Does order splitting signal uninformed order flow?
Hans Degryse∗
Frank de Jong†
Vincent van Kervel‡
August 1, 2013
Abstract
We study the problem of a large liquidity trader who must trade a fixed amount
before a deadline and wishes to minimize the expected cost of trading. We add
this trader to the Kyle (1985) framework to endogenize the price impact of trading.
Under the assumption that the informed traders have short-lived private information,
we show that the predictable component in the order flow only stems from the trades
by the large liquidity trader. In turn, the market maker perceives this predictable
component as uninformed and does not revise prices, such that the liquidity trader
enjoys lower price impacts. We thus show that order splitting is a noisy form of
preannouncing trades, i.e., sunshine trading (Admati and Pfleiderer, 1991). This
prediction runs in opposite direction to the “front-running” or predatory trading
hypothesis of Brunnermeier and Pedersen (2005).
JEL Codes: G10; G11; G14;
Keywords: Market microstructure, Kyle model, Order splitting, Optimal execution
problem
∗
KU Leuven, Tilburg University and CEPR.
Tilburg University
‡
Corresponding author, VU University Amsterdam. Email: [email protected]. The authors thank
Fabio Feriozzi, Corey Garriott, Peter Hoffmann, Katya Malinova, Andreas Park, Ioanid Rosu and seminar
participants at the University of Toronto and the Advances in Algo and HF Trading conference at the
University College London, 2013. Comments and suggestions are appreciated. All errors are ours.
†
1
1
Introduction
In the last decades equity turnover has increased sevenfold, while average order sizes have
declined tenfold (Chordia, Roll, and Subrahmanyam, 2011).1 Among other reasons, this is
a consequence of algorithmic trading and the practice of “order splitting”, where a large
quantity to be traded is split up into many small packages which are executed over time.
Order splitting is nowadays a standard practice in the investment management industry.
This paper studies the optimal execution problem of an institutional investor who must
trade a given quantity before a deadline, and wishes to minimize the expected execution
costs. The investor has liquidity motives and aims to optimally split up her to-be-traded
quantity across periods and trades. We endogenize the price impact parameter by placing
this problem in the Kyle (1985) framework, and show that the price impact (Kyle’s lambda)
is strongly affected by the strategy of this large liquidity trader. Moreover, we find that
order splitting is a noisy form of preannouncing trades, i.e., sunshine trading (cf. Admati
and Pfleiderer (1991)). In the model, the predictable component in the order flow stems
from the large liquidity trader only, which is a noisy signal of her trading interest. This
predictable order flow is uninformed and does not affect prices, which reduces the liquidity
trader’s price impact.
Our mechanism at first sight seems counterintuitive as we typically expect traders
to hide their trading interest. In particular, our prediction runs in opposite direction of
the predatory trading hypothesis by Brunnermeier and Pedersen (2005), where predatory
traders (or “front-runners”) exploit the liquidity needs of others by trading ahead of them.
Our theory thus provides a novel explanation of why a predictable trading strategy may
reduce the price impact of trading.
The model builds on the multi-period discrete-time adverse selection model of Kyle
1
Average trade sizes have decreased from $80,000 to $7,000, and monthly turnover increased from 6%
to 40% of market cap for CRSP stocks in the period 1993 - 2008 (Chordia, Roll, and Subrahmanyam,
2011).
2
(1985), where we add a discretionary large liquidity trader who optimally splits up trades
over time. The large liquidity trader, the informed trader and the noise traders submit
market orders to the competitive market maker, who observes the aggregate order flow and
determines the price to clear the market. The market maker is risk neutral and cannot
distinguish between trades from informed and uninformed investors. Therefore, the trades
by the discretionary large liquidity trader do affect the price, and because the market is
anonymous she cannot simply preannounce her trading interest (Admati and Pfleiderer
(1991)).2
The model’s key assumption is short-lived private information, meaning that in each period a new informed trader arrives who may only trade in that period. Also, in each period
a new innovation in the fundamental value of the asset occurs, such that the asymmetric
information does not resolve over time.
Our main contribution is that, under these assumptions, order splitting by the discretionary liquidity trader is a noisy form of preannouncing trading interest, i.e., sunshine
trading. Our result follows from a general result of the Kyle model that to the market
maker, the trades by the informed traders are unpredictable. Then, only the trades by
the discretionary liquidity trader cause predictability in the order flow. In equilibrium, the
market maker attaches a zero price impact to the predictable component of the order flow.
The assumption of short-lived private information creates a separating equilibrium, in the
sense that the informed traders cannot mimic the strategy of the discretionary liquidity
trader.3 Effectively, the discretionary liquidity trader sends a credible signal of her uninformed trading interest, and is rewarded by a lower price impact of trading. This result
contrasts Brunnermeier and Pedersen (2005), who show that the selling pressure of distressed traders induces predatory traders to also sell and subsequently buy back the asset
2
While an investor could say that he would buy a certain amount of shares in the future, there is no
mechanism that forces him to actually to do so, i.e., preannouncement is a non-credible commitment.
3
If the informed traders would also split up trades across periods, then the autocorrelation in the order
flow is not strictly uninformed and the equilibrium breaks down.
3
at lower prices. Their result requires a lack of liquidity provision in the market,4 whereas
we assume a large group of competitive market makers.5
We motivate our modeling assumptions for the different types of players as follows.
Informed traders have short-lived information, possibly because they are faster in processing public information than other traders. These informed traders use aggressive highfrequency trading strategies with short trading horizons of typically less then a few minutes,
such as arbitrage, structural or directional strategies (as defined by the SEC (2010)).6 The
discretionary liquidity trader is an institutional trader with a trading horizon of a day
(which is typical according to Campbell, Ramadorai, and Schwartz (2009)), and this horizon is much longer than the few minutes of the informed traders. The market maker in
the model could represents a group of high-frequency traders with a passive market making strategy, who predominantly use limit orders to earn the bid-ask spread and liquidity
rebates. Brogaard, Hendershott, and Riordan (2012) confirm that high-frequency traders
with aggressive strategies are typically informed traders, whereas the high-frequency traders
who supply liquidity get adversely selected. The informed traders in the theory of Foucault,
Hombert, and Rosu (2012) also have short-lived private information, and are motivated as
high-frequency traders who observe news faster than the rest of the market.
The degree that the market maker learns about the liquidity traders trading interest
depends strongly on the resiliency of the market, i.e., the speed with which prices converge
to the fundamental value. The resiliency is determined by the trading aggressiveness of the
informed traders, who revert the price impact of the strategic liquidity trader by trading in
the opposite direction. Therefore, the net order flow gets close to zero over time in a very
resilient market, which impairs the market makers’ learning about the liquidity trading
4
If the liquidity shock of distressed traders is small, then the predatory traders will in fact compete for
liquidity provision (see Proposition 2´, p. 1841).
5
See also Carlin, Lobo, and Viswanathan (2007), where predatory trading arises because no market
makers exist to absorb order flow.
6
The SEC defines four broad high-frequency trading strategies: passive market making, arbitrage,
structural (e.g., trading on latency and the use of flash orders) and directional (trading on fundamentals,
momentum and order anticipation).
4
interest.
The general version of the model allows for a flexible degree of resiliency, but we also
consider two special cases. The first is the version with news trading, where the informed
traders trade on the innovation in the fundamental value only, rather than the difference
between the price and the fundamental value. The trades by the liquidity trader therefore
have a permanent price impact, and the market has zero resiliency. This situation is most
relevant when the liquidity trader has a very short deadline, such that the price impacts
of the individual trades accumulate over time. In the second special case the fundamental
value is announced after each period (similar to Admati and Pfleiderer (1988)). In this case
the temporary price impact of the current trade has fully disappeared before the liquidity
trader submits the next trade. This case corresponds with a perfectly resilient market, and
is most realistic when the time between trades is large, or when the market maker may
learn from prices of highly correlated assets. In both special cases, the market maker can
learn very clearly about the trading interest of the liquidity trader, which strongly reduces
the overall execution costs.
Our model also contributes to a growing literature that studies the optimal execution
problem of a large trader7 by endogenizing the price impact parameter. The liquidity
trader must trade a fixed amount before a deadline, and has a U-shaped optimal execution
strategy: the first and last trades are large, and intermediate trades are small. The optimal
trade size in each period depends on the following tradeoff. On the one hand, a larger trade
increases the expected uninformed order flow in all future rounds, which then receive a zero
price impact. On the other hand, a larger trade increases the price of the current and all
future rounds, because prices only slowly revert to the fundamental value via informed
trading. For the initial trade the first effect is large whereas for the last trade the second
effect is small, which generates the U-shape. The price impact parameter (illiquidity)
7
See e.g., Almgren and Chriss (1999, 2000), Engle, Ferstenberg, and Russell (2012), as discussed in the
literature section.
5
comoves negatively with the quantity traded by the liquidity trader, as uninformed trading
reduces the price impact parameter (Kyle, 1985).
A more general contribution of the model is that the predictable component of the order
flow may explain why resiliency exists in electronic limit order markets, i.e., why liquidity
replenishes after a trade. In the model, if the market perceives that a certain trade belongs
to a series of liquidity motivated trades, then the liquidity consumed by the trade should
be replenished quickly. Similarly, the presence of the predictable component in the order
flow may also explain why liquidity on the bid and ask side is asymmetrical at times.8
We solve the problem numerically, as a closed form solution is not available. The
problem has many state variables because the market maker must learn not only about the
fundamental value, but also about the trading interest of the discretionary liquidity trader.
In addition, the optimization problem of the discretionary liquidity trader is constrained,
as she must trade exactly the exogenously given quantity.
The model explains several empirical findings. Griffin, Harris, and Topaloglu (2003)
estimate a VAR model with five-minute returns, institutional order imbalance and retail
order imbalance, and find that the positive autocorrelation of the institutional order imbalance over the preceding 30 minutes does not affect current returns (i.e., is uninformed).
In addition, current returns positively predict future institutional order imbalance; which
coincides with our theory as the unexpected component of the order flow affects current
returns and signals future liquidity trading interest and order imbalance. The predictions
of our model also confirm several empirical results of Chordia, Roll, and Subrahmanyam
(2002, 2004). In particular, they find that the daily order imbalance is strongly autocorrelated whereas returns have virtually zero autocorrelation, which suggests that predictable
order flow is uninformative at the daily level. Heston, Korajczyk, and Sadka (2010) argue
that predictable patterns in volume, returns and order imbalance are caused by systematic
8
Van Achter (2008) shows that asymmetric liquidity may result from heterogeneous trading horizons
of investors, which affects the decision to place limit or market orders.
6
trading patterns of institutional investors. Bessembinder, Carrion, Tuttle, and Venkataraman (2012) test the sunshine trading versus predatory trading hypothesis for an oil futures
ETF that rolls over its position on a monthly basis, and find more support for the sunshine
trading hypothesis.
Our model supports the theoretical results of Obizhaeva and Wang (2005) and Alfonsi,
Fruth, and Schied (2010). They also find an optimal U-shaped trading pattern, as a large
initial trade creates a price pressure that attracts many new limit orders to the limit order
book. Essentially, this mechanism is also present in our model as the informed trades
reduce the price pressure caused by the strategic liquidity trader. Novel in our paper
is the channel that the expected uninformed order flow has zero impact on prices, and
that the resiliency is explicitly modeled in an adverse selection framework. Chordia, Roll,
and Subrahmanyam (2004) study a two-period model with a discretionary liquidity trader.
They obtain a closed-form solution, but the discretionary trader is limited to trade either
in one period only, or to split up his trades equally across both periods. Because trading is
restricted to two rounds, they do not obtain the U-shape trading pattern and the liquidity
trader cannot update his strategy over time. While their model focusses on the relation
between order imbalance and stock returns across days, our model focusses on the optimal
intraday trading strategy.
2
Literature review
This paper is mostly related to papers that study the optimal execution problem and to
extensions of Kyle (1985) that focus on strategic liquidity traders.
Our model contributes to the literature on the optimal execution problem, which is
the problem of a large liquidity trader who must trade a given quantity before a deadline
7
who aims to minimize execution costs (Bertsimas and Lo, 1998).9 Almgren and Chriss
(1999, 2000), Engle, Ferstenberg, and Russell (2012) find the optimal execution strategy in
a mean-variance framework. Obizhaeva and Wang (2005) and Alfonsi, Fruth, and Schied
(2010) study this problem in a limit order book market, and show that the optimal strategy
strongly depends on the resiliency of the book, i.e., the speed with which the limit order
book recovers after a trade. Huberman and Stanzl (2005) add transaction costs, which is
important as in the continuous time limit the execution cost of the problem of Bertsimas
and Lo becomes in fact independent of the actual strategy.10 They also allow for a timevarying price impact function, which can be obtained in our framework easily as well
by changing the variance of the noise trades and the fundamental innovations over time.
Easley, Lopez de Prado, and O’Hara (2012) show that the order imbalance affects the
endogenously determined trading horizon and the price impact function. These papers are
partial equilibrium models in the sense that the price dynamics are exogenously determined.
We endogenize the price impact function and show how it is affected by order splitting.
The paper is closely related to extensions of Kyle (1985) with strategic liquidity traders,
i.e., discretionary traders. Admati and Pfleiderer (1988) model a group of discretionary
traders who may decide in which period to submit there entire trade, and find that in
equilibrium the informed and discretionary traders will trade in the same period. In contrast, we analyze the behavior of a single discretionary liquidity trader, and find that she
optimally splits up her trades over time. Subrahmanyam (1995) analyses the role of circuit
breakers and extends the Admati and Pfleiderer model to a two-period version, where the
discretionary trader is limited to trade either in one period only, or to split up her trades
equally across both periods.While in equilibrium the discretionary trader indeed splits up
across periods, the informed traders are restricted to trade in the first period only.11 In
9
See also Kissell, Glantz, and Malamut (2003), Chapter 15 in Hasbrouck (2007).
In continuous time, there are infinitely many trading rounds before the fixed deadline, which becomes
meaningless.
11
Chordia, Roll, and Subrahmanyam (2004) also make the assumptions of equal order splitting and a
single period with informed trading.
10
8
our model a new informed trader arrives every period, which introduces an important dynamic aspect as the price pressure from uninformed trades in the current period affects the
informed order flow in the next.
Back and Pedersen (1998) extend the Admati and Pfleiderer model by allowing for
long-lived private information, and find that market depth and volatility are constant over
time. Spiegel and Subrahmanyam (1992) extend the static Kyle model by replacing the
noise traders with risk averse and price sensitive liquidity traders who trade for hedging
motives.12 Massoud and Bernhardt (1999) extend the previous model to a two-period
version, and find that some results of Kyle get reversed. For example, the price impact
becomes steeper over time, because the liquidity traders wish to trade in earlier periods
as to avoid the pricing risk in later periods. In Spiegel and Subrahmanyam (1995) risk
averse discretionary liquidity traders also trade for hedging motives, and will trade either
at the beginning of the day, or later in the same direction as the market makers—effectively
providing liquidity. Mendelson and Tunca (2004) find that the risk aversion of liquidity
traders generally reduces informational efficiency, and that insider trading may improve
the welfare of risk averse liquidity traders because they reduce the volatility of prices.
This paper also relates to the preannouncement of trading interest (Admati and Pfleiderer, 1991), as order splitting generates a predictable and uninformed component in the
order flow. Huddart, Hughes, and Williams (2010) analyze the case where the insider
must preannounce his trades, but also has a liquidity motive to trade (e.g., risk sharing).
Suboptimal risk sharing follows, because even though an insider might trade for liquidity
reasons, the market maker revises prices because the trade may reflects private information.
Huddart, Hughes, and Levine (2001) analyze the Kyle model where informed traders must
announce their trades after submission, like employees of a corporation need to. In this
case, the insider adds noise to his strategy to jam the signal of the market maker.
12
The liquidity traders have hedging motives in Vayanos (1999) too.
9
3
Model setup
Consider a Kyle (1985) framework, where trading occurs sequentially in a number of batch
auctions. Each auction is organized as an anonymous batch market where investors submit
market orders. Trading begins at time 0 and ends at time 1, and takes place during
t = 1, ..., T periods, each of length 1/T . Time 0 represents the beginning of the trading
day for example, and time 1 the end. Three players exist in the Kyle model. There is a
risk neutral informed trader who observes the fundamental value of the asset and trades
to maximize profits. In addition, a group of noise traders trade random amounts every
period. Then, a competitive market maker observes the total order flow in a period, and
chooses the price and his quantity to clear the market.
We deviate from the standard multi-period Kyle model in two important ways. First,
we introduce a strategic liquidity trader who must trade a given amount before the deadline
at time 1. She is a “discretionary” liquidity trader who chooses the optimal quantity to
trade in each period.13 The total quantity is drawn from a normal distribution before
trading starts, and does not change afterwards.
Second, we assume that in every trading round a new informed trader arrives, who
observes the fundamental value and may trade only once. Also, in every period an innovation in the fundamental value of the asset occurs. In essence, we assume that private
information is short lived, which resembles Foucault, Hombert, and Rosu (2012) where
high-frequency traders (HFTs) respond to news (i.e., short-term information) extremely
quickly. In this setup the flow of news and asymmetric information is constant over time,
which is realistic as news might be generated by trades in correlated assets for example.
The traders are all risk neutral and submit market orders. Denote by S ∼ N (0, σS2 )
the total quantity that the strategic liquidity trader must trade before the deadline, where
13
Our definition differs from Admati and Pfleiderer (1988), where the discretionary liquidity traders
may only choose a single period to submit the entire trade.
10
a positive value represents a purchase and a negative value a sale. This quantity is exogenously determined and she cannot access other trading venues. In each periods t she
P
submits a fraction ft S, where she chooses the vector f~ = (f1 , ..., fT ) subject to Tt=1 ft = 1.
Denote by xt the strategically chosen order flow of the informed investor, and ut ∼ N (0, σu2 )
the randomly determined uninformed order flow of non-discretionary noise traders. Then,
the total order flow in each period is
yt = xt + ft S + ut .
(1)
The process of the fundamental value is given by
vt = v0 +
Xt
j=1
ej ,
(2)
where ej ∼ N (0, σe2 ) and IID.
An important element in the model is that a trade by the liquidity trader affects the
current price, which in turn affects the strategy of informed traders in future periods. In
fact, price pressures due to trades in the current period are beneficial to informed traders
in future periods as they increase the pricing error. This mechanism does not exist in the
models of Bertsimas and Lo (1998), Almgren and Chriss (2000) and Huberman and Stanzl
(2005) for example.
3.1
The market maker
We restrict attention to the recursive linear equilibrium, where the market maker’s pricing
function and the strategies of the traders are linear in the information sets. The market
maker determines the price Pt after observing the current and past order flow, which are
summarized in a vector It = (y1 , ..., yt ) that represents his information set. The information
set contains the sequence of past order flow (past prices do not contain additional informa11
tion, as they depend linearly on the order flow). Market efficiency states that the market
maker chooses the price such that Pt = E(vt |It ). Following Kyle (1985), we conjecture (and
verify in equation (18)) that the pricing schedule is
Pt = Pt−1 + λt (yt − E(yt |It−1 )),
(3)
where E(yt |It−1 ) represents the expected order flow. The price is only affected by the
unexpected component of the order flow. Given short lived private information, we show
below that E(yt |It−1 ) = ft E(S|It−1 ), i.e., the expected order flow depends only on the
market makers expectation of the quantity traded by the strategic liquidity trader.
The informed order flow is unpredictable to the market maker, which is a result obtained by Kyle. A proof by contradiction is that if informed trades were autocorrelated to
the market maker, he could immediately set a price that reflects this information which
eliminates the autocorrelation. In the model, the market maker behaves competitively and
incorporates all relevant information to set the price. The strategy of the informed trader
is described next.
3.2
The informed trader
The problem of the informed trader is identical to the one period version of Kyle (1985).
The informed trader observes vt (a perfect signal) and It−1 , and knows the pricing function
(3). He chooses xt to maximize his expected profits
max E[xt (vt − Pt )|vt , It−1 ]
xt
= max [xt (vt − Pt−1 − λt (E(yt |vt , It−1 ) − E(yt |It−1 )))] .
xt
12
(4)
Given yt = xt + ut + ft S, we have E(yt |vt , It−1 ) − E(yt |It−1 ) = xt . We set the first order
condition to zero with respect to xt ,
0 = E[(vt − Pt−1 − 2λt xt )]
xt = βt (vt − Pt−1 ),
with βt =
1
,
2λt
(5)
where βt represents the aggressiveness of the informed trader in period t. Importantly,
the market maker cannot predict the informed order flow, as E(xt |It−1 ) = 0 because
E(vt |It−1 ) = E(vt−1 |It−1 ) = Pt−1 . Note that βt does not depend on the number of trading
rounds like in the multi-period version of Kyle, because each insider trades only once.
We have fixed the pricing rule and the informed traders strategy for a single period,
and we proceed by describing the trading process across periods.
4
Recursive model representation
In this section we first present the general version of the model where the informed trader
trades with aggressiveness parameter βt on (vt − Pt−1 ). Then we describe two special cases
where she trades with βt on the innovation et only. The special cases simplify the analysis
and will prove useful benchmark models.
4.1
The general model
The strategic liquidity trader submits a fraction ft of the total quantity S each period, of
which a part is expected and a part unexpected, i.e., reveals new information about her
trading interest. In this setup, the market maker needs to learn about the fundamental
value of the asset vt and about the quantity S. Learning about S improves the prediction
of the uninformed component of the order flow and therefore also the informed component,
13
which provides a clearer signal about the true value. The market maker observes the order
flow yt , and learns about S and vt by updating the conditional expectations and variances.
We assume f~ is non stochastic and known in equilibrium. The learning process can be
derived from a Kalman filter, with state vector
 
S
Xt =   ,
vt
(6)
 
ft
ht =   .
βt
(7)
yt = h0t Xt − βt Pt−1 + ut .
(8)
and parameter vector
The observation equation is
The dynamics of the state vector are quite simple here,
 
0
Xt = Xt−1 +   .
et
(9)
Now denote the prediction equations
Xt|t−1 = E(Xt |It−1 ),
Vt|t−1 = V ar(Xt |It−1 ),
(10)
V̂t = V ar(Xt |It ).
(11)
and update equations
X̂t = E(Xt |It ),
Using the Kalman filter and update equations, we can derive the following results. First,
the prediction equations are
Xt|t−1 = X̂t−1 ,
14
(12)
and
Vt|t−1


0 0
.
= V̂t−1 + 
2
0 σe
(13)
Define the unexpected order flow
ỹt = yt − E(yt |It−1 ) = yt − ft St|t−1 .
(14)
The updating equations for the expectation and variance of the state vector then are
X̂t = Xt|t−1 +
Vt|t−1 ht
ỹt ,
h0t Vt|t−1 ht + σu2
(15)
Vt|t−1 ht h0t Vt|t−1
.
h0t Vt|t−1 ht + σu2
(16)
and
V̂t = Vt|t−1 −
The Kalman filter equations can be solved recursively, starting from

V̂0 =
σ2
 S
0
0
σv20

,
 
0
X̂0 =   ,
v0
(17)
where σv20 is the initial variance of the fundamental value. The market maker uses the
unexpected order flow to update the price and the expectation of S according to the
parameters λt and ϕt respectively, which are given from equation (15) as
 
ϕ
 t =
λt
ft
 
ft
Vt|t−1  
βt
 
.
ft
βt Vt|t−1   + σu2
βt
(18)
Thus, λt represents the price impact parameter and ϕt the speed with which the market
maker learns about S. Equations (15) and (18) show that the price equals the market
makers’ expectation of the fundamental value, which confirms the conjectured pricing rule
15
(3).
4.2
Two special cases
In this section we consider a version of the model with news trading, and a version where
the fundamental value is announced after each period. In the case of news trading, the
informed trader only observes the signal et and trades with aggressiveness βt . In the case
where the fundamental value is announced after each period, the informed trader trades
on vt − Pt−1 = vt − vt−1 = et .14 In both cases, the market maker learns about et according
to one equation which simplifies the representation.
When the informed traders trade on news only, they do not revert the price pressure
caused by the strategic liquidity trader. In this case the price impact of the strategic
liquidity trader is permanent and the market has zero resiliency. The news trading case
is most realistic if the liquidity trader has a relatively short deadline, such that the price
impacts of her trades simply accumulate over time.
In the case that the fundamental value is announced after each period, the market is
perfectly resilient as the price impact of an uninformed trade is immediately reverted. This
case would be realistic if the time between trades is relatively long, while the market maker
can infer the fundamental value by observing prices of correlated assets. In this situation
the liquidity traders’ temporary price impact fully disappears between the trades.
In the general case, the parameter βt of the informed traders strategy βt (vt − Pt−1 )
represents the degree of resiliency of the market. The special cases where the market has
zero perfect resiliency are useful benchmarks. In these cases the state vector and parameter
vector are
14
The fundamental value of period t is announced after trading round t, but before the innovation
et+1 has realized. Thus, in each period there is asymmetric information.
16
Xt = S,
ht = ft .
(19)
The observation and unexpected order flow equations are
yt = ft St + βt et + ut ,
(20)
ỹt = yt − E(yt |It−1 ) = yt − ft St|t−1 .
(21)
The two versions proceed differently.
4.2.1
News trading
In the case of news trading the filter equations are simply Vt|t−1 = V̂t−1 and St|t−1 = Ŝt−1 .
The updating equations are
Vt|t−1 ft
ỹt ,
ft0 Vt|t−1 ft + βt2 σe2 + σu2
Vt|t−1 ft ft0 Vt|t−1
V̂t = Vt|t−1 − 0
.
ft Vt|t−1 ft + βt2 σe2 + σu2
Ŝt = St|t−1 +
After each period the market maker updates the expectation of S, which allows him to form
a more precise signal of all the past innovations. For τ ≥ 0, the expectation of innovation
et after period t + τ is
E(et |It+τ ) = (yt − ft Ŝt+τ )
βt σe2
.
βt2 σe2 + σu2
(22)
This recursion is quite simple as only the expectation Ŝt+τ changes each period. The price
equals the sum of the expectations of the innovations
Pt = E(vt |It ) =
Pt
17
n=1 E(en |It ).
(23)
4.2.2
Announcement of the fundamental value
In this case, the market maker updates the expectation and variance of S based on both
the order flow and the announcement of the fundamental value. The solutions are in closed
form, and given by
Ŝt =
V̂t =
σS2
Pt
i=1 (fi (yi − βi ei ))
,
P
σu2 + σS2 ti=1 fi2
σu2 σS2
.
P
σu2 + σS2 ti=1 fi2
(24)
The updating equation of the price Pt is
Pt = E(vt |It ) = vt−1 +
4.3
βt σe2
βt2 σe2 + σu2 + ft2 V̂t−1
ỹt .
Problem of the strategic liquidity trader
The goal of the strategic liquidity trader is to find an optimal strategy f~ that maximizes the
expected trading profits (i.e., minimizes the execution costs). Furthermore, this strategy
must be a rational expectations equilibrium, in the sense that the market maker must
correctly anticipate the optimal strategy to determine his response, and given the market
makers’ response the strategy must indeed be optimal for the liquidity trader.
The strategic liquidity trader knows the realization of the quantity S, which in fact
contains two important sources of information. First, although her trades are uninformed,
they do affect prices and therefore create a pricing error. This pricing error will be exploited
by the informed traders in future periods, and therefore the liquidity trader can predict
future order flow and prices. This information is static in the sense that it is predictable
before trading starts.15
We will show in the next section that the realization of S does not affect the strategy f~t , when we
prohibit the liquidity trader to update her strategy over time (see footnote 20).
15
18
Second, after observing any period’s order flow the liquidity trader can filter out her
own trades, such that the remaining component reveals a more precise signal about the
fundamental value (as compared to the market maker). The liquidity trader uses this
expectation of the fundamental value to predict the informed order flow in future periods.
This information is dynamic and affects the optimal trading strategy depending on the
realizations of the noise trades and asset innovations.
We allow the strategic liquidity trader to repeat her static optimization every period,
which is a (quasi) dynamic solution.16 She observes the order flow and constructs the
expectation and variance of the fundamental value according to the prediction and update
equations of the market maker in Section 4.1, with the only difference that the conditional
variance of S is zero.17 Each period she recalculates her optimal trading strategy for the
remaining periods.
We find numerical solutions for the static and dynamic problem, but only the static
solution and the special cases are rational expectations equilibria (REE). In the special
cases the informed trader submits βt et (equation (21)), which is unpredictable to both the
market maker and the strategic liquidity trader. Therefore, the liquidity traders additional
knowledge of S is not relevant in predicting informed trades, and essentially she does
not have any informational advantage. For this reason the static solution is dynamically
optimal.
The dynamic model is not an REE, as the liquidity trader updates her strategy over
time which is unpredictable to the market maker. Specifically, by observing the order flow
the liquidity trader forms a more precise estimate of the fundamental value, which allows
her to predict future periods informed trades (βt (vt − Pt−1 )) and price changes. Based on
the future price changes she chooses to either accelarete or decelerate her trading program.
However, Monte Carlo simulations presented in the results section reveal that the deviations
16
17
We cannot find a solution with dynamic programming due to the large number of state variables.
We omit the equations because the problem is otherwise identical.
19
in the dynamic strategy are very small, and have a neglible effect on the expected execution
costs.
5
Numerical approach
At time 0, the discretionary liquidity trader maximizes expected profits
max
f1 ,...,fT
E0S
hXT
s.t.
t=1
i
ft S(vT − Pt ) ,
XT
t=1
(25)
ft = 1.
Typically, this type of problem is solved with dynamic programming (see e.g., Bertsimas
and Lo (1998)). However, we cannot obtain a closed form solution due to the complexity
introduced by the many state variables (the value equation contains high-degree polynomials). Therefore, we use numerical methods to find the solution. We consider a static
equilibrium strategy which is fixed before trading starts, and a repeated static strategy
that is allowed to change over time.
We start by taking numerical values for the exogenous parameters of the model, T, σu2 , σe2
and σS2 , which are the number of trading rounds, and the variances of the noise trade, the
innovations in the fundamental value and the quantity of the strategic liquidity trader.
First, the market maker makes a guess of the equilibrium strategy f~, which fixes the
parameters λt , βt and ϕt in the model (equation (18)). Then, we take the expected profit
function in equation (25) and iterate the entire problem backwards to period 1 using the
recursive equations for the price, the order flow and the Kalman filter. Now we have a single
equation that contains the total expected profits as a function of S, f~, and the realizations
of u1 , ..., uT and e1 , ..., eT which are zero in expectation. We maximize this function and
20
obtain an optimal solution f~∗ .18
Next, we iteratively find a rational expectations equilibrium, where the market maker
correctly anticipates f~ and responds by choosing λt and ϕt , and given the response of the
market maker the sequence f~ maximizes the expected trading profits of the liquidity trader.
The iteration starts by the market makers’ initial belief of the sequence f~ which fixes λt
and ϕt , for which we maximize the profit function yielding the liquidity traders optimal
strategy f~∗ . We substitute this solution into the market makers belief, and we recalculate
the liquidity traders optimal strategy. We iterate this process until convergence, i.e., until
the optimal strategy is arbitrarily close to the market makers expectation of the strategy.19
Now we have obtained the static solution, which generates fixed values for λt , βt , ϕt , V̂t
and Vt|t−1 . Note that the optimal strategy is independent of the realization of S, i.e., the
fraction submitted by the liquidity trader each period is independent of the total quantity
she must trade.20 This is a necessary requirement as the market maker cannot observe S,
and otherwise would not be able to form a rational expectation of the strategy f~.
The dynamic solution continues from the static solution. In round 1, the strategic
liquidity trader submits f1 S from the static solution (which now becomes realized), and
afterwards observes y1 . Based on y1 she updates her expectation of vT , and recalculates her
optimal strategy yielding f2∗ , ..., fT∗ . The market makers expectation of f~ from the static
solution remains unchanged, as he cannot predict the updates of the liquidity trader. In
round 2, the liquidity trader submits f2 S (which now becomes realized), and afterwards
observes y2 . This process is repeated until we reach period T . In each period we find the
solution of the sequence f~t conditional upon the liquidity traders information set, which is
18
Here we can require that all ft > 0, i.e., that she never sells when S > 0 and never buys when S < 0.
For certain parameter values however, it is optimal that some fractions become negative.
PT
19
In the results section, we assume convergence is reached when i=1 |fi∗ − fi | < 1/100, 000; where fi∗
is the optimized sequence and fi the market makers’ expectation.
20
The proof is that if we substitute the order flow and price equation into equation (25) and take the
FOC with respect to ft (for any t), the S drops out. Intuitively, the per period trading costs ft SPt are
quadratic in ft , and minimization of trading costs across periods balances the per period price impact.
21
a repeated static optimization.
The dynamic solution differs from the static solution. The reason is that over time, the
liquidity trader learns from the order flow and updates her entire (expected) future trading
strategy. Specifically, she can filter out her own (uninformed) trades from the aggregate
order flow, and thus obtains an estimate of the informed order flow and fundamental value
which is more precise than that of the market maker. Using this additional information
she predicts future informed trades, and decides to accelerate or decelerate her trading
program. However, only in the special cases of news trading and announcement of the
fundamental value does the dynamic solution coincide with the static solution.
6
Results
We first analyze the optimal strategy of the liquidity trader before trading begins, but
after the realization of S, T, σu2 , σe2 and σS2 . The liquidity trader incorporates in her strategy
the impact of her trades on the market makers expectation of S and prices, which in turn
affects future periods’ informed order flow and prices.
6.1
Static solution
We solve the model numerically for the base case first, and then change the value of one
parameter at a time to analyze the impact of that parameter on the equilibrium outcome.
For the base case we use the following parameter values. The number of trading periods T =
5, the strategic liquidity trader must trade the quantity S = 1, which has an unconditional
variance of σS2 = 3. The volatility of noise trading and innovation in the fundamental
value are σu2 = 1 and σe2 = 1 per trading round. We also set the initial variance of the
fundamental value to σe2 . We normalize the price to zero, as the initial price level has no
impact on the optimal order splitting strategy.
22
Figure 1 plots the parameters of interest over time for the base case. The upper panel
shows the U-shaped trading strategy f~, where the fractions range between 16.2% − 25.4%,
and the closely corresponding inverted U-shape for the price impact parameter λt . The
variables move in opposite direction as more uninformed trading reduces the price impact
(as in Kyle (1985)). The bottom panel shows the total order flow yt , informed order flow
xt and the market makers expectation of the uninformed order flow E(S|It ), which are
all expectations formed by the liquidity trader before trading starts. As of round 2 the
informed trader starts trading in opposite direction to the strategic liquidity trader, which
pushes the net order flow closer to zero.
Table 1 presents the results for the other parameters (upper panel), and the deviations
to the base case (lower panels). Column three shows the parameter ϕ, the speed with which
the market maker learns about S from the order flow. Most of the learning on S occurs
in the first trading round because f1 is large (column 5), and then V ar(S|I1 ) reduces from
3 to 2.76. The expected order flow y1 is 0.25, which generates a price impact 0.16 cent.
Therefore, in period 2 the informed trader sells 0.12 (column 7), which pushes the price
partly back towards the fundamental value. In period 2 the liquidity trader buys 0.178,
such that the expected net order flow y2 is 0.057. The first cell in the last column shows
the expected execution cost of a benchmark model where the market maker does not learn
about the trading interest of the liquidity trader. Then she trades a fraction 1/T = 0.2
each period, and the model is reduced to a repeated single period Kyle (1985) model. In
the benchmark case the expected execution costs are 0.22 and with learning the costs are
0.2, which is a modest reduction of 9%.
The main result of the model is that the market maker learns about the trading interest
of the uninformed liquidity trader. In particular, the initial large trade increases the market
makers expectation of S to E(S|I1 ) = 0.08, such that in period 2 the market maker subtracts the impact of the expected order flow λ2 f2 E(S|I1 ) = 0.011 from the price. The price
reduction lasts for all future periods as well, such that the remaining 74% of the total order
23
size receives a discount. After round 2, an additional λ3 f3 E(S|I2 ) = 0.008 is subtracted
from P3 and the future prices, and similar amounts in the remaining periods. Summed
over all periods, the predictable component of the order flow reduces P5 by 0.03, which is
relevant compared to the total price impact of 0.25 (P5 ). The predictable component in the
order flow reduces prices as it effectively signals the liquidity traders uninformed trading
interest, i.e., order splitting is a noisy form of sunshine trading (Admati and Pfleiderer
(1991)).
The market makers’ expectation of the uninformed order flow E(S) is small as it does
not exceed 0.09 in any period, which is a consequence of the high level of resiliency. The
resiliency is determined by the aggressiveness of the informed traders strategy, who reduces
the pricing error by half each period. In a resilient market, signalling uninformed trading
interest is quite costly to the liquidity trader for two reasons. First, the trading costs in any
period ft SPt are quadratic in ft , such that larger trades are relatively more expensive.21
Second, a larger trade by the liquidity trader in the current period causes a price pressure,
which will be followed by sales of the informed traders in future periods. The sales reduce
the net order flow in future periods which impairs the market maker’s learning about S.
The second panel of Table 1 shows a market with low resiliency, i.e., where the informed
trader trades with aggressiveness parameter βt = 1/(10λt ). When resiliency is lower, the
market maker can learn much more clearly about S: E5 (S) = 0.32, compared to 0.09 in the
base case. The reason is that the sales by the informed traders in periods 2 to 5 are smaller,
such that the net order flow reflects the liquidity traders trades more clearly. Resiliency
also has a strong impact on the optimal execution strategy, which has a strong U-shape.
The parameter values explain why the optimal trading strategy is U-shaped, which
follows from a tradeoff. On the one hand, a larger current trade provides a stronger signal
about future trades of the liquidity trader, which then enjoy a lower price impact. On the
21
The reason is that ft affects ft SPt directly, and indirectly via Pt .
24
other hand, a larger current trade increases both the current and future prices, as trades
by informed traders only slowly push the price back towards the fundamental value. The
positive effect is particularly important in the first trading period, as a large initial trade
generates a clear signal to the market maker about future uninformed trading interest. In
the last trading period, the negative effect is small because future prices are then irrelevant
to the liquidity trader.
The impact of expected uninformed order flow on prices offers a novel explanation for
resiliency in electronic limit order markets, i.e., why liquidity replenishes after a trade. If
the market perceives that a certain trade belongs to a series of trades and is likely to be
uninformed, then the market attaches a low price impact to that trade such that liquidity
will be replenished quickly. In this paper we model a batch auction where the resiliency
occurs instantaneously, i.e., that λ2 f2 E(S|I1 ) is simply subtracted from P2 , but in a limit
order market traders need some time to respond. Note that the expected order flow affects
the price level, and the slope of the limit order book λt via the reduction in the conditional
variance of S.
The mechanism of expected uninformed order flow may also explain why liquidity on
the bid and ask side of the order book can be asymmetrical at times. If the market expects
substantial order flow from uninformed buyers, the ask side should be more liquid. A trade
executed on the bid side is more likely to stem from informed traders, such that the bid
side is less liquid.
The third panel shows the case where the number of trading periods T = 3. The
liquidity trader must trade larger quantities each period, which has a strong effect on the
expected price and execution costs. The price effect gets partly mitigated by the fact that
the market maker learns more clearly about S. In the bottom panel the unconditional
variance of S is higher, σS2 = 5. In this case, the market maker can learn more about S,
which increases the expected order flow in remaining periods. For this reason, there is also
25
more curvature in the U-shape of the sequence of ft .
6.2
Special cases
The results for the case of news trading and announcement of the fundamental value are
presented in Table 2.
In the case of news trading, the informed traders do not revert the price pressure caused
by the strategic liquidity trader. This represents a market with permanent price impact
and zero resiliency. The upper panel of Table 2 shows that the market maker can form a
clearer signal about S due to the absence of resiliency: E5 (S) = 0.22, compared to 0.09
in the base case. The optimal execution strategy is to trade a slightly increasing fraction
each period, but is quite flat in general. The U-shape pattern has disappeared because the
permanent price impact makes a large initial trade very costly (as it permanently increases
the prices of all future trades). While the level of informed trading and the price impact
parameter are lower in each period, the absence of resiliency still increases overall execution
costs.
Panel 2 shows the case where the fundamental value is announced between trading
periods. In this case the market is perfectly resilient and the temporary price impact is
zero, which is perhaps reasonable when the time between the trades of the liquidity trader
is large. Now the market maker receives a very clear signal about S by observing both the
order flow and the fundamental value: E5 (S) = 0.38, compared to the base case of 0.09. In
this case the execution costs reduce by 20% compared to the case without learning (0.10
instead of 0.08). The optimal execution strategy is similar to that in the news trading case.
26
6.3
Dynamic solution
The dynamic version of the model repeats the static optimization after each period. The
strategic liquidity trader forms expectations of the current and all future parameter values
conditional on her information set, and updates her strategy accordingly. Therefore, for
each trading round t she constructs T estimates, which gives an T × T matrix for every
parameter. We are interested in how the optimal strategy depends on the realizations of
the order flow.
The dynamic results of the model are shown in Table 3. Parameter values in bold
font type have realized, whereas the other values are expectations of the strategic liquidity
trader. The left block of the table shows the parameters in an environment where the
fundamental value rises, as we set e1 = e2 = e3 = 1, e4 = e5 = 0 and all noise trades to
zero. In the right block we study an environment where only the noise traders purchase,
and set u1 = u2 = u3 = 1, u4 = u5 = 0 and all innovations in the fundamental value to
zero.
In the first panels we observe that the realizations of the order flow cause only small
dynamic adjustments to the optimal strategy of the strategic liquidity trader. The largest
change in the left panel is for example 0.01, E(f5 |I0S ) minus E(f5 |I1S ); and in the right
panel it is 0.013. Due to the buying pressure of the informed traders (left panel) and
noise traders (right panel), the strategic liquidity trader beliefs the stock is overpriced and
therefore postpones her own purchases a little.22 By postponing her trades, she anticipates
that informed traders in future rounds will reduce the pricing error—making it cheaper to
buy later. The second panel shows the expected and realized prices. The expected prices
adjust slowly to the order flow, because the market maker is uncertain whether order flow
stems from the informed or uninformed traders. The difference between expected and
realized order flow in the next block show that order flow is difficult to predict. If a current
22
The bottom panel of the left block shows that E(v1 |I1S ) = 0.544 while P1 = 0.665, i.e., the liquidity
trader believes the stock is overpriced by 0.12.
27
purchas stems from a noise trader, then the next periods’ informed trader will sell (to
correct the mispricing), whereas if a current purchase stems from the informed trader, then
the next periods’ informed trader will also purchase (because informed traders do not fully
reveal their signal). Panel four shows the liquidity traders expectation of informed trading
in future rounds, which depends on her expectation of the fundamental value.
The fifth block shows the liquidity traders expectation of the market makers expectation
of S. The market maker beliefs that the buying pressure of the informed traders (left block)
and noise traders (right block) might stem from the strategic liquidity trader, and therefore
increases the expectation of E(S|It ) over time. The last block shows the realizations of the
remaining parameters. They can be reported in a single column as the liquidity trader’s
expectations of the future values equals that of the current value.
6.4
Static versus Dynamic
Only the special cases of news trading and announcement of the fundamental value are
rational expectations equilibria (REE). If the strategic liquidity trader could commit ex
ante to her strategy, the static solution is also an REE. For example, she may use an
algorithm that does not update the strategy over time. However, in general this is not a
credible equilibrium.
The dynamic model is not an REE, as the liquidity trader updates her strategy over
time which is unpredictable to the market maker. Specifically, by observing the order flow
the liquidity trader forms a more precise estimate of the fundamental value, which allows
her to predict future periods informed trades and price changes. Based on the future price
changes she chooses to accelarete or decelerate her trading program. In the special cases
however, the informed trades are fully unpredictable to the liquidity trader and therefore
the static solution is dynamically optimal.
28
In the next section we do monte carlo simulations to show that the dynamic changes
are in fact relatively small. Thus, the main results of the model should be robust. Also,
the dynamic updates have a neglible effect on the profit of the market maker, and are not
statistically significantly different from zero for 10,000 simulations.
6.5
Monte Carlo simulations
We simulate 10,000 trading games to calculate the realized trading profits of the different
types of traders, and compare the cases where the liquidity trader uses the static or dynamic
trading strategy. We draw random values for the realizations of S and all et and ut (with
variances 3, 1 and 1 respectively). The results are shown in Table 4, and the static and
dynamic case appear virtually identical.
The upper panel shows the dynamic strategy. The market maker earns a small loss on
average (-0.0132 per game), which is slightly higher than in the static version (-0.0134).
Both numbers are tiny compared to the standard deviation of 6.1 per game, and not signifi√
cantly different from zero for 10,000 simulations (the t-statistic is −0.0132/(6.6/ 10000) =
−0.2). Intuitively, we expect that the market makers’ profits are in fact higher in the dynamic version of the model, because the liquidity trader essentially responds to fundamental
information. For example, if the liquidity trader beliefs the stock is overvalued then she
decelerates her buying program, such that prices do not get pushed away further from the
fundamental value.
The average informed traders’ profit is 4.02, and the losses to the liquidity trader and
noise traders are 0.59 and 3.42 respectively. The fifth column shows that the liquidity
traders average loss is 0.617 in case the market maker would not learn about S. The
difference of 0.03 is the expected gain when the market maker learns about uninformed
order flow. The liquidity traders profits in the static model are virtually identical to those
in the dynamic model (the difference is 0.0001).
29
The bottom panel shows the static and the average dynamic strategy of the liquidity
trader (the sequence of ft ). The absolute differences are very small (less than 0.005), which
is consistent with the small difference in expected profits. Note that the results of this
table cannot be compared directly to those of Table 1, because in the latter we fix S = 1.23
6.6
Extant literature
In this section we relate our results to the literature.
The optimal trading strategy of our liquidity trader is U-shaped, similar to the limit
order book models of Obizhaeva and Wang (2005) and Alfonsi, Fruth, and Schied (2010).
In their papers, the initial transaction is large and pushes the price above the fundamental
value to attract new limit orders.24 Then, the future trades are of equal size and exactly
consume the new limit orders that are placed. The last trade is larger again, and pushes
up the price after which the price dynamics are not relevant anymore. Our model has this
feature too, because the informed traders push back the price towards the fundamental
value each period, such that they essentially provide liquidity to the market. In addition,
our model has the mechanism that the market maker learns about the trading interest of
the uninformed liquidity trader which receives a zero price impact. This second channel
increases the curvature of the U-shape pattern. In unreported results, when we set σS2 = 0
the market maker does not learn about S,25 and we obtain an optimal strategy that is
symmetrically U-shaped similar to e.g. Obizhaeva and Wang (2005).
In models like Almgren and Chriss (2000), the optimal strategy is to trade a constant
fraction each period because this minimizes the quadratic trading costs (if the trader is
23
Although E(S) = 0, the trading strategy and profits are symmetric for positive and negative values
of S. Therefore,
the expected magnitude of the liquidity demand is the absolute value of S, which is
p
E(|S|) = σS2 2/π = 1.784. The expected trading costs are thus much larger than in the case where S = 1.
24
In Alfonsi, Fruth, and Schied (2010) the trade increases the quoted spread, which attracts new limit
orders.
25
Technically, setting σS2 = 0 implies that S = 0 because S ∼ (0, σS2 ). Then, by choosing S > 0, we force
that the market maker does not learn about S, which leads to a symmetric optimal strategy.
30
risk neutral). In these models trades have a permanent price impact, similar to our news
trading case except that signalling of uninformed order flow is absent. When the liquidity
trader is risk averse, the optimal trade sizes decline over time to reduce the exposure to
future price swings (which is also obtained by Huberman and Stanzl (2005)). Bertsimas
and Lo (1998) add an AR(1) news process, and find that trading a constant fraction each
period is no longer optimal. Compared to Bertsimas and Lo (1998) and Almgren and Chriss
(2000), the price impact parameter in our model is endogenous and directly depends on
the optimal strategy.
7
Conclusion
We study trading of a stock in a market with asymmetric information. It is shown that a
liquidity motivated trader with a long trading horizon can credibly signal her uninformed
trading interest by splitting up her order over time. Then, the predictable component
in the order flow stems only from her trades and can be clearly identified by the market.Accordingly, the predictable component of the order flow does not affect prices, such
that the liquidity trader enjoys a lower price impact.
This result is driven by the strong assumption that informed traders have short-lived
private information. However, our main result should hold if the liquidity trader has a longer
trading horizon than the informed traders. In this case, the predictability in the order flow
at frequencies lower than the horizon of the informed traders represents the trading interest
of the liquidity trader. This assumption seems reasonable, given that institutional investors
have trading horizons of sometimes several days, while high-frequency traders often have
trading horizons of less then a few minutes.
An interesting extension would be to analyze our problem with multiple long-term
liquidity traders, and for a portfolio of stocks with correlated liquidity trading.
31
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35
Table (1) Numerical example of the static model: general case
We solve the model numerically, and show the expected parameter values conditional upon
the information set of the strategic liquidity trader. We use the following parameter values
for the base case (upper panel). The number of trading periods T = 5, the strategic
liquidity trader must trade the quantity S = 1, which has variance σS2 = 3. The variance of
noise trading σu2 = 1 and the innovations in the fundamental value σe2 = 1, per period. The
first cell of column Σcost represents the total costs if the market maker would not learn
about the the trading interest of the liquidity trader, i.e., absent signalling. The last cell
shows the execution costs when the market maker does learn about the trading interest of
the liquidity trader. Each panel shows the effect of a change in one of the parameters values.
The solutions are calculated before any trade has taken place, and we set the expectation
of all u and e to zero.
T
ft
1
2
3
4
5
0.254
0.178
0.162
0.174
0.232
T
ft
1
2
3
4
5
0.353
0.210
0.140
0.132
0.164
T
ft
1
2
3
0.356
0.290
0.354
T
ft
1
2
3
4
5
0.271
0.172
0.157
0.169
0.230
λt
Base Case
T = 5, S = 1, σS2 = 3, σe2 = 1, σu2 = 1., β = 1/(2λ)
φt
Vt (S) Et (S)
yt
xt
Pt
costt
0.647
0.678
0.693
0.695
0.679
0.319
0.062
0.014
0.023
0.083
λt
φt
0.362
0.450
0.525
0.574
0.599
0.694
0.325
0.169
0.144
0.187
λt
φt
0.638
0.663
0.657
0.261
0.078
0.084
λt
φt
0.605
0.666
0.688
0.691
0.666
0.496
0.064
0.009
0.032
0.128
2.756
2.749
2.748
2.747
2.733
0.081
0.084
0.084
0.084
0.089
0.254
0.057
0.023
0.030
0.077
0.000
-0.121
-0.139
-0.144
-0.155
Low resiliency: β = 1/(10λ)
Vt (S) Et (S)
yt
xt
2.265
2.138
2.105
2.082
2.041
0.245
0.287
0.298
0.306
0.320
0.353
0.182
0.105
0.094
0.122
0.000
-0.028
-0.036
-0.038
-0.042
Close deadline: N = 3
Vt (S) Et (S)
yt
xt
2.721
2.699
2.674
0.093
0.100
0.109
0.356
0.118
0.135
High variance S:
Vt (S) Et (S)
yt
4.328
4.320
4.319
4.317
4.283
0.134
0.136
0.136
0.137
0.143
36
σS2
0.271
0.049
0.025
0.036
0.086
0.000
-0.171
-0.219
=5
xt
0.000
-0.123
-0.132
-0.133
-0.145
Σcost
0.164
0.193
0.200
0.210
0.250
0.042
0.034
0.032
0.037
0.058
0.215
Pt
costt
Σcost
0.128
0.187
0.220
0.252
0.295
0.045
0.039
0.031
0.033
0.048
0.211
Pt
costt
Σcost
0.227
0.288
0.353
0.081
0.083
0.125
0.305
Pt
costt
Σcost
0.164
0.181
0.184
0.193
0.229
0.044
0.031
0.029
0.033
0.053
0.208
0.20
0.197
0.289
0.190
Table (2) Numerical example of the static model: special cases
We solve the model numerically for the version with news trading and with announcement
of the fundamental value. The parameter values are those of the base case in Table 1. The
first cell of column Σcost represents the total costs if the market maker would not learn
about the the trading interest of the liquidity trader, i.e., absent signalling. The last cell
shows the execution costs when the market maker does learn about the trading interest of
the liquidity trader. The solutions are calculated before any trade has taken place, and we
set the expectation of all u and e to zero.
T
ft
1
2
3
4
5
0.181
0.190
0.199
0.209
0.221
T
ft
1
2
3
4
5
0.195
0.198
0.200
0.202
0.204
News trading
T = 5, S = 1, σS2 = 3, σe2 = 1, σu2 = 1, βt = 1/(2λt ).
λt
φt
Vt (S) Et (S)
yt
Pt
costt
0.477
0.476
0.475
0.474
0.472
0.248
0.246
0.245
0.244
0.243
λt
φt
0.474
0.475
0.477
0.479
0.480
0.526
0.482
0.444
0.411
0.382
2.865
2.732
2.599
2.466
2.333
0.045
0.089
0.134
0.178
0.222
0.181
0.190
0.199
0.209
0.221
Fundamental value
Vt (S) Et (S)
yt
2.691
2.434
2.217
2.033
1.875
0.103
0.189
0.261
0.322
0.375
37
0.195
0.198
0.200
0.202
0.204
Σcost
0.086
0.169
0.246
0.320
0.388
0.016
0.032
0.049
0.067
0.086
0.30
Pt
costt
Σcost
0.098
0.089
0.081
0.074
0.069
0.019
0.018
0.016
0.015
0.014
0.10
0.25
0.08
Table (3) Numerical example dynamic model.
We solve the model numerically, and show the expected parameter values conditional upon the information
set of the strategic liquidity trader. We use the parameter values of the base case: T = 5, demand
of strategic liquidity trader S = 1, with variance σS2 = 3. The variance of noise trading σu2 = 1 and
innovations in the fundamental value σe2 = 1 each period. The left block shows the case “Rise in value”,
where the fundamental value increases by 1 unit in periods 1, 2 and 3, and there is zero noise trade. The
right block shows the case “Buying noise traders”, where noise traders buy 1 unit in period 1, 2 and 3 while
the innovations in the fundamental value are zero. Each panel shows how the liquidity traders expectations
of a particular variable get updated as trading progresses. Each column shows the information set of the
liquidity trader, i.e., the order flow of the periods he has observed. Values in bold are realizations of that
variable. Therefore, the panels show how the strategic traders’ expectations are updated according to
realizations of the order flow, which depends on et and ut . The bottom panel shows the realizations of the
remaining model parameters in columns.
Rise in value
e1 = e2 = e3 = 1, e4 = e5 = 0
Buying noise traders
u1 = u2 = u3 = 1, u4 = u5 = 0
I0S
I1S
I2S
ft
I3S
I4S
I0S
I1S
I2S
ft
I3S
I4S
1
2
3
4
5
0.255
0.177
0.162
0.173
0.232
0.255
0.173
0.159
0.171
0.243
0.255
0.173
0.158
0.170
0.244
0.255
0.173
0.158
0.170
0.245
0.255
0.173
0.158
0.170
0.245
0.255
0.177
0.162
0.173
0.232
0.255
0.172
0.158
0.171
0.245
0.255
0.172
0.158
0.170
0.246
0.255
0.172
0.158
0.170
0.246
0.255
0.172
0.158
0.170
0.246
1
2
3
4
5
0.165
0.193
0.199
0.210
0.249
0.665
0.682
0.686
0.694
0.732
0.665
1.410
1.413
1.420
1.457
0.665
1.410
2.270
2.277
2.313
0.665
1.410
2.270
2.703
2.739
0.165
0.193
0.199
0.210
0.249
0.812
0.826
0.829
0.837
0.874
0.812
1.152
1.155
1.162
1.198
0.812
1.152
1.330
1.337
1.373
0.812
1.152
1.330
0.731
0.768
1
2
3
4
5
0.255
0.056
0.023
0.030
0.077
1.028
0.084
0.059
0.069
0.132
1.028
1.157
0.068
0.079
0.146
1.028
1.157
1.305
0.082
0.149
1.028
1.157
1.305
0.695
0.152
0.255
0.056
0.023
0.030
0.077
1.255
0.092
0.070
0.081
0.147
1.255
0.573
0.074
0.085
0.153
1.255
0.573
0.326
0.086
0.154
1.255
0.573
0.326
-0.786
0.151
1
2
3
4
5
0.000
-0.121
-0.139
-0.143
-0.155
0.773
-0.089
-0.099
-0.102
-0.110
0.773
0.984
-0.090
-0.091
-0.099
0.773
0.984
1.147
-0.088
-0.096
0.773
0.984
1.147
0.525
-0.093
0.000
-0.121
-0.139
-0.143
-0.155
0.000
-0.079
-0.088
-0.090
-0.098
0.000
-0.598
-0.083
-0.085
-0.092
0.000
-0.598
-0.831
-0.085
-0.092
0.000
-0.598
-0.831
-0.956
-0.095
1
2
3
4
5
0.081
0.084
0.084
0.085
0.089
0.329
0.330
0.330
0.331
0.335
0.329
0.396
0.396
0.396
0.400
0.329
0.396
0.413
0.413
0.418
0.329
0.396
0.413
0.427
0.432
0.081
0.084
0.084
0.085
0.089
0.401
0.403
0.403
0.403
0.407
0.401
0.432
0.432
0.432
0.437
0.401
0.432
0.436
0.436
0.440
0.401
0.432
0.436
0.416
0.420
vt
EtS (vt )
VtS (vt )
Vt (S)
ϕt
vt
E1S (vt )
VtS (vt )
Vt (S)
ϕt
1
2
3
3
3
0.000
0.544
1.286
2.147
2.577
2.000
1.911
1.938
1.964
1.975
2.756
2.748
2.748
2.746
2.732
0.320
0.061
0.014
0.023
0.083
0
0
0
0
0
0.000
0.704
1.037
1.212
0.601
2.000
1.911
1.938
1.964
1.975
2.756
2.748
2.748
2.746
2.732
0.320
0.061
0.014
0.023
0.083
T
Pt
Pt
yt
yt
xt
xt
Et (S)
1
2
3
4
5
Et (S)
38
Table (4) Monte Carlo simulations
We calculate the profits of all traders by simulating the model 10.000 times with random realizations
of the quantity traded by the strategic liquidity trader, the innovations in the fundamental value and the
quantities trader by the noise traders. The upper panel shows the results for the dynamic model, where the
strategic liquidity trader updates her strategy after each period. The middle panel shows the static model,
where she is not allowed to update her strategy over time. We show the mean and standard deviation
of the profits to each of the trader types summed over the five periods (the informed traders, the market
maker (MM), the noise traders and the liquidity trader (LT)). We also show the liquidity traders profits in
the naive case where the market maker does not learn about the trading interest of the liquidity trader, i.e.,
absent signalling. The last column shows the absolute quantity traded by the liquidity trader (in number
of shares). The bottom panel shows the static strategy and the average and standard deviation of the
dynamic strategy. Since we simulate 10.000 times, the standard deviation of the mean of each variables
is simply the standard deviation of a single game divided by 100. The parameter values are as described
in Table 3: T = 5, variance of S is σS2 = 3, the volatility of noise trading σu2 = 1 and innovations in the
fundamental value σe2 = 1.
Overall profits to the players:
Inf ormed
Mm
N oise
LT Strategic
LT N aive
Diff
Abs(S)
-0.6175
3.1952
0.0304
1.2991
1.3864
1.0495
Dynamic model
Mean
St dev.
4.0235
7.0916
-0.0132
6.1387
-3.4227
4.2344
-0.5876
2.7249
Static model
Mean
St dev.
4.0234
7.0863
-0.0134
6.1383
-3.4224
4.2345
-0.5876
2.7283
Strategy f-sequence
Period
Static
Dynamic
St.Dev
Diff
f1
f2
f3
f4
f5
0.255
0.177
0.162
0.173
0.232
0.255
0.177
0.161
0.171
0.236
0.000
0.036
0.021
0.019
0.062
0.0000
-0.0001
-0.0017
-0.0024
0.0042
39
0.700
0.300
ft
0.690
0.250
0.680
0.200
0.670
0.150
λt
0.660
0.100
0.650
0.050
0.640
0.000
1
2
3
4
5
0.300
0.250
0.200
yt
0.150
Et(S)
0.100
0.050
0.000
-0.050 1
2
3
-0.100
4
5
xt
-0.150
-0.200
Figure (1) The evolution of the parameters over time. The upper figure shows the
price impact parameter (λt , left axis) and the liquidity traders optimal strategy over time
(ft , right axis). The bottom figure shows the net trading volume (yt ), the volume by the
informed trader (xt ) and the market makers expectation of the uninformed trading volume
(Et (S)). These values are expectations of the strategic liquidity trader before trading starts,
based on parameter values of the base case in Table 1.
40