We analyze the role of liquidity and collection of information in order to measure the speed of revelation of information during the preopening of order-driven markets. We extend Vives (1995) model to the case where risk averse traders receive a new private signal before each round of quotation of the preopening. We show that price discovery takes place at high speed which is consistent with the empirical studies of Biais, Hillion and Spatt (1999)

Germain, Laurent

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                               This is an author-deposited version published in: http://oatao.univ-toulouse.fr/  Eprints ID: 5805  To cite this version: Germain, Laurent High Speed of Learning in Financial Markets. (2003) Finance, vol. 24 . pp. 79-86. ISSN 0752-6180 Open Archive Toulouse Archive Ouverte (OATAO)  OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible.   Any correspondence concerning this service should be sent to the repository administrator: staff-oatao@inp-toulouse.fr  High Speed of Learning in Financial MarketsLaurent GERMAIN ∗September 2003∗Groupe de Finance ESC Toulouse, SUPAERO and Europlace Institute of Finance, I ac-knowledge financial support from the European Union grant ERBFMRXCT 960054 part of thisarticle was written while I was a faculty at the London Business School. I thank Bruno Biais,Gabrielle Demange, Bertrand Jacquillat, Michel Levasseur, Eric Renault, Jean-Charles Rochetas well as seminar participants at Toulouse University, EEA, ESEM, AFFI, Oxford Universityand London Business School for their comments. I am especially undebted to Nour Meddahi. Ithank Emmanuel Schertzer for helpful research assistance. Correspondence to Laurent GermainESC Toulouse 20 Boulevard Lascrosses 31068 Toulouse Cedex, email l.germain@esc-toulouse.fr,tel + 33 561 294 943.1AbstractHigh Speed of Learning in Financial MarketsWe analyze the role of liquidity and collection of information in order to mea-sure the speed of revelation of information during the preopening of order-drivenmarkets. We extend Vives (1995) model to the case where risk averse tradersreceive a new private signal before each round of quotation of the preopening.We show that price discovery takes place at high speed which is consistent withthe empirical studies of Biais, Hillion and Spatt (1999).2Re´sume´Forte Vitesse d’Apprentissage sur les Marche´s FinanciersNous analysons le roˆle de la liquidite´ et de l’arrive´e d’informations dans lamesure de la vitesse de convergence des prix durant la pre´-ouverture de marche´sgouverne´s par les ordres. Nous e´tendons le mode`le de Vives (1995) au cas ou` desagents averses au risque rec¸oivent une nouvelle information avant chaque tour decotation. Nous montrons que la vitesse de convergence est plus rapide et qu’elleest valide´e par les e´tudes empiriques de Biais, Hillion et Spatt (1999).31 IntroductionHow do prices aggregate dispersed information in financial markets ? This is oneof the fundamental questions in finance and in the theory of market microstruc-ture. In his seminal paper Grossman (1976) proposes a theoretical framework tounderstand the role of prices as a vector of information over one period of trad-ing. In a dynamic setting, Kyle (1985) and Holden and Subrahmanyam (1992)among others study the revelation of information during the trading day. Vives(1995) focuses on the speed at which agents learn from prices during the pre-opening of order-driven markets. Indeed, the Paris Bourse, Toronto, Tokyo orMadrid stock exchanges have set up a preopening period in order to facilitate theprice discovery after the arrival of new information which occurs after a periodof non quotation (as overnight). In Paris, for example, during one hour and 45minutes the participants of the market can send buy or sell orders and observethe indicative clearing price. At any time before the opening of the market theagents can cancel their orders. At the opening of the market a price is set andthe market is cleared.In Vives (1995), risk averse traders are endowed with a noisy private signal siabout the liquidation value of a risky asset v. We extend Vives (1995) to the casewhere agents receive a new signal before each round of quotation. As a matterof fact, the flow of information received and collected by the market participantsduring the preopening of financial markets is important as the agents exhangeinformation with each other, and try to obtain as more information as possiblein order to anticipate the opening price1. Then, it is likely the case that theirprivate information evolves over the time of the preopening. At each period,traders observe the indicative price as well as their private signal and revisetheir demand accordingly. Asymptotically, for a large number of quotations, thelearning process converges to the rational expectations equilibrium and traderslearn v, the liquidation value of the risky asset. The speed of learning is definedas the speed at which prices converge to the fully revealing rational expectationsequilibrium - REE. In Vives (1995), the speed of revelation of information whenthe liquidity of the market is provided by market makers is√n where n is thenumber of round of quotations. But Biais, Hillion and Spatt (1999) in theirempirical financial study on the Paris Bourse reject this speed and find a speedof revelation of information which is much higher and is equal to√n3. In thispaper, we show that such high speed of revelation of information can be attributedto the collection of new private information2.As in Vives (1995), to characterize the learning process we focus on two pa-rameters :• the reaction of the traders to their private information,• and the speed at which prices reveal private information.With market makers supplying the liquidity in the financial market (or alter-natively competitive limit orders) - endogenous price function - our results arethe following :• the speed of learning is n 32 ,• and the reaction of the traders to their private information goes to infinity.In fact, the liquidity effect described by Vives (1995) and the informationaleffect, that we highlight in this model play the same symmetric role. It is why the2revelation of information happens very quickly when traders keep on collectinginformation and the liquidity is endogenously provided by market makers.In the next section, we characterize the equilibrium conditions. In section 3 westudy the asymptotic properties of the market. Section 4 states some concludingremarks.2 The model: financial market with competitivemarket makers or limit ordersAt each round n there is a probability δn that v be revealed to the traders anda probability 1 − δn that the tatonement process continues, those probabilitiesare independent over time. As in Vives (1995) we assume that the sequence (δn)is non decreasing. The probability that the tatonement process lasts until stagen is (1 − δn)...(1 − δ1) which goes to 0 as the sequence (δn) is non-decreasing.As Biais, Hillion and Spatt (1999) notice this is different from the Paris Boursewhere everybody knows the time of the opening. Nevertheless, this correspondsto certain financial markets as the Frankfurt Bourse where the opening time israndom3.In this market there are three types of agents - it is a competitive version ofKyle (1985):• Market makers who set the price conditionally on past volumes ωn,pn = E(v|ωn), (1)where ωn = (ω1, ..., ωn−1, ωn)4.3• A continuum of informed agents in the interval [0, 1] maximizing CARAutility U(x) = −e−ρx where ρ is the risk aversion coefficient5.Each agent i at round n is endowed with a set of signals (sni = si1, ..., sin)withsin = v + εinwhere v is the liquidation value of the normally distributed risky assetv˜ ; N(v¯, σ2v) and (εi)i ; N(0, σ2εi) are independent normally distributedvariables and independent to v too6.Even if we consider that those private signals are independent over time,we will see that a correlation does not change any of the results and thatone could consider the case of an AR(1) where εin = γ εi(n−1) + hin and(hin)i,n variables are independent with v and hin ; N(0, σ2hi).• At each round, noise traders submit inelastic demand,ut ; N (0, σ2u),where ut are independent with (εi)i and with v.Informed agents maximize their expected utility at each period conditional toIin their information both private and public. Past prices (pτ , τ < n) and pastvolumes (ωτ , τ < n) are public information. Therefore Iin is equal to (sni , pn−1)which are the past signals received from t = 1 to t = n and past prices untiln− 1 7.We define Xin as the quantity demanded by the ith agent and piin = Xin(v −pn) his profit. Each agent solves the following program :maxXinE(U(Xin(v − pn))|Iin) with U(x) = − exp−(ρx)4Therefore, the optimal quantity is:X∗in =E(v − pn|Iin)ρV ar(v − pn|Iin) (2)We focus on linear equilibria:Xin = ainfin + Fn(pn−1)where fin is the OLS estimator of v over sni (the mean of the private signals)and Fn is a deterministic function of past prices.By the SLLN we have∫ 10findi = v and therefore the aggregate volume is:ωn = anv + Fn(pn−1) + un (3)The following proposition characterizes the linear equilibrium.Proposition 1 There is a unique linear equilibrium where:Xn(sn, pn−1) = an(fin − pn−1) (4)and ωn the total volume is equal to:∫ 10Xndi+ un =∫ 10an(fin − pn−1)di+ un = an(v − pn−1) + unThe equilibrium price function is:pn = pn−1 + λnωnwhere an and λn are defined in the proof in the Appendix.53 Asymptotic properties: financial market withcompetitive market makers or limit ordersThe speed of revelation of information is captured by the asymptotic varianceτn = V ar(v|pn)−1 which is the informativeness of prices8 . When τn = +∞, v isrevealed.We compute ν such that n−ντn goes to a constant Aτ∞ as n goes to infinity.Aτ∞ is the asymptotic precision. By Amemya (1985) theorem :√n−ντn(pn − v); N (0, (Aτ∞)−1)Vives (1995) shows that the presence of competitive market makers allow thetraders to react increasingly to their private information. In this case the speedof convergence is√n, and√n(pn−v); N(0, σ2uρ2σ4ε). The following propositioncharacterizes the reaction of the traders to their signals and the speed of the pricediscovery process in our general set up.Proposition 2 The reaction (an) to private information is increasing and goesto infinity.The speed of convergence is n32 :n32 (Pn − v); N (0, σ2uρ2σ4ε)Proof: see AppendixThe speed of revelation of information is therefore the cube of that in Vives(1995). In fact, in our model informed agents have an unbiaised estimator ofv fin on top of the equilibrium price which is too an unbiaised estimator of v.Informed agents can compute a new unbiaised estimator the precision of which6is a weighted average of the two others fin et yn. As a consequence, the precisionof their information is higher than in the case where they had only one privatesignal over all the preopening period which accelerates the learning process.Whereas in Vives (1995), the reaction to the private information (an) is in-creasing, and goes to a limit (ρσ2ε)−1, in this model there is no limit to the reactionof the agents to their private information. an is not bounded because the depthof the market λn =anτuτnis of the order of n−3 whereas the order is n−1 with onlyone private signal. This shows the role of competitive market makers or limitorders in inducing revelation of information in a financial market where there isan important collection of information. (an) goes to infinity does not mean thatthe demand of the informed goes to infinity. This demand is finite and E|X|goes to 0 9. Indeed, each agent is competitive and the reaction to their privateinformation is infinite because at the limit they know v perfectly.4 ConclusionWe have shown that if the traders collect private information at each round ofquotation during the preopening period of an order-driven market then moreinformation is incorporated into prices and the speed of convergence towards thefully revealing REE is consistent with empirical studies. Indeed, in the case wherethe liquidity is endogenous the speed of learning predicted by our model is thesame as the speed estimated by Biais, Hillion and Spatt (1999) that is to say n32 .7APPENDIXPROOF OF PROPOSITION 1Using (3) the price function is given by:pn = E(v|ωn) = E(v|a1v + u1 + F1(p0), ..., anv + un + Fn(pn−1)).Let define zt = atv+ut and yt the OLS estimator of v over (zt). Vives (1995)shows that pt = E(v|zt) (consequently to the fact that the functions F (.) aredeterministic) and so that pt = E(v|yt) by definition of yt.Note that the main difference between the model presented in this article andthe one described in Vives (1995) is that the trader receives sequential signals.We know that the information collected to evaluate v can be summarized by asingle variable, which is the OLS estimator of v over the (si).We note fin this variable10.Therefore, Iin = (sni , pn−1) can be summarized by fin, the OLS estimator ofsni , and yn−1, the OLS estimator of pn−1.As in Vives (1995), we assume that the demand of a client is a linear func-tion of his private information (summarized by fin) and the public information(summarized by yn), and we find that:Xin = an(fin − pn−1)As in Vives (1995) we can write pn = pn−1+λnωn where the aggregate volumeωn = an(v − pn−1) + un and the depth of the market is λn = anτuτnwith τn =8τv + τuAn and An =∑nt=1 a2t and we show that an is defined as the solution ofthe following cubic equation :F (an) = (ρσ2εn−1an − 1)τn−1 + ρa3nτuτn= 0Morever, we will show in the next proof that this maximum is unique.QEDPROOF OF PROPOSITION 2From the proposition above, an and τn can be seen as the roots of a cubicequation, more precisely :F (an, τn) = 0Where F (an, τn) = (ρσ2εn−1an − 1)τn−1 + ρa3n τuτn .On the intervall <+∗ × <+∗, the function F is C1 and ∂F∂a is stricly positive.Applying the theorem of implicit functions, at each point (a∗n, τ∗n) solution ofthe equation above, there exists a function ψ defined on an open intervall Iτ∗n(containing τ ∗n) which goes to an open intervall Ia∗n (containing a∗n) such that :• For every τ included in Iτ∗n , if a is solution of F (τ, a) = 0, then a = ψ(τ).• ψ is C1 andψ′ = −∂F∂τ∂F∂aIt is straightforward to show that on <+∗×<+∗ : ∂F∂a > 0 and ∂F∂τ < 0. Hence,at every point (a∗n, τ∗n) the differential of the function ψ locally defined isalways positive.9Therefore an is an increasing function of τn and an is unique. Moreover, τnis obviously increasing with n, therefore (an) is increasing with n too. Hence, asτn > na1, the informativeness of prices, goes to ∞. Moreover we have :anρ(σ2εn−1 + anλn(τn−1)−1) = 1If an goes to a finite limit, the equation above is not verified when n goes to∞ as the first term goes to 0. Therefore, as (an) increases, it goes to ∞.Let f be a function of n such that an = n(ρσ2ε)−1 + f(n). By substituing anto this new expression in the equation above, it can easely be shown thatlimn→∞f(n)/n = 0Therefore,an ∼ n(ρσ2ε)−1Now let determine the speed of convergence.It is straightforward that the standard deviation of the estimator yn is equalto σu∑t=nt=1 at∑t=nt=1 a2t.After computing the equivalent of the numerator and the denominator of thisexpression when n goes to ∞ we find that the standard deviation of yn is:σyn ∼ (√3σ2ε)To apply the general central limit theorem (see Amemiya (1985)) we have tocheck that the following condition is fullfilled:limn→∞Max{ak|k = 1...n}An= 0We know that an ∼ n(ρσ2ε)−1 and An =∑t=nt=1 a2t ∼ n3 13( 1ρσ2ε )2. Because an isincreasingMax{ak|k = 1...n}An=anAn10which goes to zero as n goes to ∞Hence, we use the theorem Amemiya (1985) and we have :σ−1yn (t=n∑t=1a2t )12 (yn − fin); N(0, 1)and that√n3(yn − fin); N(0, 3ρ2σ2uσ4ε)QED11Footnotes1 This has been related to us by market practitioneers.2 Biais and Martinez (1999) study the preopening in Frankfurt and in Paris.3 It would be of interest to measure the speed of learning in Frankfurt as wellin order to see if this difference could change the result.4 In this paper we consider competitive market makers or alternatively competi-tion between traders who place limit orders. As Biais, Hillion, Spatt (1995)notice ”the conditional probability that investors place limit orders (ratherthan hitting the quotes) is larger when the bid-ask spread is larger or theorder book is thin.”5 Alternatively, Medrano and Vives (2001) consider a strategic agent.6 One could have considered that at each round of quotation the traders receivea new signal with a positive probability. This would have diminished thespeed of convergence. Nevertheless we also consider the case where theprivate signals follow an AR(1) and show that any correlation would havechanged our results.7 As prices and volumes are observationally equivalent it is not necessary toinclude the past volumes in the information set.8 τi is the inverse of the variance σ2i .9 It is straightforward that E|Xn| = (2/pi) 12anτ−1n−1 whereanτv + τuAn−1goes to0.1210 Consequently, there is is a very simple isomporphism between the two models.In fact, in this proposition, it is sufficient to substitute fin to si in Vives(1995) to derive the new results of this proof.13ReferencesAMEMIYA,T. Advanced Econometrics, Oxford: Basil Blackwell., 1985.BIAIS, B., MARTINEZ, I. ”Price Discovery Across the Rhine”, Mimeo, ToulouseUniversity, 2002.BIAIS, B., HILLION, P., and SPATT, C. ”An empirical analysis of the orderflow and order book in the Paris Bourse”, Journal of Finance, 15, 1995,1655-89.BIAIS,B., HILLION, P., and SPATT, C. “Price Discovery and Learning duringthe Preopening in The Paris Bourse”, Journal of Political Economy, 107,1999, 1218-48.GROSSMAN, S. “On the Efficiency of Competitive Stock Markets where Tradeshave Diverse Information”, The Journal of Finance, 31, 1976, 573-585.HOLDEN, C., and A. SUBRAHMANYAM, “Long-Lived Private Informationand Imperfect Competition”, The Journal of Finance, 47, 1992, 247-270.MEDRANO, L.A., and VIVES,X. ”Strategic Behavior and Price Discovery”,RAND Journal of Economics, 32, 2001, 221-48.KYLE,A. ”Continuous Auctions and Insider Trading”, Econometrica, 53, 1985,1315-35.VIVES, X. ”The Speed of Information Revelation in a Financial Market Mech-anism”, Journal of Economic Theory, 67, 1995, 178-204.14

High Speed of Learning in Financial Markets

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