Limited Attention and News Arrival in Limit Order
Markets∗
Jérôme Dugast†
August 13, 2012
Abstract
I propose a model of limit order market where agents have limited attention and
thus do not monitor the market continuously. After news on fundamentals arrive, investors limited attention delays the price change. This delay last until all limit orders
quoted at initial prices have been canceled or executed. Anticipating the trading dynamic following news arrival, investors decide to submit limit or market orders which
endogenously sets the level of liquidity supply in the limit order book. Limit orders offer
price improvement compared to market orders but bear a risk of being picked-off after
the news release. For a particular investor a higher level of attention (i.e a more frequent market monitoring) reduces her risk of being picked-off since she can react faster
to news and cancel her limit orders. At the same time it adversely worsens the risk
of being picked-off for other investors since she can execute against their limit orders
faster. Overall a global increase of all investors’ attention level could affect liquidity
supply both way. My model predicts that, when the probability of news arrival is high
enough, (i) liquidity supply before news arrival is positively related to this attention
level and (ii) the duration between news arrival and the price change is inversely related
to this attention level.
∗
The Internet Appendix is available upon request by email.
HEC, School of management, Paris. E-mail: jerome.dugast@hec.edu. I thank Thierry Foucault, William
Fuchs, Johan Hombert, Christine Parlour, Ioanid Rosu, Pierre-Olivier Weill for their very helpful comments.
†
1
1
Introduction
Investors have limited attention capacities and thus cannot monitor continuously financial
markets and its flow of information. In particular they are unable to read public news
instantaneously after their release. Before a news content is common knowledge in the
market, it must have been monitored by all investors. Consequently at short horizon public
information become private knowledge for investors who observed it first. Because of limited
attention, public information release generates a short term period of adverse selection. When
a piece of news about the asset fundamental value arrives, how do markets integrate news?
How limited attention affects this process? By studying the market reaction to these events
we can better understand the nature of price discovery in financial markets. The question of
market reaction to public information have been addressed empirically by Eredington and Lee
[1995], Fleming and Remolona [1999] and Green [2004] for instance. These papers study the
reaction of US Treasury securities markets (plus Eurodollar and Deutsche Future markets for
Ederington and Lee) to scheduled macroeconomic announcement. The two first papers show
that the market reacts to the announcement following two phases. In the first phase the price
shifts quickly to a new level in line with the main figures of the annoucemnent. The second
phase of this reaction is characterized by a high volatility suggesting that investors disagree
on the precise interpretation of the annoucement. This phase ends when the announcement
interpretations of market participants eventually converge. Green’s paper shows that these
macro announcements increase the level of adverse selection suggesting that investors with
better processing abilities can take advantage of these events. These works do not really
look for drivers of the short term dimension of market reaction (the first phase). Moreover
they only focus on news for which the release is scheduled whereas most of financial news are
not. Now these short term reactions to news become a very important issue since some High
Frequency Trading activities have grown by using intensive monitoring technology to trade
very fast on financial news. These new actors take advantage of their attention capacities
to become the short term privately informed. To better understand these recent changes in
financial market there is a need to shed light on the short term market reaction to news and
the role that limited attention plays. In addition this question must be address considering
electronic limit order markets which are the trading organizations that most equity and
derivative exchanges have now adopted over the past decades and which nowaday enable the
2
growth of automated trading.
In limit order markets investors are both supplying liquidity by sending limit orders
and consuming liquidity with market orders. Then, in this type of markets, the process of
information integration into prices relies on two underlying dynamics. After a news release,
investors aware about the associated change in the fundamental value can profit from a
transitory arbitrage opportunity and generate a directional flow of market orders that execute
against stale limit orders. Investors who have a limit order in the order book eventually
cancel these orders to react to new information. During this process limit orders providers
face the ”risk of being picked-off”. Indeed if uncancelled their limit orders offer a free option
opportunity and can be hit by the directional flow of market orders. Everything else equal
this risk of being picked-off enhances the expected loss associated to limit order submission
and has a negative impact on the liquidity supply. When investors have limited attention
capacity they cannot monitor the market continuously. Consequently the new fundamental
value become complete common knowledge only after all investors have monitored the market
and observed the piece of news. Thus the intensity (or rate) at which investors monitor the
market is a first order importance parameter to describe the dynamic of the limit order
book after a news release. Indeed it determines the flow of investors becoming aware of the
new asset fundamental value. A high market monitoring rate reduces the delay for news
observation and then increases this flow. Consequently the flow of directional market orders
and the flow of limit order cancellations are high as well.
Understanding the link between investors’ limited attention and their trading strategy is
crucial to study the dynamics of limit order markets. A priori this link has no clear direction.
To see why, let’s consider a global increase of investors monitoring intensity. On the one hand
investor can cancel their limit orders faster after news arrival which reduces the risk of being
picked-off and makes limit orders more profitable. However investors can send directional
market orders faster to execute against her stale limit orders faster which adversely worsen
the risk of being picked-off for her limit orders. Overall limit orders could be more or less
profitable after a global increase of monitoring rate. Here I identify conditions which make
limit orders overall more profitable after a global increase of investors monitoring intensity.
One can view the recent development of Algorithmic Trading (AT) in financial market as
an attempt by investors to increase their attention capacities. With the support of computers,
traders can enhance and systematize their market monitoring process. For instance High
3
Frequency Traders (a particular type of AT) aim to implement trading strategies requiring
to monitor the market almost continuously and to react to market event very fast. The
current rise of High Frequency Trading shows the importance of market monitoring. From
this viewpoint this paper also contributes to theoretically address the effect of HFT on limit
order markets.
The paper’s main contribution is to give a theoretical framework to analyse the impact
of limited attention on investors trading strategies and on limit order market dynamics. To
address this question I propose a model of a limit order market in presence of uncertainty
on the asset fundamental value. Investors’ motive for trading relies on two components of
the asset value : a common fundamental value and an idiosyncratic private value. In this
framework limited attention is equivalent to infrequent market monitoring. Investors monitor
and are present in the market with respect to a single Poisson process. This model extends
the OTC markets framework of Duffie, Garleanu and Pedersen [2005, 2007] to limit order
markets. The main difference is that it focuses on imperfect market monitoring whereas the
main imperfection in Duffie et Al. papers is the search friction for trading counterparty. In
my model, after news arrival, the diffusion of new information among investors is gradual
because of this imperfect market monitoring.
Before the fundamental value changes, investors trade due to their difference in private
value. During this phase the limit order book is in a steady-state. The level of liquidity
supply is constant and determined by the trade-off between market and limit order. Market
orders provide execution immediacy whereas limit order provide price improvement but bear
execution delay and the risk of being picked-off when the fundamental value changes. At
equilibrium the level of liquidity supply adjust so that investors indifferently use market
orders and limit orders.
When the fundamental value changes, it is publicly observable but investors do not observe
this change immediatly. After this change occured they have to wait until the first time they
monitor the market to observe the new fundamental value and eventually take an action.
This generates a transition phase in the order book where limit orders are either picked-off
by some directional market orders or removed by their owner and placed at an adequate
price level. Once limit orders at the initial price level have all been removed or executed, the
transition phase is over and the limit order book converges to a new steady-state without
fundamental value uncertainty.
4
My model provides empirical implications related to the effect of limited attention: when
the probability of a news arrival is high enough (i) the liquidity supply in the limit order book
before news arrival increases with the monitoring intensity, (ii) the duration between public
news arrival and the subsequent change in price decreases with the monitoring intensity
and (iii) the share of limit orders cancellation in the price discovery process increases with
the monitoring intensity. The paper also provides cross sectional or time-series empirical
implications related to the asset fundamental value volatility. A higher fundamental volatility
is associated with: (iv) a lower liquidity in the limit order book before news arrival, (v) a
lower share of limit orders cancellation in the price discovery process increases and (vi) a
lower duration between news arrival and the price change.
The empirical implications provided in this paper about the effect of market monitoring
are consistent with recent papers on Algorithmic Trading. Hendershott et al. [2011] use the
start NYSE autoquoting in 2003 as an exogenous instrument to measure the causal effect of
Algorithmic trading on market liquidity and find a positive impact of AT on informativeness
(ii) of quotes and a reduction of trade related price discovery (iii). Hasbrouck and Saar
[2011] construct a measure of ”low-latency” activity (a proxy for the activity of high frequency
traders in a market) and find that some measures of market quality, including depth displayed
near the bid-ask spread, are positively related to the ”low-latency” measure.
Section 2 reviews the related literature. Section 3 presents the setup and assumptions of
the model. Section 4 gives the equilibrium and its general description. Sections 5, 6 and 7
describe the properties of the different phases involved in the equilibrium dynamic of the limit
order market. They are respectively the steady state phase, the transition phase and the last
phase. Section 8 derives and explains the principal comparative statics of the equilibrium.
Section 9 discuss empirical implications of the model. Section 10 concludes.
2
Literature review
Theories of attention allocation by investor accross assets have been recently developped by
Peng and Xiong [2006], Van Nieuwerburgh and Veldkamp [2009] and Mondria [2010]. These
papers focus on the effect of limited attention on portfolio diversification and asset prices.
My paper contributes to this literature on limited attention by looking at its effect on trading
mechanism. Here limited attention constrains investors to imperfectly monitor the market.
5
Market monitoring imperfection has already been stressed as a key determinant of market
dynamics by Darrell Duffie [2010] Presidential Address. Foucault, Kadan and Kandel [2009]
address this problem in a limit order book framework. In their paper agents strategically
choose their level of market monitoring but agents are exogenously considered as limit order
or market order users. Biais and Weill [2009] and Biais, Hombert and Weill [2010] also
consider imperfect monitoring agents. Compared to other limit order market modelling
there is a unique time discount rate among investors whereas Foucault et al. [2005,2009]
and Rosu [2009,2010] assumes two types of time preference (patient and impatient traders)
to help generate different trading behaviour. The focus of my paper is different since it
deals with limit order market dynamics generated by asset value uncertainty rather than
aggregate liquidity shocks as in Biais et al. [2009,2010]. Moreover I model how fundamental
uncertainty affect trading strategies before the change in the asset fundamental value occurs.
Whereas in Biais et al. modelling the market dynamic starts with the liquidity shock. In
my model the change in the asset fundamental value is a publicly observed signal but it is
not instantaneously observed since market monitoring is imperfect. Pagnotta and Philippon
[2012] study the effect of competition between Exchanges on the market monitoring intensity
or latency they provide to their customers and identify this competition as an incentive for
investing in fast trading technologies. Biais et al. [2009,2011], Pagnotta and Philippon [2012]
as well as my paper use and adapt the model of search friction in OTC markets introduced
by Duffie, Gârleanu and Pedersen [2005, 2007] and extended by Lagos and Rocheteau [2009],
Lagos, Rocheteau and Weill [2011], Vayanos and Weill[2008] and Weill [2007,2008].
The problem of information monitoring and liquidity provision has been studied by Foucault, Roëll and Sandas [2003] in the case of a dealership market. In their model Market
Makers face adverse selection by informed traders and can reduce this risk by monitoring
public information and adjusting their quote. The choice of the monitoring rate is costly. In
my model monitoring rate is an exogeneous parameter but it affects both liquidity supply
and demand since in limit order markets they are potentially constituted of the same investors. Goettler, Parlour and Rajan [2009] design a very realistic environment of limit order
market that is not tractable and meant to be solved numerically. In their paper traders do
not continuously monitor the market and they ex-ante decide to be privatly informed or not
about the asset fundamental value.
This paper also builds on the dynamic limit order market litterature. There are quite
6
a few paper dealing with this problem when compared to its practical importance. One of
the reasons is that limit order markets are very hard to model. Foucault [1999] and Parlour
[1998] are the first models of limit order markets designed as dynamic games capturing the
inter-temporal aspect of the problem. The tractability of these models is appreciable but is
reached at the cost of strong assumptions. Both incorporate private and/or common value
as drivers of trading and price formation processes but do not allow for strategic decision
over the limit order lifetime. Foucault, Kadan and Kandel [2005] focus on the dynamic of
the liquidity supply in a limit order market. In their paper people trade for liquidity reason
and solve the market vs. limit order trade-off in function of their preference for immediacy.
Rosu [2009] generalizes Foucault, Kadan and Kandel framework and design a continuous
time model where traders can freely send limit orders at any price and can cancel them.
Rosu [2010] add a common value environnment to his previous model. The two papers by
Rosu are build on the fundamental assumption that limit orders are continuously monitored
by their owners. As in Rosu’s models I design a framework that allow for an entire freedom
of choice for investors’ order management at the exception of the zero or one unit holding
constraint (as in Rosu [2009,2010]). Pagnotta [2010] design a limit order book model with
insider trading where agents optimally choose their trading frequency. At equilibrium they
don’t trade continuously but they continuously observe the market and update their belief
accordingly.
3
3.1
Model
Preferences and asset dynamics
The economy is constituted of a continuum of investors [0, L]. They are all risk neutral and
infinitely lived, with time preferences determined by a constant discount rate r > 0.
Preferences. As in Duffie, Gârleanu and Pedersen [2005,2007], an investor is characterized by whether she owns the asset and by an intrinsic type that is ”high” or ”low” which
corresponds to high or low asset private value for the investor. A high type owner enjoys
a utility flow of v by owning this asset whereas a low type owner receives a utility flow of
v − δ. Between time t and time t + dt an investor can be affected by a change in her private
7
value and switches from one type to another (high to low or low to high) with probability ρ.dt.
Asset supply and dynamic. The asset supply is equal to S =
L
2
that is initially dis-
tributed among investors who can hold either 1 unit or 0 unit of this asset.
The dynamic of the fundamental value v is the following:
- at t = 0 the asset fundamental value is equal to v0
- at date τ > t the asset fundamental value changes. This time τ is random and follows
a Poisson distribution of intensity µ, P(µ). At date τ the fundamental value switches
to v u = v0 + ω or v d = v0 − ω with equal probabilities
1
2
- for t > τ the asset keeps the same fundamental value until the end of the game.
For 0 < t < τ the state of the world is ζ = ∅. For τ < t the state of the world is either ζ = u
if v = v0 + ω or ζ = d if v = v0 − ω.
Given the previous assumptions any investor must have a type in the set {ho, hn, lo, ln}
(h: high, l: low, o: owner, n: non-owner). And we can divide the mass of investors in 4
populations: Lho , Lhn , Llo , Lln . They verify the equations
Lho + Lhn + Llo + Lln = L,
Lho + Llo =
L
2
It is possible to extend the number of possible by taking into account the limit order
submission status of investors. Indeed in a limit order book an owner can either be out of the
market or have an order in the order book at any price reachable. As well for a non-owner.
This setting can generate many subtypes of the previous types. Let’s call T the set of all
possible types. If an investor does not have any limit order submitted in the order book she
is out. If she has a limit order submitted we have to precise at which price it is. For instance
a type ln can be ln − out or ln − B with a limit order at price B. Symmetrically a type lo
can be lo − out or lo − A with a limit order at price A.
Assumption 1. We assume that
ω >> δ
8
It means if the change of the fundamental value, with magnitude ω, is not followed by a
change in price, the gain from the arbitrage opportunity is around ω, and is bigger than gain
from trade for difference in private values, measured by δ.
3.2
Limit order market
Trading takes place in a limit order market. Prices at which trade can occur must belong too
a countable set of prices, the price grid. The minimal difference between two of these prices
is the tick size, ∆. Investors can use limit or market orders to trade. Limit orders are orders
that specifies a maximum price at which the order can be executed. They are stored in the
order book until matched with another market order. The depth of the limit order book at
price P , DP , is the volume of all limit orders submitted at price P . Market orders do not
specify a price limit. They hit most competitive limit orders and get execution immediacy.
For technical reasons we assume that this price grid is bounded. This is reasonnable since
trading will not occur at prices higher than a certain threshold if the asset fundamental value
is bounded (the corresponding strategies would be strictly dominated by a strategy in which
investors don’t trade). This assumption is also present in Parlour [1998], Foucault, Kadan
and Kandel [2005] for instance.
Each time they are in contact with the market (see below Ass.1) investors can take an
infinite number of actions in the following list.
• As an owner they can : do nothing and remain an owner, send sell limit order in the
order book and remain an owner until his/her order is executed, send a sell market
order and become a non-owner or cancel a previous sell limit order.
• As a non-owner they can : do nothing and remain a non-owner, send buy limit order
in the order book and remain a non-owner until his/her order is executed, send a buy
sell market order and become an owner or cancel a previous buy limit order.
This defines the action set of an investor that we define (with some notation abuse) as
A = {do nothing, market order, limit orders at the different prices}N
Assumption 2. Between time t and t + dt an investor can take an action if she contacts
the market. It happens in the following cases:
9
• when she monitors the market, which occurs with probability λ.dt. The market monitoring process is a Poisson process of intensity λ.
• When her private value changes (as described above), which occurs with probability ρ.dt.
Anytime an investor contacts the market she observes the asset fundamental value.
Assuming that investors are not permanently in the market is realistic since traders have
to allocate their time among trading and other activities, and in their trading time they
cannot focus all the time on a single asset. The level of presence of any investor is measured
by λ and at the limit λ = ∞ all investor are in the market at all time.
The second assumption states that investors continuously monitor their private value and
contact the market whenever this private value changes. This assumption is probably more
realistic than the opposite. It also allows to reduce the anticipation problem of the investor
that has to take into account the possibility of future idiosyncratic shock especially when
facing the decision to send a limit order. Indeed she knows that when a shock occurs she has
the possibility to cancel a previous limit order. Then it prevents from being executed while
it is not optimal anymore according to her private value.
Assumption 3. In the limit order book, limit orders are executed following a ”Pro-rata
matching” execution rule. It means that all limit orders submitted at the same price (A
or B) have the same probability of execution at any point in time, irrespectively of their
submission date.
Assumption 4. We assume that
δ
>> ∆, in particular δ − (r + 2ρ)∆ > 0
r
It means that the gain from trade for difference in private values, measured by rδ , is bigger
that the implicit cost of trading, the bid-ask spread which is related to the tick-size, ∆.
10
3.3
Value function and equilibrium concept
An investor is choosing her actions at each random time when she is contacting the market.
The strategy σ of an agent is a function
σ :H × Ξ × [0, ∞) → A
(h, ξ, t) 7→ a
Ξ is the set of all potential state variables. An element of this set ξ ∈ Ξ is defined as
ξ = (θ, v, S) where θ ∈ T is a type, v is the fundamental value of the asset and S is the
aggregate state of the limit order book, that is to say the bid and ask prices and all the
depths at these prices. H is the set of all possible histories of actions and observations of an
investor:
H = {h ∈ (at1 , . . . , atn , ξt1 , . . . , ξtn , t1 , . . . , tn ) ∈ An × Ξn × [0, ∞)n , t1 < . . . < tn , n ∈ N}
Her strategy, σ, and the strategies of all other investors, Σ, are generating her asset holding process ηt ∈ {0, 1} that is equal to 1 when she holds one unit of the asset, her type
process θt ∈ T and a process of trading prices Pt at which her orders are executed any time
she changes her holding i.e. when ηt switches from 0 to 1 or conversely.
At time t the value function of an investor playing strategy σ is given by
Z
V (ht , ξt , t; σ, Σ) = Et
∞
e−r(s−t) dUs
t
s.t dUt = ηt (v − δI{θt ∈ lo} )dt − Pt dηt
The strategy σ is a best response to the other players set of strategies Σ if and only if for all
strategy γ,
∀ht ∀ξt ∀t V (ht , ξt , t; σ, Σ) ≥ V (ht , ξt , t; γ, Σ).
In this paper I focus more specifically on Markov perfect equilibria where the strategy
does depend only on state variables, (θ, v, S).
11
3.4
Discussion of assumptions
Asset value dynamic. The modelling for the asset value dynamic is equivalent to other
dynamics where the asset pays-off at some random time in the future and does not provide a
continuous flow of utility. This interpretation of the asset dynamic is probably more relevant
to take the model at the daily frequency for instance.
• The asset pays off the cash-flow V =
v
r
at a random time that occurs with respect to a
Poisson process with intensity r. And being a low type induces a cost for holding the
asset which is equal to δ per unit of time.
• Or, at a random time that occurs with respect to a Poisson process with intensity r,
The asset pays off a cash-flow
v
r
for high types and
v−δ
r
for low types.
Pro-rata execution rule. In practice there are some markets where the ”Pro-rata matching” is implemented. However for the majority of stock markets Time Priority applies. The
reality of the Time priority is mitigated by the fact that there are multiple trading platforms
and that agents can use smart order routine for achieving best trading conditions. The flow
of market orders is split among different trading platforms. Then the time at which a limit
order executed is randomized.
4
4.1
Equilibrium result
The symmetric equilibrium
Proposition 4.1. For all intensity of news arrival, µ, there exists an equilibrium for which
bid and ask prices are symmetrical with respect to the fundamental value of the asset,
v ∗ − 2δ
r
,
1
δ
∆
1
δ
∆
B ∗ = (v ∗ − ) − , A∗ = (v ∗ − ) + .
r
2
2
r
2
2
There are 3 pairs of these prices: the one at the beginning of the game, (A0 , B 0 ), and the one
at the ends of the game, (Au , B u ) and (Ad , B d ). At these prices the equilibrium is unique and
is characterized by the depths of the limit order book at price A0 and B 0 at the beginning of
the game,
∅
DA0 = DB 0 = αeq
L
12
Proof. see Internet Appendix E.3
The equilibrium dynamic of the limit order market has the following features:
• For 0 ≤ t < τ , when the asset fundamental value is equal to v0 , trading takes place
because of investors’ private value: lo and hn trade with each other via the limit
order book, ho and ln do not trade. In particular buy limit orders are submitted by
types hn and sell limit orders are submitted by types lo.
During this first phase the dynamic of the limit order market is the following:
– the limit order book is in a steady-state phase. The liquidity provision at each
price in the order book and the order flows do not vary over time.
– lo type investors send sell limit or market orders with respect to a mixed strategy.
They choose to send a market order at price B 0 with probability m or a limit
order at price A0 with probability 1 − m. They do not send sell limit orders at
prices higher than A0
– hn type investors send buy limit or market orders with respect to a mixed strategy.
They choose to send a market order at price A0 with probability m or a limit order
at price B 0 with probability 1 − m. They do not send sell limit orders at prices
lower than B 0 .
– A sell limit order submitted at price A0 is executed in the future with respect to
a Poisson distribution of intensity l∅ . A buy limit order submitted at price B 0 is
executed in the future with respect to a Poisson distribution of intensity l∅ .
∅
– The parameter αeq
that characterize the initial depths adjusts such that the limit
order book is in a steady-state.
• Once the asset fundamental value has changed, a transition phase starts for the limit
order book. This transition phase last for a finite duration T . During this phase, for
τ < t < τ + T , trading takes place because the new fundamental value of the asset
is not in line with the market prices and thus creates an opportunity of arbitrage.
– if the new fundamental value is v0 + ω, lo investors cancel their sell limit order
and repost them at a higher price Au , hn and ln investors send buy market orders
13
to execute against stale limit orders at price A0 of lo investors and then stay out
of the order book, ho investors stay out of the order book.
– if the new fundamental value is v0 − ω, hn investors cancel their buy limit order
and repost them at a lower price B d , ho and lo investors send sell market orders
to execute against stale limit orders at price B 0 of hn investors and then stay out
of the order book, ln investors stay out of the order book.
• When all limit orders that could potentially be picked-off have been executed or cancelled the transition phase is over. Trading takes place because of investors’ private
value. During this last phase ho and ln investors do not trade, lo and hn investors
trade with each other via the limit order book. The equilibrium strategy and the limit
order book aggregate state converge to a steady-state phase that has the same
features than the first phase except that there is no uncertainty for the future fundamental value, µ = 0. Bid and ask prices are either Au and B u or Ad and B d depending
on the previous change in the fundamental value.
L.O = Limit Order; M.O = Market Order
A= Ask Price; B= Bid Price; u = « up » good news
L.O
Au
Au
Bu
Liquidity supply
at ask price A
L.O
A
L.O
L.O cancellation &
resubmission
Au
M.O
L.O
M.O
Bu
L.O
L.O cancellation &
resubmission
A
M.O
B
M.O
B
B
L.O
SELL ORDERS IN RED
Liquidity supply
at bid price B
BUY ORDERS IN BLUE
« Good » news arrival
Steady State
No limit orders left at A
Transition
Converging
to
Steady State
Figure 1: Dynamic of the limit order book in the symmetric equilibrium
14
4.2
equilibrium multiplicity
There are others equilibria than the one described above. First there are equilibria where the
three pair of prices, (A0 , B 0 ), (Au , B u ) and (Ad , B d ), are different than the prices given above.
In these case the strategies are almost identical to the one of the symmetrical equilibrium
but the equilibria do not necessarily exists for high values of µ.
But there are also some other class of equilibrium. Indeed in order to solve for the
equilibrium of this game one must proceed by guess and check. The first step is to conjecture
equilibrium strategies for all agents. The easiest is to assume that all agents play the same
strategy. Given this strategy it is possible to determine the dynamic of the limit order book.
The last step is then to check that it is not profitable to operate a one-shot deviation from
the conjectured strategy for any type, at any point in time of the game while other agents
are playing the conjectured strategy.
Because the problem solving requires that the equilibrium is guessed, defining the set of
all equilibria is difficult. The type of equilibrium described above is not unique. There is for
instance the empty limit order book equilibrium. It has the following features:
• For 0 ≤ t < τ , when the asset fundamental value is equal to v0 , trading takes place
because of investors’ private value: lo and hn trade with each other via the limit
order book, ho and ln do not trade. Investors coordinate on a trading price P where
ho and ln send (marketable) limit orders. The buy and sell order flows due to ln’s and
hn’s are exactly equal which implies that their limit orders are immediatly executed
and that the limit order book is always empty of liquidity.
• For t > τ the same type of trading dynamics takes place at price P u if v = v0 + ω and
at P d if v = v0 − ω
In this type of equilibrium there is no transition phase. The trading price immediatly adjust
when the fundamental value changes. Moreover this type equilibrium reach the maximum
welfare possible in the model. Indeed this equilibrium is such investors of type ho and ln
have incentive to trade immediatly and thus do not stay with their current position more
than an infinitesimal period of time. At the end all the asset supply is owned by ”high-type”
investors which is socially optimal.
15
In this paper I have decided to focus on the first class of equilibria rather than the second
because the reality of limit order markets is quite obviously closer from the first one. Moreover
in the empty limit order book equilibrium investors coordinate with each other to trade at
a price that they do not observe when they look at the limit order book since it is empty.
When the limit order book is not empty investors can figure out what are the equilibrium
prices by observing the state of the limit order book.
5
Limit order book in steady state
In this section I explicit the strategy that corresponds to the first phase of the game. Here
I don’t focus on the symmetric equilibrium and I extend the analysis to the whole class of
similar equilibria. In the first phase the limit order book is in a steady state until the time
of the news arrival. Liquidity is supplied at ask and bid prices A and B and the level of this
supply remains constant over time until the asset fundamental value changes.
Definition 5.1. A limit order market is in a steady state when the diplayed depth in the
order book and the different order flows are deterministic and do not change over time.
This steady state is possible in the model because there is a continuum of investors. Each
investor faces idiosyntratic uncertainty on her type. She switches from ”high” to ”low” or
”low to high” with respect to a Poisson process of intensity ρ. By the law of large number
applied to the continuum of investors, the share of these investors that are switching of type
is deterministic and equal to ρ.dt at each point in time. Fot the same reason the share of
investors monitoring the market is deterministic and equal to λ.dt.
5.1
One-tick market
Proposition 5.1. A limit order market that is in a steady state at equilibrium is necessarily
a one-tick market in the the bid-ask spread, A − B = ∆, is equal to the tick of the market.
Moreover if the fundamental value does not change over time and the limit order market
is in steady state forever then liquidity supply is concentrated at best bid and ask prices:
• all sell limit orders are sent at the price A, generating a depth DA , and there are no
sell limit order at higher prices than A
16
• all buy limit orders are sent at the price B, generating a depth DB , and there are no
buy limit order at lower prices than B
Proof. In a steady state at equilibrium limit orders and market orders are sent by agents
with respect to an optimal strategy. It generates flows of limit and market orders that are
constant and deterministic over time so that the steady state hold. Let’s consider a agent for
who it is optimal to send a buy market order at A. If there is a reachable price A < P < B it
is profitable to send a limit order at P since it would be immediatly hit by the flow of market
orders and would get price improvement compare to A. This contradicts the optimality of
the strategy.
In the steady state that last forever, a sell limit order that is send at a higher price than
A will never be executed because the liquidity supply at the price A keep the same positive
value and is never totally consumed. Symmetrically for buy limit orders. Then there is no
incentive to send limit orders further from the best quotes A and B.
The equilibrium dynamics of the limit order book that I conjecture implies that the limit
order book is going to be in a steady state before the news release about the change in
the fundamental value and some time after the prices have adjusted to the new fundamental
value. In these two steady states the fact we are in a one-tick market is an equilibrium result.
Moreover it is also true that in steady states limit orders are only sent to the best quotes
even in the initial phase where the steady state does not last forever but this is a result that
is proved in further propositions.
This one-tick market result explicitly relies on the modelling. There is a continuum of
investors and a ”zero or one unit” holding constraint. Then random idiosyncratic events affect
a deterministic share of these investors because of the law of large number and finally turn
into deterministic flow of orders and cancellations that have to be finite because of the holding
constraint. The key reason of this result is the the market order flow that is deterministic
and continuously positive which makes any limit order alone inside the bid-spread immediatly
executed. It is also the fact the instantaneous market order flow is infinitesimal and thus is
not big enough to move the prices. One might think that in a large market where trades
take place quite continuously and where market orders are small enough to not push prices,
for instance if robot optimized execution by slicing a big order into small ones, then the
occurrence of one-tick bid-ask spread could be big. Indeed the incentive to send a limit order
17
inside the best quotes rather than a market order would hold because execution could be
almost immediate.
5.2
Steady state strategy
The limit order book being in a steady state corresponds to a phase where motive for trade is
investors’s private value. Investors of type ho and ln are satisfied with their situation when
prices at which the asset is traded are between the high valuation and the low valuation.
Then they should not trade. Investors of type hn and lo would be better off by changing
their holding status. Then they have an incentive to trade. If they use a market order they
directly join the group of ”satisfied” people. If they use a limit order then they remain
”unsatisfied” until their order is executed. The consequence of this strategy is that investors
of type lo and hn who have already contacted the market are in the limit order book. And
then in the steady state (i.e. at the limit) they are all in the limit order book.
Strategy 5.1. More specifically the equilibrium strategy is defined as follows:
• ho: cancel any sell limit order and stay out of the market
• hn: send a buy limit or market order with respect to a mixed strategy : when she
contacts the market she submit a buy market order with probabilty mA ∈ [0, 1]. It is
executed at the ask price A.
• lo: send a sell limit or market order with respect to a mixed strategy : when she contacts
the market she submit a sell market order with probabilty mB ∈ [0, 1]. It is executed at
the bid price B.
• ln: cancel any buy limit order and stay out of the market
The prices A and B are assumed to be different by one tick A − B = ∆ so that we are in a
one-tick market.
This equilibrium strategy implies the one-tick market result for the value of the bid-ask
spread as well as for the fact that limit order would be sent only at best quotes.
In this equilibrium the populations Lho and Lln are not present in the limit order book.
As soon as a ho type switches to a lo type she instantaneously contacts the market: either
she instantaneously switches to a ln type by sending a sell market order or stay a lo type
18
by sending a sell limit order. Symmetrically as soon as a ln type switches to a hn type
she instantaneously contacts the market: either she instantaneously switches to a ho type
by sending a buy market order or stay a hn type by sending a buy limit order. This is
straightforward that in a steady state equilibrium, DA = Llo and DB = Lhn .
5.3
Steady state populations
In a steady state the composition of aggregate populations stays at the same level. Then the
flows of population from high type to low type and from low type to high type must be equal
to each other, ρ(Lho + Lhn ) = ρ(Llo + Lln ). Combined with the constraints due to the size of
the overall population L and the asset supply L/2 we obtain
Lho + Lhn =
L
L
, Llo + Lln =
2
2
Proposition 5.2. In a steady state equilibrium there is an α ∈ R such that the different
types of population satisfy
1
Lho = Lln = ( − α)L, Lhn = Llo = αL
2
This implies that
1
2
− α ≥ 0 and α ≥ 0.
This free parameter α is determined at equilibrium. It is equal to the liquidity supply in
the limit order book since the depths are equal to DA = Llo = αL and DB = Lhn = αL.
5.4
Micro-level dynamic of the limit order book
In the equilibrium conjecture hn and lo types are indifferent between limit and market orders
so that we observe in the same time flows of liquidity demand and supply that make the
state of the limit order book sustainable and steady. Given that we look for a steady state
equilibrium, the flows must be steady as well.
The mixed strategy implies that a share mA of the population of hn type contacting the
market at t sends a buy market order and the rest of them send a buy limit order. For the
same reason a share mB of the population of lo type contacting the market at t sends a sell
market order and the rest of them send a sell limit order.
19
Ask Side: At time t, on the ask side of the market the depth is constantly equal to DA = Llo
and the order flows going in and out of the ask side of the order book are
• Outflow due limit order executions: execution of buy market orders send by hn
type contacting the market, mA (λLhn + ρLln ).dt.
• Outflow due limit order cancellations: people switching from lo to ho, ρLlo .dt, lo
type cancelling their sell limit order to send a sell market order, mB λLlo .dt
• Inflow due to limit order submissions: people switching from ho to lo type submitting a sell limit order, (1 − mB )ρLho .dt
The steady state condition is : ρLlo + mA (λLhn + ρLln ) + mB (λLlo + ρLho ) = ρLho .
Bid Side: At time t, on the ask side of the market the depth is constantly equal to DB = Lhn
and the order flows going in and out of the bid side of the order book are
• Outflow due limit order executions: execution of sell market orders send by lo
type contacting the market mB (λLlo + ρLho ).dt.
• Outflow due limit order cancellations: people switching from hn to ln, ρLhn .dt,
hn type cancelling their sell limit order to send a sell market order, mA λLhn .dt
• Inflow due to limit order submissions:: people switching from ln to hn submitting
a sell limit order, (1 − mA )ρLln .dt
The steady state condition is : ρLhn + mB (λLlo + Lho ) + mA (λLhn + ρLln ) = ρLln .
5.5
Execution rate and liquidity provision
At any point in time t of the steady state phase, the flow of market orders execute against
a share of the limit orders in the order book. Given that the limit order are executed with
respect to the Pro-Rata rule all limit orders on the side of the book are equally likely to be
executed. Between t and t + dt this probability is equal to the share of the liquidity supply
equal to the instantaneaous flow of market order.
For instance on the ask side, the flow of market order is equal to mA (λLhn + ρLln ).dt
and the liquidity supply is equal to DA = Llo . The instantaneous probability of execution is
20
equal to
lA .dt =
mA (λLhn + ρLln )
.dt
Llo
lA is the execution rate for sell limit orders. In the same way we can define the execution
rate for buy limit orders, lB =
mB (λLlo +ρLho )
.
Lhn
Compensation for providing liquidity. When an investor submits a limit order instead of a market order, she chooses optimal execution price but renounces to execution
immediacy, bears a risk of non-execution and eventually a risk of being picked-off when the
asset fundamental value changes. Providing liquidity requires some kind of compensation
for risk taking. This compensation is obtained by an appropriate execution delay for limit
order execution. More precisely the execution rate lA and lB must incentivize liquidity provision via limit orders. In equilibrium execution rate are such that market and limit orders
are equally profitable for types hn and lo. This mechanism appears clearly in agents value
functions (next subsection).
Once the execution rates and the state of the limit order book, or equivalently the populations, are defined at equilibrium, the mixed strategy are perfectly defined. For intance
mB =
Lhn
l .
λLlo +ρLho B
Steady state liquidity provision. By incorporating the execution rates the two steady
state equations can be rewritten as
ρLhn + lB Lhn + lA Llo = ρLln
ρLlo + lA Llo + lB Lhn = ρLho
These equations are in fact equivalent and it defines the value of the steady state population, or equivalently the last free parameter α
α=
ρ
1
2 2ρ + lA + lB
The aggregate properties of the limit order market in this steady state is completely
described by the value α and the execution rates lA and lB . Indeed they define the steady
state populations, the depths and the aggregate order flow in the limit order book.
21
5.6
Value functions
The equilibrium strategy generates the following system of equations defining the different
value functions for each investor types.
Type ho: An investor of type ho stays out of the market its asset until she switches to
the lo type. Her situation is affected when the fundamental value changes. Her value function Vho−out is defined as follows
Vho = v.dt + (1 − r.dt)[(1 − ρ.dt − µ.dt)Vho−out + ρ.dtVlo
1 d
1 u
(0) + Vho−out
(0))]
+ µ.dt( Vho−out
2
2
µ u
d
⇐⇒ (r + ρ)Vho−out = v + ρVlo + [Vho−out
(0) + Vho−out
(0)]
2
u
d
The term µ2 [Vho−out
(0) + Vho−out
(0)] corresponds to the change in utility when the asset fun-
damental value changes up or down. These are the values of being a type ho during the
transition phase.
Type ln: An investor of type ln stays out of the market until she switches to the hn
type. Her value function Vln−out is defined as follows
(r + ρ + µ)Vln−out = ρVhn +
µ u
d
[Vln−out (0) + Vln−out
(0)]
2
As for the type ho, the value function of type ln is defined by the change of value function
in the case of a change of type or when the fundamental value changes. However type ln do
not receive any utility flow as ho who gets v.dt at each point in time.
Type hn: An investor of type hn send a buy market order with probability mA or limit
order with probability 1 − mA . Sending a buy market order at price A provides her with the
value function Vho−out − A. Indeed she gets execution immediacy by trading at the ask price
A and instantaneously switch to type ho. Sending a buy limit order at price B provides her
22
with the value function Vhn−B defined as follows
(r + ρ + lB + mA λ + µ)Vhn−B = ρVln−out + mA λ(Vho−out − A) + lB (Vho−out − B)
µ u
d
(0) + Vhn−B
(0)]
+ [Vhn−B
2
Once the limit order book has been submitted several event can happen: either the investor
changes of type with intensity ρ and become ln or the investor contacts again the market
with intensity λ and cancel her limit order to send a market order with probability mA or
the limit order is executed with intensity lB . Each of these events correspond to a change
in the utility function and define the value function of submitting a limit order. Types hn
become indifferent between and market orders if and only if Vhn−B = Vho − A and then the
value function of a type hn is Vhn = Vhn−B = Vho − A.
Type lo: An investor of type lo send a buy market order with probability mB or limit
order with probability 1 − mB . Sending a sell market order at price B provides her with the
value function Vln−out + B for the same reason than for type hn. Sending a sell limit order
at price A provides her with the value function Vlo−A defined as follows
(r + ρ + lA + mB λ + µ)Vlo−A = v − δ + ρVho−out + mB λ(Vln−out + B) + lA (Vln−out + A)
µ u
d
+ [Vlo−A
(0) + Vlo−A
(0)]
2
Types hn become indifferent between and market orders if and only if Vlo−A = Vln + B and
then the value function of a type lo is Vlo = Vlo−A = Vln + B.
The indifference equations define required values for lA and lB . This is easy to check that
these value do not depend on the mixed strategy mA or mB . Actually, as mentionned earlier,
these parameters adjust to make the equilibrium possible. Typically lA and lB depends on
value functions in the transition phase. These value function depend on α since the level of
liquidity provision affect the duration of the transition phase for instance. Remind that α
corresponds to the depth of the limit order book and that the transition phase lasts until
this depth has been completly executed or removed. In the end lA and lB depends on α and
reciprocally. Solving for the equilbrium of the game is equivalent to solve this fixed point
23
problem for the first phase steady state. It also requires to study the transition phase.
The transition phase is followed by a third phase where the limit order book converges to
a steady state without fundamental uncertainty. The first steps to solve for the game is then
to study the steady state equilibrium without uncertainty and its corresponding converging
equilibrium.
5.7
Steady state in the symmetric equilibrium
In the initial steady state the symmetric equilibrium the quotes are:
1
δ
∆
1
δ
∆
B 0 = (v 0 − ) − , A0 = (v 0 − ) + .
r
2
2
r
2
2
The symmetry of this equilbrium implies that the term of the trade-off between limit order
vs. market order is the same on both side of the market. Then the execution rate that makes
investors indifferent between limit and market orders are the same for sell and buy orders
∅
∅
∅
lA
∅ = lB ∅ = l
Proposition 5.3. The execution rate that makes lo and hn investors indifferent between
limit and market orders is decreasing with respect to α∅ ,
∂l∅
∂α∅
Moreover for α∅ = 0 and α∅ =
1
4
l∅ is finite and positive.
The steady state condition of the limit order book implies that the depths of the order
book, measured by α∅ and the equilibrium execution rate are linked by the formula
ρ
α =
⇐⇒ l∅ = ρ
4(ρ + l∅ )
∅
1
−1
4α∅
The execution rate implied by this formula is infinite for α∅ = 0 and nil for α∅ =
1
.
4
It
ensures that the two curves corresponding to the two relations between l∅ and α∅ and thus
that potential equilibria exist.
24
2000
1500
1000
0.0002
0.0004
0.0006
0.0008
0.0010
∅
Figure 2: Twe two functions of l∅ w.r.t αeq
(λ = 100,µ = 50, r = 1, ρ = 2, ∆ = 1, δ =
10, ω = 50).
5.8
Steady state equilibria without fundamental uncertainty
When there is no fundamental uncertainty, µ = 0, the indifference conditions can be solved
in close form. A type hn is indifferent between limit and market orders if
(r + ρ + lB )(Vho − A) = ρVln + lB (Vho − B)
A type lo is indifferent between limit and market orders if
(r + ρ + lA )(Vln + B) = v − δ + ρVho + lA (Vln + A)
Proposition 5.4. Solving the system implied by these equations gives
v − rA − ρ∆
∆
rB − ρ∆ − (v − δ)
lA =
∆
11
1 1
Vho =
(v − ρ∆) +
(v + ρ(A + B))
r2
r + 2ρ 2
11
1 1
Vln =
(v − ρ∆) −
(v + ρ(A + B))
r2
r + 2ρ 2
lB =
Vhn = Vho − A
Vlo = Vln + B
Proof. see Internet Appendix B.4.1
25
Proposition 5.5. Assuming that δ − (r + 2ρ)∆ > 0, the equilibrium bid and ask prices, B
and A, verify the inequalities
v ρ
v δ ρ
− + ∆≤B<A≤ − ∆
r r r
r r
and the one-tick market property, A − B = ∆. The equilibrium is defined by the conjectured
strategy and the equilibrium populations are characterized by the value
αeq =
1 ρ∆
2 δ − r∆
Proof. see Internet Appendix B.4.2 and B.5.1
Remark 5.1. The assumption that δ − (r + 2ρ)∆ > 0 is necessary to assure that the interval
v δ ρ v ρ − r + r ∆, r − r ∆ is non-empty and larger than ∆.
r
To understand the above inequality we can look at the subset of equilibrium prices
v δ r+ρ v δ ρ
v δ ρ v ρ
−
, −
− + ∆, − ∆
⊂
r r r + 2ρ r r r + 2ρ
r r r
r r
This inclusion is a consequence of δ − (r + 2ρ)∆ > 0.
investor to hold the asset forever and
v
r
−
δ ρ
r r+2ρ
v
r
r+ρ
− rδ r+2ρ
is the value for a ”low” type
is the value for a ”high” type investor to
hold the asset forever. These are the reserve values for these two types of investor when they
hold the asset. In the case where a low-type owner and a low-type non-owner would meet
once and leave the market afterward then the trading price would have to be between these
two reserve values.
In the steady state equilibrium of the limit order market trading also takes place between
low type owners and a high type non-owners. However the range of trading prices is wider
than the difference between the two reserve values because investors can trade more than once.
Monitoring intensity irrelevancy. An interesting feature of this equilibrium is that aggregate outcomes, as αeq , do not depend on λ, the monitoring intensity. This is an expected
outcome of the model since trades occur because of difference in private values and because
these private values are monitored continuously. This suggests that monitoring rate has a
limited role in a stable market. More specifically monitoring rate plays a role when liquidity
26
supply is, for instance, cyclical like in Foucault et al. [2009]. In my model there is no cycle
since order flows are such that the order book is steady.
Equilibria multiplicity. This is important to notice that there are multiple potential
equilibria because in the range vr − rδ + ρr ∆, vr − ρr ∆ there could be more than two prices
available. For instance if they are 3 available prices in this range there are 2 potential equilibria. The choice of an equilibrium over another should involve some coordination between
agents in a stage preceding the equilibrium or at the beginning of a game where the limit
order book is empty and then is filled to converge to a steady-state equilibrium. In this paper
I don’t address this question. In the last sections of the paper I must pick a particular pair of
prices on which investors coordinate to play the equilibrium with fundamental uncertainty.
In the case without fundamental uncertainty the choice of the equilibrium is irrelevant in
term of welfare. In each of these equilibrium the welfare of the population is equal to
11
1
− αeq L × (Vho + Vln ) + αeq L × (Vlo + Vhn ) =
(v − ρ∆) − αeq ∆ L
2
r2
which does not depend on the level of the bid and ask prices in the range of the available
equilibrium prices.
Friction of the tick size. The maximum level of welfare than can be drawn from the
asset is equal to
v1
L.
r2
It is obtained when all the asset supply 21 L is owned by high type in-
vestors whose population size is 12 L as well. In this situation each share of the asset is always
offering a utility flow equal to v and then has a value equal to vr . To reach this optimum it
requires that when hn and lo investors come to the market, after they have changed of type,
they can trade immediatly at one price that is the same for buyers and sellers.
In our the steady-state equilibria this optimum can be reached if the tick size is nul,
∆ = 0, as we can see in the formula for the welfare. In these equilibria ∆ is the friction that
prevents from reaching the optimum. First because the bid-ask spread is positive investors
have to pay an implicit trading cost ∆ if they choose execution immediacy which generates a
ρ
loss in welfare captured by the term − 2r
∆L. Second because there is a difference is the price
execution for limit and market orders, investors have an incentive to send limit orders and
to wait for execution whereas it would be socially optimal to get immediatly executed. The
27
corresponding loss is captured by the term −αeq ∆L. It shows that the presence of liquidity
supply, αeq L is suboptimal.
When ∆ = 0, the bid and the ask prices are infinitly close. Then the price improvement
of submitting limit orders is nil and the execution intensities lA and lB must be infinite to
incentivize limit order submission. Because of these infinite execution rate the limit orders
are instantaneously executed and the limit order book is always empty. We can view this
equilibrium as a situation where investors coordinate to trade with each other at a single
price P = A = B and where there is no difference between limit and market orders.
Perfectly symmetric case. The particular pair of prices that I pick in the following
of paper define what I call the perfectly symmetric case. This is the case where the term of
limit order vs. market order trade-off is the same on both side of the market. The equilibrium
prices makes the execution rates equal, lA = lB :
δ
∆
1
δ
∆
1
B = (v − ) − , A = (v − ) +
r
2
2
r
2
2
In this particular setting the value of the execution rate is lA = lB = l =
δ−(r+2ρ)∆
.
2∆
The
execution rate depends on δ, the difference between the low and high private value, which
measures the gain from trade. The bigger it is the less investors are willing to wait and
thus require a higher execution rate. This execution rate is also negatively impacted by the
tick-size since a high tick-size is a high trading cost for market orders which makes investors
more willing to wait to be executed with a limit order.
Limit order survival before cancellation in the order book. Once a limit order has
been submitted it does not stay in the in the order book forever even if it is not executed.
Let’s consider the perfectly symmetric case (the intuition does not change accross equilibria).
Each time an investor with a limit order in the book monitor the market, with intensity λ,
she cancels a limit order to send a market order with probability m =
αeq
l.
λαeq +ρ(1/2−αeq )
She
will also cancel her limit order when she changes of type, with respect to a Poisson process
of intensity ρ. Overall, conditionally on not being executed, the cancellation of limit order
follows a Poisson distibution of intensity ρ + λm.
When λ is high the intensity ρ + λm is bounded and does not reach high values. It is
28
actually lower than ρ + l. It means that, when the monitoring intensity increases, investors
do not play a strategy where they send a limit order for a very short amount of time and to
cancel it quick if not executed. This kind of strategy is growing due to HFT and has been
documented by Hasbrouck and Saar [2009]. They refer at ”fleeting orders” for these type of
short-time suvivorship limit orders.
Continuous monitoring.
The market monitoring rate λ does not impact the aggre-
gate values of the equilibrium as the value function, the population levels linked to αeq
or the limit order execution rates lA and lB . We can take the model to the limit where
investors are continuously monitoring the market, λ = ∞. Let’s consider the ask side of
the book and remind that the flow of market orders hitting the ask side at t is equal to
mA (λLhn + ρLln ).dt = mA (λαeq + ρ( 21 − αeq )).dt = lA Lhn .dt = lA αeq .dt. This flow is independant of λ. When λ → ∞ we must have mA → 0 so that this flow remains constant. And at
the limit the flow of market order is equivalent to mA λαeq .dt which implies that mA λ → lA .
For an investor of type hn, mA λ.dt is the probability that she submits a market order at
time t. Noticing that allows to describe the investors strategy in the limit case. When an
investor switches to type hn she submits a limit order at price B with probability 1 because
the probabilty to send a market order is mA that is infinitesimal. At time t her order is
either executed with probability lB .dt, or she decides to cancel it to send a market order with
respect to a mixed strategy with probability lA .dt, or she cancels it if she switches to type
ln.
For the same reason when an investor switch to type lo she submits a limit order at price
A with probability 1. At time t her order is either executed with probability lA .dt, or she
decides to cancel it to send a market order with respect to a mixed strategy with probability
lB .dt, or she cancels it if she switches to type ln.
Taking the limit case leads to an equilibrium where investors plays a Poisson mixed strategy to choose between limit and market orders. If we were to consider directly the problem
with continuous monitoring we could end up with different type of mixed strategies where
for instance investors would submit market order with positive probability at the time their
type change and then play a Poisson mixed strategy. However these strategies should be
such that the flow of market orders and the execution rates are the same as defined above
since the terms of the trade-off do not change.
29
The empty limit order book equilibrium. When there is no fundamental uncertainty
the empty limit order book equilibrium is defined by the price P in the grid at which investors
coordinate to trade. The outcome of this equilibrium is the same that the steady-state equilibrium outcome with ∆ = 0 and A = B = P . It implies for instance that (v − δ)/r < P < v/r.
The difference between this equilibrium and the previous asymptotic equilibrium for ∆ = 0 is
that the other possible trading prices are still finite whereas in the asymptotic case it become
, vr ]. In particular it implies that the set of possible deviations is
a continuum of values [ v−δ
r
smaller than in the asymptotic case and thus ensures the equilibrium result for the empty
limit order book case. In the case where there is fundamental uncertainty, µ > 0, the empty
limit order book equilibrium also correponds to the asymptotic equilibrium for ∆ = 0.
6
Limit order book in transition phase
When there is fundamental uncertainty, µ > 0, the first steady state phase does not last
forever and is followed by a transition phase. Preceding the beginning of the transition phase
the world is in the state ζ = ∅. The transition phase starts when the asset fundamental value
changes. This also corresponds to public news arrival. It arrives at some point in time τ . It
is stochastic and follows a Poisson distribution, P(µ). For times t > τ the state of the world
is either ζ = u (up) and v = v 0 + ω or ζ = d and v = v 0 − ω(down) with equal probability.
We call T u and T d the duration of the transition phases in the different states of the world.
6.1
Transition phase strategy
To define properly the strategy in the transition phase we need to decide what will be the
steady state phase after the transition is over. As mentionned in the case without uncertainty
there are multiple steady state equilibria corresponding to different level of prices. Here again
there are multiple equilibria since investors have to coordinate on the pair of ask and bid
prices that will define the final steady state.
Strategy 6.1. After the fundamental value has changed if ζ = u, for t > τ + T u , investors
coordinate on the steady state equilibrium over the bid-ask prices (Au , B u ) and if ζ = d,
for t > τ + T d , investors coordinate on the steady state equilibrium over the bid-ask prices
(Ad , B d ).
30
During the transition phase, the strategy is:
• In the case ζ = u, for τ < t < τ + T u :
- lo’s cancel any sell limit order that is not at price Au and submit a limit order at
price Au
- ho’s cancel any sell limit order and stays out of the market
- ln’s send a buy market order and immediatly behave as their new type, lo
- hn’s send a buy market order and immediatly behave as their new type, ho
• In the case ζ = d, for τ < t < τ + T d :
- hn’s cancel any buy limit order that is not at price B d and submit a limit order at
price B d
- ln’s cancel any buy limit order and stay out of the market
- ho’s send a sell market order and immediatly behave as their new type, hn
- lo’s send a sell market order and immediatly behave as their new type, ln
The idea of this strategy is that when the fundamental value changes to a high level for
instance, non-owner turn into arbitrageurs and have an incentive to buy the asset while it is
tradable at a low price, A0 , and to resell it at a high price Au later. Here we conjecture that
this strategy is optimal.
6.2
Limit order book dynamics in the transition phase
Before the transition phase begins the limit order book is filled with some limit orders and
offers liquidity. In particular liquidity provisions at best ask and bid prices are defined by
∅
the value of the depths of the limit order book at prices A0 and B 0 . These are equal to DA
0
∅
and DB
0 . During the transition phase trading occurs only on one side of the order book. On
this side limit orders give an arbitrage opportunity. On the other side investors cancel their
limit order and send a market order to hit limit order offering this opportunity.
For instance when the asset fundamental value makes a positive jump, ζ = u, sell limit
orders submitted at price A0 give a profit opportunity to buyers. Indeed A0 was an equilibrium price when the asset fundamental value was equal to v0 and it is no longer the case
31
u
after this value has moved up to v0 + ω. The liquidity supply on the ask side DA
0 (t) is
meant to disappear. It decreases with respect to two kind of actions. At each time t a mass
u
0
(λ + ρ)DA
0 (t).dt of investors contact the market and cancel their limit order at price A .
At the same time a mass (λ + ρ) × (Lhn (t) + Lln (t)).dt =
λ+ρ
L.dt
2
of investors who do not
own the asset contact the market and send buy market orders that execute at price A0 . The
u
dynamic of DA
0 (t) is given by the following proposition.
Proposition 6.1. When ζ = u during the transition phase the depth at price A0 is
1 −(λ+ρ)t
1
∅
u
L]e
DA
0 (t) = − L + [DA0 +
2
2
which is decreasing and has a unique zero, defining the duration T u .
Proof. see Internet Appendix D.2.2
When the asset fundamental value moved down trading occurs and the phenomenon is
the same as in the ”up” case but on the bid side of the order book at price B 0 .
Proposition 6.2. When ζ = d during the transition phase the depth at price B 0 is
1
1 −(λ+ρ)t
d
∅
DB
L]e
0 (t) = − L + [DB 0 +
2
2
which is decreasing and has a unique zero, defining the duration T d .
Proof. see Internet Appendix D.2.4
6.3
Equilibrium in the subgame starting with the transition phase
Once the common value v has changed, after the transition phase, for t > τ + T u/d , the
conjecture strategy defines are an equilibrium solved in section III. For instance if ζ = u at
t = T u investors play the dynamic equilibrium with the different population at time T u and
u
u
u
u
u
u
the depth DA
and DB
u (T ) at A
u (T ) = 0 at B .
Proposition 6.3. In the subgame starting at τ , whether ζ = u or d, the conjecture strategy
is an equilibrium strategy.
Proof. see Internet Appendix D.3.1 and D.3.2
32
The analysis of this transition phase allows to understand the trading mechanism that is
underlying to the dynamic of prices in a limit order market. As I develop it in the empirical
implication section we can evaluate the impact of market monitoring or fundamental volatility
on the time that it takes for prices to reflect new information. This is also possible to
determine the role that limit and market orders play in this price discovery process. In
particular we can quantify the effect of the market monitoring rate on the share of limit
order cancellations and market order executions in the erosion of the initial liquidity supply.
One would link this result to the effect of High Frequency Trading around news arrival.
Yet we know what equilibrium strategy is played after news arrival. This section does
not solve for the entire game equilibrium since the initial conditions of the limit order book
at τ depend on the features of the equilibrium preceding time τ .
6.4
Transition phase of the symmetric equilibrium
In the symmetric equilibrium the dynamic of the order book in the transition phase is given
∅
∅
d
u
∅
by the initial values of the depths, DB
∅ = DB ∅ (0) = DA∅ (0) = DA∅ = α L, and the equation
1 −(ρ+λ)(t−τ )
1
∅
d
u
]Le
∀t DB
∅ (t) = DA∅ (t) = D(t) = − L + [α +
2
2
In the symmetric equilibrium the durations of the transition phases are the same in both
states u and d
Tu = Td = T =
7
1
ln(1 + 2α∅ ).
ρ+λ
After the transition phase : convergence to a steady
state without fundamental uncertainty
In this section I explicit the strategy and the dynamic of the limit order book that corresponds to the last phase of the game after the transition phase is over. In the last phase of the
game the limit order book converges to a steady state. The asymptotic steady state of the
last phase have the same type of strategy the steady state of the first phase. Trading takes
place at prices Au and B u if v = v0 + ω or at Ad and B d if v = v0 − ω. The main difference is
that in the first phase investors know that the asset fundamental value will change at some
33
point in time which is not the case in the last phase. In this last phase µ = 0.
Here I present the general case of this dynamic equilibrium that converges to steady state
without fundamental uncertainty. In this equilibrium the terms of the trade-off between limit
and market orders do not change over time and are the same as the ones in the asymptotic
steady state. I use the same notations as in the limit order book in steady state without
fundamental uncertainty (αeq , A, B...etc, cf subsection 4.7).
Starting at t = 0 from a one tick market where the depth at prices A and B are DA (0)
and DB (0) constituted respectively by a share of the population Llo (0) and of Lhn (0), agents
follow their corresponding steady state equilibrium strategy described earlier. The rates
at which hn and lo types are sending market orders, mA (t) and mB (t), are evolving so that
the terms of the trade off are the same as in the steady state equilibrium. More precisely the
intensities at which limit orders are executed are unchanged and equal to lA and lB . In this
framework the dynamic of the different population is given by the dynamic of the parameter
α,
1
Lho (t) = Lln (t) = ( − α(t))L
2
Lhn (t) = Llo (t) = α(t)L
and the value functions for each type are the same as in the former steady-state equilibrium.
To fully characterize the level of convergence of the limit order book we look at how its
state is different from the limit steady state. First in the steady state all agents have positions
in line with their optimal strategy. For instance at the limit t = ∞ all types lo have a limit
order in the book at price A. In the dynamic game a lo type agent may have been out of the
market to start with and then has to wait for her first contact with the market to submit a
limit order. The difference Llo (t) − DA (t) measures the mass of types lo out of the market.
At time t the mass of investors out of the market who would optimaly be in the market are
given by the following equations
Llo (t) − DA (t) = (Llo (0) − DA (0))e−(λ+ρ)t
Lhn (t) − DB (t) = (Lhn (0) − DB (0))e−(λ+ρ)t
34
The second distance measure from the asymptotic steady state is the difference between
the populations level at time t and the populations in steady state equilibrium. This difference
is captured by the expression α(t) − αeq . These two dimensions of the level of convergence
is actually reduced to one. Indeed, as for the steady-state case, describing the evolution of
α(t) is enough to describe the dynamic of the order book since Lho (t), Lln (t), Lhn (t), Llo (t),
DA (t) and DB (t) are fully defined when α(t) is known.
Proposition 7.1. The dynamics of the equilbrium populations are given by the dynamic of
the parameter α,
α(t) = αeq + (α(0) − αeq )e−(2ρ+lA +lB )t
+ lA κA
with κA =
1 − e−[λ−(ρ+lA +lB )]t −(2ρ+lA +lB )t
1 − e−[λ−(ρ+lA +lB )]t −(2ρ+lA +lB )t
e
+ lB κB
e
λ − (ρ + lA + lB )
λ − (ρ + lA + lB )
Llo (0)−DA (0)
,
L
κB =
Lhn (0)−DB (0)
L
Proof. see Internet Appendix C.2
Proposition 7.2. The conjecture strategy is an equilibrium strategy. It generates a dynamic
of the limit order book that converges toward a steady state equilibrium at the same ask and
bid prices, A and B. In this dynamic equilibrium the value functions for the different agent
types are constant and equal to the value functions in the asymptotic steady state equilibrium.
8
Determinants of the liquidity supply prior to the
news arrival
Proposition 8.1. For a value of µ high enough, an increase of the monitoring rate λ has a
∅
positive impact on αeq
.
∅
∂αeq
>0
∂λ
∅
An increase of the fundamental volatility, µ or ω has a negative impact on αeq
∅
∅
∂αeq
∂αeq
< 0,
<0
∂µ
∂ω
∅
Moreover limµ→∞ αeq
= 0.
35
Proof. see Internet Appendix E.4
0.00040695
0.00040690
0.00040685
400
600
800
1000
∅
w.r.t λ ∈ [50, 1000] (µ = 50, r = 1, ρ = 2, ∆ = 1, δ = 10, ω = 50).
Figure 3: Evolution of αeq
0.0010
0.0009
0.0008
0.0007
0.0006
0.0005
0.0004
30
40
50
60
70
∅
Figure 4: Evolution of αeq
w.r.t µ ∈ [20, 70] (λ = 100, r = 1, ρ = 2, ∆ = 1, δ = 10, ω = 50).
The two last comparative statics come from the fact that the execution rate that makes
investors indifferent between limit and market order increases with µ and ω. Investors are
less willing to use limit order when the volatility of the asset fundamental value increases,
everything else equal.
∂l∅
∂l∅
> 0,
>0
∂µ
∂ω
Mechanically the depth of the order book adjusts because the relation α∅ =
verified.
36
ρ
4(ρ+l∅ )
has to be
The effect of λ on α∅ also come from the fact that l∅ decreases with respect to λ. However
this dependence of l∅ on λ comes from two different channels. Let’s consider on individual:
when λ increases, these are the monitoring rate, λ∗ , of this particular individual and the
monitoring rate of the rest of the population, λ0 , that both increase (These two monitoring
rates are equal λ∗ = λ0 = λ). These two monitoring rates have different effects on l∅ . The
expression of l∅ is the following
l∅ =
µ
δ − (r + 2ρ)∆
+
[U1 + U2 ]
2∆
2∆
with
u
u
d
0
d
U1 = Vln−out
(0) + B 0 − Vlo−A
0 (0) = Vho−out (0) − A − Vhn−B 0 (0)
Z T
δ
∆
∗
= −∆ − [ω − + r ]
h(t)e−(r+ρ+λ )t dt
2
2 0
d
d
u
0
u
U2 = Vln−out
(0) + B 0 − Vlo−A
0 (0) = Vho−out (0) − A − Vhn−B 0 (0)
=
ω + 2δ − r ∆2
(λ∗ + ρ) ωr − ρ∆ −(r+ρ+λ∗ )T
+
e
r + ρ + λ∗
r + ρ + λ∗
The dependence of l∅ on λ0 is hidden in the duration of the transition phase T and in the
function h
0
1
1 − e−(ρ+λ )t
∅
T =
ln(1
+
2α
),
h(t)
=
1
−
with h(T ) = 0
ρ + λ0
2α∅ e−(ρ+λ0 )t
U2 and U1 are the difference in value function between taking liquidity and supplying liquidity at the beginning of the transition phase. When an investor decides between a market
order and limit order she takes into account the outcome of the difference after the news
arrival. l∅ internalizes these differences in value functions to make the investor indifferent.
U2 instance is the difference between sending a market order or sending a limit order when
the piece of news arrives and when the limit order does not risk to be picked-off. When
an investor has the type lo for instance, before news arrival, she knows that if she sends a
market order and that afterward a negative piece of news arrived, ex-post it will be optimal.
That is why U2 is positive. If she sends a market order it will be ex-post suboptimal after
37
negative news. If this investors increases her monitoring rate she will be able to ”correct” this
mistake more easily before the transition phase is over. Then she will need less compensation
in term of execution rate. If the monitoring rate of other investors increases, the duration of
the transition phase decreases and the time window for sending a market order at the initial
price shrinks then she will require a higher execution rate to be indifferent. That is why whe
have
∂U2
∂U2
< 0 and
>0
∗
∂λ
∂λ0
U1 instance is the difference between sending a market order or sending a limit order when
the piece of news arrives and when the limit order does risk to be picked-off. When an
investor has the type lo for instance, before the news arrival, she knows that if she sends a
market order or a limit order at A0 and that afterward a positive piece of news arrives, she
knows that both of these actions will be ex-post sub-optimal since as a lo type she will prefer
to post a limit order at Au and as a ln type she will prefer to send a buy market order and
then post a limit order at Au . What makes this investor still choose the market order or the
limit order at A0 before the news arrival is the gains from trade due to her private value plus
the equal likelihood of negative news arrival.
U1 is negative because as a lo type with a limit order at price A0 in the worst case will
end up being executed at price A0 and become a ln during the transition phase. The most
suboptimal situations in the transition phase after a positive news is to be a ln type. Because
the value function of types lo − A0 and ln are those of suboptimal situations it is difficult
to intuitively figure out which of these two benefits more from the increase of λ∗ and suffers
more from the increase of λ0 . At the end we find that
∂U1
∂U1
> 0 and
>0
∗
∂λ
∂λ0
Overall and after calculation we obtain
∂l∅
µ ∂U1 ∂U2
∂l∅
µ ∂U1 ∂U2
=
+
< 0 and
=
+
>0
∂λ∗
2∆ ∂λ∗
∂λ∗
∂λ0
2∆ ∂λ0
∂λ0
and the total effet of the increase of λ is a decrease of l∅ for µ high enough.
38
∂l∅
∂l∅
∂l∅
=
+
<0
∂λ
∂λ∗ ∂λ0
9
Empirical implications
9.1
Liquidity supply before news arrival
Prediction 9.1. Before the asset fundamental value changes to a different high or low level
∅
L. When the probability that
the liquidity supply on both sides of the order book is equal to αeq
news arrival occurs is high (i.e. µ high enough) the liquidity supply is positively impacted
by an increase of the monitoring rate λ. It is also negatively impacted by an increase of the
volatility of the fundamental value either through µ or ω.
The overall effect of an increase of the global market monitoring is to increase the liquidity
supply in the order book which indicates that the risk of being picked-off for a limit order
is lessened. As mentionned previously this effect was not obvious since the increase of the
monitoring rate qualitatively extend one investor capacity to react fast to new information
but also enable other traders to do so.
The effect of fundamental volatility on liquidity supply is as expected. Indeed when the
fundamental value changes the execution of a limit order corresponds to a loss because it is
picked-off. The ex-ante probability of this loss increases with µ and the size of this loss is
proportionnal to ω.
9.2
Duration between news arrival and price change
The duration between the news arrival and the change in transaction prices in the limit order
book is the duration of the transition phase.
T =
1
∅
ln(1 + 2αeq
)
ρ+λ
Prediction 9.2. For high enough values of λ and µ an increase of the monitoring rate (λ)
decreases the duration of the transition phase T . An increase of the volatility of the asset
fundamental (µ or ω) decreases as well this duration.
39
Prices in the limit order book reflect the new fundamental value of the asset once there
is no arbitrage opportunity left, that is to say that the initial liquidity supply offering this
arbitrage opportunity has disappeared. The populations of potential arbitrageurs is fixed.
This is the group of now-owner if the fundamental value goes up and the group of owner if
the fundamental value goes down. Then the instantaneous flow of directional market orders
aiming to profit from the arbitrage opportunity is proportionnal to the rate at which this
population monitor the market, λ + ρ, and it does not depend on the paramaters that rule
∅
.
the dynamic of the fundamental value. µ and ω only affect the initial liquidity supply αeq
The effect of an increase of µ or ω is mechanical since it decreases the initial liquidity supply
that is consumed and removed faster in the transition phase. The effect of λ affect both the
liquidity supply and the flow of directional orders. Since the liquidity supply is a bounded
function of λ, the monitoring rate ends up reducing the duration of the transition phase at
the limit (this result needs to be further explored).
One should notice that as soon as the asset holding constraint on investors is independent
of µ or ω during the transition phase the flow of directional market orders would remain
independent of these parameters and the result would still hold. The ”zero or one unit”
assumption is not key here. However there is a need for a holding constraint otherwise
investors could send infinitly large orders and consume instantaneously the liquidity supply.
The fact that λ and µ or ω are independent is less obvious. As for model of limited
attention allocation, investors could decide of their λ depending on the asset characterictic.
This calls for further extension of the model to endogenize the choice of λ.
9.3
Order flow decompostion in the price discovery process
Corollary 9.1. In the transition phase the amount of market orders executed and limit orders
canceled are
∅
ln(1 + 2αeq
)
L,
MO =
2
"
#
∅
ln(1
+
2α
)
eq
∅
LOC = αeq
−
L
2
Moreover the ratio of limit order cancellation over executed market order is increasing
∅
with respect to αeq
:
∂ LOC
>0
∅ MO
∂αeq
40
Prediction 9.3. In the transition phase, the ratio of limit order cancellation over executed
market order is
• increasing with the monitoting rate λ when µ is high enough
• decreasing with the fundamental volatility parameters µ or ω
The mechanism behind this result is the following. As mentionned in the previous subsection the flow of directional market orders during the transition phase is proportional to
∅
, µ or ω. On the side of the liquidity supply the instanλ + ρ and does not depend on αeq
taneous probability for an investor to cancel her limit order is also (λ + ρ).dt. The mass of
∅
L at the beginning of the transition phase and equal to D(t) afterward.
these investors is αeq
Then the flow of limit order cancellations at t during the transition phase is (λ + ρ)D(t).dt
∅
∅
is increased, at each point in time the flow
. If initially αeq
which depends positively on αeq
of limit order cancellations is increased whereas the flow of directional market orders is the
same. This explains why the share of limit order increases. However the transition phase last
longer which explains why the number of market orders during the transition phase increases
as well.
This predictions is in line with recent empirical works by Hendershott et al.[2010] that
shows that Algorithmic Trading is associated with a reduction of trade related price discovery,
that is to say price change due to the action of market orders.
10
Conclusion
This paper models a limit order market where investors trade both for private value and for
fundamental value. Their monitoring of the market is imperfect which generates a risk of
being picked-off for limit orders when the asset fundamental value is uncertain. The main
findings of this paper are the following: when the probability of news arrival is high enough,
1. the liquidity supply in the limit order book before news arrival: (i) increases with the
monitoring intensity and (ii) decreases with the fundamental value volatility
2. the duration between news arrival and the price change : (i) decreases with the monitoring intensity and (ii) decreases with the fundamental value volatility
41
3. the share of limit orders cancellation in the price discovery process : (i) increases with
the monitoring intensity and (ii) decreases with the fundamental value volatility
42
References
[1] Back K. and Baruch S. (2007), ”Working orders in limit order markets and floor exchanges”, Journal of Finance 62, 1589-1621.
[2] Biais B., Hillion P. and Spatt C. (1995), ”An empirical analysis of the limit order book
and the order flow in the Paris Bourse”, Journal of Finance 50, 1655-1689.
[3] Biais B., Martimort D. and Rochet J-C. (2000), ”Competing mechanisms in a common
value environment”, Econometrica 68, 799-838.
[4] Biais B. and Weill P.-O. (2009), ”Liquidity shocks and order book dynamics”, Working
Paper.
[5] Biais B., Hombert J. and Weill P.-O. (2010), ”Trading and liquidity with limited cognition”, Working Paper.
[6] Duffie D., Gârleanu N. and Pedersen L. (2005), ”Over-the-counter markets”, Econometrica 73, 1815-1847.
[7] Duffie D., Gârleanu N. and Pedersen L. (2007), ”Valuation in over-the-counter markets”,
Review of Financial Studies 20, 1865-1900.
[8] Duffie D. (2010), ”Presidential address : asset price dynamics and slow moving capital”,
Journal of Finance 65, 1237-1267.
[9] Ederington L.H., and Lee J. (1995), ”The Short-run dynamics of the price adjustment
to new information”, Journal of Financial and Quantitative Analysis 30, 117-134.
[10] Fleming M. J. and Remolona E. M. (1999), ”Price formation and liquidity in the U.S.
treasury market: the response to public information”, Journal of Finance 54, 1901-1915.
[11] Foucault T. (1999), ”Order flow composition and trading costs in a dynamic limit order
market”, Journal of Financial Markets 2, 99-134.
[12] Foucault T., Kadan O. and Kandel E. (2005), ”Limit order book as a market for liquidity”, Review of Financial Studies 18, 1171-1217.
43
[13] Foucault T., Roëll A. and Sandas P. (2003), ”Market Making with Costly Monitoring:
An Analysis of the SOES Controversy”, Review of Financial Studies 16, 345-384.
[14] Foucault T., Kadan O. and Kandel E. (2009), ”Liquidity cycles and make/take Fees in
electronic markets”, Working Paper.
[15] Glosten L. and Milgrom P.(1989), ”Bid, ask and transaction prices in specialist market
with heterogeneously informed trader”, Journal of Financial Economics 13, 71-100.
[16] Glosten L. (1994), ”Is the electronic open limit order book inevitable?”, Journal of
Finance 49, 1127-1161.
[17] Goettler R., Parlour C. and Rajan U. (2005), ”Equilibrium in a dynamic limit order
market”, Journal of Finance 60, 2149-2192.
[18] Goettler R., Parlour C. and Rajan U. (2009), ”Informed traders and limit order markets”, Journal of Financial Economics 93, 67-87.
[19] Green T. C. (2004), ”Economic news and the impact of trading on bond prices”, Journal
of Finance 59, 1201-1233
[20] Hasbrouck J. and Saar G. (2009), ”Technology and liquidity provision: The blurring of
traditional definitions”, Journal of Financial Markets 12, 143-172.
[21] Hasbrouck J. and Saar G. (2011), ”Low-Latency trading”, Working Paper.
[22] Hendershott T., Jones C. and Menkveld A. (2011), ”Does algorithmic trading improve
liquidity?”, Journal of Finance 66, 1-33.
[23] Hendershott T. and Riordan R. (2009), ”Algorithmic trading and information”, Working
Paper.
[24] Hollifield B., Miller R. and Sandas P. (2004), ”Empirical analysis of limit order markets”,
Review of Economic Studies 71, 1027-1063.
[25] Hollifield B., Miller R., Sandas P. and Slive S. (2006), ”Estimating the gaine from trade
in limit order markets”, Journal of Finance 61, 2753-2804.
44
[26] Jovanovic B. and Menkveld A. (2010), ”Middlemen in limit-order markets”, Working
Paper.
[27] Lagos R. and Rocheteau G. (2009), ”Liquidity in asset markets with search frictions”,
Econometrica 77, 403-426.
[28] Lagos R., Rocheteau G. and Weill P.-O. (2011), ”Crises and liquidity in over-the-counter
markets”, Journal of Economic Theory 146, 2169-2205.
[29] Mondria J. (2010), ”Portfolio Choice, Attention Allocation, and Price Comovement”,
Journal of Economic Theory 145, 1837-1864.
[30] Pagnotta E. (2010), ”Information and liquidity trading at optimal frequencies”, Working
Paper.
[31] Pagnotta E. and Philippon T. (2012), ”Competing on speed”, Working Paper.
[32] Parlour C. (1998), ”Price dynamics in a limit order market”, Review of Financial Studies
11, 789-816.
[33] Parlour C. and Seppi D. (2003), Liquidity-based competition for order flow, Review of
Financial Studies 16, 301-343.
[34] Parlour C. and Seppi D. (2008), ”Limit order markets: a survey”, Handbook of Financial
Intermediation & Banking.
[35] Peng L. and Xiong W. (2006), ”Investor attention, overconfidence and category learning”, Journal of Financial Economics 80, 563-602.
[36] Rosu I. (2009), ”A Dynamic model of the limit order book”, Review of Financial Studies
22, 4601-4641.
[37] Rosu I. (2010), ”Liquidity and information in order driven markets”, Working Paper
[38] Seppi D. (1997), ”Liquidity provision with limit orders and a strategic specialist”, Review
of Financial Studies 10, 103-150.
[39] Van Nieuwerburgh S. and Veldkamp L. (2009), ”Information immobility and the home
bias puzzle”, Journal of Finance 64, 1187-1215.
45
[40] Vayanos D. and Weill P.-O. (2008), ”A Search-based theory of the on-the-run phenomenon”, Journal of Finance 63, 1351-1389
[41] Weill P.-O. (2007), ”Leaning against the wind”, Review of Economic Studies 74, 13291354.
[42] Weill P.-O. (2008), ”Liquidity premia in dynamic bargaining markets”, Journal of Economic Theory 140, 66-96.
46