. $P<P_{T}^{cl}$ ) at
$\tau=10$ than $\tau=4$ thus resulting this relation. Overall, case one
deserves more attention than case two because case two misconception self-
restricts a strategic seller’s price impacting power.

Table 2: Results of Apple Arrival Time of | Case ($-$): Lower Conjecture |
---|---|---
Strategic Seller | $MQ$ | $MQ^{0.1}$ | $MQ^{1}$ | Price Impact | $\mathbb{E}[\hat{\mu}]$ |
t=1 | 3.2765 | 1.1514 | 9.3008 | 0.0747 | 182.222 |
t=2 | 3.2963 | 1.1521 | 9.3864 | 0.1212 | 181.1093 |
t=3 | 3.3243 | 1.1529 | 9.512 | 0.1728 | 180.1473 |
t=4 | 3.3596 | 1.1538 | 9.6727 | 0.2279 | 179.2896 |
t=5 | 3.4 | 1.1549 | 9.8582 | 0.283 | 178.5356 |
t=6 | 3.4445 | 1.1561 | 10.0647 | 0.3374 | 177.8721 |
t=7 | 3.4898 | 1.1574 | 10.2765 | 0.388 | 177.3117 |
t=8 | 3.5356 | 1.1586 | 10.4933 | 0.4351 | 176.8247 |
t=9 | 3.5777 | 1.1594 | 10.6966 | 0.4763 | 176.3853 |
t=10 | 3.6161 | 1.1591 | 10.8899 | 0.5133 | 175.8701 |
Arrival Time of | Case ($+$): Higher Conjecture |
---|---|---
Strategic Seller | $MQ$ | $MQ^{0.1}$ | $MQ^{1}$ | Price Impact | $\mathbb{E}[\hat{\mu}]$ |
t=1 | 3.2763 | 1.1514 | 9.3202 | 0.1737 | 188.0503 |
t=2 | 3.2772 | 1.1516 | 9.324 | 0.2367 | 188.8119 |
t=3 | 3.2775 | 1.1516 | 9.3265 | 0.2864 | 189.388 |
t=4 | 3.2778 | 1.1515 | 9.3288 | 0.3251 | 189.8009 |
t=5 | 3.2781 | 1.1515 | 9.3309 | 0.3551 | 190.0937 |
t=6 | 3.2785 | 1.1515 | 9.3327 | 0.3779 | 190.2949 |
t=7 | 3.2787 | 1.1515 | 9.3339 | 0.3955 | 190.4307 |
t=8 | 3.2786 | 1.1514 | 9.3341 | 0.4089 | 190.5091 |
t=9 | 3.2777 | 1.151 | 9.3317 | 0.4193 | 190.5048 |
t=10 | 3.274 | 1.1496 | 9.3206 | 0.3307 | 189.337 |

Figure 2:  Case one, left graph, and case two, right graph. $MQ^{\rho}$. From
the bottom curve to the top curve, $\rho$ fixed at $0.1,0.2,0.5,0.8,1,1.2$
respectively.

##  3 Monitoring policies: transaction fees and clearing time randomization

###  3.1 Bilevel optimization between the exchange and the strategic trader

The previous section emphasized some flaws in auctions and a need to regulate
the arrival behavior of traders in this type of market. To mitigate the
conflict between the optimal arrival $\hat{\tau}$ from the strategic trader
viewpoint and $\hat{\tau}_{reg}$, the one that the exchange would prefer for
market quality reasons, we investigate and provide a quantitative analysis of
two tools: a transition fees policy and a randomization of the clearing time.
We denote by $\xi(t)$ the transaction fee function, where $t$ refers to the
arrival time of traders and $\tau^{cl}$ to be the duration of an auction
assumed to be a random variable whose parameter is controlled by stock
exchanges.

In this model, the strategic seller has to pay $\xi(t)$ if she arrives at time
$t$ in the auction per volume of shares of the asset traded. We then define
for any $(t,n,(p_{i})_{i=1}^{n})$, the set
$\hat{M}_{t,n,(p_{i})_{i=1}^{n}}^{\xi,\tau^{cl}}$ of optimizers

$\hat{\mu}(t,n,(p_{i})_{i=1}^{n};\xi,\tau^{cl})\in\arg\max_{\mu}\mathbb{E}_{g}[\mathbf{1}_{p_{\mu}\leq
P_{\tau^{cl}}^{cl}}K(P^{cl}_{\tau^{cl}}-p_{\mu})\big{(}P^{cl}_{\tau^{cl}}-P^{*}-\xi(t)\big{)}|N_{t}=n,(P_{i})_{i=1}^{N_{t}}=(p_{i})_{i=1}^{n}],$
(3.1)

where $p_{\mu}$ refers to a normal random variable
$\mathcal{N}(\mu,\sigma^{2})$. The problem of the strategic seller becomes

$V^{fee}(\xi,\tau^{cl}):=\sup_{\tau}V^{fee}_{\xi,\tau^{cl}}(\tau)$ (3.2)

with

$V^{fee}_{\xi,\tau^{cl}}(\tau)=\mathbb{E}_{g}\Big{[}\mathbf{1}_{\tau\leq\tau^{cl}}\mathbf{1}_{\hat{P}_{\tau}\leq
P_{\tau^{cl}}^{cl}}\big{\\{}K(P^{cl}_{\tau^{cl}}-\hat{P}_{\tau})(P^{cl}_{\tau^{cl}}-P^{*})-K(P^{cl}_{\tau^{cl}}-\hat{P}_{\tau})\xi(\tau)\big{\\}}\Big{]},$

where101010Note that if the set
$\hat{M}_{t,n,(p_{i})_{i=1}^{n}}^{\xi,\tau^{cl}}$ is not reduced to one
element, the choice of the optimizer does not affect the value function
$V^{fee}$.

$\displaystyle\hat{\mu}_{\tau}:=\hat{\mu}(\tau,N_{\tau},(P_{i})_{i=1}^{N_{\tau}},\xi,\tau^{cl}),\text{
and }\hat{P}_{\tau}\sim\mathcal{N}(\hat{\mu}_{\tau},\sigma^{2}).$

We denote by $\widehat{\mathcal{M}}^{\xi,\tau^{cl}}$ the set of optimizers
$(\hat{\mu},\hat{\tau})$ such that $\hat{\tau}$ is optimal for (3.2) and
$\hat{\mu}\in\hat{M}_{\hat{\tau},N_{\hat{\tau}},(P_{i})_{i=1}^{N_{\hat{\tau}}}}^{\xi,\tau^{cl}}$.

Assume that the strategic seller arrives in the auction at time $\hat{\tau}$
by proposing a price $\hat{P}_{\hat{\tau}}$. We denote by
$P^{cl,\hat{\tau}}_{\tau^{cl}}$ the clearing price set by the exchange and
defined by (2.2) with $P=\hat{P}_{\hat{\tau}}$ and $T=\tau^{cl}$. The problem
of exchange depends on where its top interest lies. If the exchange wants to
improve the actual price paid by the trader, the exchange aims at solving a
bilevel optimization

$\Lambda_{0}:=\min_{\xi}\Lambda_{0}(\xi)$ (3.3)

where

$\Lambda_{0}(\xi)=\min_{\tau^{cl},(\hat{\tau},\hat{\mu})}\mathbb{E}[(|P^{cl,\hat{\tau}}_{\tau^{cl}}-P^{*}|+\xi(\hat{\tau})\mathbf{1}_{\hat{P}_{\hat{\tau}}\leq
P^{cl,\hat{\tau}}_{\tau^{cl}}})^{2}],$

or by assuming that the exchange is risk averse

$\Lambda^{\rho}_{0}:=\min_{\xi}\Lambda^{\rho}_{0}(\xi)$ (3.4)

where

$\Lambda^{\rho}_{0}(\xi)=\min_{\tau^{cl},(\hat{\tau},\hat{\mu})}\mathbb{E}\Big{[}\exp{\Big{(}\rho\big{(}|P^{cl,\hat{\tau}}_{\tau^{cl}}-P^{*}|+\xi(\hat{\tau})\mathbf{1}_{\hat{P}_{\hat{\tau}}\leq
P^{cl,\hat{\tau}}_{\tau^{cl}}}\big{)}\Big{)}}\Big{]},$

subject to

(IC):
$\displaystyle(\hat{\tau},\hat{\mu})\in\widehat{\mathcal{M}}^{\xi,\tau^{cl}},$
(R): $\displaystyle
V^{fee}(\xi,\tau^{cl})\geq\frac{\gamma}{2}\mathbb{E}_{g}[\mathbf{1}_{\hat{P}_{\hat{\tau}}\leq
P^{cl,\hat{\tau}}_{\tau^{cl}}}K(P^{cl,\hat{\tau}}_{\tau^{cl}}-\hat{P}_{\hat{\tau}})],$

where $\gamma$ is the difference between the best bid price and the best ask
price. The constraint (IC) is called the incentive compatibility constraint
and models the best-reaction action $(\hat{\tau},\hat{\mu})$ is the strategic
seller when the exchange announced a transaction fee $\xi$ and a clearing time
rule $\tau^{cl}$. The constraint (R) is set to bound the transaction fee $\xi$
and ensure that the trader would benefit from the auction and not turn to the
continuous trading market to avoid the transaction fee in the periodic auction
market. With this reservation utility constraint, the auction is more
competitive than trading on the CLOB directly.

######  Remark 3.1.

This problem can by also seen as a “trader focused” exchange. We assume the
exchange wants to minimize the total spread for the trader, where

$\displaystyle\text{total spread}=$ $\displaystyle MQ+\text{transaction
spread}$ $\displaystyle=$ $\displaystyle\text{ spread between efficient price
and clearing price }+$ $\displaystyle\text{ spread between clearing price and
after-fee price (real executed price)}.$

We define the transaction spread of a trader in this way: suppose the market
clears at $P_{T}^{cl}$ and the transaction fee is $\xi$ per share, a buyer
would pay $P_{T}^{cl}+\xi$ to buy a share and a seller would receive
$P_{T}^{cl}-\xi$ to sell a share. Average transaction spread is thus
$\frac{1}{2}[(P_{T}^{cl}+\xi)-(P_{T}^{cl}-\xi)]=\xi$.

If the exchange focuses only on the market efficiency and wants to increase
both fee gains and market quality, the problem becomes

$\Lambda_{0}:=\min_{\xi}\Lambda_{0}(\xi)$ (3.5)

where

$\Lambda_{0}(\xi)=\min_{\tau^{cl},(\hat{\tau},\hat{\mu})}\mathbb{E}[|P^{cl,\hat{\tau}}_{\tau^{cl}}-P^{*}|^{2}-\xi(\hat{\tau})\mathbf{1}_{\hat{P}_{\hat{\tau}}\leq
P^{cl,\hat{\tau}}_{\tau^{cl}}}],$

or by assuming that the exchange is risk averse

$\Lambda^{\rho}_{0}:=\min_{\xi}\Lambda^{\rho}_{0}(\xi)$ (3.6)

where

$\Lambda^{\rho}_{0}(\xi)=\min_{\tau^{cl},(\hat{\tau},\hat{\mu})}\mathbb{E}\Big{[}\exp{\Big{(}\rho\big{(}|P^{cl,\hat{\tau}}_{\tau^{cl}}-P^{*}|-\xi(\hat{\tau})\mathbf{1}_{\hat{P}_{\hat{\tau}}\leq
P^{cl,\hat{\tau}}_{\tau^{cl}}}\big{)}\Big{)}}\Big{]},$

subject to

(IC):
$\displaystyle(\hat{\tau},\hat{\mu})\in\widehat{\mathcal{M}}^{\xi,\tau^{cl}},$
(R): $\displaystyle
V^{fee}(\xi,\tau^{cl})\geq\frac{\gamma}{2}\mathbb{E}_{g}[\mathbf{1}_{\hat{P}_{\hat{\tau}}\leq
P^{cl,\hat{\tau}}_{\tau^{cl}}}K(P^{cl,\hat{\tau}}_{\tau^{cl}}-\hat{P}_{\hat{\tau}})].$

For the numerical solutions, we set the bid-ask spread $\gamma$ by referring
to the estimation method in [Abdi and Ranaldo (2017)]: for a period of N days,
$\gamma=\frac{1}{N}\sum_{i=1}^{N}\gamma_{i}$, where

$\gamma_{i}=\sqrt{\max\\{4(c_{t}-(l_{t}+h_{t})/2)(c_{t}-(l_{t+1}+h_{t+1})/2),0\\}},$

$c_{t}$ is the daily close log price, $l_{t}$ is the daily low log price, and
$h_{t}$ is the daily high log price. By this method, we set Apple’s
$\gamma=0.0039$ and Alphabet’s $\gamma=0.0065$.

######  Remark 3.2.

Note that for both market impact or market efficiency optimization problems,
we focus solely on the spread and fee of the strategic trader. We do not
include the spread and fee of the other transferred limit orders because we
assume these traders are transferred from CLOB to the periodic auction for
execution and thus do not face the transaction fee imposed in the periodic
auction.

###  3.2 Randomization without fees

In this section, we focus on the solution to (3.3) or (3.4), (3.5) or (3.6)
when $\xi=0$. Echoing the discussion in Section 2.6, we focus on case $(-)$,
that is a misconception $\mu^{*}_{g}=\mu^{*}-\sigma$ and
$\mu^{mm}_{g}=\mu^{mm}-\sigma$ for the strategic trader. We assume that
$\tau^{cl}$ is a Bernoulli random variable taking values in $\\{9,10\\}$ with
$p=\mathbb{P}(\tau^{cl}=9)=1-\mathbb{P}(\tau^{cl}=10)$ and $p\in[0,1]$. The
optimization on $\tau^{cl}$ in both (3.3) or (3.4), (3.5) or (3.6) is reduced
to optimize $p$, that is

$MQ:=\min_{p\in[0,1]}MQ(p),$

with

$MQ(p)=\min_{(\hat{\tau},\hat{\mu})}\mathbb{E}[|P^{cl,\hat{\tau}}_{\tau^{cl}}-P^{*}|^{2}],$

subject to

(IC):
$\displaystyle(\hat{\tau},\hat{\mu})\in\widehat{\mathcal{M}}^{0,\tau^{cl}},$
(R): $\displaystyle
V^{fee}(0,p)\geq\frac{\gamma}{2}\mathbb{E}[\mathbf{1}_{\hat{P}_{\hat{\tau}}\leq
P^{cl,\hat{\tau}}_{\tau^{cl}}}K(P^{cl,\hat{\tau}}_{\tau^{cl}}-\hat{P}_{\hat{\tau}})],$

We recall that the strategic trader decides at time 0 when to arrive (i.e.,
chooses $\hat{\tau}$ before the auction starts). For both Apple and Alphabet,
we observe in Table 3 that the optimal $\hat{p}$ is around $0.09$.

With randomization set at $\hat{p}=0.09$, the stock exchange successfully
encourages a strategic trader to arrive earlier. We now have $\hat{\tau}=9$
instead of $\hat{\tau}=10$ which is the optimal arrival time without
randomization. We also observe the market quality improves from $3.6161$ to
$3.5850$ for Apple and from $5.1885$ to $5.1438$ for Alphabet comparing with
Table 2. Looking closely at the results in Table 3, we observe that even a
small $p$, as small as $0.09$ (which means that the auction only has $0.09$
probability to end at $9$ instead of $10$), would be sufficient to encourage
the strategic trader to not arrive at $\tau=10$. This explains why the optimal
$\hat{p}$ is very close to $0$: the exchange prefers the case when the
strategic trader arrives before the closing time and at $p=0.09$ there is only
$9\%$ of chance that the strategic trader arrives at the closing time of the
auction which happens to be $9$.

Table 3:  Randomization $p$ | Apple | Alphabet
---|---|---
| $MQ$ | $\hat{\tau}$ | $MQ$ | $\hat{\tau}$
0.0 | 3.6161 | 10 | 5.1885 | 10
0.07 | 3.5947 | 10 | 5.1584 | 10
0.08 | 3.5917 | 10 | 5.1541 | 10
0.09 | 3.585 | 9 | 5.1438 | 9
0.1 | 3.5858 | 9 | 5.145 | 9
0.2 | 3.5938 | 9 | 5.1561 | 9
0.3 | 3.6018 | 9 | 5.1673 | 9
0.4 | 3.6098 | 9 | 5.1785 | 9
0.5 | 3.6179 | 9 | 5.1897 | 9
0.6 | 3.6259 | 9 | 5.2009 | 9
0.7 | 3.6339 | 9 | 5.2121 | 9
0.8 | 3.6419 | 9 | 5.2233 | 9
0.9 | 3.65 | 9 | 5.2345 | 9
1.0 | 3.658 | 9 | 5.2457 | 9

###  3.3 Optimal transaction fees indexed on time to improve price impact for
the trader

We now turn to the solutions of (3.3) and (3.4). As a classical result of
contract theory and bilevel optimization, the shape of the contract $\xi$ has
to be specified in a discrete time framework. We assume that the exchange
proposes two type of exchange fees: either a linear fee indexed on the time of
arrival of the strategic trader and the efficient price $\xi_{\ell}(t)=at$ or
a square fee structure $\xi_{s}(t)=at^{2}$, with randomization of closing time
$\tau^{cl}$. By selecting either a linear fee structure or a square fee, the
bilevel optimization problem thus becomes

$\Lambda_{0}:=\min_{\xi\in\\{\xi_{\ell},\xi_{s}\\}}\Lambda_{0}(a),$ (3.7)

where for a choice of $\xi\in\\{\xi_{\ell},\xi_{s}\\}$

$\Lambda_{0}(a)=\min_{\tau^{cl},(\hat{\tau},\hat{\mu})}\mathbb{E}[(|P^{cl,\hat{\tau}}_{\tau^{cl}}-P^{*}|+\xi(\hat{\tau})\mathbf{1}_{\hat{P}_{\hat{\tau}}\leq
P^{cl,\hat{\tau}}_{\tau^{cl}}})^{2}],$

or by assuming that the exchange is risk averse

$\Lambda^{\rho}_{0}:=\min_{a}\Lambda_{0}^{\rho}(a)$ (3.8)

where

$\Lambda_{0}^{\rho}(a)=\min_{\tau^{cl},(\hat{\tau},\hat{\mu})}\mathbb{E}\Big{[}\exp{\Big{(}\rho\big{(}|P^{cl,\hat{\tau}}_{\tau^{cl}}-P_{T}^{*}|+\xi(\hat{\tau})\mathbf{1}_{\hat{P}(\xi)\leq
P^{cl,\hat{\tau}}_{\tau^{cl}}}\big{)}\Big{)}}\Big{]},$

subject to

(IC):
$\displaystyle(\hat{\tau},\hat{\mu})\in\widehat{\mathcal{M}}^{\xi,\tau^{cl}},$
(R): $\displaystyle
V^{fee}(\xi)\geq\frac{\gamma}{2}\mathbb{E}_{g}[\mathbf{1}_{\hat{P}_{\hat{\tau}}\leq
P^{cl,\hat{\tau}}_{\tau^{cl}}}K(P^{cl,\hat{\tau}}_{\tau^{cl}}-\hat{P}_{\hat{\tau}})].$

As before, we assume that
$p=\mathbb{P}(\tau^{cl}=9)=1-\mathbb{P}(\tau^{cl}=10)$ and the exchange
optimizes on the parameter $p$ to optimize $\tau^{cl}$.

The results are presented in Table 4 with the optimal fee structure for Apple
(’Stock’-’Apple’ cells) and Alphabet (’Stock’-’Alphabet’ cells). The second
column “MQ measure” represents the choice of the problem (3.7) or (3.8) for
different values of $\rho$. The third column gives the optimal fees’ structure
while the fourth column gives the optimal value of $p$. The fifth column gives
the optimal arrival of the strategic trader $\hat{\tau}$ solving (3.2) with
the corresponding optimal fees and $p$ value. The sixth column “Exchange gain”
represents the value $\Lambda_{0}$ or $\Lambda_{0}^{\rho}$ for different
values of $\rho$ corresponding to the value function of the exchange, while
the last column compared this value with the market quality $MQ(\hat{\tau})$
when the fee is $0$ (no transaction fees).

From the last two columns of the table, we could see that our transaction fee
model improves market quality since $\Lambda_{0}$ is always smaller that the
market quality when $\xi=0$. From the strategic trader’s perspective, they now
have the incentive to arrive earlier as they prefer to arrive at time $\tau=1$
instead of $\tau=10$ when there is no fee. We could thus conclude that the
added fee is effective to cure the flaws of a periodic auction system observed
in Section 2.6. Note moreover that $p=0$ is optimal, that is the randomization
of the clearing time becomes useless by adding transaction fees. As a matter
of fact, the regulator calibrated the fees optimally to make the strategic
seller arriving in the auction as early as possible. It deletes the impact of
the randomization of the clearing time between $\tau^{cl}=9$ or
$\tau^{cl}=10$.

Table 4: Selected Numerical Results Stock | MQ Measure | Optimal Fee | Randomization | $\hat{\tau}$ | Exchange’s Value | $MQ^{0}$
---|---|---|---|---|---|---
Apple | $\Lambda_{0}$ | $0.03t^{2}$ | $p=0$ | 1 | 3.3359 | 3.6161
| $\Lambda_{0}^{0.1}$ | $0.03t^{2}$ | $p=0$ | 1 | 1.1538 | 1.1591
| $\Lambda_{0}^{0.5}$ | $0.03t^{2}$ | $p=0$ | 1 | 2.4643 | 2.5777
| $\Lambda_{0}^{1}$ | $0.03t^{2}$ | $p=0$ | 1 | 9.4844 | 10.8899
| $\Lambda_{0}^{1.2}$ | $0.03t^{2}$ | $p=0$ | 1 | 18.6465 | 22.4917
Alphabet | $\Lambda_{0}$ | $0.03t^{2}$ | $p=0$ | 1 | 4.7725 | 5.1885
| $\Lambda_{0}^{0.1}$ | $0.03t^{2}$ | $p=0$ | 1 | 1.1908 | 1.1985
| $\Lambda_{0}^{0.5}$ | $0.03t^{2}$ | $p=0$ | 1 | 3.0913 | 3.2872
| $\Lambda_{0}^{1}$ | $0.03t^{2}$ | $p=0$ | 1 | 18.4440 | 22.3034
| $\Lambda_{0}^{1.2}$ | $0.03t^{2}$ | $p=0$ | 1 | 45.6256 | 58.8512

(a) Linear Fee $\hat{\tau}(a)$

(b) Square Fee $\hat{\tau}(a)$

(c) Under square Fee $\Lambda_{0}^{0.5}(a)$

Figure 3: Numerical analysis of the optimal arrival of the strategic seller
(a) and (b), the value of the exchange with risk aversion parameter $\rho=0.5$
(c). Note that $\hat{\tau}$ is a function of both $a$ and $p$; for example,
under linear fee, $\hat{\tau}(a=0.04,p=0)=7$, $\hat{\tau}(0.04,0.2)=8$ and
$\hat{\tau}(0.04,0.9)=9$. (a) and (b) show $\hat{\tau}(a,p(a))$, where
$p(a)=\arg\min_{p}\Lambda_{0}^{0.5}(a,p)$.

We now turn to a deeper study of how our transaction fee model improves the
market’s quality for the Apple’s stock price.111111Alphabet’s graphs carry the
same spirit as Apple’s graphs. For the sake of simplicity, we only analyze the
data from Apple in this part. Both the linear transaction fee structure and
square transaction fee structure encourage the strategic seller to arrive
earlier in the market. From Figure 3 (a) and (b), we see that as the fee
increases (led by increasing $a$), the strategic seller’s optimal $\tau$
gradually declines from $10$ to $1$ and the decline rate seems to coincide
with the structure of the fee model as Figure 3 (a) shows a linear declining
pattern and Figure 3 (b) shows an accelerating declining pattern.

From Figure 3, we see the competition between the spread term
$|P_{T}^{cl}-P^{*}|$ and the fee term $\xi$ in the stock exchange’s utility
function $\Lambda_{0}$ and $\Lambda_{0}^{\rho}$. When the fees increase
initially, the spread becomes better due to the fact that the strategic trader
is willing to arrive earlier. The decrease of the spread overpowers the
increase in fees. However, after fee reaches the optimal stage which is
$a=0.03$, the strategic trader picks the same $\hat{\tau}=1$ no matter how
large $a$ becomes. The exchange then feels no need to increase the transaction
fee. We could see from Figure 3 (b) that the minimum $a$ to encourage the
trader to arrive at $\tau=1$ is $a=0.03$, which explains why the optimal fee
structure is very consistent across different utility measures we test.

###  3.4 Optimal transaction fees indexed on time: improving market quality
while benefiting from the fees

We now turn to the solutions of (3.5) and (3.6). The bilevel optimization
problem thus becomes

$\Lambda_{0}:=\min_{\xi\in\\{\xi_{\ell},\xi_{s}\\}}\Lambda_{0}(a),$ (3.9)

where for a choice of $\xi\in\\{\xi_{\ell},\xi_{s}\\}$

$\Lambda_{0}(a)=\min_{\tau^{cl},(\hat{\tau},\hat{\mu})}\mathbb{E}[|P^{cl,\hat{\tau}}_{\tau^{cl}}-P_{T}^{*}|^{2}-\xi(\hat{\tau})\mathbf{1}_{\hat{P}_{\hat{\tau}}\leq
P^{cl,\hat{\tau}}_{\tau^{cl}}}],$

or by assuming that the exchange is risk averse

$\Lambda^{\rho}_{0}:=\min_{a}\Lambda_{0}^{\rho}(a)$ (3.10)

where

$\Lambda_{0}^{\rho}(a)=\min_{\tau^{cl},(\hat{\tau},\hat{\mu})}\mathbb{E}\Big{[}\exp{\Big{(}\rho\big{(}|P^{cl,\hat{\tau}}_{\tau^{cl}}-P_{T}^{*}|-\xi(\hat{\tau})\mathbf{1}_{\hat{P}(\xi)\leq
P^{cl,\hat{\tau}}_{\tau^{cl}}}\big{)}\Big{)}}\Big{]},$

subject to

(IC):
$\displaystyle(\hat{\tau},\hat{\mu})\in\widehat{\mathcal{M}}^{\xi,\tau^{cl}},$
(R): $\displaystyle
V^{fee}(\xi)\geq\frac{\gamma}{2}\mathbb{E}_{g}[\mathbf{1}_{\hat{P}_{\hat{\tau}}\leq
P^{cl,\hat{\tau}}_{\tau^{cl}}}K(P^{cl,\hat{\tau}}_{\tau^{cl}}-\hat{P}_{\hat{\tau}})].$

As before, we assume that
$p=\mathbb{P}(\tau^{cl}=9)=1-\mathbb{P}(\tau^{cl}=10)$ and the exchange
optimizes on the parameter $p$ to optimize $\tau^{cl}$.

######  Remark 3.3.

In an informal way, we can see from (3.9) or (3.10) that

$\text{Exchange value function}=\text{market quality cost }-\text{ fees},$

or in other words

$\text{Market quality cost}=\text{Exchange gain }+\text{ Fees' gain}.$

The results are presented in Table 5. The seventh column is the incomes
generating by the transaction fees, that is the part $\xi(\hat{\tau})$
calculated in an informal way with column "$MQ^{\xi}$" - "Exchange’s Gain".
The eigth column shows the market quality $MQ(\hat{\tau})$ by considering the
optimal fees $\xi$ while the last column compared this value with the market
quality $MQ(\hat{\tau})$ when the fee is $0$ (no transaction fees).

Table 5: Selected Numerical Results Stock | MQ Measure | Optimal Fee |  Random-ization | $\hat{\tau}$ | Exchange’s Gain | Fee Gain | $MQ^{\xi}$ | $MQ^{0}$
---|---|---|---|---|---|---|---|---
Apple | $\Lambda_{0}$ | $0.04t$ |  $p=0.1$ | 8 | 3.095 | 0.294 | 3.389 | 3.616
| $\Lambda_{0}^{0.1}$ | $0.04t$ |  $p=0.5$ | 9 | 1.117 | 0.038 | 1.156 | 1.159
| $\Lambda_{0}^{0.5}$ | $0.04t$ |  $p=0.5$ | 9 | 2.119 | 0.387 | 2.506 | 2.578
| $\Lambda_{0}^{1}$ | $0.04t$ |  $p=0.5$ | 9 | 7.225 | 2.802 | 10.027 | 10.890
| $\Lambda_{0}^{1.2}$ | $0.04t$ |  $p=0.5$ | 9 | 13.648 | 6.475 | 20.123 | 22.492
Alphabet | $\Lambda_{0}$ | $0.05t$ |  $p=0$ | 7 | 4.497 | 0.313 | 4.810 | 5.189
| $\Lambda_{0}^{0.1}$ | $0.05t$ |  $p=1$ | 9 | 1.145 | 0.047 | 1.192 | 1.199
| $\Lambda_{0}^{0.5}$ | $0.05t$ |  $p=1$ | 9 | 2.588 | 0.553 | 3.141 | 3.287
| $\Lambda_{0}^{1}$ | $0.05t$ |  $p=1$ | 9 | 13.575 | 5.863 | 19.438 | 22.303
| $\Lambda_{0}^{1.2}$ | $0.05t$ |  $p=1$ | 9 | 32.322 | 16.586 | 48.908 | 58.851

Given the exchange is both market quality and fee driven, we see that the
transaction fee model gives satisfying result as the previous section. From
the last two columns, we could see that market quality improves since
$MQ(\hat{\tau})$ with the optimal transaction fees $\xi$ calibrated by solving
the exchange’s problems (3.9) or (3.10) is always smaller that the market
quality when $\xi=0$. The exchange has better fee gains from the strategic
trader. From the strategic trader’s perspective, they now have the incentive
to arrive earlier at time $\tau=7,8,9$ instead of $\tau=10$ when there is no
fee. Unlike Section 3.3, we find $p=0.5$ optimal for Apple under several
$\Lambda_{0}^{\rho}$ measures. This is because under the fee structure
$0.04t$, the trader’s $\hat{\tau}$ changes across different $p$, while under
$0.03t^{2}$, the trader’s $\hat{\tau}$ stays at $1$.

(a) Under linear fee $\Lambda_{0}^{0.5}(a,p)$

(b) Under linear fee $\Lambda_{0}^{0.5}(a,p)$

(c) Under linear fee $\Lambda_{0}^{0.5}(a,p(a))$

Figure 4: The value of the exchange with risk aversion parameter $\rho=0.5$
under linear fee structure, (a), (b), and (c).

Under $\Lambda_{0}$ and $\Lambda_{0}^{\rho}$, Apple’s optimal fee structure is
$0.04t$. The reason is that when fee is larger than $0.04t$, the strategic
trader keeps arriving at $\tau=9$ or $\tau=10$ at which market quality is the
worst. However, when fee is smaller than $0.04t$, the strategic trader is
truly encouraged to arrive much earlier than $\tau=8$ or $9$ thus improving
market quality but the fee gain decreases more. Therefore, the exchange finds
a balance at $0.04t$ between better market quality and better fee gains.

##  4 Conclusions

We study the strategic arrival of a trader and show that a strategic trader
always joins an auction at the last moment to have the greatest price
manipulation power. Such behavior could impair fairness and market quality
especially if the trader has a misconception of the efficient price. We
propose two solutions: randomizing the closing time and introducing time-
dependent transaction fees. With randomization ($91\%$ of chance auction
closes at $T=10$ and $9\%$ of chance closes at $T=9$), the strategic trader
would join the auction before the last moment and market quality is improved.
With transaction fees, the strategic trader would enter the auction even
earlier which further improves the market quality. We consider two possible
interests of the exchange, improving the market quality and making fee gains.
Under either considerations, our solution provides better results than without
randomization or fees.

Our results certainly have limitations. We assume the presence of a single
strategic trader instead of allowing multiple strategic traders to compete. In
terms of model setting, [Gayduk and Nadtochiy (2020)] studies a general
optimal control and stopping problem with discrete controls and proves the
existence of an equilibrium in a game in which every player is strategic,
without studying randomization or transaction fees policy. Our paper is a
specific optimal control and stopping problem with discrete stopping time and
we assume all but one players are non-strategic. [Gayduk and Nadtochiy (2020)]
sheds light on a possible extension of our model to a more general case when
there are more than one strategic players in an auction. [Alfonsi and Blanc
(2016)] studies the optimal execution strategy of a strategic trader in a
continuous market whose order has price impact on the market and finds that
the existence of price manipulation strategies depend on the choice of models.
This reminds us that changing certain settings of our model may change the
behavior of the strategic trader and possibly our conclusion. Updating the
model to include interactions of multiple strategic traders would probably
lead to similar results of this paper. Strategic traders would very likely
still choose to arrive at the last moment to avoid sharing their views too
early and to gain information of others to take advantage. We also assume a
discrete optimal stopping control instead of a continuous one. The continuous
version of this problem is in the works. We assume the auction market clears
all order imbalance while in reality order imbalance exists. Finding a
workable model to allow order imbalance is a future direction to consider. In
addition, we model simply one round of an auction. If running the auction for
several rounds, we could possibly see a larger negative impact of the
strategic trader’s arrival-timing strategy and see a greater need to regulate
the arrival of traders.

Finally, we did not mention priority rules for a periodic auction. This is
because our model assumes zero imbalance of orders, so we do not find the need
of a priority rule. In reality, order imbalance exists; for example, if A
wants to sell 10 shares of stock, B wants to buy 4 shares of stock, and C
wants to buy 8 shares of stock, B or C or both might only receive part of what
they request. Priority rules need to be set to divide the 10 shares in this
situation. Cboe’s periodic auction market assigns price priority over size
priority over time priority; [Budish et al. (2015)] also mentions that price
priority should be given over time priority. In general, time priority should
be the last to consider. Therefore, we argue that our observation (a strategic
trader lacks an incentive to join early) would still be valid given the
presence of order imbalance and priority rules. However, learning how to set
priority rules for order imbalance could be a meaningful future study.

##  Appendix A Appendix: Numerical Methods

###  A.1 Problem of a Strategic Seller

Recall the objective of a strategic seller is
$V^{\circ}=\sup_{\tau}V^{\circ}(\tau)$. Fix $(t,n,\\{p_{i}\\}_{i=1}^{n})$,
recall the price $P(t,n,\\{p_{i}\\}_{i=1}^{n})$ submitted by the strategic
seller is a normal random variable.

Denote $q(p,x,y)$ to be the joint density function of
$P(t,n,\\{p_{i}\\}_{i=1}^{n})$, $P^{*}$, $\sum\limits_{j:j=1,\tau_{j}\geq
t}^{m}P_{j}$, where $\tau_{j}$ is the arrival time of the $j-th$ market maker
with price $P_{j}$. Each of the three random variables follows a normal
distribution. For simplicity, assume the three normal random variables are
mutually independent.

Define $N_{(t,T)}:=N_{T}-N_{t}$. Assume $N_{(t,T)}$ is independent of
$P(t,n,\\{p_{i}\\}_{i=1}^{n})$ , $P^{*}$, $\\{P_{j}\\}_{\tau_{j}\geq t}$.
Denote $f_{N_{(t,T)}}$ to be the probability density function of $N_{(t,T)}$.

Fix $N_{t}=n,\\{P_{i}\\}_{i=1}^{N_{t}}=\\{p_{i}\\}_{i=1}^{n}$,
$\hat{\mu}(t,n,\\{p_{i}\\}_{i=1}^{n})$ is the optimizer of :

$\displaystyle\sup_{\mu}\mathbb{E}\biggr{[}\mathbf{1}_{P_{\mu}\leq\frac{\sum_{i=1}^{N_{T}}P_{i}}{N_{T}}}\biggl{\\{}p_{\mu}^{2}\frac{-KN_{T}}{(1+N_{T})^{2}}+p_{\mu}\left[\frac{K\sum_{i=1}^{N_{T}}P_{i}-KN_{T}\sum_{i=1}^{N_{T}}P_{i}}{(1+N_{T})^{2}}+\frac{P^{*}_{T}KN_{T}}{(1+N_{T})}\right]+\frac{K(\sum_{i=1}^{N_{T}}P_{i})^{2}}{(1+N_{T})^{2}}$
$\displaystyle\qquad-\frac{P^{*}K\sum_{i=1}^{N_{T}}P_{i}}{(1+N_{T})}\biggl{\\}}\Big{|}N_{t}=n,(P_{i})_{i=1}^{N_{t}}=(p_{i})_{i=1}^{N_{t}}\biggr{]}$
$\displaystyle=$
$\displaystyle\sup_{\mu}\scalebox{1.0}[1.5]{$\displaystyle\int$}\mathbb{E}\biggr{[}\mathbf{1}_{P_{\mu}\leq\frac{\sum_{i=1}^{N_{T}}P_{i}}{N_{T}}}\biggl{\\{}p_{\mu}^{2}\frac{-KN_{T}}{(1+N_{T})^{2}}+p_{\mu}\left[\frac{K\sum_{i=1}^{N_{T}}P_{i}-KN_{T}\sum_{i=1}^{N_{T}}P_{i}}{(1+N_{T})^{2}}+\frac{P^{*}_{T}KN_{T}}{(1+N_{T})}\right]+\frac{K(\sum_{i=1}^{N_{T}}P_{i})^{2}}{(1+N_{T})^{2}}$
$\displaystyle\qquad-\frac{P^{*}K\sum_{i=1}^{N_{T}}P_{i}}{(1+N_{T})}\biggl{\\}}\Big{|}N_{t}=n,(P_{i})_{i=1}^{N_{t}}=(p_{i})_{i=1}^{N_{t}},N_{(t,T)}=m\biggr{]}dN_{(t,T)}(m)$
$\displaystyle=$
$\displaystyle\sup_{\mu}\sum_{m=0}^{\infty}f_{N_{(t,T)}}(m)\biggl{\\{}\mathbb{E}\biggr{[}\mathbf{1}_{P_{\mu}\leq\frac{\sum^{n}p_{i}+\sum^{m}P_{j}}{n+m}}\big{\\{}-Kp_{\mu}^{2}\frac{n+m}{(n+m+1)^{2}}+Kp_{\mu}\frac{(\sum^{n}p_{i}+\sum^{m}P_{j})(1-n-m)}{(n+m+1)^{2}}$
$\displaystyle\qquad+Kp_{\mu}P^{*}\frac{n+m}{(n+m+1)}+K\frac{(\sum^{n}p_{i}+\sum^{m}P_{j})^{2}}{(n+m+1)^{2}}-KP_{T}^{*}\frac{(\sum^{n}p_{i}+\sum^{m}P_{j})}{(n+m+1)}\big{\\}}\biggr{]}\biggl{\\}}$
$\displaystyle=$
$\displaystyle\sup_{\mu}\sum_{m=0}^{\infty}f_{N_{(t,T)}}(m)\biggl{\\{}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\mathbf{1}_{P_{\mu}\leq\frac{\sum^{n}p_{i}+y}{n+m}}\big{\\{}-Kp_{\mu}^{2}\frac{n+m}{(n+m+1)^{2}}+Kp_{\mu}\frac{(\sum^{n}p_{i}+y)(1-n-m)}{(n+m+1)^{2}}$
$\displaystyle\qquad+Kp_{\mu}x\frac{n+m}{(n+m+1)}+K\frac{(\sum^{n}p_{i}+y)^{2}}{(n+m+1)^{2}}-Kx\frac{(\sum^{n}p_{i}+y)}{(n+m+1)}\big{\\}}q(p_{\mu},x,y)d(p_{\mu},x,y)\biggl{\\}}.$

where the second equality is due to independence.

###  A.2 Appendix: Problem of the Regulator

$\displaystyle\mathbb{E}\left[\left|P^{cl,t}_{T}-P^{*}\right|^{2}\right]$
$\displaystyle=\mathbb{E}\left[\bigg{|}\frac{\sum^{N_{T}}P_{i}+1_{\left\\{P_{t}\leq\frac{\sum^{N_{T}}P_{i}}{N_{T}}\right\\}}P_{t}}{N_{T}+1_{\left\\{P_{t}\leq\frac{\sum^{N_{T}}P_{i}}{N_{T}}\right\\}}}-P^{*}\bigg{|}^{2}\right]$
$\displaystyle=\scalebox{1.0}[2.0]{$\displaystyle\int$}\mathbb{E}\left[\biggl{(}\frac{\sum^{N_{T}}P_{i}+1_{\left\\{P_{t}\leq\frac{\sum^{N_{T}}P_{i}}{N_{T}}\right\\}}P_{t}}{N_{T}+1_{\left\\{P_{t}\leq\frac{\sum^{N_{T}}P_{i}}{N_{T}}\right\\}}}-P^{*}\biggl{)}^{2}\Bigg{|}{N_{(t,T)}}\right]dN_{(t,T)}$
$\displaystyle=\scalebox{1.0}[2.0]{$\displaystyle\int$}\mathbb{E}\mathbb{E}\left[\biggl{(}\frac{\sum^{N_{T}}P_{i}+1_{\left\\{P_{t}\leq\frac{\sum^{N_{T}}P_{i}}{N_{T}}\right\\}}P_{t}}{N_{T}+1_{\left\\{P_{t}\leq\frac{\sum^{N_{T}}P_{i}}{N_{T}}\right\\}}}-P^{*}\biggl{)}^{2}\Bigg{|}\mathcal{F}_{t},{N_{(t,T)}}\right]dN_{(t,T)}$

By Disintegration Theorem [Kallenberg (2002)],

$\displaystyle\mathbb{E}\left[\biggl{(}\frac{\sum\limits^{N_{T}}P_{i}+1_{\left\\{P_{t}\leq\frac{\sum^{N_{T}}P_{i}}{N_{T}}\right\\}}P_{t}}{N_{T}+1_{\left\\{P_{t}\leq\frac{\sum^{N_{T}}P_{i}}{N_{T}}\right\\}}}-P^{*}\biggl{)}^{2}\Bigg{|}\mathcal{F}_{t},N_{(t,T)}\right]=\scalebox{1.0}[2.0]{$\displaystyle\int$}\Bigg{|}\frac{\sum\limits^{N_{t}}P_{i}+y+1_{\left\\{p_{t}\leq\frac{\sum^{N_{t}}P_{i}+y}{{N_{(t,T)}}+N_{t}}\right\\}}p}{{N_{(t,T)}}+N_{t}+1_{\left\\{p\leq\frac{\sum^{N_{t}}P_{i}+y}{{N_{(t,T)}}+N_{t}}\right\\}}}-x\Bigg{|}^{2}q(p,x,y)d(p,x,y),$

where $p$ refers to $P$, $x$ refers to $P^{*}$, $y$ refers to
$\sum\limits^{N_{(t,T)}}P_{j}$.

##  Appendix B Appendix: Illustrate Remark 2.4

To better illustrate the proof, we draw a figure 5 of

$y=-(KN_{T})x+K\sum_{i=1}^{N_{T}}P_{i}-KN_{T}P^{*}.$

For each point on the line, the $x$ coordinate would be

$x(p)=P_{T}^{cl}(p)-P^{*}=\frac{\sum^{N_{T}}P_{i}+p}{N_{T}+1}-P^{*}$

and the $y$ coordinate would be

$y(p)=K(P_{T}^{cl}(p)-p)=K(\frac{\sum^{N_{T}}P_{i}+p}{N_{T}+1}-p),$

where p is the price sent by the strategic seller. The shadow area is the
trader’s gain.

We have $\bar{\mu}(\omega)$ achieved at the middle point of the line segment
and $P_{\bar{\mu}(\omega)}$ is normally distributed as
$\mathcal{N}(\bar{\mu}(\omega),\sigma^{2})$.

Figure 5:

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