market transparency, market quality, and sunshine trading

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Market transparency, market quality, andsunshine trading$

M. Ángeles de Frutosa, Carolina Manzanob,n

aDepartment of Economics, Universidad Carlos III de Madrid, SpainbDepartment of Economics and CREIP, Universitat Rovira i Virgili, Av. de la Universitat 1, Reus 43204, Spain

Received 23 November 2012; received in revised form 3 June 2013; accepted 3 June 2013

Abstract

This paper analyzes the implications of pre-trade transparency on market performance. In competitivemarkets, transparency increases market liquidity and reduces price volatility, whereas these results may nothold under imperfect competition. More importantly, market depth and volatility might be positively relatedwith proper priors. Moreover, we study the incentives for liquidity traders to engage in sunshine trading. Weobtain that the choice of sunshine/dark trading for a noise trader is independent of his order size. The traderswith higher liquidity needs are more interested in sunshine trading, as long as this practice is desirable.& 2013 Elsevier B.V. All rights reserved.

JEL classification: D82; G12; G14

Keywords: Market microstructure; Transparency; Prior information; Market quality; Sunshine trading

1. Introduction

One of the most surprising phenomena in the microstructure of financial markets is theheterogeneity in pre-trade transparency exhibited by different trading venues.1 Although onecould argue that most of the modern stock trading platforms distribute information on depth,accessible to traders either by subscribing directly to the market feed, or by purchasing

see front matter & 2013 Elsevier B.V. All rights reserved.rg/10.1016/j.finmar.2013.06.001

uthor acknowledges financial support from Project ECO2012-34581, and the second author from Project33. We would like to thank Jordi Caballé and Montserrat Ferré for helpful comments.ding author. Tel.: +34 977 758914.dress: [email protected] (C. Manzano).ransparency refers to the wide dissemination of price quotations and orders before trade takes place.

his article as: de Frutos, M.A., Manzano, C., Market transparency, market quality, and sunshine trading.inancial Markets (2013), http://dx.doi.org/10.1016/j.finmar.2013.06.001


dx.doi.org/10.1016/j.finmar.2013.06.001





mailto:[email protected]




M.A. de Frutos, C. Manzano / Journal of Financial Markets ] (]]]]) ]]]–]]]2

a consolidated feed, it is also true that in the last ten years there has been a tendency to introduceanonymity into stock, bond, and foreign exchange markets.2 Similarly, we are nowadaysenvisioning the evolution to dark trading in exchange markets.3 An investigation on dark tradingcan be found in Bloomfield, O'Hara, and Saar (2011). They compare visible markets in which allorders must be displayed, iceberg (or reserve) markets that allow both displayed and partiallydisplayed orders, and hidden markets in which orders can be non-displayed.4

There is broad agreement that transparency matters; it affects the informativeness of the orderflow and, hence, the process of price discovery, but the key effects of transparency on securitymarkets are complex and contradictory. As pointed out by Eom, Ok, and Park (2007, p. 319) “…there is no consensus on whether an increase in pre-trade transparency results in an improvementor deterioration in market quality.” These authors study changes in pre-trade transparency in theKorea Exchange. They conclude that market quality is increasing in pre-trade transparency. Inthe same line, Boehmer, Saar, and Yu (2005) find that the introduction of OpenBook by theNYSE leads to a more active management of trading strategies and improvements in terms ofliquidity and informational efficiency. These results contrast with findings derived in Madhavan,Porter, and Weaver (2005). This paper shows that an increase in pre-trade transparency inthe Toronto Stock Exchange leads to wider spreads, lower depth, and higher volatility. Thisis consistent with the empirical evidence from the French Stock Exchange, where liquidityincreased after anonymity was introduced (Foucault, Moinas, and Theissen, 2007), the sameoccurred when brokers identification codes were removed at the Tokyo Stock Exchange(Comerton-Forde, Frino, and Mollica, 2005). Similarly, the experiments by Bloomfield, O'Hara,and Saar (2011) support the robustness of informational efficiency and liquidity in opaqueregimes too.In this paper we focus on anonymity, which is a particular aspect of pre-trade transparency.

When broker identities are displayed, investors learn about order flows.5 Thus, we investigate theeffects of disclosing information about the composition of the order flow to market participants.Depending on who makes the disclosure decision, two types of pre-trade transparency can bedistinguished: mandatory/prohibited and voluntary. In the former, the decision whether to reveal(or not to reveal) information about the composition of the order flow is taken by the exchange.In the later, the investors voluntarily decide whether to reveal the orders.One of the studies focused on the implications of the first type of pre-trade transparency is

Madhavan (1996). He compares two trading mechanisms, called opaque and transparent. In theopaque market, the exchange does not reveal any information about the composition of the orderflow, whereas in the transparent market the exchange reveals the price insensitive component thatcomes from traders who have liquidity needs. He shows that there exists an inverse relationshipbetween price volatility and market depth; for some parameter configurations, an increasein transparency delivers the desirable properties of higher liquidity and lower price volatility,whereas for others it can exacerbate volatility and decrease liquidity. These results are derived

2Anonymity was introduced into the French Stock Exchange in 2001 and into the Italian one in 2004.3Dark pools are trading systems where there is no pre-trade transparency of orders in the system. They can be split into

two types: systems such as crossing networks in which cross orders are not subject to pre-trade transparency requirements,and trading venues, such as regulated markets and MTFs, that use waivers for avoiding to display orders. By contrast, litmarkets are pre-trade transparent.

4Other issues related to pre-trade transparency are the comparison of floor versus automated trading systems and thelogic for trading halts, among others.

5Linnainmaa and Saar (2012) show that broker identities can be used to extract meaningful signals about the types ofinvestors who initiate trades.

Please cite this article as: de Frutos, M.A., Manzano, C., Market transparency, market quality, and sunshine trading.Journal of Financial Markets (2013), http://dx.doi.org/10.1016/j.finmar.2013.06.001




M.A. de Frutos, C. Manzano / Journal of Financial Markets ] (]]]]) ]]]–]]] 3

under the proviso that rational investors hold improper or non-informative priors about theliquidation value of the risky asset.6 We here propose to frame Madhavan's analysis in a morecanonical way. We assume that rational investors are endowed with proper priors.

We show that in competitive markets, transparency increases market liquidity and reduces pricevolatility,7 whereas under imperfect competition, the implications of market transparency areambiguous. More importantly, the inverse relationship between price volatility and market liquidity,obtained in competitive markets or when investors have improper priors, may not hold with imperfectcompetition and proper priors (i.e., the inverse relationship between market depth and price volatilityreported in Madhavan, 1996 may not hold if investors have proper priors).8 If market depth andvolatility are negatively related, an increase in transparency either stabilizes prices and increases marketliquidity (both of them suitable properties of a financial market) or increases volatility and reducesliquidity (both of them undesirable for a market). Since preferences for both of these market indicatorsare aligned, one could conclude that transparency is unambiguously a good or a bad property for amarket to have. However, if such a (negative) relation does not hold, then there are trade-offs amongthese two market indicators, and a clear ranking between transparent and opaque markets may not exist.

Additionally, we find that transparency increases the precision of traders' predictions aboutthe liquidation value. However, the comparison on market liquidity across market structures alsodepends on prior specification, as with proper priors the opaque market is deeper for a largerparameter specification set.

This paper also addresses the issue of voluntary disclosure of the orders, prior to trading, of someliquidity traders to the other participants, a practice known as sunshine trading. We study theincentives for liquidity traders to engage in sunshine trading. We obtain that the choice of sunshine/dark trading for a noise trader is independent of his order size. Moreover, as long as sunshinetrading is desirable, the traders with higher liquidity needs find this practice more profitable.

Sunshine trading has also been analyzed by Admati and Pfleiderer (1991).9 They find that theidentification of liquidity orders reduces the trading costs of those who preannounce, but its effects onthe trading costs and welfare of other traders are ambiguous. They also show that sunshine tradingincreases the informativeness of the price, whereas it reduces the variance of the price change. Admatiand Pfleiderer consider a continuum of informed agents, who are price-takers and their motive fortrading is information. We consider a finite number of informed traders, who behave strategically andtheir motive for trading is information and hedging. Thus, our results assess the impact of relaxing theassumption of a continuum of informed agents in their conclusions. Another difference betweenAdmati and Pfleiderer's paper and ours is that we endogenize the choice of the preannouncement.

The remainder of this paper is organized as follows. Section 2 outlines the notation and thehypotheses of the model. Section 3 characterizes the unique symmetric linear equilibrium in ageneral framework. Section 4 examines the implications of transparency when the disclosuredecision is taken by the exchange and Section 5 deals with the choice of sunshine/dark trading byindividuals. Section 6 provides some concluding comments. Finally, proofs are relegated to theAppendix.

6Bayesians believe only in proper priors as a foundation for statistics. For a discussion on the relationship betweenproper and improper prior distributions, see O'Hagan (1994).

7See also Rindi (2008) for a comparison of anonymous versus pre-trade transparent regimes under perfect competition.8Although we lack a clear-cut intuition for the need of uninformative priors to deliver Madhavan's (1996) results, we

believe that the arguments behind them are along the following lines. The use of improper priors is equivalent to the useof no prior information. By contrast, informative priors convey information on what are the relevant values of theparameters when it comes to the study of the properties that a given model satisfies.

9Dia and Pouget (2011) find evidence of sunshine trading in the West-African Bourse.






2. The model

We consider a pure exchange economy where at time 0 a risky asset is traded in an auctionmarket against a riskless bond, whose return is normalized to zero. The liquidation value of therisky asset in period 1 is represented by the random variable ~v. We assume that ~v is normallydistributed with finite mean v and variance s2v .

10 The risky asset is traded at the automaticmarket-clearing price p, so that its return is given by its liquidation value minus the price.In this economy there are N strategic traders indexed by n who are endowed with assets and

are privately informed about the liquidation value of the risky asset. Trader n holds an initialendowment of yn shares of the riskless asset and an initial endowment of ωn shares of the riskyasset. Traders have CARA preferences over final wealth of the form Uð ~WnÞ ¼ −expf−ρ ~Wng,where ρ40 denotes the common coefficient of risk aversion and ~Wn represents the final wealthfor the n-th investor, which is given by

~Wn ¼ ð~v−pÞqn þ ~vωn þ yn;

where qn denotes the units of the risky asset traded by investor n.11 Thus, traders are risk averse and

they choose their desired order quantities to hedge their risk exposures. But portfolio hedging is nottheir sole motive for trade, as strategic traders posses private information. Before trading takesplace, each strategic trader receives a private signal conveying information about the liquidationvalue of the risky asset. Let ~sn ¼ vþ ~εn denote the private signal of the n-th trader, where v is therealized value of the risky asset and ~εn is a random variable that represents the private signal errorfor the n-th investor.12 Traders use their information to speculate in the market so that part of tradeis information-motivated with transaction volume arising endogenously. In addition to the demandsof the speculative traders, there is another component of order flow, denoted by ~z, which representsthe imbalances arising from uninformed or noisy traders whose demands are price inelastic. Weassume that ~z is independent of the remainder of random variables in the model.Informed traders submit their (net) demand functions to a unique auctioneer, who aggregates

all the demand schedules and the noisy demand and calculates the price at which the marketclears. At this price, the auctioneer allocates quantities to satisfy traders' demands.We end the model assuming that all the random variables ~ε1;…; ~εN ; ~ω1;…; ~ωN are uncorrelated

with ~v, and normally and independently distributed with a mean normalized to zero and a varianceequal to s2ς for ς¼ ε;ω.13 Finally, their joint distribution is assumed to be common knowledge.In this set-up, we analyze several trading mechanisms that differ in their degree of transparency

regarding the random imbalance of uninformed liquidity traders. The first mechanism is opaque in thatrational traders do not receive any information about the composition of the order flow. In thisframework, we assume that ~z is normally distributed with a mean normalized to zero and a varianceequal to s2z . The second one is transparent so that the realization of ~z is disclosed to all market

10As in most models of market microstructure (e.g.; Hellwig, 1980; Diamond and Verrecchia, 1981; Glosten andMilgrom, 1985; Kyle, 1985, 1989; Huddart, Hughes and Levine, 2001, among others), we suppose that priors are proper.By doing so, we are able to determine how the strength of prior information affects the robustness of the results on marketmetrics provided by Madhavan (1996). Moreover, we will try to recover his results, whenever possible, by taking a largevariance of ~v. The use of limits to recover the results under improper priors is standard. See, for instance, Morris and Shin(2003) in the context of global games.

11The derivation of this expression is provided in the Appendix (see computations previous to Eq. (A.1)).12Throughout the paper, given the random variable ~x, we denote its realized value by x.13Models assuming a CARA utility function plus a normal distribution of all random variables abound in the market

microstructure literature. Moreover, they are the most commonly used even nowadays, as, for instance, in Rindi (2008).






participants prior to trading. In this setup, the expected value of ~z is the realization of this randomvariable, z, and its variance is null. Finally, we examine a trading mechanism with the possibility ofsunshine trading (i.e., a framework where some liquidity traders voluntarily preannounce the size oftheir orders).

3. The symmetric linear equilibrium in a general framework

In order to compute the optimal demand schedules in the aforementioned trading protocols, wedefine and characterize a symmetric linear equilibrium in a more general framework, assumingthat the expected value of the noise demand, denoted by z, may not be null. Then, giving valuesto the expected value and the variance of the noise demand, we obtain the characterizations ofthe symmetric linear equilibria for the different trading mechanisms. For instance, in the opaquemarket z ¼ 0 and s2z40, whereas in the transparent market, z ¼ z and s2z ¼ 0.14

Let qnð�; InÞ denote the net demand schedule of informed trader n, given that he has observed theinformation set In, where In ¼ ðsn;ωn; ynÞ (i.e., In contains the private signals observed by trader n).Thus, qn ¼ qnðp; InÞ represents the net quantity demanded by the n-th informed investor, for particularrealizations of both the price of the risky asset ~p and the vector that collects his information ~I n.

In this economy, we search for a rational expectations equilibrium under imperfect competition(REE, for short) in which traders follow identical linear demand functions. An REE definesa Bayes-Nash equilibrium in demand functions; it is a price and a set of demand schedules suchthat the market clears and each trader n, given his information set, submits demand orders thatmaximize his conditional expected utility, taking into account the effect of his trading on pricesand taking as given the strategies of other traders. We next define it formally.

Definition 1. A rational expectations equilibrium is a price p and a vector of strategiesq¼ ðq1;…; qNÞ, such that:

(i)

14

15

indethe d

PleJo

Excess demand is zero at the equilibrium price:

∑N

n ¼ 1qnðp; InÞ þ z¼ 0:

(ii)
Each trader n maximizes the expected utility of final period wealth given the strategies ofother traders,
qnðp; InÞ∈arg maxfqng

EðU½ð~v−pÞqn þ ~vωn þ yn�jp; InÞ for n¼ 1;…;N:

As mentioned above, for the tractability of the analysis, we focus on the symmetric linear
rational expectations equilibrium (SLE, for short).
Definition 2. A symmetric linear equilibrium is a rational expectations equilibrium in whichtraders' net demand functions are identical linear functions, that is,

qnðp; InÞ ¼ μþ βsn−αωn−γp; ð1Þwhere μ, β, α, and γ are constants.15

We thank an anonymous referee for this suggestion.With CARA utility, note that the solution of the optimization problem for investor n stated in Definition 1 ispendent of the endowment of the riskless asset for this trader. This is the reason why we omit yn in the expression ofemand function for investor n.

ase cite this article as: de Frutos, M.A., Manzano, C., Market transparency, market quality, and sunshine trading.urnal of Financial Markets (2013), http://dx.doi.org/10.1016/j.finmar.2013.06.001





Following Kyle (1989), in a SLE the optimal demand function for trader n is given by

qnðp; InÞ ¼Eð~vjp; InÞ−p−ρ varð~vjp; InÞωn

1ðN−1Þγ þ ρ varð~vjp; InÞ

: ð2Þ

This demand maximizes traders' utility whenever the second order condition for a maximumholds, or, equivalently, if the inequality below is satisfied:16

2ðN−1Þγ þ ρ varð~v p; InÞ40:

�� ð3Þ

To study the existence of a SLE, we first write all coefficients of speculators' demands asfunctions of the coefficient α. This coefficient is then characterized as a root of a polynomial. Ifsuch a root exists, then one might conclude that a SLE exists. These facts are formally stated inthe following results:

Lemma 1. If there exists a SLE, then:

μ¼ 2αρs2vðN−ðN−1ÞαÞ

v−

N−2N−1

−α

N−ðN−1Þα z; ð4Þ

β¼ α

ρs2ε; ð5Þ

and

γ ¼ α

ρs2ε1þ 2s2ε

ðN−ðN−1ÞαÞs2v

� �: ð6Þ

Lemma 2. If there exists a SLE, α∈ð0; ðN−2Þ=ðN−1ÞÞ. In addition, this coefficient is a root ofa polynomial of third degree QðαÞ ¼ aα3 þ bα2 þ cα−d, whose coefficients are given bya¼ ðϕþ 1ÞðN−1Þ2, b¼ ðN−ϕðN−2ÞÞðN−1Þ, c¼ ðN−1Þφ, and d¼ φðN−2Þ, with ϕ¼ ρ2s2εs

2ω

and φ¼ ρ2s2εs2z .

Lemma 2 shows that α is a function of N and of two payoff unrelated components in the trade:a liquidity component due to traders' endowments of the risky asset as endowments can bethought of as liquidity shocks, ϕ¼ ρ2s2εs

2ω, and a liquidity component due to the presence of

noise traders, φ¼ ρ2s2εs2z .17

We end this subsection providing a necessary and sufficient condition for the existence anduniqueness of a SLE.

Proposition 1. When s2z≠0, there exists a unique SLE iff N≥3. By contrast, when s2z ¼ 0, thereexists a unique SLE iff NoðN−2Þϕ.

16A complete derivation of Eqs. (2) and (3) can be found in the Appendix.17A similar decomposition is used by Naik, Neuberger, and Viswanathan (1999) to characterize the SLE in a model

with interdealer trading.






This result shows the existence and uniqueness of the SLE under general conditions. Whens2z≠0, we only require that N≥3, as in Kyle (1989).18 In addition, Proposition 1 indicates thatwhen s2z ¼ 0, a SLE may fail to exist as existence requires that the liquidity shock viaendowments is large enough. The rationale for this inequality is as follows. Whenever the price isnot a good estimator of the private information (because either s2ω or s2ε are high), or,alternatively, agents face a high inventory cost from maintaining their initial holdings of the riskyasset (the coefficient of risk aversion is high), then they find it profitable to participate inthe market as ϕ will be large enough. But if traders are well informed and/or there is littleendogenous liquidity needs (s2ω, s

2ε , and ρ are low so that ϕ is also low), then agents are

unwilling to reveal their information to others as they find the readjustment of their portfolio veryexpensive. They prefer to consume their initial endowment and this results in a marketbreakdown. When this is the case, a SLE fails to exist.

4. Transparency and its implications

In this section, we analyze and compare the opaque mechanism with the transparentmechanism. Recall that in the first one rational traders do not receive any information about thecomposition of the order flow, whereas in the second the realization of the noise demand isdisclosed to all market participants prior to trading. In what follows, we use superscripts O and Tto refer to the opaque and the transparent markets, respectively.

Recall that in the opaque market z ¼ 0 and s2z40, whereas in the transparent market z ¼ z ands2z ¼ 0. Substituting these values in Lemmas 1 and 2, and Proposition 1, we obtain the followingresults:

Corollary 1. A SLE in an opaque market exists iff N≥3. If it exists, then:

μO ¼ 2αO

ρðN−ðN−1ÞαOÞs2vv;

βO ¼ αO

ρs2ε;

and

γO ¼ αO

ρs2ε1þ 2s2ε

ðN−ðN−1ÞαOÞs2v

� �;

where αO is the unique real root belonging to ð0; ðN−2Þ=ðN−1ÞÞ of the polynomial of degreethree QðαÞ.Corollary 2. A SLE in a transparent market exists iff NoðN−2Þϕ. If it exists, then:

μT ¼ μT0−μT1 z¼

2αT

ρðN−ðN−1ÞαT Þs2vv−

N−2N−1

−αT

N−ðN−1ÞαT z;

βT ¼ αT

ρs2ε;

18When N¼2, the polynomial QðαÞ, stated in Lemma 2, simplifies to QðαÞ ¼ ðϕþ 1Þα3 þ 2α2 þ φα, which lacks ofreal strictly positive roots.






and

γT ¼ αT

ρs2ε1þ 2s2ε

ðN−ðN−1ÞαT Þs2v

� �;

where αT ¼ ðϕðN−2Þ−NÞ=ððϕþ 1ÞðN−1ÞÞ.Remark 1. Using the expression of αT , it follows that the coefficient associated with z, i.e., −μT1 ,is equal to −1=ðN þ ϕÞ. The negativeness of this coefficient indicates that in the transparentmarket, strategic traders place orders that partly accommodate the noise demand.

4.1. Transparency and equilibrium comparison

Next, we discuss the impact of transparency on the strategic behavior of investors, as itinfluences the existence of equilibrium, as well as the price intercept and slope of traders'demands.Regarding the existence of an equilibrium, note that if a SLE exists for the transparent market,

it exists for the opaque market as well. The condition for existence in the former, NoðN−2Þϕ,must meet N42 but, as N is a natural number, then it requires N≥3. Note that this is thecondition for existence in the latter. Thus, trading is more robust in the opaque market than in thetransparent market. In other words, transparency may induce a form of market failure.19

As for the shape of the demand functions, transparency has two main effects. An increment inthe price of the risky asset makes agents more optimistic about its liquidation value, which leadsto a smaller reduction in the individual demands as compared to the opaque market. Second,demands become less sensitive to traders' liquidity shocks and private signals. The rationaleis that in the transparent market, there is a higher need for camouflage, which makes demandsless sensitive to private information. Corollary 3 summarizes these results.

Corollary 3. Traders' demands are less sensitive to both private information and price in atransparent market as

αToαO; βToβO and γToγO:

In sum, transparency reduces endogenous liquidity trading creating less risk sharing (αToαO),makes orders less responsive to private information about the liquidation value (βToβO), andreduces demands' price-responsiveness ðγToγOÞ.

4.2. Transparency and market quality

In this subsection we examine the economic implications of pre-trade transparency. Specifically,we analyze the differences between the opaque and the transparent market in terms of some measuresof market quality.

4.2.1. Informational efficiencyThe market microstructure literature provides several sensible measures of the informational

efficiency of a market.20 One is the precision of the information held by informed traders,

19This form of market failure refers to the non-existence of the type of equilibrium we focus on, the SLE. For thecomparison to be meaningful, we focus on parameter values that satisfy NoðN−2Þϕ.

20See Kyle (1989) for a thorough discussion of these measures.






measured by var−1ð~vjp; InÞ. Another is the informational content of the equilibrium price, whichcaptures the information revealed by prices to uninformed traders, measured by var−1ð~vjpÞ.Propositions 2 and 3 compare these measures of the informational efficiency.

Proposition 2. At the time the trade is made and z is realized, pre-trade transparencyunambiguously increases the precision of the information held by informed traders.

In the transparent market, the information held by agents is more precise than in the opaquemarket. Next, we show that prices by themselves may not contribute to this greater precision asin equilibrium var−1ð~vjpOÞ might be larger than var−1ð~vjpT Þ.Proposition 3. (i) Prices are more informative in the transparent market iff the followinginequality holds:

αOo αT

1−NμT1¼ ðN þ ϕÞðϕðN−2Þ−NÞ

ϕðϕþ 1ÞðN−1Þ :

(ii) If ϕð2þ N2−4NÞ−N240 holds, or if s2z is low enough, then prices are always moreinformative in the transparent market.21

Notice that from the expressions of the market clearing price, given by

~pO ¼ 1NγO

NμO þ βO ∑N

j ¼ 1~sj−αO ∑

N

j ¼ 1~ωj þ ~z

!ð7Þ

and

~pT ¼ 1NγT

NμT0 þ βT ∑N

j ¼ 1~sj−αT ∑

N

j ¼ 1~ωj þ ð1−NμT1 Þ~z

!; ð8Þ

it follows that

var−1ð~vjpOÞ ¼ var−1 ~v�� 1ρs2ε

∑Nj ¼ 1sj−∑

Nj ¼ 1ωj þ z

αO

� �

and

var−1ð~vjpT Þ ¼ var−1 ~v�� 1ρs2ε

∑Nj ¼ 1sj−∑

Nj ¼ 1ωj þ

ð1−NμT1 ÞzαT

� �:

As ~v and ~z are uncorrelated, prices are more informative in the transparent market ifαOoαT=ð1−NμT1 Þ. Transparency has two opposite effects on the informativeness of prices: onthe one hand, it reduces endogenous liquidity trading (αO4αT ), making prices less informative.On the other hand, it facilitates that noise trading be accommodated (μT140), leading prices tobe more informative. The dominant effect is in general ambiguous, but there are at least twoinstances when transparency enhances price informativeness. If s2z is small enough, then αO

approaches αT making the first effect insignificant. As the second effect is independent of s2z ,transparency increases the informational content of prices. Similarly, if ϕ is large enough, thentransparency increases the informativeness of prices. The first effect is not very significant

21A particular case in which prices are more informative in the transparent market is when N converges to infinity, anoutcome consistent with what is reported in Admati and Pfleiderer (1991).






as αO−αT is small when ϕ is large. The second effect, despite being small as well, becomes thedominant one.

4.2.2. Market liquidityIn order to measure the impact of transparency on market liquidity, we now compare the

market depth in the two market structures. Following Kyle (1985), the market depth is defined asthe quantity of noise trading required to move the price of the risky asset by one unit. Formally,market depth is given by

∂ ~pO

∂~z

� �−1

¼ NγO and∂ ~pT

∂~z

� �−1

¼ NγT

1−NμT1:

From Eq. (2) and the market clearing condition, it follows that

p¼∑N

n ¼ 1Eð~vjhMn ; snÞ þ1

ðN−1ÞγM� �

zþ ðz−∑Nj ¼ 1ωnÞρ varð~vjhMn ; snÞ

N;

with

hMn ¼ βM∑j≠n

sj−αM∑j≠n

ωj þ δfM ¼ Ogz; M ¼O;T ;

where δfM ¼ Og is an indicator function that takes value one if M¼O and zero if M ¼ T . Thus,noise trading affects market price through three channels captured by the three terms in equationabove: an adverse-selection effect, a strategic-behavior effect, and a risk-bearing effect.The adverse-selection effect is captured by the first term via hMn . An increase in z increases hOn

without affecting hTn (in the transparent market this effect is absent as noise demand is displayed).Speculator n assumes that an increment might be due to his competitors receiving favorablesignals about the payoff of the risky asset. Each speculator therefore adjusts his forecast upwards,which generates a price boost. The strategic-behavior effect, the second term z=ððN−1ÞNγMÞ,measures the competitiveness of the market or its size via N, as well as the price sensitivity ofstrategic traders' demands. As they are less price-sensitive in the transparent market, this secondeffect is stronger in the transparent market.22 Finally, there is a risk-bearing effect. The market-clearing price must accommodate for inducing risk-averse speculators to trade. As speculatorsare better informed in the transparent market, this third effect is more important in the opaquemarket.Whenever N converges to infinity, the strategic-behavior effect vanishes. The equilibrium

price is unambiguously more sensitive to changes in the noise demand in the opaque market. Thetransparent market is deeper. This result may no longer hold under imperfect competition.Two sufficient conditions for obtaining a larger market depth in a transparent market are given

in Proposition 4.

22To clarify the role of the strategic-behavior effect, note that under perfect competition the optimal investors' demandsare given by


ρ varð~vjp; InÞ(e.g.; Diamond and Verrecchia, 1981). Using these demands' expressions and the market clearing condition, it followsthat the equilibrium price has only the terms associated with the adverse-selection effect and the risk-bearing effect.






Proposition 4. If

s2εð1þ ϕÞNs2v

þ 1−ϕðN2−4N þ 2Þ

N2

� �o0; ð9Þ

or, alternatively, if s2z is sufficiently small, then the transparent market is deeper.

Proposition 4 suggests that the comparison of the market liquidity between the two marketstructures is ambiguous and depends on the parameter specification. The intuition behindProposition 4 runs as follows. When s2z is small enough, αO is close to αT , making γO−γT verysmall, so that transparency increases market depth. Regarding the sufficient condition in (9), itsintuition is less clear. When N converges to infinity, it simplifies to 1oϕ. This inequalitycoincides with the condition that guarantees the existence of the SLE in very competitivemarkets. Thus, transparency increases market liquidity if the market is sufficiently competitive.

Let DL represent the difference in liquidity among the two markets. Formally,

DL¼ ∂ ~pO

∂~z

� �−1

−∂ ~pT

∂~z

� �−1

¼ NγO−NγT

1−NμT1:

Fig. 1 displays the differences in liquidity, DL, in terms of s2z for different values of N. The solidcurve corresponds to N¼10, the dashed one to N¼20, and the dotted one to N¼30. A negativevalue of DL indicates that the transparent market is deeper. Fig. 1 illustrates that as N increases,the parameter configuration set in which the transparent market is more liquid is higher. In thelimit, when N converges to infinity, the transparent market is more liquid for all s2z .

The ambiguity of the effect of changes in pre-trade transparency on liquidity derived in thiswork is consistent with the empirical evidence. Madhavan, Porter, and Weaver (2005) find thatan increase in pre-trade transparency in the Toronto Stock Exchange led to lower depth. Thisresult contrasts with the empirical investigation of pre-trade transparency at the NYSE conductedby Boehmer, Saar, and Yu (2005).

4.2.3. VolatilityThe volatility of equilibrium prices is measured by varð~v− ~pMÞ. Straightforward computations

yield23

~v− ~pO ¼ ~v−μO

γO−βO

γO∑N

j ¼ 1~sjN

þ αO

γO∑N

j ¼ 1 ~ωj

N−

1NγO

~z

and

~v− ~pT ¼ ~v−μT0γT

−βT

γT∑N

j ¼ 1 ~sjN

þ αT

γT∑N

j ¼ 1 ~ωj

N−1−NμT1NγT

~z:

Therefore,

varð~v− ~pMÞ ¼ ∂~pM

∂~z

� �2

s2z þ gðαM ; rÞ; with ð10Þ

23Formulae below can be easily derived from Expressions (7) and (8).





20 40 60 80 100 120 140 160 180 200

-14-12-10

-8-6-4-202468

101214

variance of z

DL

Fig. 1. Difference in market liquidity, DL, as a function of s2z .


gðαM ; rÞ ¼ 1−βM

γM

� �2

s2v þβM

NγM

� �2

Ns2ε þαM

NγM

� �2

Ns2ω; ð11Þ

where r stands for the quotient s2ε=s2v . Substituting the values of the equilibrium coefficients,

gðαM ; rÞ simplifies to

gðαM ; rÞ ¼ s2εð4Nr þ ðϕþ 1ÞðN−αMðN−1ÞÞ2ÞNðN−αMðN−1Þ þ 2rÞ2 :

To analyze the difference in price volatility,

DV ¼ varð~v− ~pOÞ−varð~v− ~pT Þ;we decompose it into two terms, DV1 and DV2, with

DV1 ¼∂ ~pO

∂~z

� �2

−∂ ~pT

∂~z

� �2" #

s2z

and

DV2 ¼ gðαO; rÞ−gðαT ; rÞ:The first term, DV1, shows that the difference in price volatility partly stems from the differencein market liquidity, DL. However, the presence of a second term, DV2, indicates that there areother factors affecting the difference in price volatility. Whenever DV2 vanishes or it is smallenough, then:

signðDVÞ ¼ sign∂ ~pO

∂~z

� �2

−∂ ~pT

∂~z

� �2" #

¼ −sign∂ ~pO

∂~z

� �−1

−∂ ~pT

∂~z

� �−1" #

¼ −signðDLÞ:

Thus, an inverse relationship between market depth and price volatility emerges, which indicatesthat the mechanism with greater market price volatility provides the lower market depth.As Madhavan (1996) pointed out, a framework in which this inverse relationship arises is

when priors are uninformative. To derive his result, note that Corollaries 1 and 2 imply that in






this case (i.e., when s2v converges to infinity):

ðβO=γOÞ ¼ ðβT=γT Þ ¼ 1 and ðαO=γOÞ ¼ ðαT=γT Þ ¼ ρs2ε :

Hence, gðαO; rÞ ¼ gðαT ; rÞ (see Eq. (11)) and DV2 ¼ 0 follows. Consequently, sign DV ¼−sign DL.

Another set-up in which an inverse relationship between market depth and price volatilityemerges is in large markets. Note that

limN-∞

varð~v− ~p:OÞ ¼ limN-∞

varð~v− ~pT Þ ¼ 0:

Consequently,

limN-∞

varð~v− ~p:OÞ−varð~v− ~pT Þ� ¼ 0:�

Moreover, after some algebra, it is easy to show that DV1 is of the order of 1=N2, but appealingto the Mean Value Theorem, limN-∞N2DV2 ¼ 0. Thus, DV2 converges to zero faster than DV1

and the result follows.The analytical derivation of sufficient conditions on the primitives that guarantee a direct

relationship between price volatility and market depth is not easy. However, there are someparticular cases where we can ensure that this type of relationship is feasible. For instance, if

s2εð1þ ϕÞNs2v

þ 1−ϕðN2−4N þ 2Þ

N2

� �40 ð12Þ

holds, then there exists a unique value of s2z , say s2z , such that both markets are equally liquid

(see the proof of Proposition 4), i.e., DV1 ¼ 0 and DV ¼DV2. Thus, around s2z , volatility and

liquidity are aligned. For completeness, we next include some numerical examples illustratingthis point.

Figs. 2 and 3 expose a particular case in which (12) holds. In Fig. 2, DL is displayed in termsof s2z for N ¼ 10, ϕ¼ 2, s2ε ¼ 10, s2v ¼ 1

2, and ρ¼ 1. We obtain that s2z ¼ 3:052. Thus, when

s2zo3:052, the transparent market is more liquid, whereas when s2z43:052, the opaque isdeeper. Fig. 3 represents DV as a function of s2z for the parameter configuration: N ¼ 10, ϕ¼ 2,s2ε ¼ 10, s2v ¼ 1

2, and ρ¼ 1. One can observe how for low values of s2z (s2zo4:030), the price

1 2 3 4 5 6 7 8 9 10

-5

-4

-3

-2

-1

0

1

2

3

4

5

variance of z

DL

Fig. 2. DL in terms of s2z .





1 2 3 4 5 6 7 8 9 10

-0.10

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

variance of z

DV

Fig. 3. DV in terms of s2z .

50 100 150 200 250 300

-0.14-0.12-0.10-0.08-0.06-0.04-0.020.000.020.040.060.080.100.120.14

variance of z

DV

5 10 15 20 25 30

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

variance of z

DV

Fig. 4. (A) DV for different values of N. (B) DV for different values of s2ν .







volatility is higher in the opaque market, whereas for s2z44:030, the opposite holds true.Therefore when s2z∈ð3:052; 4:030Þ, the opaque market is both more liquid and volatile.

Fig. 4A displays the differences in volatility in terms of s2z for different values of N. The solidcurve corresponds to N¼10, the dashed one to N¼20, and the dotted one to N¼30 (as Nincrements the parameter configuration set in which the transparent market is less volatilebecomes higher). In the limit, when N converges to infinity, the transparent market is less volatile(and more liquid) for all s2z . In Fig. 4B, we plot DV as a function of s2z for different values of s

2v .

The solid line corresponds to s2v ¼ 12, the dashed one to s

2v ¼ 1, and the dotted one to s2v ¼ 5. This

figure shows that as s2v increases, the parameter configuration set in which the transparent marketis less volatile is higher. In the limit, when s2v converges to infinity, the transparent market is lessvolatile (and more liquid) for all s2z .

Finally, all these results are summarized in Proposition 5.

Proposition 5. Under improper priors, there is an inverse relationship between market depthand price volatility. This result may no longer hold if priors are proper unless N is large enough.

5. Sunshine trading

In this section we examine the possibility that some liquidity traders voluntarily preannouncethe size of their orders to the other market participants, a practice known as sunshine trading.Thus, the timing of the game will be as follows: first, noise traders decide whether or not toannounce their order sizes, and second, trading takes place. We solve by backward inductionand, therefore, initially we assume that the number of noise traders who preannounce isfixed. Formally, suppose that the noise demand comes from H liquidity traders, indexed byh¼ 1;…;H. Thus, ~z ¼∑H

h ¼ 1~zh, where ~zh denotes the demand for noise trader h. Let ~zh,h¼ 1;…;H, be i.i.d. with ~zh∼Nð0;s2z=HÞ. There are two types of liquidity traders: announcersand nonannouncers. The announcers are noise traders who preannounce the size of their trades,whereas the nonannouncers do not. Let A denote the subset of liquidity traders who areannouncers. Formally,

A¼ fh∈f1;…;Hg such that h is an announcerg:Let HA represent the cardinality of this set (0≤HA≤H) and let zA denote the realization of theaggregate demand of announcers, i.e., zA ¼∑h∈Azh. Notice that HA ¼ 0 corresponds to aframework similar to the opaque mechanism, whereas HA ¼H models a setup analogous tothe transparent market. Therefore, for these two extreme values of HA, Corollaries 1 and 2characterize the SLE, respectively. For intermediate values of HA, replacing z and s2z by zA andððH−HAÞ=HÞs2z in Lemmas 1 and 2, and Proposition 1, we derive the following result:

Corollary 4. Suppose that 0oHAoH. A SLE exists iff N≥3. If it exists, then:

μ¼ μ0−μ1zA ¼2α

ρs2vðN−ðN−1ÞαÞv−

N−2N−1

−α

N−ðN−1Þα zA;

β¼ α

ρs2ε;

and

γ ¼ α

ρs2ε1þ 2s2ε

ðN−ðN−1ÞαÞs2v

� �;






where α is the unique real root belonging to ð0; ðN−2Þ=ðN−1ÞÞ of the polynomial of degree threeQðαÞ, with φ¼ ððH−HAÞ=HÞρ2s2εs2z .

Note that the equilibrium coefficients depend on the number of announcers. In what follows,in order to emphasize this fact, we will write the equilibrium coefficients γ and μ1 as γðHAÞ andμ1ðHAÞ.We now focus on the first stage of the game and we study the incentives for noise traders to

engage in sunshine trading. We assume that noise traders take this decision by comparing theirconditional expected trading costs. Let CNAðzh;HAÞ (CAðzh;HAÞ) denote the expected tradingcosts of a nonannouncer (an announcer), conditional on his trade size zh, when there are HA

announcers. Thus, the noise trader h is willing to preannounce his trade size whenever24

CAðzh;HA þ 1ÞoCNAðzh;HAÞ: ð13ÞThen, if (13) holds for all HA ¼ 0;…;H−1, and for all zh, h¼ 1;…;H, then there exists anequilibrium in which all the noise traders decide to engage in sunshine trading.Direct computations yield

CNAðzh;HAÞ ¼1

NγðHAÞz2h

and

CAðzh;HAÞ ¼1−Nμ1ðHAÞNγðHAÞ

z2h:

Let us make some comments:

1.

2

asstha

2

annoptma

PJo

The conditional expected profits of a noise trader are lower if he is announcer, wheneverzh≠0.

2.
Noise traders with higher liquidity needs will be more interested in sunshine trading as long asthis practice is desirable.
3.
The choice of preannouncement for a noise trader is independent of his order size. Thisproperty is crucial since it implies that a noise trader who has already preannounced hisorder does not regret this decision when he observes that other noise traders also display theirorders.
4.
Suppose that zh≠0. Note that CAðzh;HA þ 1ÞoCNAðzh;HAÞ if and only if ð1−Nμ1ðHA þ 1ÞÞ=ðNγðHA þ 1ÞÞo1=ðNγðHAÞÞ. This inequality shows that the preannouncement has twoopposite effects on conditional expected trading costs. First, it reduces the priceresponsiveness of traders demands (γðHA þ 1ÞoγðHAÞ),25 leading to higher conditionalexpected trading costs. Second, it facilitates that the order is partly accommodatedðμ1ðHA þ 1Þ40Þ, leading to lower conditional expected trading costs. Whenever the secondeffect dominates, the noise trader h will wish to become an announcer.
4When CAðzh;HA þ 1Þ ¼CNAðzh;HAÞ, the noise trader h is indifferent between preannouncing his trade or not. Weume that he breaks the tie by choosing not to preannounce his trade. The preannouncement can carry some fixed costst are not considered in this paper (for instance, the costs of transmitting the information to the exchange).5The intuition of this fact is as follows. Notice that the degree of market transparency increases with the number ofouncers. Then, in a market with more announcers, an increase in the price of the risky asset makes agents moreimistic about its liquidation value, which leads to a smaller reduction in the individual demands, as compared to arket with fewer announcers.

lease cite this article as: de Frutos, M.A., Manzano, C., Market transparency, market quality, and sunshine trading.urnal of Financial Markets (2013), http://dx.doi.org/10.1016/j.finmar.2013.06.001





5.
In the Appendix it is shown that whenever zh≠0, CNAðzh;HAÞ is increasing in HA, whereas theshape of CAðzh;HAÞ depends on the parameter configuration.26
Whenever CAðzh;HAÞ is decreasing (and this property is satisfied when ϕ is high enough), asCAðzh;HAÞoCNAðzh;HAÞ for all HA and zh≠0, it holds that

CAðzh;HÞoCAðzh;H−1Þo⋯oCAðzh; 1ÞoCAðzh; 0ÞoCNAðzh; 0ÞoCNAðzh; 1Þo⋯oCNAðzh;HÞ for all zh≠0: ð14Þ

Hence, (13) holds and therefore, there is an equilibrium in which all the noise traders whose sizeorder is not null preannounce.

It is important to point out that (14) is a sufficient condition for the existence of this sortof equilibrium. A weaker condition that also guarantees its existence is given byMaxHA∈f0;…;H−1gCAðzh;HA þ 1ÞoCNAðzh; 0Þ for all zh≠0, h¼ 1;…;H.

In general, other types of equilibria can exist. For instance, if there exists a value of HA suchthat (13) does not hold, then there is an equilibrium in which HA noise traders decide topreannounce and the remainder does not, where HA denotes the lowest HA such that (13) doesnot hold.

Next, we include two examples illustrating the aforementioned types of equilibria.

Example 1. H¼2, z1 ¼ 10, z2 ¼ 10, N ¼ 10, ϕ¼ 5, s2ε ¼ 10, s2v ¼ 10, ρ¼ 1;s2z ¼ 10.Fig. 5 shows that the graph of the function CNAðzh;HAÞ (solid curve) is located above the graph

of the function CAðzh;HA þ 1Þ (dotted curve), h¼1,2.27 This implies that there is an equilibriumin which all the noise traders preannounce.

Example 2. H¼2, z1 ¼ 10, z2 ¼ 10, N ¼ 4, ϕ¼ 3, s2ε ¼ 10, s2v ¼ 10, ρ¼ 1;s2z ¼ 10.Fig. 6 shows that the graph of the function CNAðzh;HAÞ (solid curve) and the graph of the

function CAðzh;HA þ 1Þ (dotted curve) intersect in a point. This implies that there is anequilibrium in which one noise trader preannounces and the other noise trader does not.

Finally, Proposition 6 shows that in large markets the decision of preannouncement isunambiguous.

Proposition 6. In large markets (N or H high enough), all the noise traders (whose order size isnot null) decide to preannounce their order size.

The logic for this result is as follows: In large markets, the action of a noise trader has anegligible impact on the economy. In particular, in this case αðHA þ 1Þ is very close to α HAð Þ,making γðHA þ 1Þ−γðHAÞ very small, so that the first effect of preannouncement has littlerelevance.28 Therefore, the second effect dominates and, consequently, all the noise traders(whose order size is not null) wish to become announcers.

26The reason of this inconclusiveness is that an increase in HA decreases both the numerator and the denominator ofCAðzh;HAÞ. By contrast, as the numerator of CNAðzh;HAÞ is independent of HA and its denominator is decreasing in HA,we unambiguously conclude that an increase in HA increases CNAðzh;HAÞ.

27Note that in this example, the size of the orders of the noise traders coincide (z1 ¼ z2). Then, Fig. 5 can be usedfor both.

28See Comment 4 of this section.





0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00

10

20

30

40

50

60

70

80

HA

C

Fig. 5. Conditional expected trading costs in terms of HA (with the parameter configuration stated in Example 1).

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

160

180

200

220

240

260

280

300

HA

C

Fig. 6. Conditional expected trading costs in terms of HA (with the parameter configuration stated in Example 2).


6. Conclusions

We have examined the effects of disclosing information about the price-insensitive componentof the order flow on the market quality. Initially, we assumed that the decision of opaqueness/transparency is taken by the exchange and, consequently, we have compared a fully opaquemarket with a fully transparent market. In large markets, we have obtained that transparencyincreases liquidity and reduces price volatility, whereas in thin markets, the implications ofmarket transparency depend on parameter specification. We have shown that the inverse relationshipbetween price volatility and market liquidity obtained in Madhavan (1996), assuming improper priors,may not hold with proper priors. The practical implication is that a change in transparency that lowersprice volatility does not always reduce the execution costs of liquidity traders.Our work might be relevant for researchers and regulators alike. From our analysis it can be

argued that transparency is beneficial for active securities, independently of the public






knowledge traders initially hold on their liquidation value. For inactive securities, one marketstructure might turn out to be “superior” in markets with a low number of analysts or in those inwhich traders have little knowledge on the securities they are trading (e.g., international traderswho often lack expertise in the financial markets in which they operate). Otherwise, a clearranking between transparent and opaque markets might not exist.

Finally, we have assumed that the decision to reveal the orders (prior to trading) is donevoluntarily by the noise traders. We show that the choice of sunshine trading/dark rooms for anoise trader is independent of his order size. Moreover, the noise traders with higher liquidityneeds are more interested in sunshine trading, as long as this practice is desirable. Our analysisalso indicates that in large markets, all the noise traders opt for sunshine trading.

Appendix A

Derivation of Equations (2) and (3): Conditional on his information set, agent n chooses mn

shares of the riskless asset and Qn shares of the risky asset so as to

maxQn;yn

Eð−e−ρðvQnþmnÞjp; InÞs:t: pQn þ mn ¼ pωn þ yn;

or, equivalently, as mn ¼ pðωn−QnÞ þ yn, to

maxQn

Eð−e−ρWn jp; InÞs:t: Wn ¼ vQn þ pðωn−QnÞ þ yn:

Let qn denote the units of the risky asset traded by investor n. Thus, qn ¼Qn−ωn. The previousoptimization problem is equivalent to

maxqn

Eð−e−ρWn jp; InÞs:t: Wn ¼ ðv−pÞqn þ vωn þ yn: ðA:1Þ

Suppose that investors, other than n, use identical linear net demands given by

qjðp; IjÞ ¼ μþ βsj−αωj−γp; for all j≠n:

Then, from the market clearing condition, it follows that investor n faces a linear residual supplycurve given by

p¼ pn þ λqn; ðA:2Þin which

pn ¼ðN−1Þμþ β∑j≠nsj−α∑j≠nωj þ z

ðN−1Þγ and λ¼ 1ðN−1Þγ :

Hence, investor n has to choose the quantity she trades so as to maximize her expected utilityconditional on ðpn; InÞ. Furthermore, as Wnjpn; In is normally distributed, it follows that

Eð−e−ρWn jpn; InÞ ¼−e−ρðEðWnjpn;InÞ−ðρ=2ÞvarðWnjpn;InÞÞ:

Her optimization problem simplifies to

maxqn

EðWnjpn; InÞ−ρ

2varðWnjpn; InÞ;






with, recall (A.1),

EðWnjpn; InÞ ¼ ðEðvjpn; InÞ−pn−λqnÞqn þ Eðvjpn; InÞωn þ yn;

and

varðWnjpn; InÞ ¼ varðvjpn; InÞðqn þ ωnÞ2:Substituting the two expressions above into the previous optimization problem and maximizingwith respect to qn, the first and the second order condition require, respectively, that the equalityand inequality stated below hold:

Eðvjpn; InÞ−pn−2λqn−ρ varðvjpn; InÞðqn þ ωnÞ ¼ 0;−2λ−ρ varðpn; InÞo0:

From (A.2) and the first equality, the value of qn is deduced, with

qn ¼Eðvjpn; InÞ−p−ρ varðvjpn; InÞωn

λþ ρ varðvjpn; InÞ:

Finally, taking into account the expression of the residual supply curve, given in (A.2), the two(in)-equalities are equivalent to Eqs. (2) and (3) in the main body of the paper, as


1ðN−1Þγ þ ρ varð~v p; InÞ

��and

2ðN−1Þγ þ ρ varð~v p; InÞ40: □

��

Proof of Lemma 1. From the market clearing condition, it follows that the vector ðp; InÞ isinformatively equivalent to the vector ðhn; snÞ,

hn ¼ β∑j≠n

sj−α∑j≠n

ωj þ z;

so that Eð~vjp; InÞ ¼ Eð~vjhn; snÞ, and varð~vjp; InÞ ¼ varð~vjhn; snÞ. Hence,Eð~vjp; InÞ ¼ v þ a1ðNðγp−μÞ−βsn þ αωn−βðN−1Þv−zÞ þ a2ðsn−vÞ

and

varð~vjp; snÞ ¼ a2s2ε ;

where

a1 ¼s2εs

2vβðN−1ÞD

;

a2 ¼s2vðβ2ðN−1Þs2ε þ α2ðN−1Þs2ω þ s2z Þ

D;

and

D¼ β2ðN−1Þs2εðNs2v þ s2εÞ þ ðα2ðN−1Þs2ω þ s2z Þðs2v þ s2εÞ:






Plugging the expression for Eð~vjp; InÞ obtained above into (2), and equating coefficientsaccording to (1), it follows that

μ¼ vð1−a2Þ−a1ðβðN−1Þv þ z þ NμÞ1

ðN−1Þγ þ ρ varð~v p; InÞ;�� ðA:3Þ

β¼ a2−a1β1


α¼ −a1αþ ρ varð~vjp; InÞ1


and

γ ¼ 1−a1Nγ1

ðN−1Þγ þ ρ varð~vjp; InÞ: ðA:6Þ

From the system above we obtain all coefficients as functions of α. To do so, we first derive twoauxiliary equations. The first one is obtained after computing ð1−αÞ=γ, using (A.5) and (A.6),and performing simple manipulations, which gives:

a1γðN−ðN−1ÞαÞ ¼N−2N−1

−α: ðA:7Þ

The second one follows from rearranging (A.6), and combining it with (A.7), resulting in

ρ varð~vjp; InÞ ¼2α

γðN−ðN−1ÞαÞ : ðA:8Þ

To determine β, we first divide (A.4) by (A.5) and, then, using the fact that varð~vjp; InÞ ¼ a2s2ε inthe resulting equation, we obtain the expression given in (5).

To derive the expression of γ, we first compute varð~vjp; InÞ. Substituting the values of a1 anda2 into (A.4), and using (A.8) in the resulting equation, we get

β¼ s2vγðα2ðN−1Þs2ω þ s2z Þ1

N−1þ 2α

N−ðN−1Þα

� �D: ðA:9Þ

Multiplying both sides of the previous equality by βs2ε and taking into account the expression ofa1, we get

β2s2ε ¼γa1ðα2ðN−1Þs2ω þ s2z Þ

1þ 2αðN−1ÞN−ðN−1Þα

;

and from (A.7), it follows that:

β2s2ε ¼N−2N−1

−α� �

ðα2ðN−1Þs2ω þ s2z ÞN þ αðN−1Þ : ðA:10Þ






Using the previous equality in the expression of varð~vjp; InÞ, we get varð~vjp; InÞ ¼ 2s2εs2v=

ððN−ðN−1ÞαÞs2v þ 2s2εÞ. Substituting this expression in (A.8) and operating, the expression for γgiven in (6) is derived. Finally, μ as given in (4), is obtained by dividing (A.3) by (A.6), whichgives

μ

γ¼ vð1−a2Þ−a1ðβðN−1Þv þ z þ NμÞ

1−a1Nγ;

which combined with (A.8) and varð~vjp; InÞ ¼ s2vð1−a1βðN−1Þ−a2Þ, it follows that:

μ¼ v2α

ρs2vðN−ðN−1ÞαÞ−γa1z;

and from (A.7), we obtain (4). □

Proof of Lemma 2. From (A.9) and (A.8) we obtain

β¼ s2vðα2ðN−1Þs2ω þ s2z Þ0@ 1ðN−1Þγ þ ρ varð~vjp; InÞ

1AD

:

Since (3) implies β40, it follows that α40 because of (5). The inequality αoðN−2Þ=ðN−1Þfollows from (A.10).29 Combining this results with (6), we have that γ40. To derive α we firstnote that (A.7) and (A.8) imply ðN−2−αðN−1ÞÞρ varð~vjp; InÞ ¼ 2ðN−1Þa1α. Sincevarð~vjp; InÞ ¼ a2s2ε , plugging the values of a1 and a2 into the previous expressions, andsubstituting the expression of β given in (5) in the resulting equation, straightforwardcomputations imply that α is a solution of a polynomial of degree three:

QðαÞ ¼ aα3 þ bα2 þ cα−d;

where a, b, c, and d are given in the statement of this lemma. □

Proof of Proposition 1. Lemmas 1 and 2 imply that to study the existence of a SLE we onlyneed to analyze the roots of QðαÞ belonging to ð0; ðN−2Þ=ðN−1ÞÞ. Next, we distinguish twocases: (1) s2z≠0 and (2) s2z ¼ 0.Case 1: s2z≠0 (or equivalently φ≠0). Notice that QðαÞ ¼ 0 can be rewritten as

ðN−1Þð1þ ϕÞðαÞ2 ϕðN−2Þ−Nðϕþ 1ÞðN−1Þ−α� �

þ φN−2N−1

−α� �

¼ 0:

As φ40 and 0oαoðN−2Þ=ðN−1Þ, we have that α4ðϕðN−2Þ−NÞ=ððϕþ 1ÞðN−1ÞÞ. Therefore, inthis case α∈I ¼ ðα; ðN−2Þ=ðN−1ÞÞ, where α ¼maxf0; ðϕðN−2Þ−NÞ=ðϕþ 1ÞðN−1Þg. Evaluatingthis polynomial in the extremes of the interval I , we have QðαÞo0 and QððN−2Þ=ðN−1ÞÞ40, whichguarantees the existence of a zero in I . Finally, uniqueness follows from the fact that QðαÞ is strictlyincreasing in I .Case 2: s2z ¼ 0 (or equivalently φ¼ 0). In this case, the coefficients c and d of the polynomial

QðαÞ are null and, therefore, we can explicitly solve the roots of QðαÞ. In this case, we get α¼ 0and α¼ ðϕðN−2Þ−NÞ=ðϕþ 1ÞðN−1Þ. Then, since we need the roots belonging to the interval I ,we conclude that in this case the existence of a SLE is guaranteed if and only if NoðN−2Þϕ andin this case α¼ ðϕðN−2Þ−NÞ=ððϕþ 1ÞðN−1ÞÞ. □

29Note that 0oαoðN−2Þ=ðN−1Þ implies that N42 must hold for a SLE to exist.






Proof of Corollary 3. In the opaque market, φ40. From the proof of Proposition 1, we obtainthat αO4ðϕðN−2Þ−NÞ=ððϕþ 1ÞðN−1ÞÞ ¼ αT . Finally, since β and γ as functions of α areincreasing, we conclude that βToβO and γToγO. □

Proof of Proposition 2. In equilibrium, we have varð~vjp; InÞ ¼ 2s2vs2ε=ðs2vðN−ðN−1ÞαÞ þ 2s2εÞ.

Thus, var−1ð~vjp; InÞ ¼ ðs2vðN−ðN−1ÞαÞ þ 2s2εÞ=ð2s2vs2εÞ. As αToαO, it follows thatvar−1ð~vjp; InÞ is higher in the transparent market. □

Proof of Proposition 3. As all random variables are normally distributed, we have that

var−1ð~vjpMÞ ¼ s2v−cov2ð~v; ~pMÞvarð ~pMÞ

� �−1

; M ¼O;T :

From Eqs. (7) and (8), we have

var−1ð~vjpOÞ ¼ 1s2vs2ε

s2ε þNs2v

ϕþ 1þ 1

NðαOÞ2φ

0BB@

1CCA

and

var−1ð~vjpT Þ ¼ 1s2vs2ε

s2ε þNs2v

ϕþ 1þ ð1−NμT1 Þ2NðαT Þ2 φ

0BBB@

1CCCA:

Notice that var−1ð~vjpOÞovar−1ð~vjpT Þ is equivalent to 1=αO4ð1−NμT1 Þ=αT . Substituting αT andμT1 by their values, this inequality simplifies to

αOo αT

1−NμT1¼ ðN þ ϕÞðϕðN−2Þ−NÞ

ϕðϕþ 1ÞðN−1Þ :

It is easy to see that αT=ð1−NμT1 Þ4ðN−2Þ=ðN−1Þ iff ϕððN−2Þ2−2Þ−N240. As αOoðN−2Þ=ðN−1Þ, then αOoαT=ð1−NμT1 Þ holds in this case, and consequently, pricesare more informative in the transparent market. Finally, this result is also satisfied providedthat αO is low enough, which is equivalent to saying that s2z is low enough. Note that if s2z is lowenough, then αO gets close to αT . Hence, as αToαT=ð1−NμT1 Þ, we have that αOoαT=ð1−NμT1 Þ. □

Proof of Proposition 4. From (7) and (8), it follows that

∂ ~pO

∂~z

� �−1

¼ NγO and∂ ~pT

∂~z

� �−1

¼ NγT

−NμT1 þ 1¼ NγT ðN þ ϕÞ

ϕ:

Substituting the equilibrium values, we have that the difference in liquidity, DL, among the twomarkets is given by

DL¼ N

ρs2εαO 1þ 2r

N−ðN−1ÞαO� �

−ðN þ rð1þ ϕÞ þ ϕÞððN−2Þϕ−NÞ

ðϕþ 1ÞðN−1Þϕ

� �;

where r¼ ðs2ε=s2vÞ. DL is strictly increasing in αO. At αO ¼ αT , it attains a negative value.At αO ¼ ðN−2Þ=ðN−1Þ it is also negative if s2εð1þ ϕÞ=ðNs2vÞ þ ð1−ϕð2−4N þ N2Þ=N2Þo0, so






that DLo0, and the transparent market is deeper. Otherwise, the result follows when s2z is lowenough as then αO is close enough to αT . □

Lemma A.1. Suppose that zh≠0. Then, CNAðzh;HAÞ is increasing in HA, whereas the shape ofCAðzh;HAÞ depends on the parameter configuration.

Proof. Applying the chain's rule, we have

∂CNA

∂HAðzh;HAÞ ¼ −ρs2ε

ðN−ðN−1ÞαÞ2 þ 2Ns2εs2v

Nα2 2s2εs2v

þ N−ðN−1Þα� �2 z

2h

∂α∂φ

∂φ∂HA

and

∂CA

∂HAðzh;HAÞ ¼ ρs2ε

ðN þ αðN−1ÞÞ2−2N s2εs2v

þ N

� �

Nα2ðN−1Þ 2s2εs2v

þ N−ðN−1Þα� �2 z

2h

∂α∂φ

∂φ∂HA

:

As ∂α=∂φ40 and ∂φ=∂HAo0, it follows that CNAðzh;HAÞ is increasing in HA, whereas

sign∂CA

∂HAðzh;HAÞ

� �¼ −sign ðN þ αðN−1ÞÞ2−2N s2ε

s2vþ N

� �� :

Next, we distinguish three cases:Case 1: 4ϕ2ðN−1Þ2=ðϕþ 1Þ2−2Nðs2ε=s2v þ NÞ≥0 (i.e., ϕ high enough). In this case

ðN þ αðN−1ÞÞ2−2Nðs2ε=s2v þ NÞ40 for all α∈ðα; ðN−2Þ=ðN−1ÞÞ, which implies thatCAðzh;HAÞ is decreasing in HA.Case 2: 4ðN−1Þ2−2Nðs2ε=s2v þ NÞ≤0. In this case ðN þ αðN−1ÞÞ2−2Nðs2ε=s2v þ NÞo0 for all

α∈ðα; ðN−2Þ=ðN−1ÞÞ, which implies that CAðzh;HAÞ is increasing in HA.Case 3: 4ϕ2ðN−1Þ2=ðϕþ 1Þ2−2Nðs2ε=s2v þ NÞo0 and 4ðN−1Þ2−2Nðs2ε=s2v þ NÞ40. In this

case CAðzh;HAÞ is not monotonic in HA. □

Proof of Proposition 6. Suppose that zh≠0. Note that CAðzh;HA þ 1ÞoCNAðzh;HAÞ isequivalent to

1−Nμ1ðHA þ 1Þo γðHA þ 1ÞγðHAÞ

: ðA:11Þ

Direct computations yield limN-∞αðHA þ 1Þ ¼ limN-∞αðHAÞ ¼ ðϕ−1Þ=ðϕþ 1Þ. Substitutingthe expressions of μ1ðHA þ 1Þ, γðHA þ 1Þ, and γðHAÞ in (A.11), and taking the limit when Nconverges to infinity, we have that 0o1. Therefore, when N is high enough, CAðzh;HA þ 1ÞoCNAðzh;HAÞ for all HA ¼ 0;…;H−1, and for all h¼ 1;…H such that zh≠0. This implies that allthe noise traders whose order size is not null decide to preannounce their order size.A similar reasoning can be done when H is large enough. Concretely, limH-∞α

ðHA þ 1Þ ¼ limH-∞αðHAÞ ¼ αO. Again, substituting the expressions of μ1ðHA þ 1Þ,γðHA þ 1Þ, and γðHAÞ in (A.11), and taking the limit when H converges to infinity, we have that

1−N

N−2N−1

−αO

N−ðN−1ÞαOo1:






As αOoðN−2Þ=ðN−1Þ, the previous inequality holds. Therefore, when H is high enough,CAðzh;HA þ 1ÞoCNAðzh;HAÞ for all HA ¼ 0;…;H−1, and for all h¼ 1;…H such that zh≠0.This implies that all the noise traders whose order size is not null decide to preannounce theirorder size. □

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