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- 1. A Comparison of Stock Market Mechanisms ∗ † Giovanni Cespa April 6, 2001 Abstract This paper studies the relationship between the amount of public infor- mation that stock market prices incorporate and the equilibrium behavior of market participants. The analysis is framed in a static, NREE setup where traders exchange vectors of assets accessing multidimensional information un- der two alternative market structures. In the ﬁrst (the unrestricted system), both informed and uninformed speculators can condition their demands for each traded asset on all equilibrium prices; in the second (the restricted system), they are restricted to condition their demand on the price of the asset they want to trade. I show that informed traders’ incentives to exploit multidimensional private information depend on the number of prices they can condition upon when submitting their demand schedules, and on the speciﬁc price formation process one considers. Building on this insight, I then give conditions under which the restricted system is more eﬃcient than the unrestricted system. Keywords: Financial Economics, Asset Pricing, Information and Market Eﬃ- ciency. JEL Classiﬁcation: G100, G120, G140. ∗ Support from the EC through the TMR grants n. ERBFMBI-CT97-2528 and n. ERBFMRX- CT96-0054, the Universit´ des Sciences Sociales de Toulouse (GREMAQ) and Ente per gli Studi e Monetari e Finanziari “Luigi Einaudi,” is gratefully acknowledged. This is a revised version of the third chapter of my Ph.D dissertation. I thank my supervisor, Xavier Vives, Bruno Biais, Sandro Brusco, Jordi Caball´, Marco Pagano, Jozs´f S´ckovics and seminar audience at the Universit` degli e e a a Studi di Salerno (CSEF) for helpful comments and suggestions. The usual disclaimer applies. † Departament d’Economia i Empresa, Universitat Pompeu Fabra, Ramon Trias Fargas, 21–22, Barcelona. E-mail: giovanni.cespa@econ.upf.es. Preliminary and incomplete. Please do not circulate . 1
- 2. 1 Introduction The literature on stock market design, recognizing the beneﬁts of market connections, has recently devoted attention to study mechanisms that allow traders the exchange of portfolios of assets. The idea behind these contributions is that the impossibility 1 of operating in more than one market at the same time may either aﬀect traders’ ability to rebalance their portfolios (Bossaerts, Fine, and Ledyard 2000) or seriously limit their possibility of taking advantage of trade relevant information (Amihud and Mendelson 1991a, Amihud and Mendelson 1991b). The former eﬀect may generate liquidity dry-ups that ultimately compromise traders’ ability to improve their posi- tions, preventing full market equilibration. The latter may be the source of price gaps triggering program trades designed to exploit arbitrage opportunities, that in turn 2 cause price oscillations (Amihud and Mendelson 1991a). A mechanism allowing the trade of portfolios of assets would thus mitigate price volatility and permit better portfolio rebalancing. From the perspective of market design it is then important to understand how to concretely implement such a trading venue. Consider, for instance, a trader sub- mitting an order to buy a given vector of assets. He may want to condition his demand not only on the price of the asset he is trading, but also take advantage of cross-conditioning possibilities. In particular, he may want to condition his decision to buy say 100 shares of company A both on the price of company A and on that of company B, to the extent that information ﬂows about the two companies are somewhat related. This type of cross-conditioning has been advocated by many au- thors on grounds of improved eﬃciency and reduced volatility (Beja and Hakansson 1979, Amihud and Mendelson 1991a, Economides and Schwartz 1993). However, little 3 theoretical analysis has assessed the desirability of its introduction. 1 A feature that characterizes virtually all of the well-known existing stock markets. 2 Probably spurred by theoretical considerations of this type, recent advances in information tech- nology are deeply modifying the way stock markets handle trading procedures. Examples include Optimark, a trading system directed to institutional traders, allowing the speciﬁcation of diﬀerent parameters to condition trade execution upon (Eric and Weber 1998); Bondconnect, a system allow- ing the exchange of portfolios of assets (Bossaerts, Fine, and Ledyard 2000); Archipelago, an open limit order book system allowing participants to submit non standard types of orders (Wall Street Letter, December 4, 2000.). For a survey see also the “Economist,” May, 18th 2000. 3 See Manzano (1997) for a comparison of the properties of a system where traders submit multi- 2
- 3. This paper analyzes the properties of two call-auction trading mechanisms in which risk averse, competitive speculators trade vectors of assets against a competi- vive, risk neutral market making sector. In the unrestricted mechanism, agents can submit multi-price contingent limit orders (and market makers can observe the vec- tor of order ﬂows). In the restricted mechanism they can only submit standard limit orders (and market makers can only observe the order ﬂow of the asset they price). Equilibrium behavior and implications for price informativeness are addressed. A system that permits multiple order ﬂows observation and multi price condition- ing has two eﬀects on the information structure of the market. On the one hand, it makes prices exploit all the correlated information contained in order ﬂows. On the other hand, it gives traders a set of signals (i.e. equilibrium prices) they can use to contrast the content of their private information. These two eﬀects are not unrelated. Indeed, a fundamental insight of the paper is that speculators’ incentives to exploit multidimensional private information crucially depend on the number of prices they can condition upon and on the number of order ﬂows market makers observe when setting the opening price. To see this, suppose a trader receives good news about two assets. This can either be a signal that both assets are valuable or the consequence of noise in his private signals. However, if signal noise terms are, say, positively correlated, then observing that both prices are, for example, lower than his private signals reinforces 4 the trader’s suspect that good news are actually the result of a noise bias. As a consequence, he revises downwards his estimation of pay-oﬀ values and optimally scales down the position in both assets. But what if noise terms are uncorrelated? In this case, knowing that both prices are lower than private signals does not help the trader to disentangle the source of this diﬀerence. It could be that signals are contemporaneously upwardly biased or that asset values are actually high. Still, one may think that pooling private and public information could be useful to the extent that pay-oﬀs are correlated. However, as market makers set prices conditionally on the vector of order ﬂows, the trader does not gain any additional insight by doing price contingent orders to those of a system where they bear single price restriction in the context of a Kyle model where many insiders exchange vectors of assets. 4 This is actually true only in a large market where, owing to the strong law of large numbers, the eﬀect of private signal noise disappears in the aggregate. Thus, when a trader compares a price with a private signal, he can be sure that the two are aﬀected by uncorrelated sources of noise. 3
- 4. so. 5 Thus, he refrains from exploiting all private and public information and submits single signal and single price contingent orders. The above example shows that pooling multidimensional public and private in- formation has added value for a trader if and only if it allows to separate noise from fundamental information. This implies that in the unrestricted system, traders ex- ploit the “whole” vector of private signals provided the noise terms aﬀecting their information are correlated. This is not the case in a system where traders (and market makers) bear single price (order ﬂow) restrictions. In this system, as traders cannot contrast the content of all private signals with the market’s opinion, they rely more on private information. To see this, consider again the above example, and suppose that the two assets are negatively correlated and that error terms are uncorrelated. What should the trader do if he observes that the price of asset 1 is lower than his private signal? On the one hand he may think the market is wrong; on the other hand, the second signal leads 6 him to revise downwards his expectation about asset 1. As a consequence he scales 7 down his position. What now if error terms too are correlated? In the paper I show that in this case his position depends on two factors: on the one hand, the strength of asset correlation vis-`-vis noise terms correlation; on the other hand, the relative a magnitude of private signals. Finally, whenever correlation across noise terms and pay-oﬀs are equal, traders are not able to disentangle the information in their signals from the noise they incorporate, and thus refrain from pooling private information. Therefore, as long as private signals are correlated either through pay-oﬀs or through noise terms, in the restricted system informed traders ﬁnd it optimal to exploit the whole vector of private signals. The bottomline is thus that a trading mechanism that disseminates a large amount of endogenous public information (i.e. equilibrium prices and order ﬂows) reduces the incentives informed traders have to exploit multidimensional private information (i.e. 5 In other words, all correlated information about pay-oﬀs is already impounded into prices. Thus, exploiting the whole vector of private signals, does not improve the trader’s estimation of asset values vis-`-vis the market makers’. a 6 This is so because a negative correlation across asset values implies that it is more likely that one of the two assets is less valuable than the other. 7 Notice the diﬀerence: in this system the fact that market makers cannot observe the order ﬂow of asset 2 induces the trader to exploit the information he has about this asset in trying to asses the true value of asset 1. 4
- 5. 8 their vector of signals). Based on this eﬀect, I show that under some conditions, the unrestricted system is less eﬃcient than the restricted one. Indeed, given that trading aggressiveness impacts on market eﬃciency, one expects prices to be more informative in the system where traders exploit more intensively private information. This intuition is correct, provided order-ﬂow correlation is due to pay-oﬀs. In this case, as I argued above, traders ﬁnd it optimal to combine pri- vate information in the restricted system but not in the unrestricted one. Thus, in the former market makers observe more correlated information and ﬁnd it easier to estimate pay-oﬀs. This ﬁndings challenges the view that a multi price contingent system should always render the market more eﬃcient: “a mechanism which enables simultane- ous conditioning of orders for diﬀerent assets (. . . ) would increase the information available to traders, improve value discovery and reduce volatility” (Amihud and Mendelson 1991b). This assertion points at the positive eﬀect that observing multi- ple sources of correlated information has. By contrast, my paper unveils the negative side of a multi price contingent system, by analysing its feedback eﬀect on traders’ incentives to exploit private information. The paper also analyses a market where, at the opening call, market makers set prices observing more than one order ﬂow while traders bear single price restrictions in their orders. An example of such an intermediate market is given by the opening 9 call auction carried out in the NYSE. Numerical simulations show that when the information structure is symmetric and only correlation across asset pay-oﬀs aﬀects order ﬂows, the restricted system delivers more informative prices than the inter- mediate system and this, in turn, is more informative than the unrestricted system. The eﬀect at work is the same I have outlined above: restricting the amount of public information that informed speculators observe forces them to exploit more intensively their private information, enhancing the informativeness of order ﬂows. There is by now a considerable literature studying the eﬀects of diﬀerent market structures on traders’ behavior. Madhavan (1992) compares the properties of quote driven systems to those of order driven systems, showing that the equilibrium in the last type of markets is more robust. Biais (1993) compares centralized and fragmented 8 However, the result is not symmetric: restricting the amount of public information traders observe does not necessarily imply they exploit more intensively private information. 9 See Lindsay and Schaede (1990) and O’Hara (1995). 5
- 6. markets showing that although expected spreads are equal in both markets, central- ized markets exhibit more volatile spreads than fragmented markets. Pagano and R¨ell (1993) assess the eﬀects of market transparency on uninformed traders’ losses. o All of these contributions are framed in a context where traders exchange a single asset. Little is known about the properties of markets where traders’ private infor- mation is multi dimensional. A notable exception is the paper of Manzano (1997) where the author, in a multi dimensional Kyle model, compares multi price and single price contingent systems. Less directly related are the analysis of Wohl and Kandel (1997) and Subrahmanyam (1991). The former studies the properties of a system where traders can condition their demand for a given asset on a market index. The latter analyzes the implications of allowing traders to exchange basket securities as, for instance, index futures contracts. In both frameworks, however, each trader’s private information is unidimensional, which does not allow the possibility to analyze the eﬀects of diﬀerent mechanisms on traders’ incentives to use private information. The paper is organized as follows. In the next section, I outline the basic notation. In the third section, I characterize the unique equilibria of the two mechanisms. In the fourth section, I compare their properties. In the ﬁfth section, I introduce the intermediate mechanism and present the results of numerical simulations. The sixth section concludes the paper. A ﬁnal appendix collects most of the proofs. 2 The General Setup In all the models there are 3 classes of traders exchanging a vector of 2 risky assets (with random liquidation value v ∈ IR2 ) and a risk less one (with unitary return). 10 In particular we have: (a) a continuum of risk-averse (CARA) informed speculators distributed in the interval [0, 1] with identical risk-aversion coeﬃcient γ −1 who receive private signals about the array of assets; (b) a sector of competitive, risk-neutral market-makers, and (c) noise traders, trading for liquidity purposes. For the notation, we indicate with bold, capital letters, 2 × 2 matrices; with bold, low case letters, 2 × 1 vectors and with low case letter, scalars. In particular, A is a 2 × 2 matrix, aij is the generic i, j-th element of this matrix and b is a 2 × 1 vector. 10 While the generalization to the K > 2 case is straightforward for the unrestricted mechanism, the same is not true for the other case. 6
- 7. Furthermore, Πx indicates the precision matrix of the random vector X, τxi is the precision of the random variable Xi and ρx is the correlation coeﬃcient of the random vector (X1 , X2 ). As far as the information structure, various assumptions are made. Most of them are standard in the rational expectations literature. For completeness we list them in order: A.1 All random variables are assumed to follow a joint, bi-variate, normal distribu- tion. In particular v ∼ N (¯ , Π−1 ). v v A.2 Prior to trading, every informed agent i ∈ [0, 1], receives a vector of signals si = v + i , where i ∼ N 0, Π−1 , i ∈ IR2 , E[ i , j] = 0, for i = j. 1 A.3 The SLLN holds: 0 i di = 0, almost surely. A.4 Noise traders’ demand is modeled as a two dimensional, normal random vector u ∼ N 0, Π−1 . u A.5 The random vectors v, u and i are mutually independent ∀i and the matrices Πv , Πu and Π are positive deﬁnite. A.6 At time 2 the vector of risky assets is liquidated. A.7 The distributional assumptions are common knowledge among the agents in the economy. In the next section we characterize the equilibria of the two models. 3 Trading Mechanisms In this section we lay down the two trading mechanisms and solve for their linear equilibria. In particular, in section 3.1 we consider the unrestricted mechanism where (a) speculators condition their demand for each asset j on the vector of private signals si and on the price of assets h = 1, 2, and (b) market makers set the price of asset j conditionally on the observation of the order ﬂow of all assets h = 1, 2; in section 3.2, we study the restricted mechanism where (a) speculators condition their demand for an asset j on the vector of private signals si and on the price of asset j only and (b) market makers set the price of asset j conditionally on the information contained in 7
- 8. order ﬂow j. 3.1 The Unrestricted System The unrestricted model is a version of the multi-asset model of Admati (1985) with the addition of a risk-neutral, competitive, market-making sector as in Vives (1995). To build the equilibrium of this model, consider the following argument. When submitting his demand, trader i has available the vector of signals si = v + i . In any linear equilibrium, due to the SLLN, private and public information are condition- ally independent, therefore an informed strategy depends both on si , and on public information p. I assume that a generic informed agent i, submits a vector of demand schedules (limit orders) Xi (si , p) , indicating the position desired in each asset k at every price vector p, contingent on the available information, and restrict attention to linear equilibria. In period 1 the MM sector observes the aggregate order ﬂow 1 L(·) = xi di + u. 0 Notice that this means that in pricing asset j the market maker can use the correlated information contained in order ﬂow h = j too. To compute the equilibrium, consider a candidate symmetric, linear equilibrium xi = Asi + φ (p) , where A and φ(·) are respectively the matrix of trading intensities and a linear func- tion of current prices. The aggregate order ﬂow is then L(·) = z + φ (p) , where z = Av + u, is the informational content of the order ﬂow. Because of compe- tition for the order ﬂow and risk neutrality, market makers set the equilibrium price equal to the conditional expectation of v given z, p = E [v|z] (3.1) = Π−1 (Πv v + A Πu z) , ¯ 8
- 9. where Π = Πv + A Πu A. Given the formula for the market price now one can solve for the equilibrium parameters of an informed trader’s demand function. Since in equilibrium the pa- rameters of the price are known, the informed can condition his estimation of v on the following random vector −1 (A Πu A) (Πp − Πv v ) |v ∼ N v, A−1 Πu (A−1 ) , ¯ and on the received signal si |v ∼ N v, Π−1 . Therefore, his demand is given by 11 xi = γΠ (si − p). 12 This proves the following Proposition 1 In the two asset model with a risk-neutral, competitive MM sector there exists a unique equilibrium in linear strategies where agents trade according to the function xi (p, si ) = γΠ (si − p), (3.2) and the prices are given by p = Λz + (I − ΛA) v , where Λ = Π−1 A Πu . ¯ Remark 1 The matrix Λ is the multi-asset extension of the Kyle (1985) measure of market depth. Note that for the equilibrium to be well-deﬁned, Λ must be invertible and, given our assumptions, this is the case. Remark 2 Market makers’ risk neutrality, pushes informed out of the market making activity. This, in turn, implies that Corollary 1 Informed speculators demand in each asset k = 1, 2, depends on the whole private signal vector si and on the whole price vector p iﬀ ρ = 0. Proof. Follows from the fact that A = γΠ . QED 11 This follows from the fact that (Var[v|p, si ])−1 = Πv + A Πu A + Π = Π + Π , see e.g. DeGroot (1969). 12 −1 The result follows immediately from normal theory, since xi = γ (Var[v|p, si ]) (Var[v|p, si ](Π si + Πp) − p) = γ(Π si + Πp − (Π + Π )p). 9
- 10. Therefore, informed multi-price conditioning is optimal if and only if the conditional precision matrix of the speculators’ private signals is not diagonal. The intuition is as follows. Given that prices are set contingently on the observation of all the order ﬂows, cross-asset public information is already totally exploited. In other words, informed traders cannot improve upon market makers in combining public information. How- ever, market makers cannot observe the signals informed receive. Therefore, to the extent that error terms are correlated, multi price conditioning can improve informed speculators’ performance. To see this, consider the following example. Example 1 Writing explicitly xi1 , one can see that the trading intensity in asset 1 is the composition of two eﬀects: a direct one stemming from the informational advantage the informed has over the rest of the market in asset 1, and an indirect one coming from the informational advantage she has on the remaining assets, to the extent that the received signals are correlated. To see this, indicate with τ j , j = 1, 2 the (conditional) signal precision in asset j. Then, the strategy of an informed in asset 1 can be written as follows √ γτ 1 γ τ 1τ 2ρ xi1 = (si1 − p1 ) − (si2 − p2 ). (3.3) (1 − ρ2 ) (1 − ρ2 ) Assume that ρ > 0 and that speculator i receives two signals si1 , si2 such that si1 > p1 and si2 > p2 . This can happen for two reasons: the ﬁrst one is that both the assets are worth more than what the market thinks (i.e. asset prices are biased downward e.g. by noise traders’ selling pressure); the second one is that both the signals that she has received are biased upward. A downward bias in equilibrium prices is good news since it gives to the trader the possibility of taking advantage of the market’s forecast error. Her demand in each asset is larger, the more precise are the signals she has received. On the other hand the hypothesis of a contemporaneous, upward bias into the speculator’s signals is strengthened by the existence of positive correlation across signal-error terms (i.e. by the fact that an error that biases upward the information contained in si2 is more likely to happen together with an error biasing upward the information about asset 1 as well). Given this, the speculator reinforces her belief that the good news she received about both the assets is due to the eﬀect of error biases and reduces her demand in both asset 1 and asset 2. Clearly, the correction that correlated information induces, is stronger (weaker) the higher (lower) is the correlation across error terms since for a bivariate normal distribution, the value of 10
- 11. Fρ ( i1 , i2 ) is increasing in ρ for all ρ ∈ [−1, 1] and all ﬁxed ( i1 , i2 ) (Tong (1990)). Thus, higher correlation across error terms make more probable the event of a joint bias in private signals. When no correlation across error terms exists, speculators have no way to reduce the bias in the private information they receive by pooling together private signals and ﬁnd it optimal to submit single-signal and single-price contingent limit orders. Notice, however, that market makers still use all the order ﬂows when pricing an asset. Indeed, their demand can be written as xM M = (Λ−1 − A)(¯ − p), i v and it is easy to see that the diagonality of Π does not imply the diagonality of (Λ−1 − A). I now turn the attention to the characterization of the restricted system. 3.2 The Restricted System In the restricted system, a speculator i can condition his demand for an asset j on the whole vector of private signals si and on the price of asset j only. In this case, we can interpret market makers as uninformed speculators. In this way, the model captures the features of the opening auction of those systems where all traders are allowed to condition their demand of an asset j on the price of asset j only Wohl and 13 Kandel 1997. In any linear equilibrium, private and public information are conditionally inde- pendent, thus the speculator’s strategy depends on both his signal and the price. In particular, we assume that a generic speculator i submits a demand schedule XRij (si , pRj ) , j = 1, 2, indicating the desired position in asset j at every price pj , contingent on the available information, and restrict attention to linear equilibria. 13 To the best of my knowledge, this is the ﬁrst attempt to characterize the equilibrium in a multi- asset framework where competitive, risk averse traders receive diﬀerent signals and bear restrictions in the number of asset prices they can condition on. Manzano (1997) studies the case where traders are risk neutral and act strategically. 11
- 12. As far as the market maker of asset j is concerned, in period 1 he observes the order ﬂow of asset j (carrying information about all the assets) but cannot observe the order ﬂows of the asset h = j. Formally he thus observes 1 LRj (·) = xRij di + uj , j = 1, 2. 0 To compute the equilibrium, consider a candidate symmetric equilibrium xRij = ARj si + φRj (pRj ), j = 1, 2, where ARj is the 1 × k vector of trading intensities and φRj (·) is a linear function of the j-th price. The aggregate order ﬂow of asset j is then LRj (·) = zRj + φRj (pRj ) , j = 1, 2, where zRj = j (AR v + u), is the informational content of order ﬂow j and j is a column vector containing a 1 in the j-th position and a zero elsewhere. Given competition and market makers’ risk neutrality, the equilibrium price of asset j is ¯ pRj = vj + λRj j (AR (v − v ) + u) , ¯ where j AR Π−1 j v λRj = j AR Π−1 (AR ) + Π−1 j v u τuj τvk γτ j + aRjk ρv τvj /τvk − ρ τ j /τ k = √ , j = k = 1, 2, τuj (aRjj )2 τvk + (aRjk )2 τvj + 2ρv aRjj aRjk τvj τvk + τvj τvk is the regression coeﬃcient of vj on zRj i.e. the usual measure of market depth. Consequently, we have the following Lemma 1 In every linear equilibrium of the restricted model, the vector of equilib- rium prices is given by ¯ pR = ΛR z R + (I − ΛR AR ) v , (3.4) where ΛR = diag(λR1 , λR2 ) is the matrix of market depths of the restricted model and z R = AR v + u, is the vector of informational contents of order ﬂows. 12
- 13. Owing to the restriction on the market makers’ information sets, in this model a market maker can learn the liquidation value of a given asset j by using correlated asset information about another asset h = j iﬀ informed speculators ﬁnd it worth to condition their demand about asset j on the signal sih , h = j. This does not happen in the unrestricted model where even if A is diagonal, the price of asset j depends on the order ﬂow of asset h = j (to the extent that Πv or Πu are not diagonal). Informed speculators’ equilibrium demand parameters are characterized in the following lemma. Lemma 2 In every linear equilibrium of the restricted model, an informed speculator i’s demand for asset j = 1, 2 is given by ¯ xRij = j AR (si − v ) − bRj (¯j − pRj ), v (3.5) where, j AR = γ (Var[vj |pRj , si ])−1 c2j , (3.6) is the vector of the sensitivities of the demand for asset j to the speculator’s private signals, bRj = −γ (Var[vj |pRj , si ])−1 (1 − c1j /λRj ) , (3.7) is the sensitivity of the demand for asset j to the equilibrium price of the asset and c1j , c2j , and Var[vj |pRj , si ] are deﬁned in the appendix. Proof. See the appendix. QED The next proposition proves existence and uniqueness of the equilibrium in the restricted system. Proposition 2 In the restricted market mechanism where speculators trade asset j submitting a limit order conditional on the signal vector si and on the price pRj and market-makers price asset j conditional on the order ﬂow zRj , there exists a unique linear equilibrium where the price vector is given by (3.4) and the demand parameters are deﬁned implicitly by (3.6–3.7). Proof. See the appendix. QED Notice that existence and uniqueness of equilibrium are not obvious results given that speculators’ equilibrium trading intensities come from the solution of a system of 13
- 14. two cubic equations. In the appendix, I show how to simplify the system, reconducing it to a solvable cubic. The next proposition characterizes the equilibrium parameters. Proposition 3 In the equilibrium of the restricted system 1. aRij > 0 if and only if ρ τ i /τ j < ρv τvi /τvj ; 2. (a) aRii = γτ i (1 − γ −1 aRij Cov[ 1 , 2 ]) > 0 and (b) λRi > 0; 3. if ρ = 0, aRii = γτ i and aRij = 0; 4. if ρ τ i /τ j = ρv τvi /τvj , aRii = γτ i , aRij = 0, and bRi = −aRii . Proof. See the appendix. QED The intuition for these results is as follows. For part 1, suppose an informed speculator trading asset 1 receives two “high” signals si1 , si2 . This may be either the eﬀect of fundamental information, or of noise in the signals. The ﬁrst possibility is more likely the stronger is the correlation of asset pay-oﬀ w.r.t. error terms and the higher is the relative dispersion of asset 2 payoﬀ w.r.t. asset 1 pay-oﬀ compared to the relative dispersion of the error in signal 2 w.r.t. the error in signal 1. In this case, indeed, the eﬀect of fundamental information dominates the eﬀect of noise in the signal vector. For part 2 (a) Suppose that aR12 > 0. This means that, given the relative size of the correlation and dispersion of the information structure, informed agents increase their speculative position in asset 1 upon receiving “good news” about asset 2. How- ever, if ρ > 0, good news about asset 1 may come from the joint eﬀect of signal noise terms. Therefore, in this case, informed speculators scale down the weight they put on si1 in trading asset 1 the more, the higher is the trading intensity they put on si2 . For 2 (b), the idea is that impossibility of observing more than one order ﬂow when pricing an asset eliminates the multicollinearity eﬀects that happen in the 14 unrestricted system. Therefore, the depth matrix is positive deﬁnite. 14 The reason for this face here is diﬀerent from the one in Caball´ and Krishnan (1992). In their e case the positive deﬁniteness of the depth matrix is a shortcoming of the hypothesis of imperfect competition across insiders that make impossible the existence of unexploited arbitrage opportuni- ties. 14
- 15. For part 3, the intuition is that a given signal sih is useful in trading an asset k = h to the extent that it carries information (a) either about vk or (b) about the error term ik . As the correlation across error terms vanishes, sih is still useful for the information it contains about vh . Therefore, speculators use it in trading asset k. Finally, for part 4, given what said above, the conclusion is not surprising. If ρ τ i /τ j = ρv τvi /τvj , there is no way for a speculator to disentangle noise from information by pooling the two signals he receives. As a consequence aRij = 0.15 Remark 3 By inspection of the cubic equation deﬁning informed trading intensities, it turns out that they do not depend on the correlation across noise traders’ demands (see the appendix). This makes sense, since both the informed and the uninformed speculators when trading asset j are observing only the orderﬂow of asset j. There- fore, they cannot use the information contained in the order ﬂow of asset h = j to disentangle noise from information as it is possible in the unrestricted model. As done for the unrestricted system, we can now study how traders form their ¯ portfolios as a function of private and public information. Suppose v = 0, then xRi1 = aR11 si1 + aR12 si2 − bR1 pR1 , xRi2 = aR22 si2 + aR21 si1 − bR2 pR2 . Using the results of proposition (3) and rearranging, τ1 τ1 xRi1 = γτ 1 (si1 − pR1 ) + aR12 v2 − ρ v1 + i2 −ρ i1 − ϕ1 (·)pR1 , τ2 τ2 τ2 τ2 xRi2 = γτ 2 (si2 − pR2 ) + aR21 v1 − ρ v2 + i1 −ρ i2 − ϕ2 (·)pR2 , τ1 τ1 where ϕj (·), j = 1, 2 is a function of aR11 , aR12 . Suppose that ρ → 1, and τ 1 = τ 2 , then the terms i2 , ρ i1 tend to cancel out (the same happens to i1 and ρ i2 ), thus a speculator’s demand is given by xRi1 γτ 1 (si1 − pR1 ) + aR12 (v2 − v1 ) − ϕ1 (·)pR1 , xRi2 γτ 2 (si2 − pR2 ) + aR21 (v1 − v2 ) − ϕ2 (·)pR2 . 15 Notice that the previous proposition does not imply that if Πv and Π are diagonal, then pRj = pj , since as long as ρu = 0, even though speculators, in the unrestricted system, do not combine the information contained in their signals, market makers can still learn from the correlation across noise terms and exploit this information when pricing assets. 15
- 16. Now, if ρv < ρ , both aR12 and aR21 are negative. Therefore, whenever si2 > si1 it is more likely that asset 2 has a higher value than asset 1. Thus, speculators decrease the intensity with which they speculate in the market for asset 1 and increase the weight they put on asset 2. The opposite happens when si1 > si2 . Thus, in the restricted system, pooled private information (rather than the combination of private and public information) allows speculators to disentangle the eﬀect of noise in their signals. I conclude the section considering the symmetric case (i.e. the case where the precision matrices are doubly symmetric). This simpliﬁes the model and gives the following corollaries of the previous propositions. Corollary 2 In the 2-asset, symmetric case, there exists a unique symmetric equi- librium of the restricted model, where informed speculators’ trading intensities are implicitly deﬁned by the following system of equations: ρ τ ((1−ρ2 )(ρ τ −(1−ρ2 )aR1 aR2 τu )+(1−ρ2 )ρv τv ) aR1 = γ τ − v τ (1−ρ2 )+(1−ρ2 )((aR2 )2 (1−ρ2 )τu +τv ) , v v (3.8) τ ((1−ρ2 )(ρ τ −(1−ρ2 )aR1 aR2 τu )+(1−ρ2 )ρv τv ) aR2 = γ −ρ τ + v τ (1−ρ2 )+(1−ρ2 )((aR2 )2 (1−ρ2 )τu +τv ) . v v Where aR1 = (AR )11 = (AR )22 and aR2 = (AR )12 = (AR )21 . Proof. See the appendix. QED In this case similar results apply to the equilibrium parameters, Corollary 3 In the equilibrium of the restricted, symmetric model, 1. aR2 ≥ 0, if and only if ρ − ρv ≤ 0; 2. aR1 = γτ − ρ aR2 > 0, λR > 0. Proof. See the appendix. QED Clearly, the intuitions given for proposition 3 carry over to the above corollary. 4 Comparisons In this section, I compare equilibrium behavior and price informativeness across the two systems. 16
- 17. 4.1 Trading Intensities The following proposition follows directly from corollary 1 and proposition 3. Proposition 4 In every linear equilibrium of the unrestricted system, τ1 0 A=γ , 0 τ2 if and only if ρ = 0. In every linear equilibrium of the restricted system τ1 0 AR = γ , 0 τ2 if and only if ρ τ 1 /τ 2 = ρv τv1 /τv2 . Proposition 4 captures the idea that disseminating a large amount of public in- formation reduces traders’ incentives to exploit private signals. However, restricting public information diﬀusion, does not necessarily imply that traders exploit more intensively their private information. Indeed, in the unrestricted model it suﬃces that ρ = 0 to make private (and public) information pooling redundant (even if ρv = 0). This is so because correlat- ed information about pay-oﬀs is already incorporated into prices. Thus, combining private signals cannot improve traders’ estimation. In the restricted system, on the other hand, traders also may refrain from pooling private information. However, the reason here is due to the impossibility to disentangle the eﬀects of noise from those of fundamental information in their signals. Before turning to the eﬀects on price eﬃciency, I show how diﬀerent equilibrium behavior aﬀects the way traders interpret private signals. Suppose that 1.56 −2.2 15.62 −3.49 1.09 −0.49 Πv = , Πu = , Π = , −2.2 3.12 −3.49 0.78 −0.49 0.21 and γ = 1. Then, computing trading intensities in the two systems, gives 1.02 −0.3 A = γΠ , AR = . 0.416 0.144 17
- 18. Notice that while A is symmetric, the matrix AR does not have any particular struc- ture. Next, consider the oﬀ-diagonal terms. Note that aR12 < 0. The intuition is as follows. We know that the sign of aR12 depends on the relative dispersion of the distribution of i2 w.r.t. i1 compared to that of v2 w.r.t. v1 and on the size and signs of the correlation eﬀects across noise and fundamentals. It is easy to see that for this √ √ parameter conﬁguration, ρ τ 1 /τ 2 ≡ .3 5 > .6 .2 ≡ ρv τv1 /τv2 , implying that aR12 < 0. Now if, for example, si1 = 1 and si2 = 4, speculators in the unrestricted system interpret this signal vector as good news about asset 1, while in the restricted system they give it the opposite interpretation. Figure 1 summarizes the diﬀerences in trading behavior across the restricted and the unrestricted system for the symmetric case. If correlation across signal noise and fundamental values is as in parts II, III, V and VI, then the sign of the weight a speculator puts on signal si2 (si1 ) when trading asset 1 (2) coincides in the 2 systems. However, if they are as in parts I and IV, then speculators behave diﬀerently in the two systems e.g. in part I, they put a positive weight on si2 when trading asset 1 in the restricted system, while they do the opposite in the unrestricted system. 4.2 Price Eﬃciency In this section, I compare the informativeness of the price vector in the two systems in the symmetric case. Price informativeness is deﬁned as the reduction in the un- conditional variance of the asset pay-oﬀ due to the observation of the vector of order ﬂows. Therefore, in the unrestricted system IpU R = Π−1 v − Var[v|z]−1 11 11 −1 −1 = Πv − Π , 11 11 while in the restricted system IpR = Π−1 v − Var[v|z R ]−1 11 11 −1 = Πv − Π−1 R 11 . 11 This deﬁnition is clearly natural in the unrestricted system. In the restricted system, it corresponds to the point of view of an econometrician interested in estimating the 18
- 19. Figure 1: Comparing trading intensities across the restricted and the unrestricted system. ρv I .... ρv = ρ .... .... .... 1 .... .... .... a2 < 0 .... VI .... aR2 > 0.... .... II a2 > 0 .. .... a2 < 0 .... aR2 > 0 .... .... aR2 < 0 ...... .... -1 ...... 0 1 ρ a2 > 0 ...... .... aR2 > 0.... .. a2 < 0 .... a2 > 0 V .... aR2 < 0 ...... aR2 < 0 .... III ...... -1 .... .... .... IV // 19
- 20. deep values of the market. Alternatively, it captures the point of view of a trader that before submitting an order for a given asset, observes the past price of the same asset and those formed on related markets. Given that trading aggressiveness impacts on market eﬃciency, one would expect that when in one system traders exploit more intensively private information, prices should also be more informative. This intuition is correct when the only source of correlation across order ﬂows is due to asset pay-oﬀs as the following proposition shows. Proposition 5 When the information structure is symmetric and ρ = ρu = 0, for ρv small IpR ≥ IpU R . Proof. See the appendix. QED This result is conﬁrmed by numerical simulations. In particular, to check it I have used the following parametrization ρv ∈ {−.9, −.8, . . . , .9}, ρu = ρ = 0 and τu , τv , τ ∈ {.2, .5, 1, 3}. In all of the analyzed cases, price informativeness was higher in the restricted system. In ﬁgure 2, I plot the price eﬃciency of the restricted system (dotted line) and the one of the unrestricted system (continuous line) when τ = τu = 1, ρ = ρu = 0, τv ∈ {.2, .5, 1, 3}. It is clear that the unrestricted system is more informative. Qualitatively the results do not change for the other parameter values. This ﬁndings is particularly important given that a major advantage a multi price contingent system should deliver, is that of rendering the market more eﬃcient. In- deed, as Amihud and Mendelson (1991b) argue “a mechanism which enables simulta- neous conditioning of orders for diﬀerent assets (. . . ) would increase the information available to traders, improve value discovery and reduce volatility.” This assertion points at the positive eﬀect that observing multiple sources of correlated information has. By contrast, proposition 5 unveils the negative side of a multi price contingent system by showing the implication of the feedback eﬀect the system has on traders’ incentives to exploit private information. The result still holds true when more sources of correlated information aﬀect order ﬂows. Proposition 6 When the information structure is symmetric, ρv = ρ = 0 and ρu = 0, IpR ≥ IpU R . 20
- 21. 4.6 1.8 4.5 1.7 Efficiencies Efficiencies 4.4 1.6 4.3 1.5 4.2 1.4 4.1 1.3 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 rv rv 0.22 0.8 0.2 0.18 Efficiencies Efficiencies 0.7 0.16 0.14 0.6 0.12 0.1 0.5 0.08 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 rv rv Figure 2: Price eﬃciencies as a function of ρv . 21
- 22. Proof. See the appendix. QED If the conditions of the proposition are met, it is easy to check that a2 = −ρv a1 . Hence, market makers in the unrestricted system observe the following vector of order ﬂows: z1 = a1 (v1 − ρv v2 ) + u1 , z2 = a1 (v2 − ρv v1 ) + u2 , where a1 = γτ /(1 − ρ2 ). As ρv increases, so does ρ , hence speculators receive v increasingly positively correlated signals. However, given that in the unrestricted system they do not take into account that correlated signal realizations may be due to fundamental information, their trading behavior has two countervailing eﬀects on the informational content of order ﬂows. Indeed, on the one hand, they speculate more intensively on each asset’s private signal; on the other hand, they reduce their trading intensity because of the eﬀects of correlated private signals’ noise terms. This reduces the weight that fundamental information has on each asset’s order ﬂow w.r.t. noise traders’ demands and makes harder the signal extraction problem of market makers. These eﬀects are not at work in the restricted system where, if ρv = ρ = 0, the impossibility of disentangling the eﬀect of noise from that of information into private signals, makes speculators unwilling to combine private information when trading an asset. As a consequence, order ﬂows end up being more informative. However, the eﬃciency ranking can be reversed with diﬀerent patterns of corre- lated information. Proposition 7 When the information structure is symmetric, the following ranking across price informativeness applies (a) if ρ = ρv = 0, IpU R = IpR ; (b) if ρv = ρu = 0, for ρ small, IpR ≥ IpU R if and only if γτ 2 τu τi ≥ τv , 3(τ + τv ) + τv where τi = τv + a2 τu + τ . 1 22
- 23. Proof. See the appendix. QED For part (a), price informativeness does not diﬀer when ρv = ρ = 0. In this case, indeed, traders’ behavior coincides in the two systems, hence the informational content of the order ﬂows is the same. For part (b) notice that in both systems, informed speculators combine private information even if asset pay-oﬀs are uncorre- lated. However, in the unrestricted system, the fact that they can condition their order on both prices, makes them speculate more aggressively on private signals than in the restricted system. This, in turn, for low values of the private signals condition- al precision, makes the informativeness of the order ﬂow higher in the unrestricted system; however, as the precision of private signals increases, the eﬀect of impounding non-correlated information in the order ﬂow, makes the signal extraction problem of 16 the market maker harder in the unrestricted system, and prices less informative. Numerical simulations were run to conﬁrm the intuition of proposition 7 using the same parameter values deﬁned above for variances and setting ρ ∈ {−.9, −.8, . . . , .9), ρu = ρv = 0 . In this case, price informativeness is higher in the restricted system for high values of τ and lower for low values of τ . In ﬁgure 3, I plot eﬃciencies as a function of ρ , when τ = τu = 3 and τv ∈ {.2, .5, 1, 3}. Given that τ is high, the restricted system is more informative. However, when one chooses, e.g. τ = 1, τu = .2, then ﬁgure 4 shows that the reverse happens. 5 An “Intermediate” System The intuition gained in the previous section, shows that allowing traders to submit multi-price contingent orders, has an impact on the informativeness of the equilibrium price vector. In particular, when only correlation across asset pay-oﬀs aﬀects order ﬂows, the restricted system equilibrium price vector is more informative than the unrestricted system one. The basic intuition, is that the ability to exploit cross-asset related information, reduces the incentives that speculators have to exploit private signals and makes them impound less information in the price vector. If this is the case, a system that partially allows cross asset information extraction, should deliver prices that on the one hand are less informative than those of the restricted system, 16 It is “as if” the fact that speculators use a signal that is not relevant from the market maker point of view, to choose their position in a given asset, increases the noise present in the order ﬂow. 23
- 24. 4.97 1.97 1.965 4.96 1.96 Efficiencies Efficiencies 1.955 4.95 1.95 1.945 4.94 1.94 4.93 1.935 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 re re 0.31 0.97 0.305 Efficiencies Efficiencies 0.96 0.3 0.295 0.95 0.29 0.94 0.285 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 re re Figure 3: Price eﬃciencies as a function of ρ . 24
- 25. 3.2 1.2 3 1 Efficiencies Efficiencies 2.8 0.8 2.6 0.6 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 re re 0.6 0.16 0.14 0.5 0.12 Efficiencies Efficiencies 0.4 0.1 0.3 0.08 0.06 0.2 0.04 0.1 0.02 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 re re Figure 4: Price eﬃciencies as a function of ρ . 25
- 26. and on the other hand are more informative than those of the unrestricted system. A similar intermediate mechanism is represented by one where market makers can observe both order ﬂows when pricing an asset, while informed speculators bear single price restrictions. The opening call auction of te NYSE gives an example of such a system. Indeed, according to Lindsay and Schaede (1990), each specialist handles more than one asset and, as a consequence, is able to make cross asset inference at 17 the moment of setting the opening price. In this intermediate system, privately informed speculators do not hold an in- formational advantage over market makers. Indeed, while market makers can only observe a linear combination of fundamentals and noisy supply, speculators observe private signals and a linear combination of order ﬂows. Therefore, it is never the case that their information set dominates the one of market makers. Formally, in the intermediate mechanism, an informed speculator i can condition her demand for a given asset j on the whole vector of private signals si and on the price of asset j only. In any linear equilibrium, private and public information are conditionally independent, thus the speculator’s strategy depends on both his signal and the price. In particular, assume that a generic speculator i submits a demand schedule XIij (si , pIj ) , j = 1, 2, indicating the desired position in asset j at every price pIj , contingent on the available information, and restrict attention to linear equilibria. As far as the market maker of asset j is concerned, in period 1 he observes the order ﬂows of assets j = 1, 2. Formally, he thus observes 1 LI (·) = xIi di + u. 0 To compute the equilibrium, consider a candidate symmetric equilibrium xIij = AIj si + φIj (pIj ), j = 1, 2, where AIj is the 1 × 2 vector of trading intensities and φIj (·) is a linear function of the j-th price. The vector of aggregate order ﬂows is then LI (·) = z I + φI (pI ) , 17 See, e.g. O’Hara (1995). Lindsay and Schaede (1990) report that in 1987 “the average number [of stocks handled by a specialist] was 3.7 stocks.” 26
- 27. where z I = AI v + u, is the informational content of the vector of order ﬂows. Given competition and market makers’ risk neutrality, the equilibrium price vector is pI = E[v|z I ] = ΛI z I + (I − ΛI AI )¯ , v (5.9) where ΛI = (ΠI )−1 (AI ) Πu , is the matrix of market depths and ΠI = Πv + (AI ) Πu AI is the precision matrix of v|z I . Consequently, we have the following Lemma 3 In every linear equilibrium of the intermediate model, the vector of equi- librium prices is given by (5.9). Notice that in this system market makers can learn cross asset information indepen- dently from informed traders’ equilibrium behavior. As a consequence, the equilibri- um price of an asset j will be informationally equivalent to the linear combination of the informational contents of all the order ﬂows. Then, the following holds Proposition 8 Linear equilibria of the intermediate system exist and are character- ized implicitly by ¯ xIij = j AI (si − v ) − j BI (¯ − pI ), v (5.10) where, j AI = γ (Var[vj |pIj , si ])−1 c2j , (5.11) is the vector of the sensitivities of the demand for asset j to the speculator’s private signals, j BI = −γ (Var[vj |pIj , si ])−1 (1 − c1j ), (5.12) is the sensitivity of the demand for asset j to the equilibrium price of the asset, and c1j , c2j , and Var[vj |pIj , si ] are deﬁned in the appendix. Proof. See the appendix. QED Uniqueness of the equilibrium is an issue. Numerical simulations have been carried out and for diﬀerent initial conditions, the solution of the ﬁxed point problem did not change. 27
- 28. Remark 4 A consequence of allowing market makers to observe diﬀerent order ﬂows (see e.g. Admati (1985) for similar results), is that multicollinearity eﬀects can arise and asset prices can be decreasing in the informational content of their order ﬂows. For example if 5.26 6.7 10.52 4.73 Πv = , Πu = , 6.7 10.5 4.73 2.6 Π = I and γ = 1, the depth matrix of the intermediate system is 0.99 0.38 ΛI = . −0.59 −0.18 Therefore, an increase in zI2 leads to a decrease in the price of asset 2. To compare price eﬃciencies, I run simulations on the three models, using the same parametrization of section 4.2. The results are summarized in ﬁgure 5, where I plot price informativeness in the three systems as a function of ρv , for τ = τu = 1 and τv ∈ {.2, .5, 1, 3} (as before the dotted and the continuous lines indicate price eﬃciencies of, respectively, the restricted and the unrestricted systems while the dashed line, indicates eﬃciency of the intermediate system) . When only correlation across asset pay-oﬀs is relevant, the restricted system is more informative than the intermediate one which, in turn, is more informative than the unrestricted system. The simulations conﬁrm the intuition of section 4.2: restricting the amount of public information that informed speculators observe when choosing their position, forces them to exploit more intensively their private information, enhancing the informativeness of order ﬂows. In the intermediate system, market makers exploit correlated information and this improves their estimation of the asset pay-oﬀs. Prices signal more information and traders have lower incentives to exploit private information. Therefore, order ﬂows end up being less informative that in the restricted system. However, given that informed cannot condition on all equilbrium prices, they have more incentives to exploit their signals than in the unrestricted sytem. As a result order ﬂows in this system are more informative. 6 Conclusions I have analyzed two trading systems where competitive speculators exploit multi di- mensional sources of private information and contrasted their eﬃciency properties on 28
- 29. 4.6 1.8 4.5 1.7 Efficiencies Efficiencies 4.4 1.6 4.3 1.5 4.2 1.4 4.1 1.3 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 rv rv 0.8 0.22 0.75 0.2 0.7 0.18 Efficiencies Efficiencies 0.65 0.16 0.6 0.14 0.55 0.12 0.5 0.1 0.45 0.08 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 rv rv Figure 5: Price eﬃciencies in the three systems as a function of ρv . 29
- 30. the basis of two diﬀerent pricing schemes. The results show that incentives to exploit private information crucially depend on the speciﬁc price formation mechanism one considers. Indeed, to the extent that private and public information are substitutable, a system allowing traders to observe more public signals, reduces their incentives to exploit private signals. This, in turn, may make a multi price contingent mechanism less eﬃcient than a single price contingent one, in stark contrast with the view that a mechanism of the ﬁrst type should render prices more informative (Amihud and Mendelson 1991b). 30
- 31. 7 Appendix Proof of lemma 2. CARA and normality of the random variables give E[vj |pRj , si ] − pRj xRij = γ . Var[vj |pRj , si ] Assume for the moment that SOC are satisﬁed (we will prove later that in equilibrium this is the case). Then, we need to compute j λ−1 (p − v ) j ¯ E[vj |pRj , si ] = vj + ¯ c1j c2j , ¯ (si − v ) where the parameter c1j and the vector c2j are deﬁned as follows c1j c2j Var[pRj , si ] = Cov[vj , {si , pRj }]. Standard normal computations give j AR Π−1 AR + Π−1 j v u j AR Π−1 v Var[pRj , si ] = , (7.13) j AR Π−1 v Π−1 + Π−1 v and j AR Π−1 j v Cov[vj , {pRj , si }] = . j Π−1 v Inverting (7.13) we obtain −1 −1 D1 −D1 j AR (Πv + Π )−1 Π −1 (Var[pRj , si ]) = , −1 −D1 (j AR (Πv + Π )−1 Π ) D2 where D1 = j AR (Πv + Π )−1 AR + Π−1 j, u and D2 = Πv (Πv + Π )−1 Π + D1 (Π−1 + Π−1 )−1 Π−1 AR jj AR Π−1 (Π−1 + Π−1 )−1 . −1 v v v v 31
- 32. Using the previous matrix and the formula Var[vj |pRj , si ] = j Π−1 j − (Cov[vj , {si , pRj }]) (Var[pRj , si ])−1 (Cov[vj , {si , pRj }]) , v after standard normal calculations one can obtain j AR (Πv + Π )−1 j c1j = , D1 c2j = j (I − c1j AR ) (Πv + Π )−1 Π , (7.14) and, Var [vj |pRj , si ] = j (I − AR c1j ) (Πv + Π )−1 j. (7.15) QED Proof of proposition 2 Equilibrium existence depends on the existence of a solution to the ﬁxed point problem (7.15). To compute the equilibrium, notice that we can rewrite the system (3.6) as follows: aR11 = γτ 1 − ρ aR12 τ 1 /τ 2 , aR22 = γτ 2 − ρ aR21 τ 2 /τ 1 , (7.16) γ h21 √ aR21 = τ2 −ρ τ 1τ 2 , 1 − ρ2 h22 γ h12 √ aR12 = τ1 −ρ τ 1τ 2 . 1 − ρ2 h11 To see this set h11 = (I − AR c11 )(Πv + Π )−1 , 1,1 h12 = (I − AR c11 )(Πv + Π )−1 , 1,2 h21 = (I − AR c12 )(Πv + Π )−1 , 2,1 h22 = (I − AR c12 )(Πv + Π )−1 . 2,2 Then, (Var[v1 |pR1 , si ])−1 c21 = 1 (h12 /h11 ) Π , (Var[v2 |pR2 , si ])−1 c22 = 1 (h21 /h22 ) Π . 32
- 33. And by explicitly expressing the equilibrium conditions, one obtains (7.16). There are now two cases to consider: the case in which ρ = 0, that gives aR11 = γτ 1 and a cubic equation in aR12 and the case in which ρ = 0. Start by considering the second (the ﬁrst is just a simpliﬁcation of it). Substituting the ﬁrst equation in (7.16) into the last one, gives the following cubic equation in aR12 (aR12 )3 (1 − ρ2 )(1 − ρ2 )φ1 + aR12 φ1 φ2 + φ3 = 0, v where √ φ1 = τ 1 τ 2 (1 − ρ2 ) + τv1 τ 2 + τv2 τ 1 + τv1 τv2 (1 − ρ2 ) − 2ρ ρv τ 1 τ 2 τv1 τv2 , v φ2 = ((1 − ρ2 )τ 2 + (1 − ρ2 )τv2 + γ 2 (1 − ρ2 )τ 1 τ 2 τu1 ), v v and √ √ √ φ3 = γ ρ τ 1 τv2 − ρv τ 2 τv1 τ 2 τv2 τv1 (τv2 (1 − ρ2 ) + τ 2 ) √ + τ 1 (τv2 + τ 2 (1 − ρ2 )) − 2ρv ρ τv2 τ 2 τ 1 τv1 . v The discriminant associated to this equation is 3 2 φ2 γφ3 ∆=4 2 )(1 − ρ2 ) + 27 2 )(1 − ρ2 )φ , (1 − ρ v (1 − ρ v 1 which can be easily proved to be positive. Therefore, the result follows. QED Proof of proposition 3. For part 1, rearranging the cubic equation deﬁning aR12 gives aR12 φ1 (1 − ρ2 )(1 − ρ2 )(aR12 )2 + φ2 +φ3 = 0. v (1) It is easy to check that (1) is positive. Therefore for a solution to exist, it must be the case that aR12 has a sign opposite to φ3 . Since φ3 > 0 ⇔ τ 1 Cov[ 1 , 2 ] > τv1 Cov[v1 , v2 ], the result follows. 33
- 34. We can now show that Var[v1 |pR1 , si ] > 0 at equilibrium. After some algebra, we can see that Var[v1 |pR1 , si ] > 0 ⇐ γτ 21 (τ 2 (1 − ρ2 ) + τv2 (1 − ρ2 )) v τv1 τ1 + 2γaR12 (1 − ρ2 )τ 1 τv2 ρv −ρ τv2 τ2 2 τv1 τ1 + (aR12 )2 (1 − ρ2 ) τ 1 (1 − ρ2 ) + τv1 (1 − ρ2 ) + τv2 ρv v v −ρ > 0. τv2 τ2 Given our characterization of aR12 , the second term of the above expression is always positive in equilibrium, while the ﬁrst and the third are easily seen to be positive. Thus, the SOC of the informed utility maximization problem are satisﬁed. For part 2, if ρ = 0 it is straightforward. Otherwise, assume that aR11 < 0, then we have aR12 = ρ−1 τ 2 /τ 1 (γτ 1 − aR11 ). If ρ > (<)0, aR12 > (<)0 always, a contradiction. Next, calculating λR gives 1 τu1 τv2 γτ 1 + aR12 ρv τv1 /τv2 − ρ τ 1 /τ 2 λR1 = √ . τu1 (aR11 )2 τv2 + (aR12 )2 τv1 + 2ρv aR11 aR12 τv1 τv2 + τv1 τv2 It is easy to see that both numerator and denominator are positive. The result follows. Part 3 and 4 follow by manipulating (7.16). QED Proof of corollary 2. In a similar way as done before, we obtain the system of equations (3.8). To solve the system, there are two cases to consider: ρ = 0 and ρ = 0. When ρ = 0, we can multiply the second equation in the system (3.8) by ρ and add it to the ﬁrst. Then aR1 = γτ (1 − ρ2 ) − ρ aR2 . (7.17) Substituting aR1 into the second equation in the system (3.8), and rearranging, we obtain γτ τv (1 − ρ2 )(ρ − ρv ) + (aR2 )3 τu (1 − ρ2 )(1 − ρ2 ) v + aR2 τ (1 − ρ2 ) + (1 − ρ2 )(τv + γ 2 τ 2 τu (1 − ρ2 )(1 − ρ2 )) = 0, v v (7.18) 34
- 35. this is a cubic equation in aR2 . Computing the associated discriminant, we get 3 τ (1 − ρ2 ) + (1 − ρ2 )(τv + γ 2 τ 2 τu (1 − ρ2 )(1 − ρ2 )) v v ∆ = 4 2 )(1 − ρ2 ) τu (1 − ρv 2 γτ τv (1 − ρ2 )(ρ − ρv ) + 27 , τu (1 − ρ2 )(1 − ρ2 ) v which is always positive. Therefore, we conclude that there exists only one real root. When ρ = 0 the ﬁrst of (3.8) gives aR1 = γτ . Substituting this solution into the second equation, after rearranging, we get (aR2 )3 τu (1 − ρ2 ) + aR2 (τv + (1 − ρ2 )(γ 2 τ 2 τu + τ )) + γ(ρ2 − 1)ρv τv τ = 0. v v v Computing the discriminant of this cubic, we obtain 3 2 τv + (1 − ρ2 )(γ 2 τ 2 τu + τ ) v γ(ρ2 − 1)ρv τv τ v ∆=4 + 27 , τu (1 − ρ2 ) v τu (1 − ρ2 ) v which is always positive. Therefore also in this case there will be a unique real root. QED Proof of corollary 3. For part 1, the argument is as follows. Rearrange the cubic equation (7.18), to get aR2 (aR2 )2 τu (1 − ρ2 )(1 − ρ2 ) + ((1 − ρ2 )(τv + γ 2 τ 2 τu (1 − ρ2 )) + τ (1 − ρ2 )) v v v +γτ τv (1 − ρ2 )(ρ − ρv ) = 0. Now, if ρ − ρv > 0, then for a real solution to exist, it must be the case that aR2 < 0 (and vice-versa). For part 2, assume that aR < 0, then rearranging (7.17), we get 1 aR2 = ρ−1 γτ (1 − ρ2 ) − ρ−1 aR1 . According to this condition, if ρ < 0, then aR2 < 0 for any ρv , which is impossible according to the proof of the previous part. If ρ > 0, then aR2 > 0, no matter ρv , again impossible. Hence the result. Computing λR , we get (aR1 + aR2 (ρv − ρ ))τu λR = . ((aR1 )2 + (aR2 )2 + 2aR1 aR2 ρv )τu + τv 35
- 36. If ρ = 0 then it is easy to see that the numerator of the fraction is positive; if ρ = 0, substituting the expression for aR1 and rearranging, the numerator of λR becomes (γτ (1 − ρ2 ) + aR2 (ρv − ρ ))τu , which is positive. The denominator of the depth is also positive since if ρv < 0 and aR2 < 0 then 2ρv aR1 aR2 > 0, while if ρv < 0 and aR2 > 0 then (aR1 )2 + (aR2 )2 + 2ρv aR1 aR2 > (aR1 − aR2 )2 > 0. A similar argument can be used in the case ρv > 0. QED Before proving proposition 6, I need the following lemma Lemma 4 When the information structure is symmetric, price informativeness in the two systems is given by (a) if ρv = ρ = 0 and ρu = 0, −1 τv + a2 τu (1 − ρ2 )3 1 v IpU R = τv − , τv + a4 τu (1 − ρ2 )5 + 2a2 τu τv (1 − ρ2 )2 2 1 2 v 1 v −1 τv + a2 τu (1 − ρ4 ) 1 v IpR = τv − 2 4 2 2 )3 + 2a2 τ τ (1 − ρ2 ) . τv + a1 τu (1 − ρv 1 u v v (b) If ρu = ρ = 0 −1 τv + a2 τu (1 − ρ2 ) 1 v IpU R = τv − 2 + a4 τ 2 (1 − ρ2 ) + 2a2 τ τ , τv 1 u v 1 u v −1 τv + (a2 + a2 )τu (1 − ρ2 ) R1 R2 v IpR = τv − 2 2 (1 − ρ2 )(a2 − a2 )2 + 2τ τ (a2 + a2 + 2ρ a a ) . τv + τu v R1 R2 u v R1 R2 v R1 R2 (c) If ρv = ρ = 0 −1 τv (1 − ρ2 ) + a2 τu u 1 IpU R = τv− 2 2 ) + τ 2 a4 + 2τ τ a2 , τv (1 − ρu u 1 u v 1 IpR = IpU R . (d) If ρu = ρv = 0, −1 τv + (a2 + a2 )τu 1 2 IpU R = τv − 2 + τ 2 (a2 − a2 )2 + 2τ τ (a2 + a2 ) , τv u 1 2 u v 1 2 −1 τv + (a2 + a2 )τu R1 R2 IpR = τv − 2 . τv + τu (a2 − a2 )2 + 2τu τv (a2 + a2 ) 2 R1 R2 R1 R2 36
- 37. Proof. From the properties of the multivariate normal random variable, Π = Πv + A Πu A, and an analogous formula holds for the restricted system. Given this, the above expressions follow from matrix algebra. QED Proof of proposition 5. Suppose ρ = ρu = 0 and perform a second order expansion of IpR and IpU R around ρv = 0. Given that both IpR and IpU R are convex in ρv and that they have a minimun in ρv = 0, it follows that ∂ 2 IpR IpR (ρv ) = ρ2 + R1 (0). v (7.19) ∂ρ2 v ρv =0 The second term in the above expression is a function of the ﬁrst partial derivative of aR2 evaluated in ρv = 0. Diﬀerentiating implicitly aR2 ∂aR2 aR1 τv = , ∂ρv ρv =0 τv + a2 τu + τ R1 substituting it in (7.19) gives ∂ 2 IpR 2a2 τu τv ((τ + a2 τu )2 + 3τv (τ + 2τ + τv )) R1 R1 =− , ∂ρ2 v ρv =0 τ 3 τi2 where τ = τv + a2 τu and τi = τ + τ . R1 In the same way one can expand IpU R around ρv = 0 and obtain that ∂ 2 IpU R IpU R = ρ2 + R2 (0), v (7.20) ∂ρ2 v ρv =0 where ∂ 2 IpU R 2a2 τu τv 1 =− . ∂ρ2 v ρv =0 τ3 Computing 2a2 τu τv ((τ + a2 τu )2 + 3τv (τ + 2τ + τv )) R1 R1 2a2 τu τv IpR − IpU R = − − − 13 τ 3 τi2 τ 2a2 τu τ 2 (4τ + a2 τu + 5τv ) = − 1 v 3τ 2 1 , (7.21) τ i which is always negative. QED Proof of proposition 6 37
- 38. Given that when ρ = ρv , aR1 = (1 − ρ )a1 = (1 − ρv )a1 , comparing the formulas for price informativeness, one can see that τv + a2 τu (1 − ρ2 )3 1 v IpR − IpU R = τv + a4 τu (1 − ρ2 )5 + 2a2 τu τv (1 − ρ2 )2 2 1 2 v 1 v τv + a2 τu (1 − ρ4 ) 1 v − 2 . τv + a4 τu (1 − ρ2 )3 + 2a2 τu τv (1 − ρ2 ) 1 2 v 1 v Notice that the denominators of both expressions are positive. Evaluating this diﬀerence at a1 = γτ /(1 − ρ2 ), and checking the sign of the numerator, it is easy to v see that Numerator(IpR − IpU R ) = 1 γ 2 ρ2 τ 2 τu (γ 4 τ 4 τu (1 − ρ2 ) + γ 2 τ 2 τu τv (2 + ρ2 ) + τv (1 − ρ2 )2 ) , v 2 v v v 1 − ρ2 v which is clearly positive. QED Proof of proposition 7 If ρ = ρv = 0, then equilibrium strategies coincide in the two mechanisms. Therefore, price informativeness is the same. If ρv = ρu = 0, then a similar reasoning as the one developed in the proof of proposition 5 leads to the inequality displayed in the proposition. QED Proof of proposition 8. The proof follows from 2 claims. Claim 1 In every linear equilibrium of the intermediate system, informed strategies are given by (5.10). Proof. In this model market makers observe each order ﬂow, therefore, pI = E[v|z I ] = ΛI z I + (I − ΛI AI )¯ , v where, z I = AI v + u. Notice that this implies that each price pIj is informationally equivalent to the linear combination j ΛI z I , where ΛI = (ΠI )−1 (AI ) Πu is the matrix of market depth in the intermediate model. 38
- 39. CARA preferences and normality of the distributions give that E[vj |pIj , si ] − pIj xIij = γ . Var[vj |pIj , si ] Standard normal calculations give Var[pIj , si ] = j ΛI AI Π−1 (AI ) + Π−1 (ΛI ) j j ΛI AI Π−1 v u v , (7.22) j ΛI AI Π−1 v Π−1 + Π−1 v and j ΛI AI Π−1 j v Cov[vj , {pIj , si }] = . j Π−1 v Step-wise inversion of (7.22) gives (Var[pIj , si ])−1 = −1 −1 D1 −D1 j ΛI AI (Πv + Π )−1 Π , −1 −D1 (j ΛI AI (Πv + Π )−1 Π ) D2 where D1 = j ΛI AI (Πv + Π )−1 (AI ) + Π−1 (ΛI ) j, u and D2 = Πv (Πv + Π )−1 Π + D1 (Π−1 + Π−1 )−1 Π−1 (AI ) (ΛI ) jj ΛI AI Π−1 (Π−1 + Π−1 )−1 . −1 v v v v Using the previous matrix and the formula Var[vj |pIj , si ] = j Π−1 j − (Cov[vj , {si , pIj }]) (Var[pIj , si ])−1 Cov[vj , {si , pIj }], v we obtain Var [vj |pIj , si ] = j (I − ΛI AI c1j ) (Πv + Π )−1 j. (7.23) 39
- 40. Finally, indicating with c1j and c2j the weights that a trader i puts on pIj and si in his estimation of vj , standard normal computations give: j ΛI AI (Πv + Π )−1 j c1j = , D1 c2j = j (I − c1j ΛI AI ) (Πv + Π )−1 Π . (7.24) QED Claim 2 A linear equilibrium of the intermediate system exists. Proof. Notice that equilibrium existence hinges upon the existence of a solution to the ﬁxed point problem (7.24). To compute the equilibrium, note that we can rewrite the system (5.11) as follows: aI11 = γτ 1 − ρ aI12 τ 1 /τ 2 , aI22 = γτ 2 − ρ aI21 τ 2 /τ 1 , (7.25) γ h21 √ aI21 = τ2 −ρ τ 1τ 2 , 1 − ρ2 h22 γ h12 √ aI12 = τ1 −ρ τ 1τ 2 . 1 − ρ2 h11 To see this set h11 = (I − ΛI AI c11 )(Πv + Π )−1 , 1,1 h12 = (I − ΛI AI c11 )(Πv + Π )−1 , 1,2 h21 = (I − ΛI AI c12 )(Πv + Π )−1 , 2,1 h22 = (I − ΛI AI c12 )(Πv + Π )−1 . 2,2 Then, (Var[v1 |pI1 , si ])−1 c21 = 1 (h12 /h11 ) Π , −1 (Var[v2 |pI2 , si ]) c22 = 1 (h21 /h22 ) Π . And by explicitly expressing the equilibrium conditions, one obtains (7.25). 40
- 41. Consider the ﬁrst equation in (7.25) and suppose that h12 ≥ 0. Then for ρ ∈ [0, 1), γτ 2 h12 0 ≤ aI12 ≤ , 1 − ρ2 h11 while for ρ ∈ (−1, 0] γτ 2 h12 γτ 2 h12 √ ≤ aI12 ≤ − 2ρ τ 1τ 2 . 1 − ρ2 h11 1 − ρ2 h11 Suppose that h12 ≤ 0, then for ρ ∈ [0, 1), γτ 2 h12 √ − 2ρ τ 1τ 2 ≤ aI12 ≤ 0, 1 − ρ2 h11 while for ρ ∈ (−1, 0] γτ 2 h12 γτ 2 h12 √ ≤ aI12 ≤ − 2ρ τ 1τ 2 . 1 − ρ2 h11 1 − ρ2 h11 A similar reasoning can be made to show that for any combination of signs of h21 , ρ , aI21 lays in a compact interval. Deﬁne Φh12 ,h21 ,ρ as the function that for each com- bination of signs of h12 , h21 and ρ , maps the vector (aI12 , aI21 ) from and to the appropriate compact set in IR2 . The function Φ is component wise continuous (it is a polynomial in aI12 , aI21 ), thus existence of a ﬁxed point follows from Brower’s ﬁxed point theorem. Therefore, for an initial guess of the trading intensities, we can calculate numeri- cally the ﬁxed point for each set of parameters. QED 41
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