1

An Integrated Order Submission Strategy Model

of Uninformed and Informed Traders in an Order-Driven Market

Pei-Han Hsin

Department of International Business, Cheng Shiu University , Taiwan (R.O.C.)

Ming-Chang Wang

Department of Business Administration, National Chung Cheng University, Taiwan (R.O.C.)

Chin-Shun Wu

Department of Business Management, National Sun Yat-Sen University, Taiwan (R.O.C.)

Only for

EUROPEAN FINANCIAL MANAGEMENT ASSOCIATION 2010 ANNUAL MEETING

Declaration:

Agree to be published in a regular issue of the European Financial Management journal

Corresponding author: Pei-Han Hsin, Department of International Business, Cheng Shiu University, No.840,

Chengcing Rd., Niaosong Township, Kaohsiung County 833, Taiwan (R.O.C.); Tel.: +886-929-073303; fax:

+886-7-3926930; E-mail: phhsin@anet.net.tw

2

An Integrated Order Submission Strategy Model

of Uninformed and Informed Traders in an Order-Driven Market

Abstract

This paper presents a subgame perfect Nash equilibrium model in which risk-averseuniformed and informed traders select their optimal combination of market and limit ordersin an order-driven market .Traders have own private information while informed traders cantake advantage of uniformed traders. Our findings are: (1) The decision on the optimalproportion of market orders depends on the order aggressiveness on own and opposite side,the volatility of asset value, the bid-ask spread ,the degree of risk aversion, the reservationvalue and the market perceptions relating to the arrival rate of limit and market orders.(2)Thestrategies consist of : complete market buy orders, a combination of market and limit buyorders, complete limit buy orders , a combination of limit buy and sell orders, complete limitsell orders , a combination of market and limit sell orders, the complete market sell orders.(3)The critical value of boundary depends on the bid-ask spread, the order aggressiveness onown and opposite side, the non-execution risk, the adverse selection risk and the marketperceptions relating to the arrival rate of limit and market orders.

Keywords: Market Microstructure, order strategy, an order-driven market

JEL classifications: G12, G14, G15

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1. IntroductionOrder strategy of market orders and limit orders are the most important issue to trading

mechanism. Over the past few years, a considerable number of studies have been made on theorder strategies of informed traders, uninformed traders and market makers in quote-drivenmarkets. Those literatures can be divided into two main categories, theoretical and empiricalliteratures.

In theoretical literatures, researchers generally suggest that informed and liquidity tradersare aggressive traders, so they prefer to submit market orders .For instance, Glosten (1994)suggests that informed traders are rational and risk averse ,and they maximize theirquasi-concave utility function formed by their cash and share position, and that risk-neutraluninformed traders submit limit orders and provide the source of bids and asks.

Handa and Schwartz (1996) explain why investors trade via limit orders and assume thatpatient uninformed traders supply liquidity to the market, and impatient informed andliquidity traders always trade immediately. Handa, Schwartz and Tiwari (2003) suggest thatthe private information is short-lived so that informed traders only submit market orders.Holden and Subrahmanyam (1992,1994) show that many informed traders aggressivelycompete with each other and then cause their private information to be revealed rapidly, sothey prefer to submit market orders.

In empirical literatures, for example, Lo, McKinlay and Zhang (2002) suggest that traderswith private information might also submit limit orders. Ellul, Holden, Jain and Jennings(2003) examine the determinants of the trader’s decision, suggesting that favorable(unfavorable) private information simultaneously increase likelihood of buy (sell) market andlimit orders. Nevmyvaka, Kearns, Papandreou, and Sycara (2005) demonstrate that under afixed order size, a fixed trading time and a pre-committed liquidity, a strategy of combiningmarket and limit orders can provide a superior performance than a strategy only using eithermarket orders or limit orders,

The researchers described above generally assume that informed traders and liquiditytraders only use market orders for trading, and that uninformed traders select either limitorders or market orders. Limit orders supply liquidity to the market and market ordersdemand liquidity from the market .Hence, the limit order of uninformed traders plays animportant role in the public’s liquidity provision which make the order-driven tradingmechanism be viable.

The question arises whether the restrictive assumptions of informed traders seem plausibleand whether we can release those restrictions. Even, informed traders might have alternateactive order strategy for their investment. How interactive could order strategy be betweenuninformed traders and informed traders? This is one of the purposes in this study.

Besides, in a quote-driven model, Angel (1990), Easley and O’Hara (1991), and Harris (1994) set an additional requirement that informed investors must to select market orders orlimit orders ,but never both , for trading outside the spread. Until now, few marketmicrostructure models analyze the order strategy in order-driven markets. Thus, thosemysteries of order strategy in an order-driven market still remain to be explored. This paperserves as a first attempt to address the order strategy of informed and uniformed traders in anorder-driven market.

As for uninformed traders, the previous studies portray them as patient investors withoutprivate information. For example, Handa, Schwartz and Tiwari (2003) develop a model inwhich uninformed traders face adverse-selection problem due to the presence of informedtraders. In their model, the informed traders and liquidity traders only submit market ordersand uninformed traders only submit limit orders. Uninformed traders could monitor limitorder book information to conjecture the arrival rates of market and limit order to optimally

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adjust quote prices and order strategy. If incoming order aggressiveness increases (decreases),the uninformed traders expect that informed traders are present (absent). In the tradingbeginning, informed traders take advantage of their information to maximize their profit andresults in rapidly consuming limit orders which cause quote prices change, and then leak outprivate information. Thus, the uninformed traders are likely to submit the market orders afterthe private information partly leak out. In a quote-driven market, the uninformed tradersresemble the dealers because they provide liquidity and immediacy to the market andencounter the problems from asymmetric information, therefore their bid-ask spreads shouldgenerate enough return to cover the loss of adverse selection. However, uninformed liquidityproviders have no obligation to keep the quote-driven market viable. Biasis, Hillion and Spatt(1995), Al-Suhaibani and Kryzanowski (2001), Ahn, Bae and Chan (2001) and Ranaldo(2004) suggest that the market orders represent the arrival of fundamental information whichsignals uninformed traders that the fundamental information leak out to the market. Therefore,we infer that the uninformed traders may submit market orders after they observe the comingof private information.

For this reason, we assume that the uniformed traders have their own private informationeven if the private information of uniformed traders is less accurate than informedtraders’ .Therefore, risk-averse uniformed traders also use a combination of limit and marketorders to maximize their expected utility.As for informed traders’ order submission behaviors, the risk-averse informed traders are in

a dynamic economic ecology where market participants could aggressively monitor the limitorder book to extract partial fundamental information. For instance, if informed buyersubmits market orders, the high arrival rate of market buy order will increase the bid and askprice formed by uninformed traders. Uninformed traders recognize that a higher arrival rateof market buy order represents favorable fundamental information, so it reflects on the pricequotation. If informed buyer submits a limit order, a higher arrival rate of buy limit orderwon’t increase the bid and ask prices. Hence, an informed trader originally submits marketorder in order to immediacy, but this behavior shrink the size of his profits through leakinginformation out. Otherwise, they submit limit orders to keep the current size of profitswithout information delivery, but the drawback is to suffer non-execution risk .We suggestthat informed traders would consider the trade-off relationship for selecting their optimalorder strategy. In a word, risk-averse informed traders also use a combination of limit andmarket orders to maximize their expected utility. Besides,they always take advantage ofuniformed traders.

After relieving above restrictive assumptions, we set a synthetic model in which bothuninformed and informed traders can choose their strategy, including pure market orders ,pure limit order, a combination of market and limit orders.

Our model has some features: hybrid order strategies, investors with risk-averse preferenceand incorporating states of limit order book. We discuss them as follows.

Firstly, we incorporate hybrid order strategies for informed and uninformedtraders.Holden and Subrahmanyam (1992) show that privately informed traders actaggressively and then their private information is revealed very rapidly, so they prefer tosubmit market orders. Angel (1994) and Harris (1998) ague that informed traders are morelikely to use market orders when the realized asset value is farther away form its expectedvalue. Harris (1998) describes that informed traders are more likely to use market orders ifthey believe information is short-lived. Handa et al. (2003) show that the short-lived privateinformation makes informed traders only trade by market orders. However, Lo, McKinlay

5

and Zhang (2002) find that traders with private information might also submit limit orders.1

Moreover, Anand et al. (2005) and Bloomfield et al. (2005) find that informed traders tend totake liquidity earlier in the trading day, while acting as liquidity supplier later in the day.Hence, the traders possibly change their order strategy in the trading day. In a word, completemarket or limit order strategy is a special case of a hybrid combination of market or limitorders. Therefore, we use a proportion of submitting market orders to merge differentarguments.

Secondly, we adopt risk-averse preference utility .The majority of limit order models userisk-neutral preference to analyze order strategies of market participants.2 A risk aversionnature refines the results from former researchers. Wald and Horrigan (2005) use theexpected utility maximization of a risk-averse investor to analyze optimal orderdecision.They analyze optimal order-submission choice in a mean-variance framework. Wang,Zu and Kuo (2007a) also analyze the optimal order-submission behaviors of risk-averseuninformed traders. Notwithstanding those papers try to include risk-averse preference foranalyzing order strategy, they still fail to consider order submission behaviors of traders toexplain hybrid strategies in different situation.

Thirdly, we incorporate the states of limit order book. The market participants supplyliquidity according to their beliefs on asset valuation. Uninformed traders have scant privateinformation, and thus they pay attention to the market to capture information flow forimproving their investment performance. Information content of the limit order book is themain source of real-time information for all market participants to revise their belief on assetvaluation. Seppi (1997) suggests that agents’decision on submitting limit or market ordersdepends on the state of the limit order book. Parlour (1998) displays that the decisions onsubmitting market or limit orders depends on the states of limit order book and on the trader’s position in the limit order queue. Besides,Wang, Zu and Kuo (2007b) show that theparticipants incorporate order book information into the bid-ask spread to manage adverse-selection risk and non-execution risk. In addition, empirical studies show that the state of thelimit order book affects a trader’s strategy.3 For example, Ranaldo (2004) examines theinformation content of a limit order book in a purely order-driven market and shows thatpatient traders become more aggressive when the own (opposite) side book is getting thicker(thinner), the bid-ask spread is getting wider or temporary volatility increases.

To the best of our knowledge, order book information has significant power to explaintrader’s degreeof aggressiveness.4 Foucault, Kadan and Kandel (2005) suggest that theproportion of patient traders and the order arrival rate are the key determinants of limit order

1 Ellul, Holden, Jain and Jennings (2003) examine the determinants of the trader’s order choice decision. Theysuggest that favorable (unfavorable) private information simultaneously increase likelihood of buy (sell) marketand limit order. Nevmyvaka, Kearns, Papandreou, and Sycara (2005) empirically demonstrate that acombination of market order and limit orders is able to obtain a superior average price than the one by onlyusing market orders or the one by only using limit orders, under a fixed order size, a fixed trading time and apre-committed liquidity.2 This risk-neutral assumption is imposed in the researches by Copeland and Galai (1983), Glosten (1994),Handa and Schwartz (1996), Foucault (1999), Handa, Schwartz and Tiwari (2003).3 Biais et al. (1995) empirically analyze supply and demand of liquidity and interaction between order book andorder flow in the Paris Bourse. Hasbrouck (1991a, 1991b) examines the information indicators of limit orderbook. Cao et al. (2005) propose that traders should use available information on the state of the book. Ahn et al.(2001) analyze the interaction between transitory volatility and order flow composition to shed light on the roleof limit orders in liquidity provision in a limit order market. Al-Suhaibani and Kryzanowski (2001) show thatthe probability of placing a market order is negatively related to inside spread and order size, and is positivelyrelated to order imbalance and previous same-side market orders. Furthermore, Chan (2005) demonstrates thattraders become more aggressive in buying and more patient in selling after previous positive stock return.4 Coppejans and Domowitz ( 2002), Pascual and Veredas (2004), Ranaldo (2004) and Hall and Hautsch (2006)

6

book dynamics in equilibrium. Thereby, the incoming order aggressiveness represents thestate of limit order book.5

In sum, our theoretical analysis corroborates the findings: (1) The decision on the optimalproportion of market orders depends on the order aggressiveness on own and opposite side,the volatility of asset value, the bid-ask spread ,the degree of risk aversion, the reservationvalue and the market perceptions relating to the arrival rate of limit and market orders.(2)Thestrategies consist of : complete market buy orders, a combination of market and limit buyorders, complete limit buy orders , a combination of limit buy and sell orders, complete limitsell orders , a combination of market and limit sell orders, the complete market sell orders.(3)The critical value of boundary depends on the bid–ask spread, the order aggressiveness onown and opposite side, the non-execution risk, the adverse selection risk and the marketperceptions relating to the arrival rate of limit and market orders.

This article is organized as follows. In Section 2, we model the order submission strategy oftraders and find out the optimal equilibrium strategies. Section 3 we discuss the features ofNash equilibrium. Section 4 concludes the paper.

2. The Model2.1 Exponential Utility FunctionConsider a decision, either placing a market order for a security with certainty trading at an

uncertainty price or placing a limit order for that security with uncertainty trading at acertainty price. We assume that all traders allocate the identical initial wealth 1W into twoassets at time 1: a bond, which pays a certain payoff 1TR at time T , and a single riskyasset x with stochastic liquidation value, denoted by Txv ,

~ , which is realized at time T . All

traders make order strategy to maximize the expected utility of terminal wealth TW at timeT . Besides, we assume that all traders are with constant absolute risk aversion (riskparameter )and have a negative exponential utility function.

TT WWu exp1 (1)

There are three types of agents in our model. The first type agents termed liquidity tradersL requires an immediate transaction via market order without order strategy due tosubjective liquidity preference. The second type agents termed informed traders I haveprivate exclusively advantageous information. The third type agents termed uninformedtraders U are scant private information. In such a hypothesis, informed and uninformedtraders make their optimal order decisions, including limit order, market order and acombination of limit and market orders.

Previous theoretical researches assume that informed traders only submit market order andthat uninformed traders only submit limit order. In this paper, we relax this assumption. Therisk-averse uninformed and informed traders maximize their profit according to theirinformation, therefore they demand (supply) liquidity when the value of their information ishigh (low). Thus, as long as the gap between trading prices and true value is large (small),

5 Wang, Zu and Kuo (2007b) examine the dynamic order-submission behavior via market order aggressivenessas a proxy for the information precision of limit order book. In this paper, we define the relative arrival rate ofmarket versus limit orders as order aggressiveness.

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both of traders are much likely to submit market (limit) orders. Moreover, we assume thatinformed traders have more advantage in getting extra information than uninformed ones.

In our model, there are two groups of uninformed traders, buyer group and seller groupsuppressed asUh and Ul .They have different interpretation of the same public informationfor the risky asset value. Let the fundamental value of the risky asset be a random variable

TUhv ,~ ( TUlv ,

~ ) at time T for uninformed buyers (sellers). While an uninformed buyer (seller)

trade for one share of the risky asset at a specified price xP , the terminal wealth TUhW , ( TUlW , )

is also a random variable.( , , 1( )Uh T Uh T x TW v W P R and , 1 ,( )Ul T x T Ul TW W P R v )Similarly,we suppress Ih ( Il ) subscripts for two groups of informed traders.

The timing of events is depicted in Figure 1. Accordingly, if buyers (sellers) submit limitorder at time 1, it will be executed when their opposite traders place market orders. Theexpected utilities of uninformed buyers and sellers are , ,( )Uh t Uh TE u W and

, ,(Ul t Ul TE u W .Those of informed buyers and sellers are , ,( )Ih t Ih TE u W and ( , ,(Il t Il TE u W .

For the buyer group, TUhv ,~ have normal distribution 2,Uh UhN at time 1 and t . Uh

is interpreted as the reservation price with the variance 2Uh . Similarly, for the seller group,

Ul is interpreted as the reservation price with the variance 2Ul for the true value of the

risky asset TUlv ,~ , with normal distribution 2,Ul UlN at time 1 and t .The investors

summit mixed strategies at time 1 and then market orders are traded. As for limit order, itdepends on the other traders being in the opposite side. To simplify the immensely complexdistribution, we assume Uh Ih H and Ul Il L .All importance inference remainsin this study.

2.2 Proportion of Submitting Market OrderIn this section, we set a model for analyzing the order-submission strategy. Recall the basic

distinction between market order and limit order for traders: (1) a market order traded in acertain execution at an uncertain price, and (2) a limit order traded in an uncertain executionat a certain price. Besides, remember that (1) market orders leak out partial fundamentalinformation to the participants, which result in price volatility, and (2) limit orders don’t leakout private information, so the prices are unchanged, but they are with non-execution risk.Let ˆ( ) be the proportion of market orders submitted by uninformed buyers (sellers) and

ˆ( ) be the proportion of market orders submitted by informed buyers (sellers) .Let 1 ˆˆ(1 ;1 ;1 ) be the proportion of limit orders and ( )a bP P be the certain price of limit

sell (buy ) orders. and are real value in the boundary of [0 , 1].We adopt a stackelberg

Time

Figure. 1. The Timing of events.

time1 time t timeT

Order-submission Decision Asset LiquidationAction

Execution of market order Execution of limit orderExecution

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model for this study. The leader, informed traders have a higher selling price or a lowerbuying price than uninformed traders. That is because informed traders have advantage ingetting precise information. For simplicity, we assume they face nearly same prices. All mainconclusion remains. Here, we assume that a market order seller trades with a market orderbuyer by a trading price.

Ranaldo (2004) and Hall and Hautsch (2006) define that high buy (sell) orderaggressiveness is a higher (lower) price of orders set by buyer (seller).6 Alternatativedefinition is that a higher arrival rate of market (limit) buy order as higher (lower) level oforder aggressiveness of buy side and a higher (lower) arrival rate of market (limit) sell orderas higher (lower)level of order aggressiveness of sell side. We assume that all tradershomogeneously recognize that the arrival rate of market sell (buy) orders denoted by mk

( Mk ) is urgently sold (bought) and the arrival rate of limit sell (buy) orders denoted by 1 mk

( 1 Mk ) is patiently sold (bought).Besides, informed seller’s(buyers’) arriving ratio( )i Ip p is an exogenous specified proportion , liquidity trader’s is ( )l Lp p , and uninformed

trader’s is ( )u Up p .

Therefore, an uninformed buyers’ limit order can be executed when they are againstinformed sellers’ market orders, liquidity sellers’ market orders, and uninformed sellers’ market orders, and fails to execute when against the limit sell orders submitted byuninformed and informed sellers. An uninformed buyers’market order can immediately beexecuted when they are against informed sellers’ market and limit orders, liquidity sellers’ orders, and uninformed sellers’ market and limit orders.As for uninformed seller’behaviors,we don’t give unnecessary details again.

An informed buyers’ limit order can be executed when they are against liquidity sellers’ orders and against uninformed sellers’ market and limit orders. An informed buyers’ market order can immediately be executed when they are against informed sellers’ market and limit orders and against liquidity sellers’ orders. Most importantly, informed buyers never meetinformed sellers and vice versa. As for informed sellers’behaviors, we don’t giveunnecessary details again.

Some prior theoretical models (Glosten, 1994; Handa and Schwartz, 1996; Foucault, 1999;Handa, Schwartz and Tiwari, 2003) assume that uninformed traders only submit limit ordersin their equilibrium. Here, we correct those defects.

2.3 Order Strategy of Uninformed TradersThe uniformed traders’counterparts are the informed traders, uninformed traders, liquidity

traders. We describe them in the following.

A. Informed Traders in the opposite Side:

Handa, Schwartz and Tiwari (2003) suggest that limit order submitters face an adverseselection problem due to the presence of privately informed traders. While limit ordersubmitter’s counterpart is informed trader, limit order submitter suffer the adverse-selectionlosses. Copeland and Galai (1983) show that adealer’s quote decision is to write a put or calloption to informed traders. In other word, the trader who places a buy (sell) limit order is to

6 Ranaldo (2004) define the mort aggressive order as a market order that demands more trading volume than isavailable at the prevailing quote. The second type of aggressive order is a market order that demands lessvolume that demands less volume than the quoted depth. The third and fourth order types are limit orders withinand at the prevailing quotes, respectively. The least aggressive category is an order cancellation.

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write a free put (call) option with a execution price of the best bid price bP (the best ask

price aP ) to the informed trader. Ivf TUhtUh ,,~ and Ivf TUltUl ,,

~ is the condition probabilitydensity function ofuninformed traders’ terminal wealth where uninformed traders trade withinformed trades (denoted by I ).

For an uninformed buyer, if informed seller’smarket order is in the opposite side, theuninformed buyers’ utility is

, , , 1 , , ,( ) 1 exp (1 )T bR P

Uh t Uh T Uh T b T Uh t Uh T Uh TE u W I v W P R f v I dv

(2)

If informed seller’slimit order is in the opposite side, the uninformed buyers’ utilityis

, , , 1 , , ,( ) 1 exp ) (1 )aP R

Uh t Uh T Uh T a b Uh t Uh T Uh TE u W I v W P R P r f v I dv

(3)

As for the utility of uninformed seller, there are two cases as follows.

If informed buyer’smarket order is in the opposite side, theuninformed sellers’ utility is

, , , 1 , , ,( ) 1 exp (1 )Ul t Ul T Ul T a T Ul t Ul T Ul TRE u W I v W P R f v I dv

(4)

If informed buyer’slimit order is in the opposite side,the uninformed sellers’ utilityis

, , , 1 , , ,ˆ( ) 1 exp ( ) (1 )

bUl t Ul T Ul T b a Ul t Ul T Ul TP R

E u W I v W P R P r f v I dv

(5)

B. Uninformed traders in the Opposite Side:

For uninformed traders’ terminal wealth conditioning on uninformed trades (denoted by Uhor Ul ) accompanying with the condition probability density function , ,Uh t Uh Tf v U

( , ,Ul t Ul Tf v U ),their expected utility is as below:

For an uninformed buyer, if uninformed seller’smarket order is in the opposite side, theuninformed buyer’sutility is

, , , 1 , , ,( ) 1 exp (1 )Uh t Uh T Uh T b T Uh t Uh T Uh TE u W U v W P R f v U dv

(6)

If an uninformed seller’slimit orders is in the opposite side, the uninformed buyer utility is

, , , 1 , , ,( ) 1 exp ) (1 )Uh t Uh T Uh T a b Uh t Uh T Uh TE u W U v W P R P r f v U dv

(7)

For an uninformed seller ,if an uninformed buyer’smarket order is in the opposite side, theuninformed seller’sutility is

, , , 1 , , ,( ) 1 exp (1 )Ul t Ul T Ul T a T Ul t Ul T Ul TE u W U v W P R f v U dv

(8)

If an uninformed buyer’slimit orders is in the opposite side, uninformed seller utility is is

10

, , , 1 , , ,ˆ( ) 1 exp ( ) (1 )Ul t Ul T Ul T b T a Ul t Ul T Ul TE u W U v W P R P r f v U dv

(9)

C. Liquidity traders in the Opposite Side:

Glosten (1994) and Handa and Schwartz (1996) suggest that limit order traders gain fromliquidity traders and lose from informed traders .However, Foucault (1999) suggests that thesetting price of limit order is fixed over time and the payoff would immediately becomenegative while new unfavorable public information arrives. Execution of limit orders isuncertain and the asset value fluctuation creates an adverse selection problem .

For uninformed traders’ terminal wealth by submitting limit buy (sell) orders conditioningon liquidity trade (denoted by L ) accompanying with the condition probability densityfunction Lvf TUhtUh ,,

~ ( Lvf TUltUl ,,~ ), their expected utility is as below.

An uninformed buyer’s expected utility is

, , , 1 , , ,( ) 1 exp (1 )Uh t Uh T Uh T b T Uh t Uh T Uh TE u W L v W P R f v L dv

(10)

An uninformed seller’s expected utility is

, , , 1 , , ,( ) 1 exp (1 )Ul t Ul T Ul T a T Ul t Ul T Ul TE u W L v W P R f v L dv

(11)

Figure 2. Counterparties and the related probability of uninformed buyers’ order strategy

Counterparties and the related probability of uninformed buyers’ order strategyare listed infigure 2. Therefore, the uninformed buyers’ optimal problem is as follows.

, ,{ }

max ( )Uh t Uh TArg E u W

(12)

, 1 , , ,1 exp (1 )TR P

m i Uh T b T Uh t Uh T Uh Tk p v W P R f v I dv

+ , 1 , , ,1 exp (1 )m u Uh T b T Uh t Uh T Uh Tk p v W P R f v U dv

Uninformed Buyers’

Order Strategy

Market SellOrders（Km）

Limit SellOrders（1-Km）

InformedSellers（Pi）

UninformedSellers（Pu）

Informed Sellers（Pi）

UninformedSellers（Pu）

LiquiditySellers（Pl）

, ,

, ,1 b

execution

execution P

, ,

, ,1 b

execution

execution P

, ,

, ,1 b

execution

execution P

,

, , a

,

β execution P

1-βnon-execution, , a

, ,

β execution P

1-βnon-execution

11

+ , 1 , , ,1 exp (1 )m l Uh T b T Uh t Uh T Uh Tk p v W P R f v L dv

+ , 1 , , ,(1 ) 1 exp ) (1 )aP R

m i Uh T a b Uh t Uh T Uh Tk p v W P R P r f v I dv

+ , 1 , , ,(1 ) 1 exp ) (1 )m u Uh T a b Uh t Uh T Uh Tk p v W P R P r f v U dv

Figure 3. Counterparties and the related probability of uninformed seller’ order strategyCounterparties and the related probability of uninformed sellers’ order strategyare listed infigure 3.Similarly, the uninformed sellers’ optimal problem is as follows.

, ,ˆ{ }

max ( )Ul t Ul TArg E u W

(13)

, 1 , , ,ˆ ˆ1 exp (1 )M I Ul T a T Ul t Ul T Ul TR

k p v W P R f v I dv

+ , 1 , , ,ˆ ˆ1 exp (1 )M U Ul T a T Ul t Ul T Ul Tk p v W P R f v U dv

+ , 1 , , ,ˆ ˆ1 exp (1 )M L Ul T a T Ul t Ul T Ul Tk p v W P R f v L dv

+ , 1 , , ,ˆ ˆ ˆ(1 ) 1 exp ( ) (1 )

bM I Ul T b T a Ul t Ul T Ul TP R

k p v W P R P r f v I dv

+ , 1 , , ,ˆ ˆ ˆ(1 ) 1 exp ( ) (1 )M U Ul T b T a Ul t Ul T Ul Tk p v W P R P r f v U dv

2.4 Order Strategy of Informed TradersThe order strategy of informed traders is similar to that of uninformed traders. We briefly

describle as follows.For informed buyer, there are three possible situations.

If an uninformed seller’smarket order is in the opposite side, theinformed buyer’s expectedutility is

Uninformed Sellers’

Order Strategy

Market BuyOrders（Km）

Limit Buy Orders（1-Km）

Informed Buyers（PI）

UninformedBuyers （PU）

Informed Buyers（PI）

UninformedBuyers（PU）

Liquidity Buyers（PL）

, ,

, a

β executionˆ1-βexecution,P

, ,

, a

β executionˆ1-βexecution,P

, ,

, a

β executionˆ1-βexecution,P

, , b

, ,

β execution Pˆ1-βnon-execution ,

, , b

,

β execution Pˆ1-βnon-execution

12

, , , 1 , , ,( ) 1 exp (1 )Ih t Ih T Ih T b T Ih t Ih T Ih TRE u W U v W P R f v U dv

(14)

If a liquidty seller’s market order is in the opposite side, the informed buyer’s expected utility is

, , , 1 , , ,( ) 1 exp (1 )Ih t Ih T Ih T b T Ih t Ih T Ih TRE u W L v W P R f v L dv

(15)

If an uninformed seller’slimit order is in the opposite side,the informed buyer’s expected utility is

, , , 1 , , ,( ) 1 exp ) (1 )a

Ih t Ih T Ih T a b Ih t Ih T Ih TP RE u W U v W P R P r f v U dv

(16)

Figure 4. Counterparties and the related probability of informed buyers’ order strategy

Counterparties and the related probability of informed buyers’ order strategyis list in figure4.Thus, informed buyers’ optimal problem is

, ,{ }

max ( )Ih t Ih TArg E u W

(17)

, 1 , , ,1 exp (1 )m u Ih T b T Ih t Ih T Ih TRk p v W P R f v U dv

, 1 , , ,1 exp (1 )m l Ih T b T Ih t Ih T Ih TRk p v W P R f v L dv

+ , 1 , , ,(1 ) 1 exp ) (1 )a

m Ih T a b Ih t Ih T Ih TP Rk v W P R P r f v U dv

As for informed seller, there are three possible situations. If uninformed buyer’smarket orderis in the opposite side,the informed seller’s expected utility is

, , , 1 , , ,ˆ ˆ( ) 1 exp (1 )R

Il t Il T Il T a T Il t Il T Il TE u W U v W P R f v U dv

(18)

If a liquidty buyer’s market order is in the opposite side, the informed seller’s expected utility

Informed Buyers’ OrderStrategy

Market Sell Orders（Km） Limit Sell Orders（1-Km）

Uninformed Sellers（Pi） Uniformed Sellers（1）Liquidity Sellers（Pl）

, , a

, ,

α execution P1-αnon-execution

, ,

, b

α execution

1-αexecution,P

, ,

, b

α execution

1-αexecution,P

13

is

, , , 1 , , ,ˆ ˆ( ) 1 exp (1 )R

Il t Il T Il T b T Il t Il T Il TE u W L v W P R f v L dv

(19)

If an uninformed buyer’slimit order is in the opposite side, the informed seller’s expected utility is

, , , 1 , , ,ˆ ˆ ˆ( ) 1 exp ( ) (1 )bP R

Il t Il T Il T b T a Il t Il T Il TE u W U v W P R P r f v U dv

(20)

Figure 5. Counterparties and the related probability of informed sellers’ order strategy

Counterparties and the related probability of informed sellers’ order strategyis list in figure 5.So,informed sellers’ optimal problem is

, ,ˆ{ }

max ( )Il t Il TArg E u W

(21)

, 1 , , ,ˆ ˆ1 exp (1 )R

M U Il T a T Il t Il T Il Tk p v W P R f v U dv

, 1 , , ,ˆ ˆ1 exp (1 )R

M L Il T b T Il t Il T Il Tk p v W P R f v L dv

+ , 1 , , ,ˆ ˆ ˆ(1 ) 1 exp ( ) (1 )bP R

M Il T b T a Il t Il T Il Tk v W P R P r f v U dv

3. Equilibrium Analysis

3.1 Equilibrium in buy Side

3.1.1 The Equilibrium for Uninformed Traders

We solve all optimal problems, which are displayed in Appendix A-D. According the results,we have some proposal as follows.

Proposition 1: The uninformed buyer’s optimal order strategy ( ) , a solution for equation

Informed Sellers’ Order

Strategy

Market Buy Orders（Ks） Limit Buy Orders（1-Ks）

Uninformed Buyers（PU） Uninformed Buyers（1）Liquidity Buyers（PL）

, , b

, ,

α execution Pˆ1-αnon-execution

, ,

, a

α executionˆ1-αexecution,P

, ,

, a

α executionˆ1-αexecution,P

14

(12) , consists of the state of the limit order book, the volatility of asset value, the bid-askspread ,the degree of risk aversion, the reservation value ,the market perceptions relating tothe arrival rate of limit and market orders, and the equilibrium of price quotation in theelectronic limit order market.

2 2

2 2 2

2 2

(1 )exp (1 )( ) ( )

2

(1 )[1 (1 ( , ))]( ( ) )

[1 (1 ( , ))]b

a

UhH b a

m i P R H Uh Uh H a a b Uh

m i P R H Uh Uh b

P P R

k p Z P P P r

k p Z P R

(22)

where

222 2

22

222 2

22

1( , )

22

1( , )]

22

b

b

a

a

p R H Uhp R H Uh Uh

UhUh

p R H Uhp R H Uh Uh

UhUh

vZ e d

vZ e d

(23)

Appendix A shows the proof .In the left side of equation, we define that term as marginalincrease of utility resulting from mixed strategy .Some are from market orders expressed as( )aP R , the others are from limit orders expressed as ( )H bP . In the right side ofequation, we define that term as marginal decrease of utility resulting from the loss of adverseselection and the opportunity cost of non-execution, 2( ( ) )H a a b UhP P P r . Theincrease of market orders decrease the non-execution cost, but increase the adverse selectioncost.

3.1.2 The Equilibrium for Informed Traders

Proposition 2: The informed buyer’s optimal order strategy ( ) , a solution for equation(17) , consists of the state of the limit order book, the volatility of asset value, the bid-askspread ,the degree of risk aversion, the reservation value ,the market perceptions relating tothe arrival rate of limit and market orders ,and the equilibrium of price quotation in theelectronic limit order market.

2 2

2 2 2

2 2

(1 )exp (1 )( ) ( )

2

(1 )[(1 ( , ))]( ( ) )

[(1 ( , ))]a

IhH b a

m P R H Ih Ih H a a b Ih

m R H Ih Ih b

P P R

k Z P P P r

k Z P R

(24)

where

15

222 2

22

222 2

22

1( , )

22

1( , )]

22

a

a

P R H IhP R H Ih Ih

IhIh

R H IhR H Ih Ih

IhIh

vZ e d

vZ e d

(25)

Appendix B shows the proof. Similar interpretations given for eq.(22) can be given to theoptimal solution of eq. (24).Assume that

2 2

1

(1 )( ) (1 )( ) ( )

2Uh

H b aY P P R

2 2 2

2 2

(1 )[1 (1 ( , ))]( ( ) )( )

[1 (1 ( , ))]b

a

m i P R H Uh Uh H a a b Uh

m i P R H Uh Uh b

k p Z P P P rG

k p Z P R

2 2

2

(1 )( ) (1 )( ) ( )

2Ih

H b aY P P R

2 2 2

2 2

(1 )[(1 ( , ))]( ( ) )( )

[(1 ( , ))]am P R H Ih Ih H a a b Ih

m R H Ih Ih b

k Z P P P rH

k Z P R

(26)

where2 2 2 2

2 2 2 2

1 (1 ( , )) 1 ( , )1 0

1 (1 ( , )) 1 ( , )b a

a

i P R H Uh Uh P R H Ih Ih

i P R H Uh Uh R H Ih Ih

p Z Z

p Z Z

(27)

, then we can picture figure 6 .If others being equal, an informed trader trades less market orders than an uninformed

trader. The result shows in figure 1.Because market buy orders increase the ask price whichdecrease the profit of trader, the smart informed buyer take this into account. In a stackelberggame, when the depth of market is small, the uninformed trader face higher sell price ,afterinformed agent consumes the sell orders by submitting market buy orders. Therefore,uninformed trader gains less profit than informed trader. Besides, informed trader can takeadvantage from uninformed ones.From figure 6, we can infer that

(1).The higher the mean of asset value, the more the buy market orders. That results in theincrease of and .（The line ( )G or ( )H shifts to the right）That means that thestocks have high reservation value, therefore the traders place more market orders .Then theproportion of market orders increases.

(2).The more subsidies an adverse-risk trader asks, the less amount the market orders are.That results in the decrease of and .

(3).If the gap of asymmetric information between informed and uninformed traders is large,the value of is probably larger than the value of.In such a situation, the uninformed islikely to submit the limit orders only. That is a special case in our model which is consistentwith the results of previous literatures.

16

Figure 6

(4).The state of limit order book affects order strategies of traders, the ratio of market orders.

(5). ( , ) is a subgame perfect Nash equilibrium, if and only if equation (22) and (24) areheld.

3.2 Equilibrium in Sell Side

3.2.1 The Equilibrium for Uninformed Traders

Proposition 3: The uninformed seller’s optimal order strategy ˆ( ) , a solution for equation(13), consists of the state of the limit order book, the volatility of asset value, the bid-askspread ,the degree of risk aversion, the reservation value ,the market perceptions relating tothe arrival rate of limit and market orders ,and the equilibrium of price quotation in theelectronic limit order market.

2 2

2 2

2 2 2

ˆ(1 ) ˆ ˆexp [ ( ) (1 )( )]2

(1 ( , ))[ˆ ˆ(1 )(1 ( , ))( )[ ( ) ]

b

Ulb a L

M I R L Ul Ul a T

M I P R L Ul Ul L Ul b a b

P R P

k p Z P R

k p Z P P P r

(28)

where

17

222 2

22

22

2 222

1( , )

22

ˆ1ˆ( , )]22

b

b

R L UlR L Ul Ul

UlUl

p R L Ul

p R L Ul UlUlUl

vZ e d

vZ e d

(29)

Appendix C shows the proof .In the left hand of equation, we define that as marginal utilityincrease resulting from mixed strategy expressed as ( )a LP and ( )bP R .In the righthand of equation, we define that as marginal utility decrease resulting from the loss ofadverse selection and the opportunity loss of non-execution 2 ˆ( ( ) )L Ul b a bP P P r .

3.2.2The Equilibrium for Informed Traders

Proposition 4: The informed seller’s optimal order strategy ˆ( ) , a solution for equation( 21), consists of the state of the limit order book, the volatility of asset value, the bid-askspread ,the degree of risk aversion, the reservation value , the market perceptions relatingto the arrival rate of limit and market orders ,and the equilibrium of price quotation in theelectronic limit order market.

2 2

2 2

2 2 2

ˆ(1 ) ˆ ˆexp [ ( ) (1 )( )]2

[(1 ( , ))]ˆ(1 )[(1 ( , ))][( ( ) )]

b

Ilb a L

M R L Il Il a

M p R L Il Il L Il b a b

P R P

k Z P R

k Z P P P r

(30)

where

222 2

22

222 2

22

1( , )

22

ˆ1ˆ( , )]22

P RB

B

R L IlR L Il Il

IlIl

L IlP R L Il Il

IlIl

vZ e d

vZ e d

(31)

Appendix D shows the proof . Similar interpretations given for eq. (28) can be given to theabove optimal solution of eq. (30).

Further, we assume that

18

2 2

1

ˆ(1 )ˆ ˆ ˆ( ) [ ( ) (1 )( )]2

Ulb a Ly P R P

2 2

2 2 2

(1 ( , ))[ˆ( ) ˆ ˆ(1 )(1 ( , ))( )[ ( ) ]b

M I R L Ul Ul a T

M I P R L Ul Ul L Ul b a b

k p Z P Rg

k p Z P P P r

2 2

2

ˆ(1 )ˆ ˆ ˆ( ) [ ( ) (1 )( )]2

Ilb a Ly P R P

2 2

2 2 2

[(1 ( , ))]ˆ( )

ˆ(1 )[(1 ( , ))][ ( ) ]b

M R L Il Il a

M p R L Il Il L Il b a b

k Z P Rh

k Z P P P r

(32)

where2 2 2 2

2 22 2

1 ( , ) 1 ( , )0 1ˆ ˆ1 ( , )1 ( , )

bb

I R L Ul Ul R L Il Il

p R L Il IlI P R L Ul Ul

p Z Z

Zp Z

(33)

,then we can picture figure 7 .

Figure 7

Others being equal, an informed trader will trade less market order than an uninformed trader.The result shows in figure 1.Because market buy orders increase the ask price, whichdecrease the profit of trader, the smart informed buyer take this into account.

In a stackelberg game ,when the depth of market is small, the uninformed seller face lowerbuy price ,after informed consume the buy orders by submitting market sell orders. Therefore,uninformed trader gains less profit than informed ones. Besides, uninformed trader can takeadvantage from uninformed ones. From figure 7, we can infer that(1).The lower the mean of asset value, the more the sell market orders. That results in theincrease of and .（The line ˆ( )g or ˆ( )h shifts to the right）That means that the

19

traders consider the stocks small reservation value, therefore they place more market sellorders .Then the proportion of market orders increases.

(2).The more subsidies an adverse-risk trader asks, the less amount the market orders are.That results in the decrease of and .

(3).If the gap of asymmetric information between informed and uninformed traders is large,the value of is probably larger than the value of .In such a situation, the uninformed islikely to submit the limit orders only. That is a special case in our model which is consistentwith the results of previous literatures.

(4).The state of limit order book affects order strategies of traders, the ratio of market orders.

(5). ˆˆ( , ) is a subgame perfect Nash equilibrium, if and only if both equation( 28) and( 30)are held.

3.3 comparative staticsConsider the impact on market order ratio of a change in the amount of order aggressiveness,bid-ask spread, price volatility and risk aversion of agents. All results are showed in appendixE and F.3.3.1 Order aggressiveness

Proposition 5a. An decrease in prior order aggressiveness on the ask side leads to high aP ,then decrease(increase) the order aggressiveness on the sell(bid) side, which results in thedecrease (increase) in the optimal proportion of sellers’（buyers’） market ordersubmission.Proposition 5b. An increase in prior order aggressiveness on the bid side leads to high bP ,then increase（decrease） the order aggressiveness on the bid （sell）side, which results inthe increase（decrease） in the optimal proportion of buyers’ （sellers’） market ordersubmission.

Al-Suhaibani and Kryzanowski (2001) document that the history of trade pattern providesinformation to incoming traders and affects the order flow. They show the positivelycorrelation between past and current market orders. Biasis, Hillion and Spatt (1995) suggestthat the positive autocorrelated order aggressiveness come from the executed strategic orders.Parlour (1998) suggest that after the opposite side market orders consume the depth of theown side quotes, the probability of submitting the own side order increases, and then thesame side traders are willing to submit limit orders. Hence, we infer that the stronger the pastorder aggressive of the own side, the stronger the current order aggressive, and the strongerthe past order aggressive of the opposite side, the weaker the current order aggressive.

3.3.2 Bid-ask spread

Proposition 6. An increase in information asymmetries between buyers and sellers leads tohigh a bP P 7, then results in the decrease in the optimal proportion of buyers’ （sellers’）market order submission.The bid-ask spread provides the uninformed traders with a proxy of the informationasymmetry. Sine the widening of the bid-ask spread represents an increase in the asymmetry

7 The bid-ask spread is a proxy of the information asymmetry (see, Madhavan, 1992; Glosten, 1994; Handa etal., 2003).

20

of information among different investors, the size of the bid-ask spread is proportional to theadverse selection costs. (Madhavan, 1992; Glosten, 1994; Handa, Schwartz and Tiwari, 2003)Lee, Mucklow, and Ready (1993) find that liquidity providers are sensitive to change in therisk of information asymmetry and manage this risk by using bid-ask spread. Biasis, Hillionand Spatt (1995) find that traders’ strategies vary with market conditions. They place morelimit orders when spreads are wide and place more market orders when spreads are narrow.Chung, Van Ness, and Van Ness (1999) find that in the NYSE, more investors enter limitorders when the spread is wide, and more investors hit the quotes when the spread is tight.Ranlado (2004) and Chan (2005) examine how a trader’sorder aggressiveness responds to achange of the bid-ask spread size. They find that the wider the spread, the weaker the orderaggressiveness. Therefore, according to information theory, the size of spread reveals thedegree of asymmetric information. In summary, while the bid-ask spread in limit order bookis wider, uninformed traders become more patient.

3.3.3 Price volatility

Proposition 7. An increase in asset volatility results in the change in the optimal proportionof buyers’ （sellers’） market order submission uncertainly.

Foucault (1999) finds the price volatility is a key determinant of making order decision.When the price volatility increases, limit order traders are exposed to the picked- off risk. Forthis reason, limit order traders ask for a larger compensation. Therefore, market order tradingis more costly, and then limit orders are placed more than market orders. However, Harris(1998) and Handa, Schwartz and Tiwari, (2003) argue that higher fundamental volatilitymakes limit orders less attractive since volatility increases limit orders’option value. Wesuggest that the higher volatility caused from information increases the picked-off risk. In thelong run, the limit order traders ask a larger compensation for the relatively high volatility.Some studies argue that liquidity needs cause price volatility. Handa and Schwartz (1996)suggest that traders gain from liquidity which drives temporary and reversible price changes,the short-run price volatility; therefore, an increase in short-term price volatility increases theplacement of limit orders. Ahn, Bae, and Chan (2001) report that an increase in transitoryvolatility is followed by an increase in market depth; an increase in market depth is followedby a decrease in transitory volatility. Ranaldo (2004) shows that the higher the price volatility,the weaker the order aggressiveness. Yet, Al-Suhaibani and Kryzanowski (2001) offercontrastive evidences that buyers (sellers) are more likely to submit market (limit) orders asthe transitory volatility increase. Apparently, our model suggests that volatility effect isclearly a controversial issue for its impact on order submission behaviors as disused above.

3.3.4 Others

According to appendix E and F, we have the following proposition.Proposition 8. An increase in risk aversion of agents leads to a high value of , thenresults in the decrease in the optimal proportion of buyers’ （sellers’） market ordersubmission.Proposition 9. An increase （A decrease） in the valuation of asset leads to a high value of

H （low L）, then results in the increase in the optimal proportion of buyers’ （sellers’）market order submission.

21

3.4 Order Submission StrategiesAccording to equation (22) and (28), we can infer that the uninformed traders’strategiesdepend on the reservation value. Most importantly, the states of limit order book, riskaversion and risk premium affect the critical boundary.

(1). When 2[ ( ) , )a a b Uhv P P P r , the uninformed traders completely submit marketbuy orders.

(2). When 2[ ( ) , ( ) ]a a b a a b Uhv P P P r P P P r , the uninformed traders submit mixedorders of market and limit buy orders.

(3). when2

, ( )2

Uha a a bv P P P P r

, the uninformed traders submit limit buy orders.

(4). When2 2

,2 2

Uh Uhb av P P

, the uninformed traders submit limit buy or/and sell

orders.

(5). when2

,( ) ,2

Uhb a b bv P P P r P

,the uninformed traders submit limit sell orders.

(6). When 2 ( ) , ( )b Uh a b b a bv P P P r P P P r , the uninformed traders submit mixed

orders of market and limit sell orders.

(7). when 20, ( )b Uh a bv P P P r ,the uninformed traders completely submit market sell

orders.

22

In a similar scenario, according to equation (24) and (30), we also infer the informed traders’strategies which depend on the reservation value. Restate that the states of limit order book,risk aversion and risk premium affect the critical boundary.

(1). When 2[ ( ) , )a a b Ihv P P P r , the uninformed traders completely submit marketbuy orders.

(2). When 2[ ( ) , ( ) ]a a b a a b Ihv P P P r P P P r , the uninformed traders submit mixedorders of market and limit buy orders.

(3). when2

, ( )2

Iha a a bv P P P P r

, the uninformed traders submit limit buy orders.

(4). When2 2

,2 2

Ih Ihb av P P

, the uninformed traders submit limit buy or /and sell

orders.

(5). when2

,( ) ,2

Ihb a b bv P P P r P

the uninformed traders submit limit sell orders.

(6). When 2 ( ) , ( )b Ih a b b a bv P P P r P P P r , the uninformed traders submit mixed

orders of market and limit sell orders.

(7). when 20, ( )b Ih a bv P P P r the informed traders completely submit market sell

orders.

Because of 2 2Ul Il , an informed trader places orders quickly than an uninformed trader

when the execution prices decrease （increase）or the bad （good）news come along. Thetraders change the market order ratio when they change their valuation of asset. Theirdifferent order submission behavior results in the evolution of liquidity. In sum, an informedtrader submits market buy order quickly than an uniformed trader does when a good or badnews come. Therefore, we have the following propositions.

Proposition 10. The investors submit market buy (sell) orders only when the reservationvalue is greater (smaller) than the ask (bid) prices.

23

Proposition 11. The investor submits limit buy or/and sell order only when the reservationvalue is between ask and bid prices.

4. ConclusionIn this study we present an integrated microstructure model to depict investors’order

submission behaviors in an order-driven market.

We assume that the uniformed traders submit not only limit order but also market order bytheir own information set which is less precise than informed traders’. Insuch assumptions,our model serves as a first attempt to model the integrated equilibrium of informed traders’ and uninformed traders’ order strategies. According to their precise private information, theinformed traders’ promptly submit market orders to gain the profit, which leak their position.For hiding their position, the informed traders submit limit order when the market priceclosed to the true value.

To sun up, our theoretical inference provides some important results.

First, the decision on the optimal proportion of market orders depends on the orderaggressiveness on own and opposite side, the volatility of asset value, the bid-ask spread ,thedegree of risk aversion, the reservation value and the market perceptions relating to thearrival rate of limit and market orders.

Secondly, the strategies consist of : complete market buy orders, a combination of marketand limit buy orders, complete limit buy orders , a combination of limit buy and sell orders,complete limit sell orders , a combination of market and limit sell orders, the completemarket sell orders.

Thirdly, the critical value of boundary depends on the bid –ask spread, the orderaggressiveness on own and opposite side, the non-execution risk, the adverse selection riskand the market perceptions relating to the arrival rate of limit and market orders.

A continuous game between informed and uninformed traders can be divided intonumerous independent stackelberg games. The results of former game affect the initialcondition of the next game. Therefore, our inferences about order submission behaviors havehigh value in microstructure market.

24

ReferencesAhn, H., Bae, K., Chan, K., 2001. Limit orders, depth, and volatility: evidence from the Stock Exchange ofHong Kong. Journal of Finance 56, 769-790.

Al-Suhaibani, M., Kryzanowski, L., 2000. An exploratory analysis of the order book, and order flow andexecution on the Saudi Stock Market. Journal of Banking and Finance 24, 1323-1357.

Anand, a., Chakravarty, S., Martell, T., 2005. Empirical evidence on the evolution of liquidity: choice of marketversus limit orders by informed and uninformed traders. Journal of Financial Markets 8, 289-309.

Angel, J.J., 1994. Limit versus market orders. Unpublished working paper, Georgetown Universisty, School ofBusiness Administration.

Biais, B., Hillion, P., Spatt, C., 1995. An empirical analysis of the limit order book and the order flow in theParis Bourse. Journal of Finance 50, 1655-1689.

Bloomfield, R., O’Hara, M., Saar, G., 2005. The ‘Make or Take’decision in an electronic market: evidence onthe evolution of liquidity. Journal of Financial Economics 75, 165-199.

Chakravarty, S., Holden, C., 1995. An integrated model of market and limit orders. Journal of FinancialIntermediation 4, 213-241.

Chan, Y. C., 2005. Price movement effects on the state of the electronic limit-order book. The Financial Review40, 195-221.

Copeland, T., and D. Galai, 1983. Information effects on the bid-ask spreads. Journal of Finance 38, 1457-69.

Coppejans, M., Domowitz, I., 2002. An empirical analysis of trades, orders, and cancellations in a limit ordermarket, Discussion paper, Duke University.

Ellul, A., Holden, C. W., Jain, P., Jennings, R., 2003. Determinants of order choice on the New York StockExchange, Kelley School of Business working paper.

Foucault, T., 1999. Order flow composition and trading costs in a dynamic order driven market. Journal ofFinancial Markets 2, 99-134.

Foucault, T., Kadan, O., Kandel, E., 2005. Limit order book as a market for liquidity. Review of FinancialStudies, 18, 1171-1217,

Glosten, L.R., 1994. Is the electronic open limit order book inevitable? Journal of Finance, 49, 1127-1161.

Glosten, L.R., Milgrom, P.R., 1985. Bid, ask and transaction prices in a specialist market with heterogeneouslyinformed traders. Journal of Financial Economics 14, 71-100.

Hall, A. D., Hautsch, N., 2006. Order aggressiveness and order book dynamics. Empirical Economics 30,973-1005.

Handa, P., Schwartz,, R., 1996. Limit order trading, Journal of Finance 51, 1835-1861.

Handa, P., Schwartz, R., Tiwari, A., 2003. Quote setting and price formation in an order driven market. Journalof Financial Markets 6, 461-489.

Harris, M., Raviv, A., 1993. Differences of opinion make a horse race. Review of Financial Studies 6, 473-506.

Harris, L, 1998. Optimal dynamic order submission strategies in some stylized trading problems. FinancialMarkets, Institutions, and Instruments 7, 26-74.

Hasbrouck, J., 1991a. Measuring the information content of stock trades. Journal of Finance 46, 179-207.

Hasbrouck, J., 1991b. The summary informativeness of stock trades: an econometric analysis. Review ofFinancial Studies 4, 571-595.

Holden, C. W., Subrahmanyam, A., 1992. Long-lived private information and imperfect competition. Journal ofFinance 47, 247-270.

Kaniel, R., Liu, H., 2004. So what orders do informed traders use? Working paper, Duke University.

Lo, A., MacKinlay, C., Zhang, J., 2002. Econometric models of limit-order executions. Journal of FinancialEconomics 65, 31-71.

Nevmyvaka, Y., Kearns, M., Papandreou, A., Sycara, K. P., 2005. Electronic trading in order-driven markets:

25

efficient execution, Proceedings of the Seventh IEEE International Conference on E-Commerce Technology(CEC’05) - 00, 190-197.

Parlour, C., 1998. Price dynamics in limit order markets. Review of Financial Studies 11, 789-816.

Pascual, R., Veredas, D., 2004. What pieces of limit order book information are informative?, Working PaperSeries 2004, FIN-04-004, Leonard N. Stern School of Business, New York University.

Ranaldo. A., 2004. Order aggressiveness in limit order book markets, Journal of Financial Markets 7, 53-74.

Rock, K., 1990. The specialist’s order book and price anomalies. Unpublished working paper, HarvardUniversity, Graduate School of Business.

Seppi, D., 1997. Liquidity provision with limit orders and s strategic specialist, Review of Financial Studies 10,103-150.

Wald, J. K., Horrigan, H. T., 2005. Optimal limit order choice. Journal of Business 78, 597-619.

Wang M.C,Lon,Zu L.P. ,Kuo C.J.,(2008a). The state of the electronic limit order book, order aggressiveness andprice formation, Asia-Pacific Journal of Financial Studies 37, 2, 245-296.

Wang M.C, Lon, Zu L.P., Kuo C.J. (2008b). Risk aversion, order strategy, and price formation , AppliedEconomics40, 1-14.

26

Appendix A:Uninformed buyer’sexpected utility function, eq.(12)：

, ,{ }

max ( )Uh t Uh TArg E u W

, 1 , , ,1 exp (1 )T bR P

m i Uh T b T Uh t Uh T Uh Tk p v W P R f v I dv

+ , 1 , , ,1 exp (1 )m u Uh T b T Uh t Uh T Uh Tk p v W P R f v U dv

+ , 1 , , ,1 exp (1 )m l Uh T b T Uh t Uh T Uh Tk p v W P R f v L dv

+ , 1 , , ,(1 ) 1 exp ) (1 )aP R

m i Uh T a b Uh t Uh T Uh Tk p v W P R P r f v I dv

+ , 1 , , ,(1 ) 1 exp ) (1 )m u Uh T a b Uh t Uh T Uh Tk p v W P R P r f v U dv

= 2

,, 1 ,22

11 exp (1 ) exp

22

Uh T Hm Uh T b Uh T

UhUh

vk v W P R dv

- 2

,, 1 ,22

11 exp (1 ) exp

22b

Uh T Hm i Uh T b Uh TP R

UhUh

vk p v W P R dv

+ 2

,, 1 ,22

1(1 ) 1 exp ) (1 ) exp

22

Uh T Hm Uh T a b Uh T

UhUh

vk v W P R P r dv

- 2

,, 1 ,22

1(1 ) 1 exp ) (1 ) exp

22a

Uh T Hm i Uh T a b Uh TP R

UhUh

vk p v W P R P r dv

27

= 2

2 2 21[1 (1 ( , ))] exp (1 ) [1 (1 ( , ))]

2b b

Uhm i P R H Uh H b i P R H Uh Uhk p Z W P R p Z

+ 2 2

2 2 21(1 )[1 (1 ( , ))] exp ) (1 ) [1 (1 ( , ))]

2a a

Uhm i P R H Uh H a b i P R H Uh Uhk p Z W P R P r p Z

where

222 2

22

222 2

22

1( , )

22

1( , )]

22

b

b

a

a

p R H Uhp R H Uh Uh

UhUh

p R H Uhp R H Uh Uh

UhUh

vZ e d

vZ e d

Then F.O.C. is

2

2 21exp (1 ) [1 (1 ( , ))] ]

2 b

Uhm H b i P R H Uh Uh bk W P R p Z P R

+ 2 2

2 2 21(1 ) exp ) (1 ) [1 (1 ( , ))]( )

2 a

Uhm H a b i P R H Uh Uh H Uh a bk W P R P r p Z P R P r

=0

=0Therefore,

2 2

2 2 2

2 2

(1 )exp [(1 )( ) ( ) ]

2

(1 )[1 (1 ( , ))] ( )]

[1 (1 ( , ))] ]b

a

UhH b a

m i P R H Uh Uh H Uh a b

m i P R H Uh Uh b

P P R

k p Z P R P r

k p Z P R

28

Appendix B:Informed buyer’s expected utility function,eq.(17)：

, ,{ }

max ( )Ih t Ih TArg E u W

2 , 1 , , ,1 exp (1 )m u Ih T b T Ih t Ih T Ih TRk p v W P R f v U dv

2 , 1 , , ,1 exp (1 )m l Ih T b T Ih t Ih T Ih TRk p v W P R f v L dv

+ , 1 , , ,(1 ) 1 exp ) (1 )a

m Ih T a b Ih t Ih T Ih TP Rk v W P R P r f v U dv

= 2 2

, ,, 1 ,2 22 2

1 1exp exp (1 ) exp

2 22 2

Ih T H Ih T Hm Ih T b T Ih TR R

Ih IhIh Ih

v vk v W P R dv

+ 2 2

, ,, , 1 ,2 22 2

1 1(1 ) exp exp ) (1 ) exp

2 22 2A

Ih T H Ih T Hm Ih T Ih T a b Ih TP R R

Ih IhIh Ih

v vk dv v W P R P r dv

= 2

2 2 21[1 ( , )] exp (1 ) [1 ( , )]

2Ih

m R H Ih H b R H Ih Ihk Z W P R Z

+ 2 2

2 2 21(1 )[1 ( , )] exp ) (1 ) [1 ( , )]

2a a

Ihm P R H Ih H a b P R H Ih Ihk Z W P R P r Z

where

29

222 2

22

222 2

22

1( , )

22

1( , )]

22

a

a

P R H IhP R H Ih Ih

IhIh

R H IhR H Ih Ih

IhIh

vZ e d

vZ e d

Then F.O.C. is

2

2 21[exp (1 ) [1 ( , )]

2Ih

m H b R H Ih Ih bk W P R Z P R

+ 2 2

2 2 21(1 )[exp ) (1 ) [1 ( , )]( )

2 a

Ihm H a b P R H Ih Ih H Ih a bk W P R P r Z P R P r

=0

Therefore,

2 2

2 2 2

2 2

(1 )exp [(1 )( ) ( ) ]

2

(1 )[(1 ( , ))] ( )]

[(1 ( , ))] ]a

IhH b a

m P R H Ih Ih H Ih a b

m R H Ih Ih b

P P R

k Z P R P r

k Z P R

30

Appendix C:

Uninformed seller’s expected utility function ,eq.(13) is：

, ,ˆ{ }

max ( )Ul t Ul TArg E u W

, 1 , , ,ˆ ˆ1 exp (1 )M I Ul T a T Ul t Ul T Ul TR

k p v W P R f v I dv

+ , 1 , , ,ˆ ˆ1 exp (1 )M U Ul T a T Ul t Ul T Ul Tk p v W P R f v U dv

+ , 1 , , ,ˆ ˆ1 exp (1 )M L Ul T a T Ul t Ul T Ul Tk p v W P R f v L dv

+ , 1 , , ,ˆ ˆ ˆ(1 ) 1 exp ( ) (1 )

bM I Ul T b a Ul t Ul T Ul TP R

k p v W P R P r f v I dv

+ , 1 , , ,ˆ ˆ ˆ(1 ) 1 exp ( ) (1 )M U Ul T b a Ul t Ul T Ul Tk p v W P R P r f v U dv

= , 1 , , ,ˆ ˆ1 exp (1 )M Ul T a Ul t Ul T Ul Tk v W P R f v dv

- , 1 , , ,ˆ ˆ1 exp (1 )

R

M I Ul T a Ul t Ul T Ul Tk p v W P R f v I dv

+ , 1 , , ,ˆ ˆ ˆ(1 ) 1 exp ) (1 )M Ul T b a Ul t Ul T Ul Tk v W P R P r f v dv

- , 1 , , ,ˆ ˆ ˆ(1 ) 1 exp ) (1 )

bP R

M I Ul T b a Ul t Ul T Ul Tk p v W P R P r f v I dv

= 2

2,

1 ,22

1ˆ ˆ[1 exp (1 ) exp2 22

Ul T LUlM L a Ul T

UlUl

vk W P R dv

31

- 2

2 2 21ˆ ˆ[ ( , ) exp (1 ) ( , )

2Ul

M I R L Ul L a R L Ul Ulk p Z W P R Z

+ 2

22 2,

1 ,22

ˆˆ 1ˆ ˆ ˆ(1 )[1 exp ( ) (1 ) exp2 22

Ul T L UlUlM L b a Ul T

UlUl

vk W P R P r dv

- 2 2

2 2 21

ˆˆ ˆ ˆ ˆ(1 ) [ ( , ) exp ( ) (1 ) ( , )

2b b

UlM I P R L Ul L b a P R L Ul Ulk p Z W P R P r Z

where

222 2

22

22

2 222

1( , )

22

ˆ1ˆ( , )]22

b

b

R L UlR L Ul Ul

UlUl

p R L Ul

p R L Ul UlUlUl

vZ e d

vZ e d

Then F.O.C. is

= 2

2 21ˆ ˆ(1 ( , ))[exp (1 )

2Ul

M I R L Ul Ul L a ak p Z W P R P R

+ 2 2

2 2 21

ˆˆ ˆ ˆ ˆ ˆ ˆ(1 )(1 ( , ))[exp ) (1 ) ( )( )

2b

UlM I P R L Ul Ul L b a L Ul b ak p Z W P R P r P R P r

Therefore,

2 2

2 2

2 2 2

ˆ(1 ) ˆ ˆexp [ ( ) (1 )( )]2

(1 ( , ))[ˆ ˆ(1 )(1 ( , ))( )[ ( ) ]

b

Ulb a L

M I R L Ul Ul a T

M I P R L Ul Ul L Ul b a b

P R P

k p Z P R

k p Z P P P r

32

Appendix D:

Informed seller’s expected utility function,eq.(21) is：

, ,ˆ{ }

max ( )Il t Il TArg E u W

2 , 1 , , ,ˆ ˆ1 exp (1 )R

M U Il T a T Il t Il T Il Tk p v W P R f v U dv

2 , 1 , , ,ˆ ˆ1 exp (1 )R

M L Il T a T Il t Il T Il Tk p v W P R f v L dv

+ 2 , 1 , , ,ˆ ˆ ˆ(1 ) 1 exp ( ) (1 )bP R

M Il T b a Il t Il T Il Tk v W P R P r f v U dv

= 2 , 1 , , ,ˆ ˆ1 exp (1 )R

M Il T a T Il t Il T Il Tk v W P R f v U dv

+ 2 , 1 , , ,ˆ ˆ ˆ(1 ) 1 exp ( ) (1 )bP R

M Il T b a Il t Il T Il Tk v W P R P r f v U dv

= 2 2

, ,, 2 , 1 ,2 22 2

1 1ˆ ˆexp exp (1 ) exp2 22 2

R RIl T L Il T LM Il T Il T a Il T

Il IlIl Il

v vk dv v W P R dv

+ 2 2

, ,, 2 , 1 ,2 22 2

1 1ˆ ˆ ˆ(1 ) exp exp ( ) (1 ) exp2 22 2

b bP R P RIl T L Il T LM Il T Il T b a Il T

Il IlIl Il

v vk dv v W P R P r dv

= 22 22

,,, 2 1 ,2 22 2

1 1ˆ ˆexp exp (1 ) exp2 2 22 2

R R Il T L IlIl T L IlM Il T L a Il T

Il IlIl Il

vvk dv W P R dv

+

33

22 22 2

,,, 2 1 ,2 22 2

ˆˆ1 1ˆ ˆ ˆ(1 ) exp exp ) (1 ) exp2 2 22 2

b bP R P R Il T L IlIl T L IlM Il T L b a Il T

Il IlIl Il

vvk dv W P R P r dv

= 2

2 2 222 1 2ˆ ˆ[ ( , ) exp (1 ) ( , )

2Il

M R L Il L a R L Il Ilk Z W P R Z

+2 2

2 2 222 1 2

ˆˆ ˆ ˆ ˆ(1 )( ( , ) exp ( ) (1 ) ( ( , )

2b b

IlM P R L Il L b a P R L Il Ilk Z W P R P r Z

Where

222 2

22

222 2

22

1( , )

22

ˆ1ˆ( , )]22

P RB

B

R L IlR L Il Il

IlIl

L IlP R L Il Il

IlIl

vZ e d

vZ e d

Then F.O.C. is

2

2 222 1 2 2ˆ ˆ[exp (1 ) ( , )

2Il

M L a R L Il Il ak W P R Z P R

+2 2

2 2 222 1 2 2 2

ˆˆ ˆ ˆ ˆ ˆ(1 )exp ( ) (1 ) ( ( , )

2 b

IlM L b a P R L Il Il L Il b ak W P R P r Z P R P r

=0

Therefore,

2 2

2 2

2 2 2

ˆ(1 ) ˆ ˆexp [ ( ) (1 )( )]2

[(1 ( , ))]ˆ(1 )[(1 ( , ))][ ( ) ]

b

Ilb a L

M R L Il Il a

M p R L Il Il L Il b a b

P R P

k Z P R

k Z P P P r

34

Appendix E:For eq 26,the results of partial derivative are listed as follows.

/F A aP bP 2 2( ) a bP P H

1Y

G

2Y

H

21 2, , , , , , ,a b a b HF Y G Y H A P P P P

It is not difficult to obtain the sign condition described above, therefore we omit the proof.

Appendix F:For eq 32,the results of partial derivative are listed as follows.

/F A aP bP 2 2( ) a bP P L

1y

g

2y

h

21 2, , , , , , ,a b a b LF y g y h A P P P P

It is not difficult to obtain the sign condition described above, therefore we omit the proof.