Dark Pool Trading Strategies 2011

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Dark Pool Trading Strategies

Sabrina Butiy Barbara Rindiz

Ingrid M. Werner

x

June, 2011

We thank Albert Menkveld and Dan Li for useful comments as well as participants at 6th Early Career

Women in Finance Conference (2010), 4th IIROC DeGroote Conference on Market Structure and MarketIntegrity (2010), Northern Finance Association Meeting (2010), and seminar participants at Bocconi Univer-sity and Toronto University. The usual disclaimer applies. We acknowledge …nancial support from IGIER,Bocconi University.

yUniversity of Toronto, Rotman School of ManagementzBocconi University and IGIERxFisher College of Business, Ohio State University.

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Dark Pool Trading Strategies

Abstract

We model a dynamic …nancial market where traders submit orders either to a limit orderbook (LOB) or to a Dark Pool (DP). We show that there is a positive liquidity externalityin the DP, that orders migrate from the LOB to the DP, but that overall trading volumeincreases when a DP is introduced. We also demonstrate that DP market share is higherwhen LOB depth is high, when LOB spread is narrow, when the tick size is large and whentraders seek protection from price impact. Further, while inside quoted depth in the LOBalways decreases when a DP is introduced, quoted spreads can narrow for liquid stocks andwiden for illiquid ones. We also show that traders’ interaction with both LOB and DPgenerates interesting systematic patterns in order ‡ow: di¤erently from Parlour (1998), theprobability of a continuation is greater than that of a reversal only for liquid stocks. In

addition, when depth decreases on one side of LOB, liquidity is drained from DP. When aDP is added to a LOB, total welfare as well as institutional traders’ welfare increase butonly for liquid stocks; retail traders’ welfare instead always decreases. Finally, when ‡ashorders provide select traders with information about the state of the DP, we show that moreorders migrate from the LOB to the DP, and DP welfare e¤ects are enhanced.

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1 Introduction

According to the U.S. Securities and Exchange Commission (SEC), Dark Pools (DP) are

Alternative Trading Systems (ATS) that do not provide their best-priced orders for inclusionin the consolidated quotation data. DP o¤er trading services to institutional investors thattry to trade in size while minimizing adverse price impact. While undisplayed liquidity hasalways been a feature of U.S. equity markets, it is only recently that DP have been singledout for regulatory scrutiny. In 2009, the SEC proposed DP-related rule changes rangingfrom a ban of ‡ash orders to increased pre- and post-trade transparency for DP venues.Moreover, the recent SEC 2010 Concept Release on Equity Market Structure shows concernson the e¤ect of undisplayed liquidity on market quality as well as on fair access to sourcesof undisplayed liquidity.

Unfortunately, there is to date very limited academic research that sheds light on these

issues. Existing models focus on the comparison between a dealer market and a crossingnetwork (e.g. Degryse, Van Achter and Wuyts, 2009), thus overlooking the features thatdrive the strategic interaction of DP with limit order books (LOB). We extend this literatureby building a theoretical model of a dynamic limit order market where traders can chooseto submit orders either to the fully transparent LOB or to a DP. We derive the optimaldynamic trading strategies and characterize the resulting market equilibrium. Speci…cally,we show how stock liquidity, volatility, tick size and price pressure a¤ect DP market share.We also demonstrate how the introduction of a DP a¤ects overall trading volume and LOBmeasures of market quality. Finally, in an extension of our model, we show how ‡ash ordersa¤ect DP market share.

There are over thirty active DP in U.S. equity markets according to the SEC. A growing

number of DP also operates in European equity markets. DP are characterized by limited orno pre-trade transparency, anonymity and derivative (almost exclusively mid-quote) pricing.However, they di¤er in terms of whether or not they attract order ‡ow through IndicationsOf Interest (IOI)1 and whether or not they allow interaction with proprietary and black boxorder ‡ow. DP report their executed trades in the consolidated trade data, but the tradereports were until very recently not required to identify the ATS that executed the trade. Asa result, it is di¢cult to accurately measure DP trading activity. Recent estimates suggestthat DP represent over 10% of matched volume (Rosenblatt Securities , December 2009).As illustrated in Figure 1, there are four broad categories of DP, namely Public CrossingNetworks, Internalization Pools, Exchange-Based Pools and Consortium-Based Pools.2

[Insert Figure 1 here]

1 IOI are sales messages re‡ecting an indication of interest to either buy or sell securities. They cancontain security names, prices and order size.

2 Exchanges o¤er dark liquidity facilities that represent another 4% of matched volume (Rosenblatt Secu-

rities , December 2009).

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As mentioned above, there are several concerns associated with DP growth. A mainconcern of the SEC relates to the possible migration of volume from transparent to darkmarkets, and hence to the e¤ect of DP trading on the execution quality of those retail andinstitutional investors who display their orders in the lit markets. Another relevant concernof regulatory authorities is the fair access to DP liquidity. While there are several aspects of the fair access issue, the problem that the SEC has focused on in their rule making is IOImessages. Indeed "actionable IOI" messages3 work as public quotes with implicit pricingand, by creating a leakage of privileged information to select investors, they can unfairlydiscriminate against public investors. Hence on October 21, 2009 SEC Chairman Schapironoted that “DP now represent a signi…cant source of liquidity in U.S. stocks”, creating a“two-tiered market”, and announced a SEC proposal for DP regulatory change.4

We use our model of a dynamic limit order market with a DP to shed light on theseconcerns raised by the SEC. We …rst determine factors that drive DP market share. We …nd

that there is a positive liquidity externality in the DP so that DP orders beget more DPorders. In our model, orders migrate from the LOB to the DP, but overall trading volumecan actually increase when a DP is introduced. We also demonstrate that DP market shareis higher for stocks with higher inside order book depth and for stocks with narrow orderbook spreads. Intuitively, this can be explained as follows. Traders optimally trade o¤ theexecution uncertainty and the midquote price in the DP against the trading opportunities inthe LOB. For stocks with larger depth at the inside or narrower spread, an order submittedto the LOB has to be more aggressive to gain priority over existing orders in the order book.As a result, the alternative of a midquote execution in the DP becomes relatively moreattractive. We also demonstrate that DP market share is higher when the tick size is larger.This follows since the wider inside spread makes market orders more expensive and hence

DP orders a more attractive alternative. Our model shows that traders use DP orders toreduce the price impact of their large orders. In particular we prove that when large ordersgenerate price pressure, traders either reduce their order size or they resort to DP orders.

We also use our model to gain insights on the e¤ect of DP trading activity on LOBmarket quality. We …nd that inside quoted depth and volume in the LOB always decreaseswhen a DP is introduced because orders migrate from the LOB to the DP. However, total

3 According to the SEC (2009), IOI messages are “actionable” if they explicitly or implicitly conveyinformation on: the security’s name, the side of the order, a price that is equal or better than the NBBO,and a size that is at least equal to one round lot. IOI are typically targeted to speci…c institutional customersand not broadcasted more widely.

4 More precisely, the SEC proposal addresses 3 issues: 1) actionable IOI: amendment of the de…nition of “bid” or “o¤er” in Rule 600(b)(8) of Regulation NMS to apply explicitly to actionable IOI, and exclusionof “size-discovery IOI”, i.e. actionable who are reasonably believed to represent current contra-side tradinginterest of at least $200,000; 2) lower substantially the trading volume threshold (from the current 5% to0.25%) in Rule 301(b) of Regulation ATS that triggers the obligation for ATS to display their best-pricedorders in the consolidated quotation data; 3) require real-time disclosure of the identity of ATS on the reportsof their executed trades.

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volume in the LOB and DP combined actually increases. The results also show that theintroduction of a DP is associated with tighter quoted spreads for liquid stocks but widerquoted spreads for illiquid stocks. The explanation is subtle and takes into account both themigration of orders to the DP and the switch between limit and market orders in the LOB.When the initial LOB depth is high, both market and limit orders switch to the DP, leavingspreads tight in the LOB. By contrast, when the initial LOB depth is low, competitionfrom DP decreases limit orders execution probability and hence increases the use of marketorders, thus widening the spread. Furthermore, we analyze the dynamic pattern in order‡ow: di¤erently from Parlour (1998), only liquid stocks exhibit a probability of continuationwhich is higher than that of a reversal, and the opposite holds for illiquid stocks. Also, we…nd an externality originating from the interaction of a LOB with a DP, whereby the latteracts as a liquidity bu¤er. In terms of traders’ welfare we show that, when a DP is added to aLOB, total welfare and institutional traders’ welfare increase for liquid stocks, and decrease

for illiquid ones. Retail traders’ welfare instead always decreases.Finally, we use our model to understand how introducing IOI messages such as ‡ash

orders a¤ects the equilibrium. We model ‡ash orders as a mechanism that provides selecttraders with information about the state of the DP before they submit orders. In this setting,we show that more orders migrate from the LOB to the DP: the reason is that everyone knowsthat informed institutions will use the DP. This means that the execution probability of DPorders increases, which reinforces the already existing liquidity externality for the DP. As aconsequence, ‡ash orders have the overall e¤ect of enhancing DP e¤ects on market qualityand traders’ welfare. Indeed, compared to the market without asymmetric information,private information on the state of the DP reduces the execution risk of DP trading, thusmaking market orders less competitive than DP orders. The result is an improvement of

both order book spread and depth, a reduction of LOB volumes but a further increase of total trading volume. Noticeably, we …nd that institutional traders bene…t of ‡ash orderswhereas retail traders bear extra losses.

The paper is organized as follows. Section 2 reviews the related literature. Section 3presents and discusses the general framework of the model, the benchmark cases with botha Limit Order Book (LOB) and a Dealer Market (DM), as well as the protocol with a DP.The equilibrium is derived in Section 4. Section 5 reports results on market quality, on thedynamic pattern in order ‡ow and on traders’ welfare. In Section 6, we extend the modelto include asymmetric information on the state of the DP. Section 7 discusses the model’sempirical implications and Section 8 summarizes the results. All proofs are in the Appendix.

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2 Literature on Dark Pools

Most of the existing theory on undisplayed liquidity focuses on the interaction between

crossing networks (CN) and dealer markets. Hendershott and Mendelson (2000) model theinteraction between a CN and a DM and show costs and bene…ts of order ‡ow fragmen-tation. Donges and Heinemann (2004) model intermarket competition as a coordinationgame among traders and investigate when a DM and a CN can coexist; Foster, Gervaisand Ramaswamy (2007) show that a volume-conditional order-crossing mechanism next toa DM market Pareto improves the welfare of additional traders. The model we proposedi¤ers from these as it considers the interaction between a LOB and a DP rather than aDM and a CN; furthermore, it focuses on the dynamic, rather than static, order submissionstrategies of traders. More recently, Ye (2009) uses Kyle’s model to …nd the insider’s optimalstrategic use of a DP and to show that DP harm price discovery especially for stocks with

high volatility; however Ye assumes that only the insider can strategically opt to trade inthe DP and he models uninformed traders as noise traders. An opposite result on pricediscovery is obtained by Zhu (2011) who models the interaction among insiders, constrainedand unconstrained noise traders by using a Glosten and Milgrom type framework. Kratzand Schoeneborn (2009) prove existence and uniqueness of optimal trading strategies fora traders who can split orders between an exchange and a DP, but assume that the priceimpact and the DP’s liquidity are exogenously given.

The paper which is closest to ours is that by Degryse, Van Achter, and Wuyts (DVW,2009), who investigate the interaction of a CN and a DM and show that the compositionand dynamics of the order ‡ow on both systems depends on the level of transparency. Ourpaper di¤ers from DVW (2009) in that it considers the interaction between a LOB -rather

than a DM- and a DP: this means that in our model traders can use both market orders andlimit orders, and it is precisely the e¤ect of competition from limit orders that drives theresults we obtain compared to those of DVW. Modelling competition between a LOB and aDP entails considering a price grid where traders can choose to place their orders. We alsoextend our model to include asymmetric information on the state of the DP.

Other strands of the academic literature are relevant for understanding the role of DP intoday’s markets. DP are characterized by limited or no pre-trade transparency, and issues of anonymity and transparency are therefore important.5 DP also coexist with more transpar-ent venues, which suggests a link with the literature on multimarket trading.6 Finally, DP

5 See for example the theoretical works by Admati and P‡eiderer (1991), Baruch (2005), Fishman and

Longsta¤ (1992), Forster and George (1992), Madhavan (1995), Pagano and Röell (1996), Röell (1991),and Theissen (2001). Several empirical papers have recently explored the signi…cance of anonymity andtransparency in experimental settings and real data: Bloom…eld and O’Hara (1999, 2000), Boehmer, Saar,and Yu (2005), Flood, Huisman, Koedijk and Mahieu (1999) and Foucault, Moinas and Theissen (2007).

6 See among the others: Barclay, Hendershott, and McCormick (2003), Baruch, Karolyi and Lemmon(2007), Bennett and Wei (2006), Bessembinder and Kaufman (1997), Boehmer and Boehmer (2003), Easley,

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are currently competing with other dark options o¤ered by exchanges to market participants,which builds a connection with the recent literature on hidden orders.7

Empirical work on crossing networks is relatively limited. Gresse (2006) …nds thatPOSIT’s crossing network (CN) has a market share of one to two percent of share volumeand, by investigating the relation between the CN trading and the liquidity of the SEAQquote-driven segment of the LSE, …nds no negative e¤ect of the CN on the dealership market.Gresse (2006) results show also that there isn’t any signi…cant increase in adverse selectionor inventory risk, but rather a spread decrease due to increased competition and risk sharing.Conrad, Johnson, and Wahal (2003) …nd that realized execution costs are generally loweron alternative trading systems and that institutional orders sent to traditional brokers havehigher execution costs than those executed in the CN. Naes and Odegaard (2006) provideevidence that orders from large institutional investors have lower realized execution costsfor the component of the orders sent to the CN, but higher costs of delay if one considers

the entire orders and includes the component sent to standard exchanges. Fong, Madhavanand Swan (2004) …nd no evidence that competition from the upstairs market and the CNhas an adverse e¤ect on the limit order book of the Australian Stock Exchange. To ourknowledge, there is still limited empirical analysis of DP in the academic literature. Ready(2009) studies monthly volume by stock in three DP: Liquidnet, POSIT, and Pipeline duringthe period June 2005 to September 2007. The data suggests that these three DP executeroughly 2.5 percent of consolidated volume (third quarter 2007) in stocks where they wereactive during a month, but only 1 percent of market consolidated volume. Moreover, he …ndsthat these three DP execute roughly 20 percent of “potential institutional volume” de…nedas the minimum of quarterly buying and selling activity by institutions estimated using 13F…lings. While his results are preliminary, he …nds that DP execute most of their volume

in liquid stocks (low spreads, high share volume), but they execute the smallest fraction of share of volume in those same stocks. Buti, Rindi and Werner (2010) examine a uniquedataset on dark pool activity for a large cross section of US securities and …nd that liquidstocks are characterized by more dark pool activity. They also …nd that dark pool volumeincreases for stocks with narrow quoted spreads and high inside bid depth suggesting that ahigher degree of competition in the limit order book enhances DP activity.

Kiefer, and O’Hara (1996), Goldstein, Shkilko, Van Ness and Van Ness (2008), Karolyi (2006), Lee (1993),Nguyen, Van Ness, and Van Ness (2007), Pagano (1989), Reiss and Werner (2004) and Subrahmanyam(1997).

7 There is little theoretical work on hidden orders: Buti and Rindi (2011), Esser and Mönch (2007)and Moinas (2006). The empirical literature is instead rather extensive: Bessembinder, Panayides andVenkataraman (2009), De Winne and D’Hondt (2007), Frey and Sandas (2008), Hasbrouck and Saar (2004),Pardo and Pascual (2006) and Tuttle (2006).

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3 The Model

In this Section we present a model for three di¤erent market organizations. We start with a

limit order book with both retail and institutional traders and use it as a benchmark model.We then add a Dark Pool which allows us to consider a market organization where traders canchoose between the two platforms. Finally, we study the market structure formed by both adealership market and a DP, that has extensively been modelled by previous literature (e.g.Degryse, Van Achter, and Wuyts, 2009). We …nd remarkable di¤erences when we comparethis market structure with the LOB plus DP mechanism.

M arket Structure

We consider a discrete time protocol that, as in Parlour (1998), features a limit orderbook for a security, which pays v at each period and is assumed constant through the tradingperiods. Trading occurs during a day that is divided into T periods: t = 1;:::;T . In eachperiod t a new risk neutral trader arrives who can be with equal probability either a largeinstitutional trader or a small retail trader. Large traders can trade j = [0; 2] shares, whereassmall traders can only trade up to 1 share at a time. Upon arrival at the market the traderselects both a trading venue and an order type, and his optimal trading strategy cannot bemodi…ed thereafter: small traders can only trade in the LOB, while large traders can chooseto trade either in the LOB or in the DP. Traders’ personal evaluation of the asset, t; isdrawn from a uniform distribution with support [0; 2]: traders with a high value of areimpatient to buy the asset, while traders who arrive at the market with a low value theasset very little and therefore are impatient to sell it; traders with a next to 1 are patientas their evaluation of the asset is close to the common value.

The LOB is characterized by a set of four prices and associated quantities, denoted as pB

i &q Bi ; pAi &q Ai

, where A (B) indicates the ask (bid) side of the market and i = f1; 2g the

level on the price grid. Hence, prices are de…ned relative to the common value of the asset,v:

pAi = v + i

pBi = v i

where is the minimum price increment that traders are allowed to quote over the existingprice, and hence it is the minimum spread that can prevail on the LOB. The associatedquantities denote the number of shares that are available at that price. Following Parlour

(1998) and Seppi (1997), we assume that a trading crowd absorbs whatever amount of therisky asset is demanded or o¤ered at pA2 and pB

2 . Hence at the second level of the book depthis unlimited and traders can only demand liquidity, whereas the number of shares availableat pA

1 ( pB1 ) forms the state of the book that characterizes time t and is de…ned as bt = [q A1 q B1 ]:

The DP operates next to the LOB; it allows market participants to enter unpricedorders to buy or sell the asset, and it is organized as a crossing network. In this trading

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venue orders are crossed at the end of time T at the spread midquote prevailing on the LOBin that period, pM id: The novelty of the DP compared to the standard crossing networks,however, is that traders have no access to any information regarding the orders previouslysubmitted by the other market participants to the DP. It follows that they can only infer itsdepth by monitoring the LOB. If a trader submits an order to the DP, he will have this orderexecuted provided that there will be su¢cient depth to match it. As will be discussed morein detail below, we consider the last three periods of the trading game and we assume thatat T 2 agents assign equal probabilities to the following three states of the DP’s depth:

DP T 2 =

8><

>:

+6 with prob = 13

0 with prob = 13

6 with prob = 13

(1)

This means that at time T 2 traders believe that either the DP is empty, or that it isfull on one or the other side of the market.8 We also assume that traders strictly monitor thebook and that when they do not observe any market or limit orders, they Bayesian updatetheir expectations on the state of the DP. So traders at T can face a double uncertainty asthey have to make inference on the state of the DP at both T 2 and T 1.

It is straightforward to extend the model discussed so far to include a dealership market(DM) that competes with a DP. Technically, this is the case when the trading crowd is movedto the …rst level of the LOB and is precisely the market structure discussed in DVW (2009).

Order Submission Strategies

Upon arrival at the market each trader decides his optimal trading venue as well as the

optimal order type. To this end he compares the expected pro…ts from the di¤erent ordertypes he can choose. The feasibility and pro…tability of these orders depend on the traders’type ( t) as well as on both the state of the LOB (bt) and the state of the DP (gDP t) at thetime of the order submission.

For example, if an impatient large seller arriving at time t opts for the LOB, he willsubmit a market order of size j , '( j; pB

i ), that gives expected pro…ts equal to et ['( j; pB

i )] = j( pB

i tv) if it is completely executed at the best price available, pBi . If instead j exceeds the

depth associated with the best opposite price, the order '( j; pB) will walk down the LOB andin this case the trader’s expected payo¤ will be equal to: e

t ['(2; pB)] = ( pB1 + pB

2 ) 2 tv.9

A more patient large seller can instead choose to submit a limit sell order of size j to pA1 on

the LOB, '( j; p

A

1 ), so that his expected payo¤ is:8 As at T 2 there are only three periods left in the trading game, if for example six shares to sell are

already standing on the ask side of the DP (6), the execution probability of any other share posted to theask side is zero, the reason being that at the most two shares can be executed at each trading round.

9 Clearly in this case j = 2, and, as the order hits di¤erent prices, we do not use an index i for the levelof the book as we do for the other order types.

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et ['( j; pA

1 )] = E

(( pA

1 tv) j

Pwt+1=1

wt+1 Prwt+1

( pA1 jt+1)+

I t T P

l=t+2

( pA1 tv)

1PW =0

jW Pwl=1

wlPrwl

( pA1 jl)Pr(

l1Pm=t+1

wm = W jl1)

where t= fbt;v;gDP tg, Prwl( pA

1 jl) is the probability that wl shares will be executed at t = l,W is the number of shares executed up to t = l 1, and I t is an indicator function equal to0 for t = T 1 and 1 otherwise. Notice that when submitting a limit order at time t; thetrader will have to compute the probability that each unit will be executed from time t + 1to T :

The large seller can also decide to submit a j-order to sell to the DP that will be executedat the end of the trading game at the spread midpoint. This strategy, '( j; pM id), has thefollowing expected payo¤:

et ['( j; pM id)] = E [ j( pM id tv)Pr

j( pM id jT )]

where Pr j

( pM id jT ) is the probability that j shares to sell will be executed in the DP. Finally

a large seller can also decide not to trade so that his expected payo¤ will be equal to zero,e

t ['(0)] = 0. Specular strategies are available to a large buyer. Notice also that a smalltrader has access to analogous strategies, with the exception of DP orders.

4 Market for Liquidity

The model is solved under three speci…cations that correspond to three di¤erent marketstructures. First, we present a benchmark model that describes the working of a pure LOB;then we focus on the protocol with an LOB competing with a DP (LOB&DP), and wecompare the results obtained to the case where a DM, rather than a LOB, competes withthe DP (DM&DP). This analysis allows us to discuss the driving factors of DP trading,and …nally the e¤ects that the price impact generated by blocks can have on traders’ choicebetween a LOB and a DP.

4.1 Benchmark Model

We focus on a three-period trading game that ends at T : Figure 2 shows an example of theextensive form of the trading game: the market opens at T 2 with two units on the best bidand o¤er, bT 2 = [22]; then nature selects a small or a large trader with equal probability,who, in turn, chooses his order among the available strategies. Given the opening state of theLOB assumed in Figure 2, the equilibrium strategies for large traders include a limit orderat either the best ask or the best bid, or a market order that hits the limit order standing at

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the …rst level of the book. Still referring to the example presented in Figure 2, suppose thatnature selects at T 2 a small trader who is rather patient and decides to submit a limitorder at pA

1 , '(1; pA

1 ); then at T 1 the book will open with 3 units on pA

1 ; and if at T 1

nature still selects another small trader who decides not to trade, then the book will open atT unchanged. It follows that at T incoming traders will choose among market buy, marketsell and no trade. The reason why traders do not submit limit orders at time T is that theirexecution probability is zero as the market closes. At time T 1 and T 2, instead, traderscan submit limit orders as their execution probability can be positive.


At each trading round, the risk-neutral large trader will choose the optimal order sub-mission strategy, 'LT;t;bt

, which maximizes his expected pro…ts conditional on the state of the LOB, bt, and his type, t. A large trader (LT ) thus chooses:

max'LT;t;bt;j

et ['( j; pB

i ); '( j; pB); '( j; pA1 ); '( j; pA

i ); '( j; pA); '( j; pB1 ); '(0)] (2)

and the small trader (ST ) chooses:

max'ST;t;bt

et ['(1; pB

i ); '(1; pA1 ); '(1; pA

i ); '(1; pB1 ); '(0)] (3)

We …nd the solution of this game by backward induction and by assuming that thetick size, ; is equal to 0:1. We start from the end-nodes at time T and compare tradingpro…ts for both large and small traders. This allows us to determine the probability of the

equilibrium trading strategies at T that can be market orders, as well as no trading. We canhence calculate the execution probabilities of limit orders placed at T 1 that allow us tocompute the equilibrium order submission strategies in that period. Given the probability of market orders at T 1; we can …nally compute the equilibrium order submission strategiesat T 2.

The framework described so far can also be simpli…ed to analyze a pure dealershipmarket; this can be accomplished by moving the trading crowd to the …rst level of the book,thus allowing traders to only submit market orders or not to trade as in DVW (2009).

4.2 Intermarket Competition: LOB&DP vs. DM&DP

Once a DP is added to the LOB, large traders have the option to submit an order to buy orto sell to the DP, and, provided that there will be enough depth to match it, the order willbe executed at the end of time T . All else equal, the optimization problem now adds thisnew order type to the strategies of large traders.


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At each trading round the risk-neutral large trader chooses the optimal order submissionstrategy, '

LT;t;bt;gDP t, which maximizes his expected pro…ts conditional on the state of the

LOB, bt, his type, t;and the state of the DP, gDP t. A large trader thus chooses the orderthat leads to the largest pro…ts:

max'LT;t;bt;j;

gDP t

et ['( j; pB

i ); '( j; pB); '( j; pA1 ); '( j; pA

i ); '( j; pA); '( j; pB1 ); '( j; pM id); '(0)] (4)

Small traders still solve problem (3), however they will now condition their strategies notonly on their type and on the state of the LOB, but also on the state of the DP.If instead a DP is added to a DM, the optimization problem for large traders simpli…es to:

max'LT;t;j;

gDP t

e

t

['( j; pB

1 ); '( j; pA

1 ); '( j; pM id); '(0)] (5)

as in the DM&DP protocol traders cannot submit limit orders.


A relevant issue in market design is to establish whether by adding a new trading oppor-tunity to a limit order book more volume is created, or whether volume is simply divertedto the new trading venue. The results from this model show that when a DP is added toa LOB, volumes shift to the DP and there is no trade creation. Conversely, when a DPis added to a DM, it indeed induces some traders to enter the market. The latter result

replicates the case studied by DVW (2009). The intuition is rather simple: in the dealershipmarket some patient traders, who do not have the possibility to compete for the provisionof liquidity by using limit orders, refrain from trading and do not enter the market to avoidpaying the spread; however, when they are o¤ered the opportunity to submit orders to theDP, they take that option: these orders can gain positive pro…ts as their possible executiontakes place at the midquote rather than at the best bid-o¤er. In the LOB instead there is notsuch e¤ect as patient traders are allowed to submit limit orders. The following Propositionsummarizes the results obtained by comparing the two protocols.

Proposition 1 .

When a Dark Pool is added to a limit order book, it induces order migration to the dark

market. When a Dark Pool is added to a dealership market, it produces trade creation.Order migration is more intense for highly liquid stocks where competition for the pro-

vision of liquidity is strong;

Dark Pools generate a liquidity-externality e¤ect: as existing dark liquidity begets future

liquidity, it increases the execution probability of dark orders.

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Table 1 reports results on equilibrium trading strategies of large traders for b(T 2;T 1;T ) =[22]: Notice that at t = T traders cannot submit limit orders and in this case, where thebook opens with two shares at the inside spread, the LOB&DP framework converges to theDM&DP one. As traders cannot submit market orders for a size greater than two shares,the LOB is full and works like a DM where dealers o¤er unlimited liquidity at the BBO (asin DVW, 2009). In the previous periods instead traders can compete for the provision of liquidity by submitting limit orders to the LOB, and the role of these orders is crucial tounderstand the di¤erences between the LOB&DP and the DM&DP frameworks. Clearly, thelonger the time to the end of the trading game, the more relevant the role of limit orders, astheir execution probability increases; hence the comparison between the equilibrium tradingstrategies at T 2 and those at T is the most appropriate to capture the di¤erences betweenthe two models.

Results show that at T 2 by moving from the LOB to the LOB&DP, the probability

that traders submit limit orders decreases from :0314 to :0109 and large traders opt forDP orders with probability :0279 (Table 1); this means that there is no trade creation butonly order migration to the DP. The same comparative static exercise performed at time T ,when traders cannot submit limit orders, results in trade creation exactly as in DVW (2009).Actually at T , when a DP option is o¤ered to market participants, those traders who werenot willing to enter the market move to the DP with probability equal to :0375. However,as discussed above, at T the LOB converges to the DM and hence trade creation takes placeonly because in a DM traders cannot submit limit orders.

The overall e¤ect of intermarket competition also depends on the state of the LOB.Table 2 reports results obtained by assuming that at T 2 the LOB opens empty. Clearly,when the LOB is empty, there is more room for limit orders as traders can post at the top of

the queue and have their limit orders executed more quickly. Hence, when the LOB is emptyat T 2, competition from limit orders is so intense that crowds out the DP. In this case, byallowing traders to choose between a LOB and a DP, they opt for the former. However therelative probability of market to limit orders increases as, due to competition from the DP inthe following periods, limit orders’ execution probability decreases and hence market ordersbecome more pro…table. As a result, the overall e¤ect of DP competition is less intense forless liquid stocks where limit orders represent a more pro…table trading strategy than for adeep LOB. Hence for less liquid stocks trade migration to DP is less severe, whereas tradecreation from DM to DM&DP is more intense.

Finally Table 1 shows that traders’ perception of DP liquidity in‡uences the execution

probability of DP orders and hence their use. When at T 2 traders do not observe anychange in the LOB, they assume that either no trade occurred or that an order was submittedto the DP. As they perceive that liquidity is building in the DP, they update their estimateof the DP depth and assign a higher probability of execution to their DP orders; the resultis that they opt for DP more frequently. As an example, this e¤ect can be observed bycomparing the results for T 1 presented in Table 1 for the case “visT 2”, where traders

13



observe a change in the LOB at T 2, and “invT 2”, where instead traders observe nochange. In the latter case, they submit orders to the DP more intensively (:050) than whenthey have no uncertainty (:0379): Analogous results are shown for the DM&DP market wherethe probability of DP orders increases from :0379 to :0444 when traders do not observe anychange in the LOB at T 2. We can therefore conclude that the positive liquidity-externalitye¤ect produced by a DP intensi…es when traders perceive that DP volume is growing.

4.3 Dark Pool Drivers

We have shown that the state of the LOB a¤ects traders’ choice between disclosed and DPtrading. We now extend this analysis by discussing more in depth the main factors relatedto the state of the LOB, i.e. depth, spread and tick size, that a¤ect traders’ choice to submitorders to the dark market. The following Proposition summarizes the results.

Proposition 2 The probability that traders submit orders to the DP:

increases with market depth and the tick size, and

decreases when the inside spread widens.

Opposite results relative to depth and inside spread are obtained for a DM&DP.

To investigate the e¤ects of depth, spread and tick size on traders’ choice, we compare, aswe did before, the equilibrium strategies at T 2 and at T . Once again, the longer the timeto the end of the game, the higher the probability of limit order execution and the strongerthe e¤ect of limit order competition. Table 3 (Panel A) shows that at T 2 an increase indepth on the top of the book from [11] to [22] reduces competition from limit orders andincreases the probability that traders opt for the DP: '(2; pA

1 ) and '(2; pB1 ) decrease (from

:0832 to :0109) and, even though market orders increase from :4168 to :4612, traders now usethe DP. If instead the same comparative static exercise is performed at time T , where thereis no competition from limit orders and the LOB resembles a dealership market, we obtainthe same result as in DVW (2009). Indeed, Table 4 shows that when depth on the ask sideincreases from [11] to [21], market orders to buy increase from :4250 to :4625 and crowd outDP buy orders, which decrease from :0750 to :0375. The same results are obtained whendepth increases only on the bid side and on both sides of the market (from [11] to [12] and to

[22]). We conclude that when market participants can compete for the provision of liquidityby using limit orders and can also opt for DP orders, the deeper the limit order book, thelonger the queue for their limit orders -due to time priority-, and the greater the probabilitythat they opt for DP orders. In dealership markets competition from limit orders is absentand greater depth instead fosters traders’ aggressiveness thus increasing market orders tothe detriment of DP orders.

14



Results obtained from comparative statics on the inside spread con…rm those from mar-ket depth. For the LOB&DP framework, the more liquid the market, the more intensecompetition in limit orders, and the higher the probability that traders opt for the DP. Toisolate the e¤ect of a spread variation, one has to control for market depth: this can beachieved by comparing two states of the book that have enough liquidity at the BBO to ab-sorb large orders. Table 3 and 5 show results for both time T 2 and T : Starting again fromperiod T 2, when the inside spread increases, i.e. the state of the book changes from [22]to [00], the increased competition for liquidity provision crowds out DP orders, even thoughthe probability of market orders decreases. We thus …nd that the wider the inside spread,the more convenient are limit orders submitted at the top of the book, and the greater isthe probability that traders choose limit instead of DP orders. Opposite conclusions can bedrawn from the same simulation performed at time T for the DM&DP framework: Table 4shows that when the spread increases (from [22] to e.g. [00]) competition from market orders

decreases and, because at T there is no competition from limit orders, the probability thattraders opt for DP increases from :0375 to :1125.

Proposition 2 also informs us about the e¤ects of a change in the tick size on theprobability of DP orders: when the tick size increases traders become more willing to supplyliquidity. An example for the book [11] is shown in Table 3 where, following an increasein the tick size, market orders decrease, while limit and DP orders increase. The intuitionfor this result is that an increase in the tick size produces two e¤ects: it widens the insidespread, and hence makes market orders more expensive, and it increases the minimum pricechange, thus making it more convenient for traders to supply liquidity. The outcome is thatmore patient traders will opt for limit orders whereas less patient (but not so impatient)traders will choose to trade in the DP.

4.4 Price Impact, Price Pressure and Dark Pool Trading

A widespread view shared by market participants in the …nancial community is that largeinstitutional traders submit orders to a Dark Pool to reduce price impact. A price impactcan arise both when impatient traders submit a large order to the top of a LOB that isnot deep enough to absorb the order, and when a patient trader submits a limit order thatproduces a price pressure, thus temporarily moving the asset value against the trader’s order.Price impact resulting from trades has been extensively investigated: for example, Engle andPatton (2004) analyze the price impact of 100 NYSE stocks strati…ed by trade frequency

and …nd strong evidence of short-run price impact for trades initiated by both buyers andsellers.10 Price pressure11 arising from passive order placement through limit orders hasbeen recently explored by Hendershott and Menkveld (2010), who estimate the price impact

10 See also Hasbrouck (1991) and Dufour and Engle (2000).11 See Gabaix et al. (2006), Brunnermeier and Pedersen (2009) and Parlour and Seppi (2008).

15



arising from liquidity supply. They …nd a large daily transitory volatility in returns for stockslisted at the NYSE due to price pressure.

In our model what drives large institutional traders to operate in a DP is their wish tobuy or sell large blocks with the lowest price impact. Consider …rst impatient traders whoare concerned about the price impact that can be generated by a market order. These tradersface the standard trade-o¤ between price risk (i.e. bearing a price impact) and executionrisk. If they choose a market order, they will obtain immediate execution but will pay agreater price impact, which is increasing in the lack of depth available on the opposite side of the market. If instead traders opt for the DP, their order will be executed with a lower priceimpact at the spread midquote; however, the order execution will be uncertain and it willdepend on the state of the DP. Hence the trader will choose the DP only if the price impactof his order is large. Our model shows this e¤ect in period T when traders are naturallyimpatient due to the proximity of the end of the game. Table 5 shows that the impatient

trader will more probably opt for a DP order when depth on the other side of the marketis shallow and therefore his price impact is large: for example, when the book opens withonly 1 share on top of the ask side, bT =[12]; instead of 2 shares, bT =[22]; large traders useDP orders more intensively (with probability :0750 instead of :0375) as in the former casetheir order will move the price up to obtain execution.

We now extend our model to embed the temporary price impact that can be generatedby passive traders submitting limit orders. To this end we assume that large limit ordersproduce a short term price pressure that lasts for one period, as shown in Figure 5.


Suppose that a large seller arrives at the market at T 2 and submits a large limit orderat pA

1 ; following this submission, the asset value jumps down by 1 tick and the next periodthe market opens with vT 1 = v : Clearly at time T the temporary price e¤ect vanishesand the asset value jumps back to v: We consider two di¤erent speci…cations: a benchmarkmodel with no DP (LOB&PP) and a model that allows for DP trading (LOB&DP&PP).The results are summarized in the following Proposition.

Proposition 3 When large orders generate price pressure, traders either reduce the size of

their order, or, if available, switch to DP orders.

Notice that, when price pressure is introduced, the execution probability of a large limit

order decreases for two reasons. First, the initial order is now on the second level of thebook and hence further away from the asset value; second, it can be easily front-run in thefollowing period by an incoming trader posting a limit order at the now empty …rst levelof the book. The result is that traders switch to those order types that protect from priceimpact either because they are small in size, or because they are undisclosed. In Table 5one can indeed notice that moving from the standard LOB protocol to the speci…cation

16



with price pressure, traders reduce their price impact by switching from large to small limitorders. When instead we introduce price pressure in the model with a DP, traders actuallyminimize their price impact by submitting DP buy and sell orders with larger probability(from :0279 to :0316).

5 Market Quality, Systematic Pattern in Order Flow

and Welfare

So far we have shown how traders react when a DP is added to a LOB and concluded thatDP produces order migration. The next relevant issue is to investigate how the introductionof a DP a¤ects the quality of the LOB, and how it impacts the dynamic pattern in order‡ow and traders’ welfare.

5.1 Market Quality

To evaluate the e¤ect of DP trading on market quality we consider inside spread (S t) marketdepth (Dt) and volume (V t). We compute expected spread and depth in period t + 1 byweighing the value that characterizes a particular state of the book with the correspondingorder submission probabilities in the previous period t:

yet+1 =

Xa=ST;LT

Pr(a) E bt

Z 20

yt+1('

a;bt;gDP tj t) f ( t) d t

(6)

where yt+1 = fS t+1; Dt+1g and '

a;bt;gDP tis the optimal trading strategy of agent a, conditional

on bt and gDP t.We calculate the expected LOB volume in each period t in a similar way and adequately

weigh the market orders submitted to the LOB by both retail and institutional traders bytheir size:

V et =X

a=ST;LT

Pr(a) E bt

Z 20

q t('

a;bt;gDP tj t) f ( t) d t

(7)

where q t('

a;bt;gDP t) is the traded quantity which is a function of the agent’s type a, the state

of both the LOB and the DP. Proposition 4 summarizes the results.

Proposition 4 When a Dark Pool is added to a Limit Order Book, market quality changes

as follows:

market depth at the best bid-o¤er decreases;

17



the inside spread decreases when the LOB opens deep, the opposite holding when it

opens empty;

LOB volume decreases, whereas total volume increases.

We have previously proved that the DP can attract orders from the LOB; more precisely,Table 1 shows that when the book opens deep (bT 2 = [22]) by moving from the LOB tothe LOB&DP protocol, the probability of both limit and market orders decrease, as tradersswitch to the DP. Clearly the e¤ect of the order migration on liquidity and volume depends onthe proportion of limit vs market orders that leave the book. A reduction of the probabilitythat traders post limit orders to the LOB decreases the provision of liquidity and henceworsens market depth and inside spread; a reduction of the demand for liquidity, i.e. theprobability that traders submit market orders, instead, certainly decreases volume but canhave positive e¤ects on both depth and inside spread, as market orders subtract liquidityfrom the book.

Table 6 shows that when the book opens with two shares on both sides, bT 2 = [22];the introduction of the DP decreases average depth by 1:30% and volume by 2:39%, butit improves the inside spread by :41%: this means that the positive e¤ect on spread of thereduction of market orders more than outweighs the negative e¤ect of the reduction of limitorders. The two opposite e¤ects on liquidity can also be explained by the fact that if theopening book is already very deep at the inside spread, then the proportion of orders thatmove to the DP leaves the best bid-o¤er very tight.12 When instead the book opens emptyat T 2 or with only one share at the best bid-o¤er, all the three measures of liquidityworsen on average. For the case with an empty book, for example, Table 2 shows that the

introduction of a DP makes limit orders less attractive so that traders opt for market orders,and as a result the inside spread increases.

Overall Proposition 4 shows that by moving from the LOB to the LOB&DP, depthand volume decrease, whereas the e¤ect on the inside spread depends on the depth initiallyavailable at the top of the book. When the book is empty or has only 1 share available,then the migration makes the inside spread wider and the whole market quality deteriorates.When instead the book opens with 2 shares at the inside, then the e¤ect on the averageinside spread is positive and the overall e¤ect on liquidity is mixed. Finally, Table 6 showsthat the overall e¤ect of the introduction of a DP on total volume is positive: the sum of theLOB&DP and the DP volume is in fact systematically greater than the amount of volumetraded in the benchmark LOB.

12 Technically, when the book is deep and tight and market orders move to the DP, the probability thatthe spread remains small increases; the opposit happens when the initial spread is wide.

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5.2 Systematic Pattern in Order Flow

Traders’ strategic interaction with the two sides of the LOB and with the DP allows us to

draw conclusions on the systematic pattern of the order ‡ow, which are summarized in thefollowing Proposition.

Proposition 5 The following systematic pattern typi…es order ‡ows in the LOB and in the

LOB&DP market:

when the book is deep, the probability of a continuation is greater than that of a reversal,

whereas when the book is shallow the opposite holds;

the DP has a positive externality on the limit order book: if depth decreases on one

side, competition for limit orders increases and liquidity gets drained from the DP to

the LOB. Hence, volumes show a smaller decline in the LOB&DP protocol.

Parlour (1998) shows that the interaction of traders with the two sides of the bookentails a probability of a continuation greater than that of a reversal, and this is consistentwith Biais, Hillion and Spatt (1999). We …nd that this e¤ect only holds when the book isdeep, whereas it is not supported by the model when the top of the book is shallow andtraders have to walk up (or down) the book in search of execution. The di¤erence in theresults originates from the fact that in Parlour’s LOB the trading crowd is positioned at thetop of the book, whereas in our model there is a two-level price grid and the trading crowdis not posted at the …rst level, but rather at the highest (second) level. This means that inour model traders have to walk up (or down) the book when there is not enough liquidity

at the top. An example will help understanding why the need to walk up the book entails aprobability of a continuation smaller than that of a reversal.

Consider Table 7 where the equilibrium strategies at T 2 are reported for two di¤erentstates of both the LOB and the LOB&DP, bT 2 = [20] and bT 2 = [10]: Comparison betweenthese two books allows us to compute the equilibrium trading strategies of a buyer arrivingat the market at T 2 and facing a book either with 2 shares at the best ask, or withonly 1 share; the latter state of the book can occur if, for example, at time T 3 the bookopens with 2 units [20] and a market buy order arrives leaving the book with only 1 shareon the ask side. Consider …rst a small buyer who has to decide whether to submit a limitor a market order. The observed reduction of the depth on the opposite side of the bookinforms him that future sellers will rather post a limit order to sell than a market order tosell:13 Table 7 shows for example that moving from [20] to [10] the probability to observelimit sell orders increases in percentage by 1:950 and by :8120; respectively for the two caseswith and without a DP, and that the probability to observe market sell orders decreases

13 Here we refer to the average probability of limit orders and market orders submitted by both large andsmall traders.

19



by :0859 and :1207. This shift from market to limit sell orders implies that the probabilityof execution of any eventual limit order to buy decreases, thus inducing the small traderto submit more market than limit buy orders. As a result, the continuation probability of a small buy order becomes greater than that of a reversal. Indeed, Table 7 shows that,after observing a reduction of the depth at the best ask, the probability that a small traderat T 2 submits '(1; pA

1 ) increases by:0128 and :0151 respectively in the market with andwithout a DP.

Notice that for the small trader in both cases the top of the ask side of the book is deepenough to have a buy order executed without walking up the book. If instead we considerthe choice of a large buyer arriving at the market at T 2, we observe that in a book [10];despite the lower execution probability of a limit buy order, he will submit fewer rather thanmore market buy orders. The reason is that when the book changes from [20] to [10], thelarge trader will have to walk up the book to have his order executed, thus paying a higher

price. As the reversal e¤ect for large traders is stronger than the continuation e¤ect for smalltraders, the average probability of a continuation is smaller than that of a reversal for boththe LOB and the LOB&DP markets: :0739 and :0702: Clearly, if after the arrival of amarket buy order the …nal state of the book at T 2 were always deep enough even for alarge trader (e.g. moving from [30] to [20]), then the ‘Parlour e¤ect’ would still hold andthe probability of a continuation would be higher than that of a reversal.14 Conversely, if the …nal state were [00]; thus forcing even small traders to walk up the book in search of liquidity, then the probability of market orders to buy would decrease for a small trader too:this is evident by comparing Table 7 with Table 3 and noticing that for the LOB&DP casethe probability of a market buy order decreases to :2142.

Table 7 also shows that when depth decreases on one side of the book, e.g. from [20]

to [10]; trading volume decreases as both market orders to sell and market orders to buydecrease. However, one should notice that the decrease in volume is more contained for theLOB&DP framework: orders decrease in total by :1946 in the LOB, whereas they diminish by:1561 in the presence of a DP. When depth decreases on the ask side of the book, competitionfor '(2; pA

1 ) increases by 8:2881 in the LOB&DP as large traders move from the DP to thebook: this means that when the book needs liquidity to attract market orders, this is drainedfrom the DP, which functions like a liquidity bu¤er.

Proposition 5 o¤ers at least two empirical implications for the dynamic pattern of theorder ‡ow: …rst, the model predicts that liquid stocks should exhibit a probability of acontinuation which is higher than that of a reversal, whereas for illiquid stocks the opposite

should hold; second, the model foresees an externality originating from the coexistence of a limit order book with a DP. When market depth on the former decreases (increases), itcreates a liquidity injection (drain) from the DP to the limit order book.

14 For brevity we do not report these results here.

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5.3 Welfare

In this Section we present results for welfare, measured by the gains from trade of market

participants. Formally, welfare for a small (W t;ST ) and a large trader (W t;LT ) arriving at themarket at time t is de…ned as:

W t;a = E bt

Z 20

t('

a;t;bt;gDP t)f ( t) d t

(8)

The expected value of welfare at time t is computed over all the possible equilibriumstates of the book. We measure total expected welfare as the sum of all agents expectedgains from trade, which includes welfare of both institutional and retail traders:

W t = 12

W t;LT + 12

W t;ST (9)

Results are summarized in the next Proposition.

Proposition 6 The introduction of a DP on a limit order book a¤ects traders’ welfare. In

particular, when a DP is added to a LOB we …nd that:

total welfare and institutional traders’ welfare increases in liquid stocks whereas it de-

creases in illiquid stocks;

retail traders’ welfare always decreases.

We compute welfare for both the benchmark LOB and the LOB&DP market and report

the results obtained from comparisons across di¤erent books and over time (Table 8). Resultsfor time T , where limit orders play no role, refer to a dealership market, in the spirit of DVW(2009). As expected, total welfare increases with the liquidity of the opening book for boththe benchmark and the LOB&DP market. All traders clearly bene…t from a more liquidmarket as, when the book opens deeper, trading volume and market depth are higher andinside spread is narrower (Table 6).

Liquidity also drives the change in traders’ welfare after the introduction of a DP: whena DP is added to a liquid stock (bT 2 = [22]) total welfare increases (:0295%), whereaswhen it is added to an illiquid stock (bT 2 = [00]), it decreases (:4735%). As stated inProposition 1, DP trading generates trade migration which harms the market when the stockis illiquid by further reducing the already thin liquidity provision. Trade migration is instead

bene…cial when the stock is liquid as it decreases the competitive pressure in the LOB andhence enhances orders submission. As a result, institutional and especially retail traders areharmed by DP when the stock is illiquid, whereas in the case of liquid stocks, the increaseof institutional traders’ welfare outweighs the losses of retail traders. Indeed large investorsbene…t from a DP added to a liquid stock as they move their orders to the DP to avoid

21



competition in the LOB. Retail traders, instead, are harmed by DP trading: even when thestock is liquid and DP narrows the inside spread, they cannot access the DP market andhence cannot fully bene…t of the decrease in competition. We can therefore conclude thatfor a LOB, when a DP is added to a liquid stock, it creates an improvement in total welfare,whereas when it is added to an illiquid stock it harms both large and small traders.

6 Asymmetric Information on the State of the DP

The Security and Exchange Commission has recently proposed various changes in the reg-ulation of non-public trading interest that have been grouped under the SEC release No.34-60997. This proposal takes its move from the widespread use of IOI messages by DPmanagers that creates a leakage of privileged information to only some select investors. IOIand Alert messages risk creating a two-tiered market that can deprive the public of infor-mation about stock prices and liquidity. In this Section we extend the model to includeasymmetric information on the state of the DP to show the e¤ects on liquidity of the result-ing two-tiered market.

Assume that, all else equal, one group of large traders receives IOI or Alert messages,such as ‡ash orders, about the state of the DP. This feature can be embedded in the modelby assuming that at each trading round nature selects with probability 1=2 a small trader,with probability 1=4 a large uninformed trader and with probability 1=4 a large informedone. If a trader arrives at the market and is informed, then he knows the state of the DPand trades accordingly. The following Proposition summarizes the results obtained for thistwo-tiered market.

Proposition 7 When some large traders receive private information on the state of the

Dark Pool,

the probability that large traders, whether informed or uninformed, choose to trade in

the Dark Pool increases and hence orders move from the LOB to the Dark Pool;

the quality of the LOB measured by depth and best bid-o¤er improves, trading volume

in the LOB decreases, whereas total trading volume increases;

in terms of welfare, institutional traders bene…t from asymmetric information, whereas

retail traders bear extra losses.

Panels A and B of Table 9 summarize the results obtained in this extended version of the model with two types of large traders: Panel A reports the equilibrium trading strategiesof large informed and uninformed traders, and Panel B reports those of small traders. Themodel has been solved by starting at T 2 with 2 shares on both sides of the LOB: this is

22



the regime with greater access to the DP and hence it is more interesting to discuss the roleof informed messages. By comparing the results reported in this Table for T 2 with thosefrom the model with only one type of large trader (Table 1), it can be noticed that wheninformation on the state of the DP is asymmetric, large traders use the DP rather intensively(:1079). When a large trader knows that the DP is full on one side, he submits an order tothe other side with probability greater that 1=2 (:5034), whereas when he notices that the DPis empty, he submits an order with a tiny probability (:0123). If instead a large uninformedtrader arrives, he submits an order to the DP with probability equal to :0437, which meansthat, compared to the case with only one type of large traders (:0279), he uses the DP moreintensively. This is due to the fact that at T 2 he anticipates that large informed traderswill submit their orders to the DP more frequently, and that, for this reason, the DP volumewill be enhanced (externality e¤ect), with the result that the execution probability of theorders submitted to the DP will increase. And if at T 2 he does not observe any trade

(Table 9, Panel A.2), the probability that at T 1 he submits to the DP increases evenfurther (:052) than in the case without informed messages as he knows that the probabilityof DP trading is higher under asymmetric information.

It follows that if IOI and Alert messages create a two-tiered market, with some largetraders holding precise information about the state of the DP, then liquidity moves fromthe LOB to the DP. In fact all traders anticipate that the informed will use the DP moreintensively and this increases the probability of execution of DP orders thus reinforcing theDP externality e¤ect.

Table 10 shows that when traders move to the DP more frequently, spread and depthin the LOB improve. Compared to the protocol without asymmetric information, here notonly the spread improves but also market depth increases; the reason is that Alert and IOI

messages have the overall e¤ect of reducing the execution risk of DP trading, thus makingmarket orders less attractive than DP orders. Considering again the probability of ordersubmission under asymmetric information, Table 9 (Panel A.1) shows that with IOI andAlert messages the probability to observe market orders decreases by 18:1%, whereas with-out asymmetric information the reduction is tiny (1:6%). Further, limit order submissionsdecrease less with asymmetric information, even though the di¤erence in di¤erence is muchsmaller. The result of this change in order submission probabilities is that volume in theLOB decreases even more than in the case without information leakage; however, due tothe heavier use of the DP, total volume executed in both the limit order book and the DPincreases to 4:1549 (Table 10).

In terms of welfare the overall consequence of asymmetric information is to amplify thee¤ects of the introduction of the DP. When the DP is added to a LOB and DP managerscan use ‡ash messages to disseminate information on the state of the DP the welfare of all large traders increases: indeed, not only large informed traders can pro…t from theirprivileged information, but also uninformed traders can take advantage of their more preciseinference on the state of the DP and use the DP more intensively. Table 11 shows that large

23



uninformed traders’ welfare increases with the addition of a DP, and it further increaseswhen IOI and Alert messages are disseminated. Quite the opposite occurs to retail traderswhose welfare at T 2 decreases by an extra :04%: as all large traders move to the DPmore frequently, the execution probability of small traders’ limit orders posted to the LOBdecreases thus generating extra losses.

In conclusion, the main upshot of asymmetric information is to reduce execution riskfrom DP trading which enhances the use of DP orders at the expense of market orders thatare usually submitted by traders sensitive to this type of risk. With less market orders,LOB volume decreases, but LOB depth as well as total volume increase. This is the reasonwhy, when the DP is added to the LOB, large traders bene…t from asymmetric information,whereas small traders bear higher losses.

7 Empirical ImplicationsA great deal of empirical implications can be derived from our model, that we summarizein this Section. First of all, our results show that when a DP is added to a LOB, volumemigrate to the DP so that volume in the LOB decreases; yet, the sum of the volume traded onboth the LOB and the DP increases. Second, we show that the overall e¤ect of intermarketcompetition crucially depends on how deep and tight the LOB is. We expect that trademigration is more intense in liquid than in illiquid stocks as in the latter competition fromlimit orders crowds out DP orders. Following the volume migration, depth at the top of thebook deteriorates, whereas the e¤ect on inside spread depends on the liquidity of the stock.For liquid stocks where the book is very deep, the relative proportion of market to limit

orders that move to the DP leaves the inside spread very tight, whereas for illiquid stocks itwidens the spread. Our results also show that when traders believe that liquidity is growingin the dark pool, dark trading intensi…es so that we expect DP volume in liquid stocks toincrease more intensively than in illiquid stocks.

Beside depth and spread, our model has also suggestions for a third determinant of DPtrading, as it shows that DP volume increases with the tick size. An increase in the tick sizeon the one hand increases the inside spread, thus making market orders more expensive, onthe other hand it makes limit orders more convenient with the result that patient traders, whoare not so patient to submit limit orders, will trade more in the DP. This is an interesting andintriguing empirical prediction of our model that, however, should be tested with caution.The type of dark pools that our model features are di¤erent from those internalization poolsthat are used by broker-dealers to internalize trades. Also in the latter an increase of the ticksize raises dark volume, but the e¤ect is driven by broker-dealers’ pro…ts from sub-pennytrading. Consequently, to separate the e¤ect that a tick size change can have on di¤erentdark venues, empiricists should control for the average order size that in internalization poolsis much smaller than in traditional dark pools (Rosenblatt Securities , October 2010).

24



Our benchmark model also quali…es standard results on systematic pattern of LOB order‡ows, as it shows that Parlour’s (1998) main …nding that the probability of a continuation islarger than that of a reversal only characterizes liquid stocks, the opposite holding for illiquidones. Furthermore, the model predicts an externality originating from the coexistence of alimit order book and a DP, as it shows that when market depth on the former decreases,it creates a liquidity injection from the DP to the limit order book. Finally, the modelshows that IOI messages tend to move volume from the LOB to the DP and to reinforcethe liquidity-externality e¤ect. However, it also shows that spread and depth in the LOBimprove following the introduction of IOI messages, as they reduce execution risk from DPtrading and hence crowd out market orders that are submitted by traders who are sensitiveto this type of risk. Hence even though volume in the LOB decreases, depth and spreadimprove.

8 Conclusions

The dynamic microstructure model presented in this paper solves for the equilibrium tradingstrategies of di¤erent agents who can choose to trade either in a Limit Order Book (LOB)or in a Dark Pool (DP). A DP is an Alternative Trading System that does not provideits best-priced orders for inclusion in the consolidated quotation data. The existing theoryshows that dark crossing networks increase liquidity. Our model shows that this is true onlywhen a DP is added to a dealership market where traders cannot compete for the provisionof liquidity by submitting limit orders. When a DP is added to a LOB, orders migrate awayfrom the LOB to the dark market. The model thus demonstrates that the dark option o¤ered

to market participants produces order migration rather than order creation as in Degryse,Van Achter and Wuyts (2009).

We also show that current DP orders stimulate the arrival of future DP orders thusincreasing their execution probability (liquidity-externality e¤ect). Traders’ choice betweenLOB and DP depends on the state of the LOB as well as on agents’ expectations on thestate of the DP: the model shows that high depth and small spread increase traders’ use of DP, and that a reduction in the tick size reduces the pro…tability of liquidity provision andhence the use of DP orders. According to the model, when large visible limit orders produceprice pressure in the LOB, traders resort to DP orders even more extensively to reduce priceimpact.

In terms of market quality, when a DP is added to a LOB we …nd that depth and volumedeteriorate on the LOB, whilst total volume increases. The e¤ect of the introduction of aDP on the inside spread of the LOB instead depends on the state of the book, improvingwhen it is deep and worsening when it is shallow.

The model also o¤ers new insights on the systematic patterns of order ‡ow that can arisefrom traders’ interaction with the LOB: for liquid stocks the probability of a continuation is

25



greater than that of a reversal, the opposite being true for illiquid stocks. Furthermore, DPact as liquidity bu¤ers by supplying liquidity after a reduction of market depth on the LOB.In terms of welfare, we show that total welfare and institutional traders’ welfare increaseonly when a DP is added to a liquid stock, and that a DP always harms retail traders.

Finally, we show how asymmetric information on the state of the DP creates a two-tieredmarket that moves liquidity from the LOB to the DP. When traders know that other tradersare informed on the state of the DP, they anticipate that the informed will use the DPmore intensively and that this will increase the execution probability of DP orders. Hence,consistently with the recent SEC Proposal on Regulation of Non-Public Trading Interest(SEC Release No. 34-60997), the model shows that IOI and Alert messages, that informsome traders on the state of the DP, can draw orders away from the transparent market.However, compared to the protocol with a DP and without asymmetric information, we showthat the use of ‡ash orders can improve not only order book spread but also market depth

when they are allowed to be used for liquid stocks.

26



Appendix

Proof of Proposition 1

Consider …rst the benchmark case. The model is solved by backward induction, startingfrom t = T . The T -trader solves a simpli…ed version of program (2), if large, or (3), if small:

max'LT;T ;bT ;j

eT

'( j; pB

i ); '( j; pB); '(0); '( j; pAi ); '( j; pA)

(2’)

max'ST;T ;bT

eT

'(1; pB

i ); '(0); '(1; pA1 )

(3’)

Without loss of generality, assume that depending on T the trader selects among n 2 N T

possible equilibrium strategies. These strategies are ordered so that ';n1;';n

T;a < ';n;';n+1

T;a ,where a = fST;LT g and the -thresholds are given by:

';n1;';n

T;a : eT (';n1

a;T ;j j T ) eT (';n

a;T ;j j T ) = 0

The ex-ante probability that a trader submits a certain order type at T is determined asfollows:

PrT

(';na;T ;j j T ) = F ( '

;n;';n+1

T;a ) F ( ';n1;';n

T;a )

Consider now period t = T 1. The incoming trader solves program (2) or (3) if large or smallrespectively, and uses PrT (';n

a;T ;j j T ), 8n 2 N T , to compute the execution probabilitiesof his limit orders. Given the optimal strategies '

a;T 1;bT 1;j , the -thresholds and theorder type probabilities at T 1 are derived as shown for period T . This procedure isthen reiterated for period T 2. The solution of the LOB&DP model follows the same

methodology, but now the large trader solves program (4).

We provide examples for the LOB&DP protocol for the three trading periods analyzed, thatbelong to the case where the book opens as bT 2 = [22]. From now onwards we assumethat in program (4) j = max

j['LT;T ;j j T ], since @ t('LT;T ;j)=@j 0 due to agents’ risk

neutrality.

Consider the following books at T : (a) bT = [20], visT 2 visT 1, (b) bT = [20], invT 2 visT 1.In the …rst case traders observe a change in the LOB in both periods, while in the secondone only at T 1. We focus on the large trader’s pro…ts that for (a) are:

e

T

['(2; pB

2 ) j

[20;vv]

T

] = 2( pB

2

T

v) = 2(13

2

T

)

eT ['(2; pA

1 ) j [20;vv]T ]= 2( T vpA

1 ) = 2( T 1 2 )

eT ['(2; pM id) j

[20;vv]T ]= 2(

pA1 + pB22 T v)Pr

2(

pA1 + pB22 jT ) = 2(1

2 T )13

eT ['(+2; pM id) j

[20;vv]T ]= 2( T v

pA1 + pB22

)Pr+2

( pA1 + pB2

2 jT ) = 2( T 1+

2)1

3

27



where yt 2 fv; ig, with v = vist and i = invt, and [xx;yy]T : [xx;yy] = fbT ; yT 2yT 1g.

By solving program (2’) for this case it is straightforward to show that all strategies areoptimal in equilibrium (N T = 4) and that for the LT ';1

LT;[20;vv] = '(2; pB

2 ), ';2

LT;[20;vv] =

'(2; pM id), ';3LT;[20;vv] = '(+2; pM id) and ';4

LT;[20;vv] = '(2; pA1 ). As an example we compute

the probability of ';1LT;[20;vv]:

';1[20;vv]

;';2[20;vv]

T;LT : eT ['

;1LT;[20;vv]] e

T [';2LT;[20;vv]] = 0 !

';1[20;vv]

;';2[20;vv]

T;LT = 1 2

PrT

(';1LT;[20;vv]) = F (

';1[20;vv]

;';2[20;vv]

T;LT ) = 12(1 2 )

In case (b), pro…ts for DP orders di¤er:

eT ['(2; pM id) j

[20;iv]T ]= 2(

pA1 + pB2

2

T v)(1

3

1+1

3

PrT 2('(+2;pMid))

PrT

2('

(+2;pMid))+PrT

2('

(

2;pMid))+PrT

2('

(0))

)

eT ['(+2; pM id) j

[20;iv]T ]= 2( T v

pA1 + pB22

)(13

1+13

PrT 2('(2;pMid))

PrT 2('(+2;pMid))+PrT 2('(2;pMid))+PrT 2('(0)))

where we omit that all probabilities at T 2 are conditional to T 2. In this case boththe -thresholds and order probabilities are a function of equilibrium order strategies atT 2, '

a;T 2;bT 2;j , that are rationally computed by the T -trader. For example, if we de…ne

'1LT;[20;iv] = '(2; pB

2 ) and '2LT;[20;iv] = '(2; pM id):

'1[20;iv]

;'2[20;iv]

T;LT : eT ['

1LT;[20;iv]] e

T ['2LT;[20;iv]] = 0

'1[20;iv]

;'2[20;iv]

T;LT = (27 )PrT 2 '(+2;pMid)+4(12 )[PrT 2 '(2;pMid)+PrT 2('(0)]2PrT 2 '(+2;pMid)+4(PrT 2 '(2;pMid)+PrT 2('(0))

PrT

'1

LT;[20;iv]

= F (

'1[20;iv]

;'2[20;iv]

T;LT ) =12

'1[20;iv]

;'2[20;iv]

T;LT

To determine the equilibrium strategies ';n

LT;[20;iv] for n 2 N T , the model has to be solved up

to period T 2. We anticipate that '(2; pB2 ) is indeed an equilibrium strategy, and that the

corresponding probability is: PrT (';1LT;[20;iv]) = (25 )

4 .

For T 1 and T 2 we only specify the pro…t formulas, as the derivation of the -thresholds and order probabilities follow the same steps presented for period T . Considerthe case bT 1 = [20], visT 2. Small trader’s pro…ts are as follows:

eT 1['(1; pB

2 ) j [20;v]T 1 ] = ( pB

2 T 1v)

eT 1['(1; pA

1 ) j [20;v]T 1 ] = e

T 1['(0)] = 0

eT 1['(1; pB

1 ) j [20;v]T 1 ] = ( T 1v pB

1 ) 12 [Pr

T ('(1; pB

1 ) j [21;vv]T ) + Pr

T ('(2; pB

1 ) j [21;vv]T )]

eT 1['(1; pA

1 ) j [20;v]T 1 ] = ( T 1v pA

1 )

28



where [xx;y]T 1 : [xx;y] = fbT 1; yT 2g. Large trader’s strategies are similar, the only di¤erence

being that j = 2, with the exception of DP orders:

eT 1['(2; pM id)] = E [( pM id T 1v) Pr

2( pM idjT )]

eT 1['(+2; pM id)] = E [( T 1v pM id) Pr

+2( pM idjT )]

We specify the …rst one:

eT 1['(2; pM id)] = 1

3 2 1

2(

pA1 + pB22

T 1v)PrT

('(+2; pM id) j [20;vi;0]T )

+ 13 2 1

2f( pA2 + pB2

2 T 1v)PrT

('(2; pA1 ) j

[20;vi;+6]T ) + (

pA1 + pB22 T 1v)

[1 + PrT

('(+2; pM id) j [20;vi;+6]T ) + Pr

T ('(2; pM id) j

[20;vi;+6]T ) + Pr

T ('(2; pB

2 ) j [20;vi;

T

where [xx;yy;z]T : [xx;yy;z] = fbT ; yT 2yT 1; DP T g. At T 2 we consider the book bT 2 = [22]

and present pro…t formulas only for the sell side of the market, the buy side being symmetric:

eT 2['(2; pB

1 )] = 2( pB1 T 2v)

eT 2['(0)] = 0

eT 2['(2; pA

1 )] = ( pA1 T 2v)f1

2 PrT 1

('(1; pA1 ) j

[42;v]T 1 )[ 12Pr

T ('(2; pA

1 ) j [32;vv]T )]

+12 PrT 1

('(2; pA1 ) j

[42;v]T 1 )[ 1

2Pr

T ('(1; pA

1 ) j [22;vv]T )+1

22 Pr

T ('(2; pA

1 ) j [22;vv]T )]g

eT 2['(2; pM id)] = E f( pM id T 2v)[Pr

2

( pM idjT 1) + (1 Pr2

( pM idjT 1))Pr2

( pM idjT )]g

where to economize space we do not specify eT 2['(2; pM id)]. Results from Proposition

1 are derived by comparing equilibrium strategies for the LOB&DP case and for the pureLOB case presented in Table 1 and Table 2. In Figures A1-A4 we provide graphical plotsfor the large trader’s pro…ts at T 2 as a function of that relate to the four main pointsof the Proposition. We consider only selling strategies, the picture being symmetric for thebuy side.


Results from Proposition 2 are obtained by straightforward comparison of the equilibriumstrategies derived in the proof of Proposition 1, for di¤erent states of the LOB. Also for thisproof we provide graphical plots. Consider …rst the DP&LOB: compare Figures A1 and A5for the e¤ect of market depth, A1 and A3 for the e¤ect of spread, and A5 and A6 for thetick size. For DM&DP, consider Figure A7.

29



30




The benchmark and the LOB&DP model are solved following the procedure already illus-trated in Proposition 1. To clarify the di¤erence between the case with and without pricepressure, consider an opening book bT 2 = [22] where a large limit sell order, 'T 2(2; pA

1 );is submitted. With no price pressure, at T 1 the book is full, bT 1 = [42], so the feasiblestrategies are only market orders, dark pool orders or no trade. With price pressure, in-stead, the new asset value becomes vT 1 = v , and, as shown in Figure 5, the price gridshifts accordingly so that the book turns empty, bT 1 = [00]. This implies that limit orders,'T 1( j; pA

1 ) and 'T 1( j; pB1 ), are now included within the available strategies. Large traders

arriving at T 2 will rationally anticipate the pro…t reduction in 2-units limit orders due toprice pressure, so j = 2 is not necessarily optimal anymore. We hence specify as an examplethe pro…t formulas for both e

T 2['(2; pA1 )] and e

T 2['(1; pA1 )]:

eT 2['(2; pA

1 )] = ( pA1 T 2v

T )f1

2 PrT 1

('(1;pAd2 ) j

[00;v]T 1 )[ 12Pr

T ('(2; pA

1 ) j [20;vv]T )]

+ 12 Pr

T 1('(2; pAd

2 ) j [00;v]T 1 )[ 12Pr

T ('(1; pA

1 ) j [22;vv]T )+1

22 PrT

('(2; pA1 ) j

[20;vv]T )]g

eT 2['(1; pA

1 )] = ( pA1 T 2v

T )f1

2 PrT 1

('(1; pA1 ) j

[32;v]T 1 )[ 1

2Pr

T ('(2; pA

1 ) j [22;vv]T )]

+ 12 PrT 1

('(2; pA1 ) j [32;v]

T 1 )[ 12

PrT

('(1; pA1 ) j [12;vv]

T )+12

PrT

('(2; pA) j [12;vv]T )]g

In Figure A8 we plot large trader’s pro…ts for the model with price pressures, that can becompared to Figure A1. Numerical values are reported in Table 5.

31




Results, presented in Table 6, are obtained by comparing the three market quality mea-

sures for both the benchmark and the LOB&DP protocol. As an example, we consider theLOB&DP model with an opening book equal to bT 2 = [22] and specify formulas for the esti-mated spread and depth at T 1 and for the executed volume at T 2. Similar computationsmake it possible to derive the market quality measures for all the other cases. We de…neequilibrium strategies at T 2 for a LT as follows: ';1

LT;[22]= '(2;pB2 ), ';2

LT;[22]= '(2; pM id),

';3LT;[22]= '(2; pA

1 ), ';4LT;[22]= '(2; pB

1 ), ';5LT;[22]= '(+2; pM id) and ';6

LT;[22]= '(2;pA1 ). The ones

for a ST are: ';1ST ;[22]= '(1;pB

2 ), ';2ST ;[22]= '(1; pA

1 ), ';3ST ;[22]= '(1; pB

1 ) and ';4ST ;[22]= '(2;pA

1 ).

S eT 1 = 12

f PrT 2

(';1LT;[22])

pA

1 pB2

+ Pr

T 2(';6

LT;[22]) pA

2 pB1

+ [ PrT 2('

;2

LT;[22]) + PrT 2('

;3

LT;[22]) + PrT 2('

;4

LT;[22]) + PrT 2('

;5

LT;[22])] pA

1 p

B

1 g+ 1

2 [ PrT 2

(';1ST ;[22]) + Pr

T 2(';2

ST ;[22]) + PrT 2

(';3ST ;[22]) + Pr

T 2(';4

ST ;[22])] pA

1 pB1

De

T 1 = 12

f2 PrT 2

(';1LT;[22])

pA

1 pB2

+ Pr

T 2(';6

LT;[22]) pA

2 pB1

+ 6 [ Pr

T 2(';3

LT;[22]) + PrT 2

(';4LT;[22])] + 4 [ Pr

T 2(';2

LT;[22]) + PrT 2

(';5LT;[22])]g

+ 12f3 [ Pr

T 2(';1

ST ;[22]) + PrT 2

(';4ST ;[22])] + 5 [ Pr

T 2(';2

ST ;[22]) + PrT 2

(';3ST ;[22])]g

V eT 2 = 1

2

[ PrT 2

(';1ST ;[22]) + Pr

T 2

(';4ST ;[22])]+1

2

2[PrT 2

(';1LT;[22]) + Pr

T 2

(';6LT;[22])]


Results are derived by comparing equilibrium trading strategies for two new starting books:bT 2 = [20] and bT 2 = [10]. As the solutions of these two cases follow the same steps as theone presented in the proof of Proposition 1 for the book bT 2 = [22], they are omitted andavailable upon the authors at request. In Figure A9-A10 we plot pro…ts for large and smalltraders respectively. Numerical values are provided in Table 7.


Results presented in Table 8 are obtained by comparing welfare levels for both the benchmarkand the LOB&DP protocol. As an example we consider again the LOB&DP case openingwith bT 2 = [22]. Notice that it is su¢cient to compute the welfare for the ask side, astraders’ strategies on the other side are perfectly symmetric. We start with period T 2,equilibrium strategies for both a LT and a ST are de…ned in the proof of Proposition 4.

32



W T 2;LT = 2f PrT 2

(';1LT;[22])(2 pB

1 ';1[22]

;';2[22]

T;LT

2 v) + PrT 2

(';2LT;[22])(2 pM id

';2[22]

;';3[22]

T;LT +

';1[22]

;';2[22]

T;LT

2 v)

+ PrT 2

(';3LT;[22])(2 pA

1 ';3[22]

;';4[22]

T;LT +

';2[22]

;';3[22]

T;LT

2 v)f2 Pr

2( pA

1 jT 1)

+ Pr1

( pA1 jT 1)[Pr

0( pA

1 jT ) + 2 Pr1

( pA1 jT )] + Pr

0( pA

1 jT 1)[Pr1

( pA1 jT ) + 2 Pr

2( pA

1 jT )]gg

W T 2;ST = 2f PrT 2

(';1ST ;[22])( pB

1 ';1[22]

;';2[22]

T;ST

2 v) + Pr

T 2(';2

ST ;[22])( pA1

';2[22]

;';3[22]

T;ST +

';1[22]

;';2[22]

T;ST

2 v)[Pr

1( pA

1 jT 1)

+ Pr0

( pA1 jT 1)Pr

1( pA

1 jT )]g

Welfare values for the following periods are computed similarly, but taking into accountthat a trader faces di¤erent states of the book depending on the orders submitted in theprevious periods. For example, still for the case bT 2 = [22], the welfare of trader a, witha = fST;LT g, arriving at T 1 is computed as follows:

W T 1;a = 12f Pr

T 2(';1

LT;[22])W [20;v]

T 1;a + PrT 2

(';6LT;[22])W

[02;v]T 1;a + Pr

T 2(';3

LT;[22])W [42;v]

T 1;a

+[PrT 2

(';2LT;[22]) + Pr

T 2(';5

LT;[22])]W [22;i]

T 1;a + PrT 2

(';4LT;[22])W

[24;v]T 1;ag + 1

2f PrT 2

(';1ST ;[22])W

[21;v]T 1;a

+ PrT 2(';4

ST ;[22])W

[12;v]

T 1;a + PrT 2(';2

ST ;[22])W

[32;v]

T 1;a + PrT 2(';3

ST ;[22])W

[23;v]

T 1;ag


This proof follows the same methodology presented in the proof of Proposition 1. To ease thecomparison with the previous framework, we provide again as an example the case bT = [20],

33



invT 2 visT 1, for both uninformed (u) and informed (i) large traders.15 Consider …rst the u-trader, pro…ts from market orders are unchanged and omitted, but pro…ts from 'u(2; pM id)become:

eT ['u(2; pM id)j

[20;iv]T ]= 2(

pA1 + pB22 T v)Pr

2(

pA1 + pB22 jT )

=2( pA1 + pB2

2 T v)[1

31+1

3

Pd=i;u

PrT 2('d(2;pMid))

PrT 2('d(+2;pMid))+PrT 2('d(2;pMid))

]

where we omit that all probabilities at T 2 are conditional to T 2. Notice that the traderrationally anticipates that i-traders observe the state of the DP when selecting their optimalstrategies. Consider now i-traders: their pro…ts for 'i(2; pM id) depend on the actual stateof the DP that in this case can be DP T = f2; 4g. We provide as an example pro…ts for'i(2; pM id):

eT ['i(2; pM id)j

[20;iv;+2]T ] = e

T ['i(2; pM id)j[20;iv;+4]T ] = 2(

pA1 + pB22 T v)

eT ['i(2; pM id)j

[20;iv;2]T ] = e

T ['i(2; pM id)j[20;iv;4]T ] = 0

So the i-trader faces no uncertainty and perfectly knows before submitting the order whetherhe will be executed or not. We omit the discussion on periods T 1 and T 2 as the intuitionis similar. Pro…ts at T 2 for both the u- and i-trader are presented in Figures A11 andA12-14 respectively.

15 The case bT = [20], visT 2 visT 1 is not interesting as no one plays on the DP.

34



Results are derived by comparing numerical values for equilibrium strategies with thoseobtained in Proposition 1. Results for market quality and welfare are obtained by computingthe corresponding numerical values using the formulas presented respectively in the proofsof Proposition 4 and 6, with the addition of i-traders.

35



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38



Tuttle, L., 2006. Hidden orders, trading costs and information, Working Paper, AmericanUniversity of Sharjah.Ye, M., 2009. A Glimpse into the Dark: Price Formation, Transaction Cost and MarketShare of the Crossing Network.Working Paper, Cornell University.Zhu, H., 2011. Do Dark Pools Harm Price Discovery? Working Paper.

39



F i g u r e 1 - C o n s o l i d a t e d U S e q u i t y v o l u m e , F e b r u a r y - O c t o b e r 2 0 0 9

40



F i g u r e 2 - B e n c h m a r k M o d e l o f L i m i t O r d e r B o o k ( L O B )

: E x a m p l e f o r j =

2 o f

t h e E x t e n s i v e F o r m o f t h e G

a m e . T h e t r a d i n g c r o w d i s i n

d i c a t e d b y T C , w h i l e

s m a l l a n d l a r g e t r a d e r s a r e n

a m e d S T a n d L T r e s p e c t i v e l y

. W e p r e s e n t o n l y t h e

e q u i l i b r i u m s t r a t e g i e s .

41



F i g u r e 3 - L i m i t O r d e r B o o k a n d

D a r k P o o l M a r k e t ( L O B & D P

) : E x a m p l e o f t h e E x t e n s i v e

F o r m o f t h e G a m e f o r j =

2 .

42



F i g u r e 4 - D e a l e r s h i p M a r k e t w i t h a D a r k P o o l ( D M & D P ) :

E x a m p l e o f t h e E x t e n s i v e

F o r m o f t h e G a m e f o r j =

2 .

43



F i g u r e 5 - T h i s F i g u r e r e p o r t s t h e m e c h a n i c s o f t h e L O B w h e n a

l i m i t o r d e r t o s e l l 2 s h a r e s i s s u b m i t t e d a t T

2 t h

a t g e n e r a t e s a

t e m p o r a r y

p r i c e p r e s s u r e o f 1 t i c k d o w n w

a r d .

44



T a b l e 1 - O r d e r S u b m i s s i o n P r o b a b i l i t i e

s - b T

2 ; T

1 ; T =

[ 2 2 ]

b T

2 = [ 2 2 ]

b T

1 = [ 2 2 ] v i s T

2 ( i n v T

2 )

b T = [ 2 2 ] v i s T

2

v i s T

1

( i n v T

2

v i s T

1 )

L a r g e T r a d e r

L O B

L O B & D P

D M & D

P

L O B

L O B & D P

D

M & D P

L O B

L O B & D

P

D M & D P

' ( 2 ; p

B 2 )

' ( 2 ; p

B 1 )

: 4 6 8 6

: 4 6 1 2

: 4 6 1 7

: 4 7 5 0

: 4 6 2 1 ( : 4 5 0 0 )

: 4 6 2 1 ( : 4 5 5 6 )

: 4 7 5 0

: 4 6 2 5 ( : 4 5 0 0 )

: 4 6 2 5 ( : 4 5 5 8 )

' ( 2 ; p

A 1 )

: 0 3 1 4

: 0 1 0 9

' ( j ; p M i d )

: 0 2 7 9

: 0 3 8 3

: 0 3 7 9 ( : 0 5 0 0 )

: 0 3 7 9 ( : 0 4 4 4 )

: 0 3 7 5 ( : 0 5 0 0 )

: 0 3 7 5 ( : 0 4 4 2 )

' ( 0 )

: 0 5 0 0

: 0 5 0 0

' ( j ; p M i d )

: 0 2 7 9

: 0 3 8 3

: 0 3 7 9 ( : 0 5 0 0 )

: 0 3 7 9 ( : 0 4 4 4 )

: 0 3 7 5 ( : 0 5 0 0 )

: 0 3 7 5 ( : 0 4 4 2 )

' ( 2 ; p

B 1 )

: 0 3 1 4

: 0 1 0 9

' ( 2 ; p

A 1 )

: 4 6 8 6

: 4 6 1 2

: 4 6 1 7

: 4 7 5 0

: 4 6 2 1 ( : 4 5 0 0 )

: 4 6 2 1 ( : 4 5 5 6 )

: 4 7 5 0

: 4 6 2 5 ( : 4 5 0 0 )

: 4 6 2 5 ( : 4 5 5 8 )

' ( 2 ; p

A 2 )

T a b l e 1 : O r d

e r S u b m i s s i o n P r o b a b i l i t i e s - b T

2 ; T

1 ; T =

[ 2 2 ] : T h i s T

a b l e r e p o r t s l a r g e t r a d e r s ’ s u b m i s s i o n

p r o b a b i l i t i e s f o

r t h e o r d e r s l i s t e d i n c o l u m n 1 f o r t h e b e n c h m a r k ( L O B ) , f o r t h e L O B & D P a n d f o r t h e D

M & D P

m o d e l r e s p e c t i v e l y . E x e c u t i o n p r o b a b i l i t i e s

a r e r e p o r t e d f o r t h e s t a t e o f t h e b o o k w i t h 2 s h a r e s o n b o

t h s i d e s

o f t h e o r d e r b o

o k a n d f o r a l l p e r i o d s T ;

T

1 a n d T

2 . F o r e x a m p l e , l a r g e a g e n t s a r r i v i n g a t t h e m

a r k e t a t

T

1 a f t e r h a v i n g o b s e r v e d a c h a n g e i n t h e

o r d e r b o o k a t T

2 ( v i s T

2 ) ; s u b m i t m a r k e t o r d e r s t o s e

l l a t B

1

w i t h p r o b a b i l i t

y : 4

7 5 0 i f t h e y t r a d e i n t h e L O B , w i t h p r o b a b i l i t y : 4

6 2 1 i f t h e y t r a d e i n t h e L O B & D P a n d : 4

6 2 1

i n t h e D M & D P

p r o t o c o l . I f i n s t e a d t h e y d o

n o t o b s e r v e a n y c h a n g e i n t h

e o r d e r b o o k a t t i m e T

2 (

i n v T

2 ) ;

t h e p r o b a b i l i t i e s a r e : 4

5 0 0 f o r L O B & D P a n d

: 4 5 5 6 f o r D M & D P .

45



T a b l e 2 - O r d e r

S u b m i s s i o n P r o b a b i l i t i e s

- b T

2 ; T

1 ; T =

[ 0 0 ]

b T

2

=

[ 0 0 ]

b T

1

=

[ 0 0 ] v i s T

2 ( i

n v T

2 )

b T =

[ 0 0 ]

v i s T

2

v i s T

1

( i n v T

2

v i s T

1 )


L O B

L O B & D P

D M & D

P

L O B

L O B & D P

D

M & D P

L O B

L O B & D

P

D M & D P

' ( 2 ; p

B 2 )

: 2 8 5 7

: 2 9 1 2

: 3 6 9 7

: 3 7 2 6 ( : 3 5 0 0 )

: 4 2 5 0

: 3 8 7 5 ( : 3 5 0 0 )

' ( 2 ; p

B 1 )

: 4 6 1 7

: 4 6 2 1 ( : 4 5 5 6 )

: 4 6 2 5 ( : 4 5 5 8 )

' ( 2 ; p

A 1 )

: 2 1 4 3

: 2 0 8 8

: 1 3 0 3

: 1 2 7 4 ( : 0 4 9 1 )

' ( 2 ; p M i d )

: 0 3 8 3

( : 1

0 0 9 )

: 0 3 7 9 ( : 0 4 4 4 )

: 1 1 2 5 ( : 1 5 0 0 )

: 0 3 7 5 ( : 0 4 4 2 )

' ( 0 )

: 1 5 0 0

' ( 2 ; p M i d )

: 0 3 8 3

( : 1

0 0 9 )

: 0 3 7 9 ( : 0 4 4 4 )

: 1 1 2 5 ( : 1 5 0 0 )

: 0 3 7 5 ( : 0 4 4 2 )

' ( 2 ; p

B 1 )

: 2 1 4 3

: 2 0 8 8

: 1 3 0 3

: 1 2 7 4 ( : 0 4 9 1 )

' ( 2 ; p

A 1 )

: 4 6 1 7

: 4 6 2 1 ( : 4 5 5 6 )

: 4 6 2 5 ( : 4 5 5 8 )

' ( 2 ; p

A 2 )

: 2 8 5 7

: 2 9 1 2

: 3 6 9 7

: 3 7 2 6 ( : 3 5 0 0 )

: 4 2 5 0

: 3 8 7 5 ( : 3 5 0 0 )

T a b l e 2 : O r d

e r S u b m i s s i o n P r o b a b i l i t i e s - b T

2 ; T

1 ; T =

[ 0 0 ] : T h i s T

a b l e r e p o r t s l a r g e t r a d e r s ’ s u b m i s s i o n


r t h e o r d e r s l i s t e d i n c o l u m n 1 f o r t h e b e n c h m a r k ( L O B ) , f o r t h e L O B & D P a n d f o r t h e D

M & D P

m o d e l r e s p e c t i v e l y . E x e c u t i o n p r o b a b i l i t i e s a r e r e p o r t e d f o r t h e s t a t e o f t

h e b o o k w i t h n o s h a r e s o n b o

t h s i d e s

o f t h e o r d e r b o

o k a n d f o r a l l p e r i o d s T ;

T

1 a n d T

2 . F o r e x a m p l e , l a r g e a g e n t s a r r i v i n g a t t h e m a r k e t a t T

a f t e r h a v i n g o b

s e r v e d a c h a n g e i n t h e o r d e r b o o k b o t h a t T

1 a n d a t T

2 ( v i s T

2 ; v

i s T

1 ) ; s u b m i t o

r d e r s t o

s e l l t o t h e D P w i t h p r o b a b i l i t y : 1

1 2 5 i n t h e L O B & D P m a r k e t a n d : 0

3 7 5 i n

t h e D M & D P . I f i n s t e a d t h e y

o b s e r v e

a c h a n g e i n t h e

o r d e r b o o k o n l y a t t i m e T

1 a n d n o c h a n g e a t T

2 ( i n v T

2 ; v

i s T

1 ) ; t h e p r o b a b i l i t i e s a

r e : 1

5 0 0

a n d : 0

4 4 2 r e s p e c t i v e l y .

46



T a b l e 3 - M a r k e t D e p t h , I n s i d e S p r e a d

: T i m e T - 2

P a n e l A - L a r g e T r a d e r

M a r k e t [ b T

2 ] ( t i c k s i z e )

L O B & D P [ 0 0 ]

L O B & D P [ 2 2 ]

L O B & D P

[ 1 1 ] ( : 1 )

L O B & D P [ 1 1 ] ( : 1 5 )

D M & D P

' ( 2 ; p B 2 )

: 2 9 1 2

' ( 2 ; p B 1 )

: 4 6 1 2

: 4 6 1 7

' ( 2 ; p B )

: 4 1

6 8

: 3 7 8 5

' ( 2 ; p A 1 )

: 2 0 8 8

: 0 1 0 9

: 0 8

3 2

: 1 1 5 8

' ( 2 ; p M i d )

: 0 2 7 9

: 0 0 5 7

: 0 3 8 3

' ( 2 ; p

M i d )

: 0 2 7 9

: 0 0 5 7

: 0 3 8 3

' ( 2 ; p B 1 )

: 2 0 8 8

: 0 1 0 9

: 0 8

3 2

: 1 1 5 8

' ( 2 ; p A )

: 4 1

6 8

: 3 7 8 5

' ( 2 ; p A 1 )

: 4 6 1 2

: 4 6 1 7

' ( 2 ; p A 2 )

: 2 9 1 2

P a n e l B - S m a l l T r a d e r

' ( 1 ; p B 2 )

: 2 1 4 2

' ( 1 ; p B 1 )

: 4 6 5 6

: 4 3

3 3

: 4 0 5 4

: 4 7 5 0

' ( 1 ; p A 1 )

: 2 8 5 8

: 0 3 4 4

: 0 6

6 7

: 0 9 4 6

' ( 0 )

: 0 5 0 0

' ( 1 ; p B 1 )

: 2 8 5 8

: 0 3 4 4

: 0 6

6 7

: 0 9 4 6

' ( 1 ; p A 1 )

: 4 6 5 6

: 4 3

3 3

: 4 0 5 4

: 4 7 5 0

' ( 1 ; p A 2 )

: 2 1 4 2

T a b l e 3 : M a r k e t D e p t h a n d I n s i d e S p

r e a d - T i m e T - 2 .

T h i s T a b l e r e p o r t s l a r g e t r a d e r s ’ ( P

a n e l A )

s u b m i s s i o n p r o

b a b i l i t i e s t o t h e L O B & D P a t

t i m e T

2

f o r t h e o r d e r s l i s t

e d i n c o l u m n 1 ; c o l u m n s 2 t o

5 r e p o r t

r e s u l t s f o r d i ¤ e

r e n t s t a t e s o f t h e b o o k a n d c o l u m n 6 g i v e s t h e e q u i l i b r i u m

o r d e r s u b m i s s i o n p r o b a b i l i t i e s f o r t h e

D M & D P p r o t o

c o l . P a n e l B r e p o r t s t h e o r d e r s u b m i s s i o n p r o b a b i l i t i e s f o r

s m a l l t r a d e r s . T h e o r d e r s u b m i s s i o n

p r o b a b i l i t i e s r e

p o r t e d i n t h i s T a b l e a l l o w t o c o m p a r e d i ¤ e r e n t v a l u e s o f m a

r k e t d e p t h a n d i n s i d e s p r e a d .

47



T a b l e 4 - L a r g e T r a d e r - T i m e T : v i s T

2

v i s T

1

( i n v T

2

v i s T

1 )

O r d e r S u b m i s s i o n P r o b a b i l i t i e s t o c o m p u t e M

a r k e t D e p t h

M a r k e t [ b T ]

L O B & D P [ 2 2 ]

L O B

& D P [ 1 2 ]

L O B & D P [ 2 1 ]

L O B & D P [ 1 1 ]

D M & D P

' ( 2 ; p B 2 )

' ( 2 ; p B 1 )

: 4 6 2 5 ( : 4 5 0 0 )

: 4 6 2 5 ( : 4 5 0 0 )

: 4 6 2 5 ( : 4 5 5 8

)

' ( 2 ; p B )

: 4 2 5 0 ( : 4

0 0 0 )

: 4 2 5 0

' ( 2 ; p M i d )

: 0 3 7 5 ( : 0 5 0 0 )

: 0 3 7 5 ( : 0 5 0 0 )

: 0 7 5 0 ( : 1

0 0 0 )

: 0 7 5 0

: 0 3 7 5 ( : 0 4 4 2

)

' ( 2 ; p M i d )

: 0 3 7 5 ( : 0 5 0 0 )

: 0 7 5 0 ( : 1 0 0 0 )

: 0 3 7 5 ( : 0

5 0 0 )

: 0 7 5 0

: 0 3 7 5 ( : 0 4 4 2

)

' ( 2 ; p A )

: 4 2 5 0 ( : 4 0 0 0 )

: 4 2 5 0

' ( 2 ; p A 1 )

: 4 6 2 5 ( : 4 5 0 0 )

: 4 6 2 5 ( : 4

5 0 0 )

: 4 6 2 5 ( : 4 5 5 8

)

' ( 2 ; p A 2 )

O r d e r S u b m i s s i o n P r o b a b i l i t i e s t o c o m p u t e I

n s i d e S p r e a d

M a r k e t [ b T ]

L O B & D P [ 0 0 ]

L O B

& D P [ 2 0 ]

L O B & D P [ 0 2 ]

L O B & D P [ 2 2 ]

D M & D P

' ( 2 ; p B 2 )

: 3 8 7 5 ( : 3 5 0 0 )

: 4 0 0 0 ( : 3 7 5 0 )

' ( 2 ; p B 1 )

: 4 5 0 0 ( : 4

2 5 0 )

: 4 6 2 5 ( : 4 5 0 0 )

: 4 6 2 5 ( : 4 5 5 8

)

' ( 2 ; p M i d )

: 1 1 2 5 ( : 1 5 0 0 )

: 0 7 5 0 ( : 1 0 0 0 )

: 0 7 5 0 ( : 1

0 0 0 )

: 0 3 7 5 ( : 0 4 9 0 )

: 0 3 7 5 ( : 0 4 4 2

)

' ( 2 ; p M i d )

: 1 1 2 5 ( : 1 5 0 0 )

: 0 7 5 0 ( : 1 0 0 0 )

: 0 7 5 0 ( : 1

0 0 0 )

: 0 3 7 5 ( : 0 4 9 0 )

: 0 3 7 5 ( : 0 4 4 2

)

' ( 2 ; p A 1 )

: 4 5 0 0 ( : 4 2 5 0 )

: 4 6 2 5 ( : 4 5 0 0 )

: 4 6 2 5 ( : 4 5 5 8

)

' ( 2 ; p A 2 )

: 3 8 7 5 ( : 3 5 0 0 )

: 4 0 0 0 ( : 3

7 5 0 )

T a b l e 4 : E s t i m a t e d D e p t h a n d I n s i d e S

p r e a d - T i m e T . T h i s T a b l e r e p o r t s l a r g e t r a d e r s ’ o r d e r s u b m i s s i o n

p r o b a b i l i t i e s t o

t h e L O B & D P m a r k e t a t t i m

e T f o r t h e o r d e r s l i s t e d i n c o

l u m n 1 ; c o l u m n s 2 t o 5 r e p o r t r e s u l t s

f o r d i ¤ e r e n t s t a

t e s o f t h e b o o k a n d c o l u m n 6

g i v e s t h e e q u i l i b r i u m o r d e r s u b m i s s i o n p r o b a b i l i t i e s f o r t h e D

M & D P

p r o t o c o l . V a l u e s i n p a r e n t h e s i s r e f e r t o t h e c a s e w h e r e t r a d e r s o n l y o b s e r v e a c h a n g e i n t h e o r d e r b o o k a

t T

1 ,

w h e r e a s t h e y d

o n o t o b s e r v e a n y v a r i a t i o n a

t T

2 ( i n v T

2 ; v

i s T

1 ) . T h e T a b l e a l l o w s f o r c o m p a r i s o n s a m o n g

d i ¤ e r e n t l e v e l s

o f m a r k e t d e p t h a n d d i ¤ e r e n t

i n s i d e s p r e a d s .

48



T a b l e 5 - P r i c e I

m p a c t a n d P r i c e P r e s s u r e

: b T

2

=

2 2


M a r k e t [ b T

2

]

L O B

L O B & P P

L O B & D P L

O B & D P & P P

' ( 2 ; p

B 1 )

: 4 6 8 6

: 4 7 0 4

: 4 6 1 2

: 4 6 1 2

' ( 1 ; p

A 1 )

: 0 2 9 6

' ( 2 ; p

A 1 )

: 0 3 1 4

: 0 1 0 9

0 0 7 2

' ( j ; p M i d )

: 0 2 7 9

: 0 3 1 6

' ( j ; p M i d )

: 0 2 7 9

: 0 3 1 6

' ( 2 ; p

B 1 )

: 0 3 1 4

: 0 1 0 9

: 0 0 7 2

' ( 1 ; p

B 1 )

: 0 2 9 6

' ( 2 ; p

A 1 )

: 4 6 8 6

: 4 7 0 4

: 4 6 1 2

: 4 6 1 2

T a b l e 5 : P r i c e I m p a c t a n d P r i c e P r e s s u r e - b T

2

=

[ 2 2 ] : T h i s T a b l e r e p o r t s l a r g e t r a d e r s ’ s u b m i s s i o n


r t h e o r d e r s l i s t e d i n c o l u m n 1 f o r t h e b e n c h m a r k ( L O B ) a n

d f o r t h e L O B & D P m o d e l , b o t h w i t h

a n d w i t h o u t p r i c e p r e s s u r e .

49



T a b l e 6 - M a r k e t Q u a l i t y

B o o k

E s t i m a t e d S p r e a d

E s t i m a t e d D e p t h

E s t i m a t e d V o l u m e

b T

2

=

[ 0 0 ]

L O B

L O B & D P

%

L

O B

L O B & D P

%

L

O B

L O B & D P

%

D

P

L O B & D P + D P

A v e r a g e

: 2 6 5 1

: 2 6 6 4

: 0 0 5 2

: 7 4 8 3

: 7 1 7 9

: 0

4 0 6

1 : 0 7 3 6

1 : 0

6 6 5

: 0

0 6 6

T o t a l

1 : 4 9 6 6

1 : 4

3 5 8

: 0

4 0 6

3 : 2 2 0 8

3 : 1

9 9 6

: 0

0 6 6

: 0 8 0 4

3 : 2

8 0 0

T

: 2 4 6 9

: 2 4 8 8

: 0 0 7 7

: 7 6 5 0

: 7 3 2 5

: 0

4 2 5

1 : 3 0 7 4

1 : 2

3 8 6

: 0

5 2 6

T - 1

: 2 4 8 3

: 2 5 0 5

: 0 0 8 9

: 7 3 1 6

: 7 0 3 3

: 0

3 8 7

1 : 1 4 5 0

1 : 1

6 4 3

: 0 1 6 9

T - 2

: 3 0 0 0

: 3 0 0 0

: 7 6 8 4

: 7 9 6 7

: 0 3 6 8

b T

2

=

[ 1 1 ]

A v e r a g e

: 1 6 8 6

: 1 6 8 9

: 0 0 2 0

1 : 2 2 8 2

1 : 2

0 5 0

: 0

1 8 9

1 : 2 7 8 8

1 : 2

5 2 0

: 0

2 0 9

T o t a l

2 : 4 5 6 3

2 : 4

0 9 9

: 0

1 8 9

3 : 8 3 6 3

3 : 7

5 6 1

: 0

2 0 9

: 0 9 2 5

3 : 8

4 8 6

T

: 2 2 1 2

: 2 2 1 8

: 0 0 2 7

1 : 0 6 2 4

1 : 0

2 6 9

: 0

2 6 6

1 : 3 2 0 1

1 : 2

5 2 5

: 0

5 1 2

T - 1

: 1 8 4 6

: 1 8 5 0

: 0 0 2 2

1 : 3 9 3 9

1 : 3

8 3 0

: 0

0 5 4

1 : 2 5 5 6

1 : 2

3 6 7

: 0

1 5 1

T - 2

: 1 0 0 0

: 1 0 0 0

1 : 2 6 0 6

1 : 2

6 6 9

: 0 0 5 0

b T

2

=

[ 2 2 ]

A v e r a g e

: 1 4 4 9

: 1 4 4 3

: 0

0 4 1

2 : 2 6 4 2

2 : 2

3 4 9

: 0

1 3 0

1 : 3 6 5 8

1 : 3

3 3 1

: 0

2 3 9

T o t a l

4 : 5 2 8 4

4 : 4

6 9 7

: 0

1 3 0

4 : 0 9 7 4

3 : 9

9 9 3

: 0

2 3 9

: 1 0 4 3

4 : 1

0 3 6

T

: 1 8 7 8

: 1 8 6 8

: 0

0 5 3

1 : 8 3 2 6

1 : 8

0 1 6

: 0

1 6 9

1 : 3 4 7 4

1 : 2

9 1 7

: 0

4 1 3

T - 1

: 1 4 6 9

: 1 4 6 1

: 0

0 5 4

2 : 6 9 5 8

2 : 6

6 8 1

: 0

1 0 3

1 : 3 4 7 9

1 : 3

1 9 6

: 0

2 1 0

T - 2

: 1 0 0 0

: 1 0 0 0

1 : 4 0 2 1

1 : 3

8 8 0

: 0

1 0 1

T a b l e 6 : M a r

k e t Q u a l i t y . T h i s T a b l e c o m

p a r e s e s t i m a t e d i n s i d e s p r e a d

( c o l u m n s 2 t o 4 ) , e s t i m a t e d d e p t h a t

t h e B B O ( c o l u

m n s 5 t o 7 ) a n d e s t i m a t e d v o l u m e s ( c o l u m n s 8 t o 1 2 ) f o r t h e L O B a n d t h e L O B & D P p

r o t o c o l ,

a n d r e p o r t s t h e p e r c e n t a g e v a r i a t i o n s ( %

=

L O B & D P L O B

L O B

) a c r o s s d i ¤ e r e n t b o o k s a n d p e r i o d s .

50



T a b l e 7 -

S y s t e m a t i c P a t t e r n i n O r d e r F l o w s

b T

2

=

[ 2 0 ]

b T

2 =

[ 1 0 ]

%

:

( b T

2

= [ 1 0 ]

)

( b T

2

= [ 2 0

] )

b T

2

= [ 2 0

]

L a r g e T r a d

e r

L O B

L O B & D P

%

:

L O B

L O B & D P

%

:

L O B

L O B & D P

L O B

& D P L O B

L O B

L O B & D P L O B

L O B

' ( 2 ; p

B 2 )

: 4 1 3 2

: 3 8 7 8

: 0

6 1 5

: 3 8 0 5

: 3 8 2 4

: 0 0 5 0

: 0

7 9 1

: 0

1 3 9

' ( 2 ; p

A 1 )

: 0 6 9 2

: 0 1 1 8

: 8

2 9 5

: 1 1 1 6

: 1 0 9 6

: 0

1 7 9

: 6 1 2 7

8 : 2

8 8 1

' ( 2 ; p M i d )

: 0 8 2 7

' ( 2 ; p

B 1 )

: 1 1 7 4

: 1 1 3 9

: 0

2 9 8

: 1 6 7 9

: 1 6 2 5

: 0

3 2 2

: 4 3 0 2

: 4 2 6 7

' ( 2 ; p

A )

: 3 4 0 0

: 3 4 5 5

: 0 1 6 2

: 1

4 4 4

' ( 2 ; p

A 1 )

: 4 0 0 1

: 4 0 3 8

: 0 0 9 2

: 1

5 0 2

S m a l l T r a d

e r

' ( 1 ; p

B 2 )

: 4 0 6 3

: 4 0 7 6

: 0 0 3 2

: 3 4 0 1

: 3 4 4 7

: 0 1 3 5

: 1

6 2 9

: 1

5 4 3

' ( 1 ; p

A 1 )

: 0 7 7 6

: 0 7 6 2

: 0

1 8 0

: 1 5 4 4

: 1 5 0 0

: 0

2 8 5

: 9 8 9 7

: 9 6 8 5

' ( 1 ; p

B 1 )

: 1 7 2 5

: 1 5 6 2

: 0

8 9 9

: 1 5 6 6

: 1 4 1 5

: 0

9 6 4

: 0

9 2 2

: 0

9 8 7

' ( 1 ; p

A 1 )

: 3 4 3 6

: 3 6 0 1

: 0 4 5 4

: 3 4 8 8

: 3 6 3 8

: 0 4 3 0

: 0 1 5 1

: 0 1 2 8

A v e r a g e P r o b a

b i l i t y

1 2P j = 1 ; 2

' ( j ; p

B 2 )

: 4 0 9 8

: 3 9 7 7

: 0

2 9 4

: 3 6 0 3

: 3 6 3 6

: 0 0 9 0

: 1

2 0 7

: 0

8 5 9

1 2P j = 1 ; 2

' ( j ; p

A 1 )

: 0 7 3 4

: 0 4 4 0

: 4

0 0 5

: 1 3 3 0

: 1 2 9 8

: 0

2 4 1

: 8 1 2 0

1 : 9

5 0 0

1 2 ' ( 2 ; p M i d )

: 0 4 1 4

1 2P j = 1 ; 2

' ( j ; p

B 1 )

: 1 4 5 0

: 1 3 5 5

: 0

6 5 5

: 1 6 2 3

: 1 5 2

: 0

6 3 2

: 1 1 9 3

: 1 2 2 2

1 2P j = 1 ; 2

' ( j ; p

A 1 )

: 3 7 1 9

: 3 8 1 5

: 0 2 6 0

: 3 4 4 4

: 3 5 4 7

: 0 2 9 8

: 0

7 3 9

: 0

7 0 2

T a b l e 7 : S y s t

e m a t i c P a t t e r n i n O r d e r F

l o w s . T h i s T a b l e r e p o r t s t h e

s u b m i s s i o n p r o b a b i l i t i e s o f t h

e o r d e r s

l i s t e d i n c o l u m n 1 f o r b o t h l a r g e a n d s m a l l t r a

d e r s , a s w e l l a s t h e a v e r a g e p r o b a b i l i t y o f m a r k e t o r d e r s , l i m i t o r d e r s

a n d D P o r d e r s . R e s u l t s a r e r e p o r t e d f o r t h e

t w o s t a t e s o f t h e b o o k b T

2 =

[ 2 0 ] a n d b T

2

=

[ 1 0 ] : F o r e a

c h s t a t e

v a l u e s a r e s h o w

n f o r b o t h t h e L O B a n d t h e

L O B & D P , a n d f o r t h e d i ¤ e r e n c e o f t h e t w o ( %

=

L O B & D

P L O B

L O

B

) :

C o l u m n s 8 a n d 9 s h o w h o w t h e s u b m i s s i o n p r o b a b i l i t i e s o f e a c h o r d e r

c h a n g e m o v i n g f r o m

b T

2 =

[ 2 0 ] t o

b T

2

=

[ 1 0 ] ; b o

t h f o r t h e L O B a n d t h e L O B & D P p r o t o c o l ( =

( b T

2

= [ 1 0 ] )

( b T

2

= [ 2 0

] )

b T

2

= [ 2 0

]

) :

51



T a b l e 8 - W e l f a r e

b T

2

=

[ 0 0 ]

b T

2

=

[ 1 1 ]

b T

2

=

[ 2 2 ]

L O B

L O B & D P

%

:

L O B

L O B & D P

%

:

L O B

L O B

& D P

%

:

L O B & D P L O B

L O B

1 0 0

L O B & D P L O B

L O B

1 0 0

L O B & D P L O B

L O B

1 0 0

T o t a l

1 : 7

4 4 3

1 : 7

4 3 5

: 0 4 3 0

1 : 8

2 2 4

1 : 8

2 5 5

: 1 7 2 9

1 : 9

2 1 3

1 : 9 2 5 8

: 2 3 4 2

T - 2

: 5 9 1 4

: 5 8 8 6

: 4 7 3 5

: 6 3 7 4

: 6 3 7 1

: 0

5 4 9

: 6 7 7 8

: 6 7 8 0

: 0 2 9 5

T - 1

: 5 8 1 8

: 5 8 0 6

: 2 0 6 3

: 6 0 2 6

: 6 0 3 0

: 0 7 4 7

: 6 3 6 4

: 6 3 7 8

: 2 2 7 9

T

: 5 7 1 1

: 5 7 4 4

: 5 6 9 1

: 5 8 2 4

: 5 8 5 5

: 5 2 3 7

: 6 0 7 1

: 6 1 0 0

: 4 6 9 4

I n s t i t u t i o n a l

2 : 3

0 2 3

2 : 3

0 7 1

: 2 0 8 5

2 : 3

7 5 5

2 : 3

8 2 7

: 3 0 3 1

2 : 5

4 7

2 : 5 5 5 5

: 3 3 3 7

T - 2

: 7 7 6 8

: 7 7 4 6

: 2 8 3 2

: 8 1 9 7

: 8 1 9 2

: 0

6 1 0

: 9 0 3 5

: 9 0 4 0

: 0 5 5 3

T - 1

: 7 6 8 5

: 7 6 8 1

: 0 5 2 0

: 7 8 7 6

: 7 8 9 0

: 1 7 7 8

: 8 4 1 1

: 8 4 3 8

: 3 2 1 0

T

: 7 5 7 0

: 7 6 4 4

: 9 7 7 5

: 7 6 8 2

: 7 7 4 5

: 8 2 0 1

: 8 0 2 4

: 8 0 7 7

: 6 6 0 5

R e t a i l T r a d e r s 1 : 1

8 6 2

1 : 1

7 9 9

: 5 3 1 1

1 : 2

6 9 2

1 : 2

6 8 3

: 0

7 0 9

1 : 2

9 5 5

1 : 2 9 6

: 0 3 8 6

T - 2

: 4 0 6 0

: 4 0 2 6

: 8 3 7 4

: 4 5 5 1

: 4 5 4 9

: 0

4 3 9

: 4 5 2 1

: 4 5 2 0

: 0

2 2 1

T - 1

: 3 9 5 0

: 3 9 3 0

: 5 0 6 3

: 4 1 7 5

: 4 1 7 0

: 1

1 9 8

: 4 3 1 6

: 4 3 1 8

: 0 4 6 3

T

: 3 8 5 2

: 3 8 4 3

: 2 3 3 6

: 3 9 6 6

: 3 9 6 4

: 0

5 0 4

: 4 1 1 8

: 4 1 2 2

: 0 9 7 1

T a b l e 8 : W e l f a r e . T h i s T a b l e r e p o r t s t h e r e s u l t s f o r t o t a l w e l f a r e a s w

e l l a s i n s t i t u t i o n a l a n d r e t a i l t r a d e r s

w e l f a r e a c r o s s d

i ¤ e r e n t p e r i o d s a n d s t a t e s o f t

h e o r d e r b o o k . F o r e a c h p e r i o d a n d e a c h s t a t e o f t h e b o o k r e s u l t s a r e

r e p o r t e d f o r b o t h t h e L O B a n d t h e L O B & D P ,

t o g e t h e r w i t h t h e d i ¤ e r e n c e b e t w e e n t h e t w o , ( %

=

L O B & D

P L O B

L O

B

) :

52



T a b l e 9 - L O B & D P & I O I - A s y m m e t r i c I n f o r m a t i o n o n t h e s t a t e o f t h e D P - L a r g e T r a d e r s

P a n e l A . 1

b T

2 = [ 2 2 ]

L O B

L O B & D P

L O B & D P & I O I

L a r g e

U n i n f

I n f

I n f

I n f

W e i g h t e d A v e r a

g e

T r a d e r

( D P = 0 )

( D P = + 6 )

( D P = - 6 )

' ( 2 ; p

B 1 )

: 4 6 8 6

: 4 6 1 2

: 4 5 4 2

: 4 6 6 7

: 4 7 2 1

: 3 8 3 6

' ( 2 ; p

A 1 )

: 0 3 1 4

: 0 1 0 9

: 0 0 2 1

: 0 2 1 0

: 0 2 4 5

: 0 1 3 3

' ( 2 ; p M i d )

: 0 2 7 9

: 0 4 3 7

: 0 1 2 3

: 5 0 3 4

: 1 0 7 9

' ( 0 )

' ( 2 ; p M i d )

: 0 2 7 9

: 0 4 3 7

: 0 1 2 3

: 5 0 3 4

: 1 0 7 9

' ( 2 ; p

B 1 )

: 0 3 1 4

: 0 1 0 9

: 0 0 2 1

: 0 2 1 0

: 0 2 4 5

: 0 1 3 3

' ( 2 ; p

A 1 )

: 4 6 8 6

: 4 6 1 2

: 4 5 4 2

: 4 6 6 7

: 4 7 2 1

: 3 8 3 6

P a n e l A . 2

b T

1 = [ 2 2 ] i n v T

2

L O B

L O B & D P

L O B & D P & I O I

L a r g e

U n i n f

I n f ( D P =

I n f ( D P =

I n f ( D P =

W e i g h t e d A v e r a

g e

T r a d e r

+ 2 , ( - 2 ) )

+ 8 o r + 4 )

- 8 o r – 4 )

' ( 2 ; p

B 1 )

: 4 7 5 0

: 4 5 0 0

: 4 4 8 0

( : 4

7 5 0 )

: 4 7 5 0

: 3 8 2 4

' ( 2 ; p

A 1 )

' ( 2 ; p M i d )

: 0 5 0 0

: 0 5 2 0

: 5 0 0 0

: 0 5 0 3

: 1 9 3 1

' ( 0 )

: 0 5 0 0

: 0 2 5 0 ( : 0 2 5 0 )

: 0 2 2 0

: 0 2 2 0

: 0 1 9 9

' ( 2 ; p M i d )

: 0 5 0 0

: 0 5 2 0

( : 5

0 0 0 )

: 0 5 0 3

: 1 9 3 1

' ( 2 ; p

B 1 )

' ( 2 ; p

A 1 )

: 4 7 5 0

: 4 5 0 0

: 4 4 8 0

: 4 7 5 0

: 4 7 5 0

: 3 8 2 4

T a b l e 9 : L O B

& D P & I O I - A s y m m e t r i c I n f o r m a t i o n o n t h e s t a t e o f

t h e D P . T h i s T a b l e r e p o r t s

i n P a n e l

A l a r g e t r a d e r s ’ s u b m i s s i o n p r o b a b i l i t i e s f o r

t h e o r d e r s l i s t e d i n c o l u m n 1 f o r t h r e e p r o t o c o l s , L O B , L O B & D P

a n d L O B & D P & I O I . F o r t h e l a t t e r , i . e . t h e

m a r k e t w i t h I O I m e s s a g e s , t h e T a b l e a l s o r e p o r t s r e s u l t s f o r b o t h

u n i n f o r m e d a n d i n f o r m e d l a r g e t r a d e r s ; f o r t h e i n f o r m e d t r a d e r s r e s u l t s a r e r e p o r t e d f o r t h e t h r e e s t a t e s o f t h e

D P , n a m e l y , e m

p t y a n d f u l l o n t h e s e l l a n d o n t h e b u y s i d e . C o l u m n 8 r e p o

r t s t h e a v e r a g e s u b m i s s i o n p r o b a b i l i t y

o f e a c h o r d e r t y

p e c o m p u t e d a s t h e w e i g h t e d a v e r a g e o f t h e o r d e r s u b m i s s i o n p r o b a b i l i t i e s o f b o t h u n i n f o r m e d a n d

i n f o r m e d t r a d e

r s . F o r e x a m p l e , t h e w e i g h t e d

a v e r a g e o f t h e s u b m i s s i o n p r

o b a b i l i t y o f D P o r d e r s t o s e l l

i s e q u a l

t o :

1 2 : 0

4 3 7 +

1 2 [

1 3 : 0

1 2 3 +

1 3 : 5

0 3 4 ] : I n P a n e l B s m a l l t r a d e r s ’ s u b m i s s i o n p r o b a b i l i t i e s a r e r e p o r t e d f o r t h e L O B & D P

a n d L O B & D P & I O I p r o t o c o l s .

53



T a b l e 9 -

L O B & D P & I O I - A s y m m e t r i c I n f o r m a t i o n o n t h e s t

a t e o f t h e D P - S m a l l T r a

d e r

P a n e l B

b T

2

=

[ 2 2 ]

b T

1

=

[ 2 2 ]

( i n v T

2 )

S m a l l T r a d

e r

L O B & D P

L O B & D

P & I O I

L O B & D P

L O B & D P & I O I

' ( 1 ; p

B 1 )

: 4 6 5 6

: 4 6

7 5

( : 4

7 5 0 )

( : 4

7 5 0 )

' ( 1 ; p

A 1 )

: 0 3 4 4

: 0 3

2 5

' ( 0 )

( : 0

5 0 0 )

( : 0

5 0 0 )

' ( 1 ; p

B 1 )

: 0 3 4 4

: 0 3

2 5

' ( 1 ; p

A 1 )

: 4 6 5 6

: 4 6

7 5

( : 4

7 5 0 )

( : 4

7 5 0 )

54



T a b l e 1 0 - A s y m m

e t r i c I n f o r m a t i o n a n d M a r k e t Q u a l i t y

B o o k

E s t i m a t e d S p r e a d

b T

2

=

[ 2 2 ]

L O B

L O B & D P

L O B & D P

% : L O B

v s

% : L O B & D P v s

& I O I

L O B & D P

L O B & D P & I O I

A v e r a g e

: 1 4 4 9

: 1 4 4 3

: 1 3 8 0

: 0

0 4 1

: 0

4 3 4

T o t a l T

: 1 8 7 8

: 1 8 6 8

: 1 7 5 7

: 0

0 5 3

: 0

5 9 4

T - 1

: 1 4 6 9

: 1 4 6 1

: 1 3 8 4

: 0

0 5 4

: 0

5 2 7

T - 2

: 1 0 0 0

: 1 0 0 0

: 1 0 0 0

E s t i m a t e d D e p t h

A v e r a g e

2 : 2

6 4 2

2 : 2

3 4 9

2 : 4

0 1 2

: 0

1 3 0

: 0 7 4 4

T o t a l

4 : 5

2 8 4

4 : 4

6 9 7

4 : 8

0 2 3

: 0

1 3 0

: 0 7 4 4

T

1 : 8

3 2 6

1 : 8

0 1 6

1 : 9

8 7 2

: 0

1 6 9

: 1 0 3 0

T - 1

2 : 6

9 5 8

2 : 6

6 8 1

2 : 8

1 5 1

: 0

1 0 3

: 0 5 5 1

T - 2

E s t i m a t e d V o l u m e

A v e r a g e

1 : 3

6 5 8

1 : 3

3 3 1

1 : 1

9 6 3

: 0

2 3 9

: 1

0 2 6

T o t a l

4 : 0

9 7 4

3 : 9

9 9 3

3 : 5

8 9 0

: 0

2 3 9

: 1

0 2 6

T

1 : 3

4 7 4

1 : 2

9 1 7

1 : 1

6 9 5

: 0

4 1 3

: 0

9 4 6

T - 1

1 : 3

4 7 9

1 : 3

1 9 6

1 : 1

8 4 9

: 0

2 1 0

: 1

0 2 1

T - 2

1 : 4

0 2 1

1 : 3

8 8 0

1 : 2

3 4 6

: 0

1 0 1

: 1

1 0 5

D P

L O B & D P +

D P

D P & I O I

L O B & D P & I O

I + D P

T o t a l

: 1 0 4 3

4 : 1

0 3 6

: 5 8 1 2

4 : 1

5 4 9

T a b l e 1 0 : A s y m m e t r i c I n f o r m a t i o n a n d

M a r k e t Q u a l i t y . T h i s T a

b l e c o m p a r e s e s t i m a t e d i n s i d e

s p r e a d ,

e s t i m a t e d d e p t h a t t h e B B O a n d e s t i m a t e d v

o l u m e s f o r t h e t h r e e p r o t o c o l s , L O B , L O B & D P a n d L O B & D

P & I O I .

C o l u m n 5 r e p o

r t s t h e c o m p a r i s o n o f t h e t h r e e i n d i c a t o r s o f m a r k e t q u a l i t y

b e t w e e n t h e L O B a n d t h e L O B & D P

m a r k e t s ( %

=

L O B & D P L O B

L O B

) , w h e r e a s c o

l u m n 6 r e p o r t s r e s u l t s f o r t h

e c o m p a r i s o n b e t w e e n L O B &

D P a n d

L O B & D P & I O I ( %

=

L O B & D P & I O I L O B & D P

L O B & D P

) : R e s u l t s a r e r e p o r t e d f o r b T

2

=

[ 2 2 ] a n d a c r o s s d i ¤ e r e n t

p e r i o d s .

T h e l a s t r o w r e p o r t s t o t a l v o l u m e s f o r e a c h m a r k e t p r o t o c o l .

55



T a b l e 1 1 - A s y m m e t r i c I n f o r m a t i o n o n t h e s t a t e o f

t h e D P

- W e l f a r e

b T

2

=

[ 2 2 ]

L O B

L O B & D P

L O B & D P & I O I

% : L O B v s

% : L O B & D P

v s

L O B & D P

L O B & D P & I O

I

T o t a l

1 : 9

2 1 3

1 : 9

2 5 8

1 : 9

6 8 6

: 2 3 4 2 %

2 : 2

2 2 5 %

T - 2

: 6 7 7 8

: 6 7 8 0

: 6 8 7 8

: 4 6 9 4 %

1 : 4

4 5 4 %

T - 1

: 6 3 6 4

: 6 3 7 8

: 6 4 8 8

: 2 2 7 9 %

1 : 7

2 4 7 %

T

: 6 0 7 1

: 6 1 0 0

: 6 3 2 0

: 0 2 9 5 %

3 : 6

0 6 6 %

R e

t a i l T r a d e r s

1 : 2

9 5 5

1 : 2

9 6

1 : 3

0 3 9

: 0 3 8 6 %

: 6 0 9 6 %

T - 2

: 4 5 2 1

: 4 5 2 0

: 4 5 1 8

: 0

2 2 1 %

: 0

4 4 2 %

T - 1

: 4 3 1 6

: 4 3 1 8

: 4 3 5 0

: 0 4 6 3 %

: 7 4 1 1 %

T

: 4 1 1 8

: 4 1 2 2

: 4 1 7 1

: 0 9 7 1 %

1 : 1

8 8 7 %

I n s t i t u t i o

n a l ( U n i n f o r m e d )

2 : 5

4 7

2 : 5

5 5 5

2 : 5

7 5 1

: 3 3 3 7 %

: 7 6 7 0 %

T - 2

: 9 0 3 5

: 9 0 4 0

: 9 0 5 5

: 0 5 5 3 %

: 1 6 5 9 %

T - 1

: 8 4 1 1

: 8 4 3 8

: 8 5 3 8

: 3 2 1 0 %

1 : 1

8 5 1 %

T

: 8 0 2 4

: 8 0 7 7

: 8 1 5 8

: 6 6 0 5 %

1 : 0

0 2 8 %

I n s t i t u t

i o n a l ( I n f o r m e d )

2 : 6

9 1 3

T - 2

: 9 4 2 1

T - 1

: 8 7 1 4

T

: 8 7 7 8

e 1 1 : A s y m m e t r i c I n f o r m a t i o n o n

t h e s t a t e o f t h e D P

- W

e l f a r e .

T h i s T a b l e r e p o r t s r e s u l t s

o t a l w e l f a r

e a s w e l l a s i n s t i t u t i o n a l a n d

r e t a i l t r a d e r s w e l f a r e a c r o s s d

i ¤ e r e n t p e r i o d s a n d m a r k e t p r o t o c o l s .

l t s a r e a l s

o r e p o r t e d f o r b o t h u n i n f o r m

e d a n d i n f o r m e d i n s t i t u t i o n a l t r a d e r s . F o r e a c h p e r i o d r e s u l t s a r e

f o r t h e L

O B , t h e L O B & D P , a n d t h e L O B & D P % I O I t o g e t h e r w i t h

t h e d i ¤ e r e n c e b e t w e e n t h e L

O B a n d

L O B & D P

( c o l u m n 5 : %

=

L O B & D P L

O B

L O B

) ; a n d t h e L O B & D P a n d t h e L O B & D P & I O I ( c o l u m n 6 : %

=

D P & I O I L O B & D P

L O B & D P

) :

Dark Pool Trading Strategies 2011

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