Trading and Liquidity with Limited Cognition

Transcription

1 Trading and Liqidity with Limited Cognition Brno Biais, Johan Hombert, and Pierre-Olivier Weill December 11, 21 Abstract We stdy the reaction of financial markets to aggregate liqidity shocks when traders face cognition limits. While each financial instittion recovers from the shock at a random time, the trader representing the instittion observes this recovery with a delay, reflecting the time it takes to collect and process information abot positions, conterparties and risk exposre. Cognition limits lengthen the market price recovery. They also imply that traders who find that their instittion has not yet recovered from the shock place market sell orders, and then progressively by back at relatively low prices, while simltaneosly placing limit orders to sell later when the price will have recovered. This generates rond trip trades, which raise trading volme. We compare the case where algorithms enable traders to implement this strategy to that where traders can place orders only when they have completed their information processing task. Keywords: Liqidity shock, limit-orders, asset pricing and liqidity, algorithmic trading, limited cognition, sticky plans J.E.L. Codes: G12, D83. We are gratefl to the editor Harald Uhlig, and for anonymos referees for insightfl comments. Many thanks, for helpfl discssions and sggestions, to Andy Atkeson, Dirk Bergemann, Darrell Dffie, Emmanel Farhi, Thierry Focalt, Xavier Gabaix, Alfred Galichon, Christian Hellwig, Hgo Hopenhayn, Vivien Lévy Garboa, Johannes Horner, Boyan Jovanovic, Ricardo Lagos, Albert Menkveld, John Moore, Stew Myers, Henri Pages, Thomas Philippon, Gary Richardson, Jean Charles Rochet, Gillame Rochetea, Ioanid Ros, Larry Samelson, Tom Sargent, Jean Tirole, Aleh Tsyvinski, Jsso Välimäki, Dimitri Vayanos, Adrien Verdelhan, and Glen Weyl; and to seminar participants at the Daphine-NYSE-Eronext Market Microstrctre Workshop, the Eropean Smmer Symposim in Economic Theory at Gerzensee, École Polytechniqe, Stanford Gradate School of Bsiness, New York University, Northwestern University, HEC Montreal, MIT, and UCI. Palo Cotinho and Kei Kawakami provided excellent research assistance. Brno Biais benefitted from the spport of the Financial Markets and Investment Banking Vale Chain Chair sponsored by the Fédération Bancaire Française and from the Eroplace Institte of Finance, and Pierre-Olivier Weill from the spport of the National Science Fondation, grant SES Tolose School of Economics (CNRS-CRM, IDEI), [email protected] HEC Paris, [email protected] University of California Los Angeles and NBER, [email protected]

2 1 Introdction The perception of the intellect extends only to the few things that are accessible to it and is always very limited (Descartes, 1641) We analyze trades and price dynamics when investors face cognition limits and the market is hit by an aggregate liqidity shock. The shock indces a transient drop in the willingness and ability of financial instittions to hold assets sch as stocks or bonds. It can be triggered by changes in the characteristics of assets, e.g., certain types of instittions, sch as insrance companies or pension fnds, are reqired to sell bonds that lose their investment grade stats, or stocks that are delisted from exchanges or indices (see, e.g., Greenwood, 25). Alternatively, an aggregate shock can reflect events affecting the overall financial sitation of a category of instittions, e.g., fnds experiencing large otflows or losses (see Coval and Stafford, 27), banks incrring large losses (see Berndt, Doglas, Dffie, Fergson, and Schranz, 25), or specialists bilding extreme positions (see Comerton Forde, Hendershott, Jones, Molton, and Seasholes, 21). 1 To recover from a shock, instittions seek to nwind their positions in mltiple markets (e.g., credit defalt swaps (CDS) corporate bonds or mortgage backed secrities (MBS)). Instittions can also raise new capital, secre credit lines or strctre derivative trades to hedge their positions. These processes are complex and take time. However, once an instittion arranges enogh deals, it recovers from the liqidity shock. In this context, traders mst collect and process a large flow of information abot asset valations, market conditions and the financial stats of their own instittion. They mst obtain and aggregate information from several desks, markets and departments abot gross and net positions, the reslting risk exposre, and compliance with reglations. When traders have limited cognition, completing these tasks is challenging and reqires significant time and effort. We address the following isses: How do traders and markets cope with negative liqidity shocks? What is the eqilibrim price process after sch shocks? How are trading and prices affected by cognition limits? Do the conseqences of limited cognition vary with market mechanisms and technologies? 1 A striking example of a liqidity shock and its conseqences on instittions and market pricing is analyzed by Khandani and Lo (28). These observe that, dring the week of Agst 6 th 27, qantitative fnds sbject to margin calls and losses in credit portfolios had to rapidly nwind eqity positions. This reslted in a sharp bt transient drop in the S&P 5. Bt, by Agst 1 th 27 prices had in large part reverted. 2

3 We consider an infinite horizon, continos time market with a continm of rational, risk netral competitive financial instittions, deriving a non linear tility flow from holding divisible shares of an asset, as in Lagos and Rochetea (29) and Gârlean (29). To model the aggregate liqidity shock, we assme that at time this tility flow drops for all instittions, as in Dffie, Gârlean, and Pedersen (27) and Weill (27). Then, as time goes by, instittions progressively switch back to a high valation. More precisely, each instittion is associated with a Poisson process and switches back to high-valation at the first jmp in this process. Unconstrained efficiency wold reqire that low-valation instittions sell to high-valation instittions. However, in or model, sch asset reallocation is delayed becase of cognition limits. In line with the rational inattention models of Reis (26a and 26b), Mankiw and Reis (22) and Gabaix and Laibson (22), we assme that each trader engages in information collection and processing for some time and, only when this task is completed, observes the crrent valation of her instittion for the asset. We refer to this observation as an information event. When this event occrs, the trader pdates her optimal asset holding plan, based on rational expectations abot ftre variables and decisions. 2 The corresponding demand, along with the market clearing condition, gives rise to eqilibrim prices. Ths, while Mankiw and Reis (22) and Gabaix and Laibson (22) analyze how inattention affects consmption, we stdy how it affects eqilibrim pricing dring liqidity shocks. In the spirit of Dffie, Gârlean, and Pedersen (25), we assme that information events, and correspondingly trading decisions, occr at Poisson arrival times. 3 Ths in or model, each instittion is exposed to two Poisson processes: one concerns changes in its valation for the asset, the other the timing of its trader s information events. For simplicity, we assme these two processes are independent. Frther, for tractability, we assme that these processes are independent across instittions. Ths, by the law of large nmbers, the aggregate state of the market changes deterministically with time. 4 Correspondingly, the eqilibrim price process is 2 Ths, in the same spirit as in Mankiw and Reis (22), traders have sticky plans bt rationally take into accont this stickiness. 3 Note however that the interpretation is different. Dffie, Gârlean, and Pedersen (25) model the time it takes traders to find a conterparty, while we model the time it takes them to collect and process information. This difference reslts in different otcomes. In Dffie, Gârlean, and Pedersen investors do not trade between two jmps of their Poisson process. In or model they do, bt based on imperfect information abot their valation for the asset. In a sense or model can be viewed as the dal of Dffie, Gârlean, and Pedersen: they assme that traders continosly observe their valation bt are infreqently in contact with the market, while we assme that traders are continosly in contact with the market bt infreqently refresh their information abot their valation. 4 We also analyze an extension of or framework where the market is sbject to recrring aggregate liqidity shocks, occrring at Poisson arrival times. While, in this more general framework, the price is stochastic, the 3

4 deterministic as well. We show eqilibrim existence and niqeness. In eqilibrim the price increases with time, reflecting that the market progressively recovers from the shock. Limits to cognition lengthen the time it takes market prices to flly recover from the shock. Yet they do not necessarily amplify the initial price drop generated by that shock. Jst after the shock, with perfect cognition, the marginal trader knows that her instittion has a low valation, while with limited cognition sch trader is imperfectly informed abot her instittion s valation, and realizes that, with some probability, it may have recovered. We also show that the eqilibrim allocation is an information constrained Pareto optimm. This is becase in or setp there are no externalities, as the holding constraints on holdings imposed by cognition limits on one trader, do not depend on the actions of other traders. While we first characterize traders optimal policies in terms of abstract holding plans, we then show how these plans can be implemented in a realistic market setting, featring an electronic order book, limit and market orders, and trading algorithms. The latter enable traders to condct programmed trades while devoting their cognitive resorces to investigating the liqidity stats of their instittion. In this context, traders who find ot their instittion is still sbject to the shock, and correspondingly has a low valation for the asset, sell a lmp of their holdings, with a market sell order. Sch traders also program their trading algorithms to then gradally by back the asset, as they expect their valation to revert pward. Simltaneosly, they sbmit limit orders to sell the asset, to be exected later when the eqilibrim price will have recovered. To the extent that they by in the early phase of the aggregate recovery, and then sell towards the end of the recovery, sch traders act as market makers. 5 The corresponding rond trip transactions reflects the traders optimal response to cognition limits. These transactions can raise trading volme above the level it wold reach nder perfect cognition. We also stdy the case where trading algorithms are not available and traders mst implement their holding plans by placing limit and market orders when their information process jmps. With increasing prices, this prevents them from bying in between jmps of their information process. When the liqidity shock is large, this constraint binds and redces the efficiency of the eqilibrim allocation. It does not necessarily amplify the price pressre of the liqidity shock, however. Since traders anticipate they will not be able to by back ntil qalitative featres of or eqilibrim are pheld. 5 In doing so they act similarly to the market makers analyzed by Grossman and Miller (1988). Note however that, while in Grossman and Miller agents are exogenosly assigned market making or market taking roles, in or model, agents endogenosly choose to spply or demand liqidity, depending on the realization of their own shocks. 4

5 their next information event, they sell less when they observe that their valation is low. Sch a redction in spply limits the selling pressre on prices. Pt differently, banning the se of algorithms cold help alleviate the initial price pressre created by the liqidity shock. Yet, this policy wold redce welfare: indeed, in or model the eqilibrim with algorithmic trading is information constrained Pareto optimal. 6 The order placement policies generated by or model are in line with several stylized facts. Irrespective of whether algorithms are available, we find that sccessive traders place limit sell orders at lower and lower prices. Sch nderctting is consistent with the empirical reslts of Biais, Hillion, and Spatt (1995), Griffiths, Smith, Trnbll, and White (2) and Elll, Holden, Jain, and Jennings (27). Frthermore, or algorithmic traders both spply and consme liqidity, by placing market and limit orders, consistent with the empirical findings of Hendershott and Riordan (21) and Brogaard (21). Brogaard also finds that algorithms i) tend not to withdraw from the market after large liqidity shocks; ii) tend to provide liqidity by prchasing the asset after large price drops; and iii) in doing so profit from price reversals. Frthermore, Kirilenko, Kyle, Samadi, and Tzn (21) find that algorithms tend to by after a rise in price and that these prchases tend to be followed by a frther increase in price. They also find that algorithmic traders engage in high freqency rond trip trades, bying the asset and then selling it to other algorithmic traders. All these findings are in line with the implications of or model. Or analysis of the dynamics of markets in which traders choose whether to place limit or market orders is related to the insightfl papers of Parlor (1998), Focalt (1999), Focalt, Kadan, and Kandel (25), Ros (29), and Goettler, Parlor, and Rajan (25, 29). However, we focs on different market frictions than they do. While these athors stdy strategic behavior and/or asymmetric information nder perfect cognition, we analyze competitive traders with symmetric information nder limited cognition. This enables s to examine how the eqilibrim interaction between the price process and order placement policies is affected by cognition limits and market instrments. The next section describes the economic environment and the eqilibrim prevailing n- 6 In Section II of or spplementary appendix we analyze the case when traders can place only market orders when their information process jmps, i.e., limit orders and trading algorithms are rled ot. In sch case the price reverts to its pre shock level sooner. Indeed, when traders can place limit orders, the sell orders stored in the book exert a downward pressre on prices towards the end of the recovery. Bt, then again, the efficiency of the allocation is higher when traders can se limit orders, and even higher they can se algorithms. Indeed these market instrments enable traders to condct mtally beneficial trades that wold be infeasible with market orders only. 5

6 1 v(h, q) tility flow v(, q) qantity Figre 1: The tility flows of high (in ble) and low valation (in red) investors, when σ =.5. der nlimited cognition. Section 3 presents the eqilibrim prevailing with limited cognition. Section 4 discsses the implementation of the abstract eqilibrim holding plans with realistic market instrments sch as limit and market orders and trading algorithms. Section 5 concldes. Proofs not given in the text are fond in the appendix, and a spplementary appendix offers additional information abot the model, along with proofs and analyses. 2 The economic environment 2.1 Assets and agents Time is continos and rns forever. A probability space (Ω, F, P)is fixed,as well as an information filtration satisfying the sal conditions (Protter, 199). 7 There is an asset in positive spply s (, 1) and the economy is poplated by a [, 1]-continm of infinitely-lived agents that we call financial instittions (fnds, banks, insrers, etc.) disconting the ftre at the same rate r>. Each instittion can be in one of two states. Either it derives a high tility flow ( θ = h ) from holding any qantity q oftheasset,oritderivesalowtilityflow( θ = ), as illstrated in Figre 1. For high valation instittions, the tility flow per nit of time is v(h,q) = q, for all q 1, and v(h,q) = 1, for all q > 1. For low valation instittions, it is v(, q) = q δ q1+σ, for all q 1, and v(, q) = 1 δ/(1 + σ), for all q > 1+σ 1.8 The two parameters δ (, 1] and σ > captretheeffectofthelowstateontilityflows. The 7 To simplify the exposition, for most stated eqalities or ineqalities between stochastic processes, we sppress the almost srely qalifier as well as the corresponding prodct measre over times and events. 8 The short selling constraint is withot loss of generality in the following sense. If we extend the tility fnctions to q< in any way sch that they remain concave, then the eqilibrim otcomes we characterize are naffected. In particlar, q< never arises. 6

7 parameter δ controls the level of tility: the greater is δ, theloweristhemarginaltilityflow of low valation instittions. The parameter σ, on the other hand, controls the crvatre of low valation instittions tility flows. The greater is σ, the less willing sch instittions are to hold extreme asset positions. 9 Becase of this concavity, it is efficient to spread holdings among low valation instittions. This is similar to risk sharing between risk averse agents, and as shown below will imply that eqilibrim holdings take a rich set of vales. 1 This is in line with Lagos and Rochetea (29) and Gârlean (29). Note that, even in the σ limit, low valation investors tility flow is redced, by a factor 1 δ, btinthatcasethe tility flow is piecewise linear. 11 The difference between the high and the low states can be interpreted as a holding cost, or a capital charge or shadow cost associated with positions. As discssed in the introdction and in Dffie, Gârlean, and Pedersen (27) a variety of instittional factors can generate sch costs, e.g., reglatory constraints on holdings (see, e.g., Greenwood, 25), need for cash (see Coval and Stafford, 27, Berndt et al., 25), position limits (see Comerton Forde et al., 21, or Hendershott and Seasholes, 27), or tax considerations. Lastly, in addition to deriving tility from the asset, instittions can prodce (or consme) a non-storable nméraire good at constant marginal cost (tility) normalized to Liqidity shock To model liqidity shocks we follow Dffie, Gârlean, and Pedersen (27) and Weill (27). Before the shock, each instittion is in the high valation state, θ = h, andholdss shares of the asset. Bt, at time zero, the liqidity shock hits all the instittions, and they make a switch to low valation, θ =. Note, however, that the shock is transient. As discssed in the introdction, to cope with the shock, instittions seek to nwind positions, raise capital, secre credit lines, or hedge positions. All this process is complex and takes time. However, once the instittion is able to arrange enogh deals, it recovers from the liqidity shock. To captre the recovery process we assme that, for each instittion, there is a random time at which it reverts 9 The crvatre of low valation tilities contrasts with the constant positive marginal tility high valation instittions have for q<1. One cold have introdced sch crvatres for high valations as well, as in Lagos and Rochetea (29) or Gârlean (29) at the cost of redced tractability, withot qalitatively altering or reslts. 1 Note also that the holding costs of low valation instittions are homothetic. This reslts in homogenos asset demand and, as will become clear later, this facilitates aggregation. 11 For the σ limit, see or spplementary appendix, (Biais, Hombert, and Weill, 21b), Section III. 7

8 to the high valation state, θ = h, andthenremainsthereforever. Forsimplicity,weassme that recovery times are exponentially distribted, with parameter γ, and independent across investors. Hence, by the law of large nmbers, the measre µ ht of high valation investors at time t is eqal to the probability of high tility at that time conditional on low tility at time zero. 12 Ths µ ht =1 e γt, (1) and we denote by T s the time at which the mass of high tility instittions eqals the spply of the asset, i.e., µ hts = s. (2) 2.3 Eqilibrim withot cognition limits Consider the benchmark case where instittions continosly observe their θ t. To find the competitive eqilibrim, it is convenient to solve first for efficient asset allocations, and then find the price path that decentralizes these efficient allocations in a competitive eqilibrim. 13 In the efficient allocation, for t>t s, all assets are held by high valation instittions, and all marginal tilities are eqalized. Indeed, with an (average) asset holding eqal to s/µ ht < 1, the marginal tility is 1 for high valation instittions, while with zero asset holdings marginal tility is v q (, ) = 1 for low valation instittions. In contrast, for t T s,wehaveµ ht s, and each high valation instittion holds one nit of the asset while the residal spply, s µ ht, is held by low valation instittions. The asset holding per low valation instittion is ths: q t = s µ ht 1 µ ht. (3) This is an optimal allocation becase all high valation instittions are at the corner of their tility fnction: redcing their holdings wold create a tility loss of 1, while increasing their holdings wold create zero tility. Low valation instittions, on the other hand, have holdings 12 For simplicity and brevity, we do not formally prove how the law of large nmbers applies to or context. To establish the reslt precisely, one wold have to follow Sn (26), who relies on constrcting an appropriate measre for the prodct of the agent space and the event space. 13 Note that, with qasi linear tilities and nlimited cognition, in all Pareto efficient allocation of assets and nméraire goods, the asset allocation maximizes, at each time, the eqally weighted sm of the instittions tility flows for the asset, sbject to feasibility. 8

9 in [, 1), so their marginal tility is strictly positive and less than 1. For t T s,assoonasaninstittionswitchesfromθ = to θ = h, itsholdingsjmpfromq t to 1, while as long as its valation remains low, it holds q t,givenin(3),whichsmoothlydeclines with time. This decline reflects that, as time goes by, more and more instittions recover from the shock, switch to θ = h and increase their holdings. As a reslt, the remaining low valation instittions are left with less shares to hold. Eqilibrim prices reflect the cross section of valations across instittions. In or setting, by the law of large nmbers, there is no aggregate ncertainty and this cross section is deterministic. Hence, the price also is deterministic. For t T s,itiseqaltothepresentvaleofa low valation instittion s marginal tility flow: p t = t e r(z t) v q (, q z ) dz, where q z is given in (3). Taking derivatives with respect to t, wefindthatthepricesolvesthe Ordinary Differential Eqation (ODE): v q (, q t )=rp t ṗ t ξ t. (4) The left-hand side of (4) is the instittion s marginal tility flow over [t, t + dt]. The right-hand side is the opportnity cost of holding the asset: the cost of bying a share of the asset at time t and reselling it at t + dt, i.e., the time vale of money, rp t,minsthecapitalgain,ṗ t. Finally, when t T s, v q (, q z )=v q (, ) = 1 and the price is p t =1/r. Ths, the price increases deterministically towards 1/r, astheholdingsoflowvalation instittions go to zero and their marginal tility increases. Instittions do not immediately bid p this predictable price increase becase the demand for the asset bilds p slowly: on the intensive margin, high valation instittions derive no tility flow if they hold more than one nit; and, on the extensive margin, the recovery from the aggregate liqidity shock occrs progressively as instittions switch back to high tility flows. Ths, there are limits to arbitrage in or model, in line with the empirical evidence on the predictable patterns of price drops and reversals arond liqidity shocks. 14 Throghot this paper we illstrate or reslts with nmerical comptations based on the parameter vales shown in Table 1. We take the discont rate to be r =.5, in line with Dffie, 14 See, e.g., for short lived shocks the empirical findings of Hendershott and Seasholes (27), Hendershott and Menkveld (21) and Khandani and Lo (28). 9

10 Panel A: Proportion of high valation investors 1 µ ht s.5 T s Panel B: Price 1 perfect cognition limited cognition time (days) Figre 2: Poplation of high valation investors (Panel A) and price dynamics when σ =.3 (Panel B). T s T f Gârlean, and Pedersen (27). We select the liqidity shock parameters to match empirical observations from large eqity markets. Hendershott and Seasholes (27) and Hendershott and Menkveld (21) find liqidity price pressre effects of the order of 1 to 2 basis points, with dration ranging from 5 to 2 days. Dring the liqidity event described in Khandani and Lo (28), the price pressre sbsided in abot 4 trading days. Adopting the convention that there are 25 trading days per year, setting γ to 25 means that an instittion takes on average 1 days to switch to high valation. Setting the asset spply s to.59 then implies that with nlimited cognition the time it takes the market to recover from the liqidity shock (T s ) is approximately 9 days, as illstrated in Figre 2, Panel A. Lastly, for these parameter vales, setting δ =1impliesthattheinitialpricepressregeneratedbytheliqidityshockis between 1 and 2 basis points, as illstrated in Figre 2, Panel B Dffie, Gârlean and Pedersen (27) provide a nmerical analysis of liqidity shocks in over the conter markets. They choose parameters to match stylized facts from illiqid corporate bond markets. Becase we focs on more liqid electronic exchanges, we chose parameter vales different from theirs. For example in their analysis the price takes one year to recover while in ors it takes less than two weeks. 1

11 Table 1: Parameter vales Parameter Vale Intensity of information event ρ 25 Asset spply s.59 Recovery intensity γ 25 Discont rate r.5 Utility cost δ 1 Crvatre of tility flow σ {.3,.5, 1.5} 3 Eqilibrim with limited cognition We now trn to the case where agents have limited cognition. In the first sbsection below, we present or assmptions on cognition limits. In the second sbsection we solve for eqilibrim. To do so, we follow several standard steps: we first compte the vale fnction of the traders, then we maximize this fnction to pin down demands, finally we sbstite demands into the market clearing condition and we obtain the eqilibrim price. In the third sbsection we present the properties of the eqilibrim, regarding welfare, holding plans, trading volme, and prices. In the forth sbsection we show that or qalitative reslts are pheld in an extension of the model where recrring preference shocks lead to stochastic eqilibrim prices. 3.1 Assmptions Limited cognition Each instittion is represented in the market by one trader. 16 To determine optimal asset holdings, the trader mst analyze the liqidity stats of her instittion, θ t.thistaskiscognitively challenging. As mentioned in the previos section, to recover from the shock, the instittion engages in several financial transactions in a variety of markets, some of them complex, opaqe and not compterized. Evalating the liqidity stats of the instittion reqires collecting, analyzing and aggregating information abot the reslting positions. Or key assmption is that, becase of limited cognition and information processing constraints, the trader cannot continosly and immediately observe the liqidity stats of her instittion. 17 Instead, we assme there is a conting process N t sch that the trader observes θ t at each jmp of N t (and only 16 For simplicity we abstract from agency isses and assme that the trader maximizes the inter-temporal expected tility of the instittion. 17 Reglators have recently emphasized the difficlty of devising an integrated measrement of all relevant risk exposres within a financial instittion (see Basel Committee on Banking Spervision, 29). Academic research has also nderscored the difficlties associated with the aggregation of information dispersed in mltiple departments of a financial instittion (see Vayanos, 23). 11

12 then). 18 At the jmps of her information process N t the trader sbmits a new optimal trading plan, based on rational expectations abot {N,θ : t}, andherftredecisions. This is in line with the rational inattention model of Mankiw and Reis (22). For simplicity, the traders information event processes, N t, are assmed to be Poisson distribted, with intensity ρ, andindependentfromeachotheraswellasfromthetimesatwhichinstittionsemergefrom the liqidity shock Conditions on asset holding plans and prices When an information event occrs at time t>, a trader designs an pdated asset holding plan, q t,,forallsbseqenttimes t ntil the next information event. Formally, denoting D = {(t, ) R 2 + : t }, weletanasset holding plan be a bonded and measrable stochastic process q : D Ω R + (t,, ω) q t, (ω), satisfying the following two conditions: Condition 1. For each t, the stochastic process (t, ω) q t, (ω) is F t -predictable, where {F t } t is the filtration generated by N t and θ t. Condition 2. For each (t, ω), the fnction q t, (ω) has bonded variations. Condition 1 means that the plan designed at time t, q t,,candependonlyonthetrader s time-t information abot her instittion: the history of her information-event conting process, and of her instittion tility stats process p to, bt not inclding, time t. 2 Condition 2 is a technical reglarity condition ensring that the present vale of payments associated with q t, 18 The time between jmps creates delays in obtaining fresh information abot θ t, which can be interpreted as the time it takes the risk management nit or head of strategy to aggregate all relevant information and disseminate it to the traders. 19 For simplicity, we don t index the information processes of the different traders by sbscripts specific to each trader. Rather we se the same generic notation, N t, for all traders. 2 We add the not inclding qalifier becase the asset holding plans are assmed to be F t -predictable instead of F t -measrable. This predictability assmption is standard for dynamic optimization problems involving decisions at Poisson arrival times (see Chapter VII of Brémad, 1981). For mch of this paper, however, we need not distingish between F t predicability and F t measrability. This is becase the probability that the trader type switches exactly at the same time an information event occrs is of second order. Therefore, adding or removing the type information accring exactly at information events leads to almost srely identical optimal trading decisions. 12

13 is well defined. To simplify notations, from this point on we sppress the explicit dependence of asset holding plans on ω. At this stage of the analysis, we assme that traders have access to a sfficiently rich men of market instrments to implement any holding plan satisfying Conditions 1 and 2. We examine implementation in Section 4, where we analyze which market instrments are needed to implement eqilibrim holding plans, and the eqilibrim that arises when the men of market instrment is not sfficiently rich. The last technical condition concerns the price path: Condition 3 (Well-behaved price path). The price path is bonded, deterministic and continosly differentiable (C 1 ). As in the nbonded cognition case, becase there is no aggregate ncertainty, it is natral to focs on deterministic price paths. Frther, in the environment that we consider the eqilibrim price mst be continos, as formally shown in or spplementary appendix (see Biais, Hombert, and Weill, 21b, Section VI). The economic intition is as follows. If the price jmps at time t, alltraderswhoexperienceaninformationeventshortlybeforet wold want to arbitrage the jmp: they wold find it optimal to by an infinite qantity of asset and re-sell these assets jst after the jmp. This wold contradict market clearing. Finally, the condition that the price be bonded is imposed to rle ot bbbles (see Lagos, Rochetea, and Weill, 27, for a proof that bbbles cannot arise in a closely related environment). 3.2 Eqilibrim Intertemporal payoffs For any time, let τ denote the time of the last information event before, withthe convention that τ =ifnoinformationeventoccrred. Correspondinglyq, represents the holdings of a trader who had no information event by time and ths no opportnity to pdate her holding plan. Given that all traders start with the same holdings at time zero, we have q, = q, = s for all. The trader s objective is to maximize the inter-temporal expected vale of tility flows, net of the cost of bying and selling assets. With the above notations in mind, this can be written 13

14 as: E e v(θ r,q τ,)d p dq τ,, (5) where v(θ,q τ,) d is the tility enjoyed, and p dq τ, is the cost of asset prchases dring [, + d], given the holding plan chosen at τ,thelastinformationeventbefore. Given or distribtional assmptions for the type and information processes, and given technical Conditions 1 to 3 we can rewrite this objective eqivalently as: Lemma 1. The inter-temporal payoffs associated with the holding plan q t, is, p to a constant: V (q) =E where ξ = rp ṗ. e rt t e (r+ρ)( t) E t [v(θ,q t, )] ξ q t, d ρdt, (6) The interpretation of eqation (6) is as follows. The oter expectation sign takes expectation over all time t histories. The ρdt term in the oter integral is the probability that an information event occrs dring [t, t + dt]. Conditional on the time t history and on an information event occrring dring [t, t + dt], the inner integral is the disconted expected tility of the holding plan ntil the next information event. At each point in time this involves the difference between a trader s time t expected valation for the asset, E t [v(θ,q t, )], and the opportnity cost of holding that asset, ξ. This is similar to the reslt in Lagos and Rochetea (29) that an investor s demand depends on his crrent marginal tility from holding the asset as well as his expected marginal tility in the ftre. Finally, the discont factor applied to time is adjsted by the probability e ρ( t) that the next information event occrs after Market clearing In all what follows we focs on the case where all traders choose the same holding plan, which is natral given that traders are ex ante identical. 21 Of corse, while traders choose ex ante the same holding plan, ex post they realize different histories of N t and θ t,andhencedifferent asset holdings. The market clearing condition reqires that, at each date, the cross-sectional average asset holding be eqal to s, theper-capitaassetspply. Bythelawoflargenmbers,andgiven 21 By ex ante identical we mean that traders start with the same asset holdings and have identically distribted processes for information event and tility stats. 14

15 ex ante identical traders, the cross-sectional average asset holding is eqal to the expected asset holding of a representative trader. Hence, the market clearing condition can be written: E [q τ,] =s. (7) for all. Integrating against the distribtion of τ,andkeepinginmindthatq, = s, leads to or next lemma: Lemma 2. The time- market clearing condition, (7), writes: ρe (1 ρ( t) µ ht )E [q t, θ t = ]+µ ht E [q t, θ t = h] s dt =. (8) This lemma states that the aggregate net demand of traders who experienced at least one information event is eqal to zero. The first mltiplicative term in the integrand of (8), ρe ρ( t),isthedensityoftime t traders, i.e., traders whose last information event occrred at time t (,]. The first and second terms in the crly bracket are the gross demands of time t low and high valation traders respectively. The last term in the crly bracket is their gross spply. It is eqal to s becase information events arrive at random, which implies that the average holding of time t traders jst prior to their information event eqals the poplation average Eqilibrim existence and niqeness We define an eqilibrim to be a pair (q, p) sbjecttoconditions1,2and3andschthat:i) given the price path, the asset holding plans maximize V (q) givenin(6),andii)theholding plans are sch that the market clearing condition (8) holds at all times. In this sbsection we first present, in Lemmas 4 to 6, properties of holding plans implied by i) and ii). Then, based on these properties we obtain or first proposition, which states the niqeness and existence of eqilibrim and gives the eqation for the corresponding price. Going back to the vale V (q), in eqation (6), and bearing in mind that a trader can choose any fnction q t, sbject to Conditions 1 and 2, it is clear that the trader inter-temporal problem redces to point-wise optimization. That is, a trader whose last information event occrred at time t chooses her asset holding at time, q t,, in order to maximize the difference 15

16 between her expected valation for the asset and the corresponding holding cost: E t [v(θ,q t, )] ξ q t,. (9) Now, for all traders, tilities are strictly increasing for q t, < 1 and constant for q t, 1. So, if one trader finds it optimal to hold strictly more than one nit at time, thenitmstbethat ξ, implying that all other traders find it optimal to hold more than one nit. Inspecting eqation (8), one sees that in that case the market cannot clear since s<1. We conclde that: Lemma 3. In eqilibrim, ξ > and q t, [, 1] for all traders. To obtain frther insights on holding plans, consider first a time t high valation trader, i.e., a trader who finds ot at time t that θ t = h. Schtraderknowsthathervalationforthe asset will stay high forever. Hence E t [v(θ,q t, )] = v(h, q t, ), t. (1) Next, consider a time t low valation trader, i.e. a trader who finds ot at time t that θ t =. Thistraderanticipatesthathertilitystatswillremainlowbytime with probability (1 µ h )/(1 µ ht ). Hence: E t [v(θ,q t, )] = q t, δ 1 µ h qt, 1+σ 1 µ ht 1+σ, q t, [, 1] (11) Comparing (1) and (11), one sees that, for all asset holdings in (, 1), high valation traders have a niformly higher marginal tility than low valation traders. Now let S ρe ρ( t) (s µ ht ) dt, (12) the gross asset spply broght by all traders mins the maximm (nit) demand of high valation traders, integrating across all traders with at least one information event. Given the definition of S, and based on the above ranking of marginal tilities, one sees that the economy can be in one of two regimes. The first regime arises if S > : in that case, since q t, 1 for high valation traders, market clearing implies that q t, > for some low valation trader. Bt then all high valation traders mst hold one nit, since they have niformly higher marginal tility for holdings in [, 1]. The second regime arises if S <. In this case market clearing implies that some high valation trader mst find it optimal to hold strictly less than 16

17 one share: sch trader either strictly prefer to hold zero share, or is indifferent between any holding in [, 1]. Bt since high valation traders have niformly higher marginal tility for holdings in [, 1], this implies that all low valation traders hold zero shares. Smmarizing: Lemma 4. Let T f be the niqe strictly positive soltion of S =. Then: if (,T f ) then, for all t (,], θ t = h implies q t, =1; if [T f, ) then, for all t (,], θ t = implies q t, =. Next, consider the demand of high valation traders when >T f. We know from the previos lemma that low valation traders hold no asset. Ths, high valation traders mst hold the entire asset spply. Moreover, since S <, the market-clearing condition implies that some high valation traders mst hold strictly less than one share. Keeping in mind that high valation traders have the same linear tility flow over [, 1], this implies they mst be indifferent between any holding in [, 1]. Ths we can state the following lemma. Lemma 5. For all >T f, the average asset holding of a high valation trader is ρe ρ( t) sdt ρe ρ( t) µ ht dt, bt the distribtion of asset holdings across high valation traders is indeterminate. Now trn to the demand of low valation traders when <T f. Taking first-order conditions when θ t = in (9), we obtain, given q t, [, 1]: q t, = ifξ 1 (13) q t, =1 ifξ 1 δ 1 µ h (14) 1 µ ht q t, =(1 µ ht ) 1/σ Q if ξ 1 δ 1 µ 1/σ h 1 ξ, 1, where Q. (15) 1 µ ht δ(1 µ h ) Eqation (13) states that low valation traders hold zero nit if the opportnity cost of holding the asset is greater than 1, their highest possible marginal tility, which arises when q =. Eqation (14) states that low valation traders hold one nit if the opportnity cost of holding the asset is below the lowest possible marginal tility, which arises when q =1. Lastly,eqation (15) pins down a low valation trader s holdings in the intermediate interior case by eqating to the derivative of (11) with respect to q t,. 17

18 As discssed above, prior to time T f the holdings of some low valation traders mst be strictly greater than zero: ths, holdings are determined by either (14) or (15) and ξ >. By the definition of Q, ξ 1 δ(1 µ h )/(1 µ ht )ifandonlyif(1 µ ht ) 1/σ Q 1. Hence, the asset demand defined by (14) and (15) can be written as q t, =min{(1 µ ht ) 1/σ Q, 1}. (16) Sbstitting the demand from (16) into the market-clearing condition (8) and sing the definition of S in (12), the following lemma obtains. Lemma 6. If (,T f ), then for all t (,], θ t = implies q t, =min{(1 µ ht ) 1/σ Q, 1} where: (1 µ ht )min{(1 µ ht ) 1/σ Q, 1}ρe ρ( t) dt = S. (17) Eqation (17) is a one-eqation-in-one-nknown for Q that is shown in the proof appendix to have a niqe soltion. Taken together, Lemmas 4 throgh 6 imply: Proposition 1. There exists an eqilibrim. The eqilibrim asset allocation is niqe p to the distribtion of asset holdings across high valation traders after T f, and is characterized by Lemma 4-6. The eqilibrim price path is niqe, is increasing ntil T f, constant thereafter, and solves the following ODE: (,T f ): rp ṗ =1 δ(1 µ h )Q σ (18) [T f, ) : p = 1 r. (19) As in the perfect cognition case, the price deterministically increases ntil it reaches 1/r. One difference is that, while nder perfect cognition this recovery occrs at time T s (defined in eqation (2)), with limited cognition it occrs at the later time T f >T s. For <T f,the time low valation traders are the marginal investors, and the eqilibrim price is sch that their marginal valation is eqal to the opportnity cost of holding the asset, as stated by (18). For >T f, the entire spply is held by high valation investors. Ths the eqilibrim price is eqal to the present vale of their tility flow, as stated by (19). 22 This proposition is illstrated 22 They mst be indifferent between trading or not. This indifference condition implies that 1 rp +ṗ =. And, p =1/r is the only bonded and positive soltion of this ODE. 18

19 in Figre 2, Panel B, which plots the eqilibrim price nder limited cognition. Note that for this nmerical analysis we set the intensity of information events ρ to 25, which means that traders observe refreshed information on θ on average once a day. This is a plasible freqency, given the time it takes to collect and aggregate information across desks, departments and markets in a financial instittion. 3.3 Eqilibrim properties In this sbsection we discss the properties of the eqilibrim price and trades and compare them to their nbonded cognition conterparts Welfare To stdy welfare we define an asset holding plan to be feasible if it satisfies Conditions 1 and 2 as well as the resorce constraint, which is eqivalent to the market-clearing condition (7). Frthermore, we say that an asset holding plan q Pareto dominates some other holding plan q if it is possible to generate a Pareto improvement by switching from q to q while making time zero transfers among traders. Becase tilities are qasi linear, q Pareto dominates q only if W (q) >W(q ), where if and W (q) =E e r v(θ τ,q τ,) d. (2) The next proposition states that in or model the first welfare theorem holds: Proposition 2. The holding plan arising in the eqilibrim characterized in Proposition 1 maximizes W (q) among all feasible holding plans. This proposition reflects that, in or setp, there are no externalities, in that the holdings constraints imposed by limited cognition for one agent (and expressed in conditions 1 and 2) do not depend on the actions of other agents. These constraints translate into simple restrictions on the commodity space (conditions 1 and 2), allowing s to apply the standard proof of the first welfare theorem (see Mas-Colell, Whinston, and Green, 1995, Chapter 16) Holdings As set forth in eqation (16), for a trader observing at t that her valation is low, the optimal holdings at time >tare (weakly) increasing in Q.Relyingonthemarketclearingcondition, 19

20 the next proposition spells ot the properties of Q. Proposition 3. The fnction Q is continos, sch that Q + = s and Q Tf s σ 1+σ =. Moreover, if (21) Q is strictly decreasing with time. Otherwise, it is hmp-shaped. The economic intition is as follows. At time + the mass of traders with high valation is negligible. Therefore low valation traders mst absorb the entire spply. Hence, Q + = s. At time T f high valation traders absorb the entire spply. Hence, Q Tf =. When the per capita spply of assets affected by the shock s is low, so that (21) holds, the incoming flow of high valation traders reaching a decision at a given point in time is always large enogh to accommodate the spply from low valation traders. Correspondingly, in eqilibrim low valation traders sell a lmp of their assets when they reach a decision, then smoothly nwind their inventory ntil the next information event. In contrast, when s is so large that (21) fails to hold, the liqidity shock is more severe. Hence, shortly after the initial aggregate shock, the inflow of high valation traders is not large enogh to absorb the sales of low valation traders who crrently reach a decision. In eqilibrim, some of these sales are absorbed by early low valation traders who reached adecisionattimet<and have not had another information event. Indeed, these early low valation traders anticipate that, as time goes by, their instittion is more likely to have recovered. Ths, their expected valation (in the absence of an information event) increases with time and they find it optimal to by if their tility is not too concave, i.e., if σ is not too high. Correspondingly, near time zero, Q is increasing, as depicted in Figre 3 for σ =.3 and.5. Combining Lemma 4, Lemma 5, Lemma 6 and Proposition 3, one obtains a fll characterization of the eqilibrim holdings process, which can be compared to its conterpart in the nlimited cognition case. When cognition is not limited, as long as an instittion has not recovered from the shock, its holdings decline smoothly, and, as soon as it recovers, its holdings jmp to 1. Trading histories are qite different with limited cognition. First an instittion s holdings remain constant ntil its trader s first information event. Then, if at her first information event the trader learns that her instittion has a low valation, the trader sells a lmp. After that, if (21) does not hold, the trader progressively bys back, then eventally sells ot 2

21 1.5 σ =.3 σ =.5 σ = T f time (days) Figre 3: The fnction Q for varios vales of σ at the next jmp of her information process. This process contines ntil the trader finds ot her valation has recovered, at which point her holdings jmp to 1. Sch rond trip trades do not arise in the nbonded cognition case Trading volme Becase they reslt in rond trip trades, hmp shaped asset holding plans generate extra trading volme relative to the nlimited cognition case. Specifically, consider a trader who, at two consective information events t 1 and t 2,discoversthatshehasalow valation. Dring the time period (t 1,t 2 ]shetradesanamontofasseteqalto t2 t 1 q t1, d + q t 2,t 2 q t1,t 2. (22) The first term in (22) is the flow of trading between time t 1 and time t 2 dictated by q t1,, the time t 1 holding plan. The second term is the lmpy adjstment at time t 2. Note that (16) implies that, whenever q t1,/ is not, it has the same sign as Q. Note also that, becase at time t 2 the trader observes that the instittion has still not recovered, q t2,t 2 <q t1,t 2. Hence, if Q is decreasing, (22) is eqal to q t1,t 1 q t2,t 2. In contrast, if Q is increasing, the amont traded between t 1 and t 2 is q t1,t 2 q t1,t 1 + q t1,t 2 q t2,t 2 =2(q t1,t 2 q t1,t 1 ) +q t1,t 1 q t2,t 2 (23) rond trip trade 21

22 Since, q t1,t 2 >q t1,t 1,(23)isgreaterthanq t1,t 1 q t2,t 2.Thefirsttermintheeqationistheextra volme created by the rond trip trade: the prchase of q t1,t 2 q t1,t 1 asaleofthesameqantityattimet 2. dring (t 1,t 2 )followedby The comparison between trading volme with nbonded cognition and with limited cognition depends on two effects: i) On one hand, the above discssion shows that cognition limits generate rond-trip trades that tend to raise trading volme. ii) On the other hand, since information events are infreqent, agents ability to realize mtally beneficial trades is limited (e.g., if ρ = there are no trades.) For the parameter vales we consider, Figre 4 shows that, for early times, effect i) dominates so that trading volme is greater with limited cognition. However, as shown in the figre, for later times, effect ii) dominates so that trading volme is lower with limited cognition. Indeed, rond trip trades become less prevalent, and stop after the time T ψ when Q reaches its maximm. When ρ goes to infinity, there is only effect i) so that, as stated in the next proposition, trading volme is at least as large with limited cognition. Proposition 4. When ρ goes to infinity, the eqilibrim price and allocation converge to their nlimited cognition conterparts. Relative to nlimited cognition, the extra volme at time t with limited cognition converges to s µht γ max (1 s),. σ When condition (21) does not hold, this asymptotic extra volme is strictly positive for small t. The proposition reveals the crcial role of the crvatre parameter, σ: as illstrated in Figre 3, when σ is small and tility is close to linear, the hmp shaped pattern in holdings is qite prononced. This leads to very large extra trading volme Prices Proposition 1 implies that, from time T s to time T f,thepricenderlimitedcognitionislower than its conterpart with nlimited cognition, 1/r. By continity, the price is lower nder cognition limits jst before T s.thenextpropositionstatesthatthisrankingofpricesholdsat all times if s<σ/(σ +1),btnotnecessarilyotherwise. Proposition 5. If (21) holds, then at all time the price is strictly lower with limited cognition. Bt if s is close to 1 and σ is close to, then at time the price is strictly higher with limited cognition. 22

23 perfect cognition limited cognition T ψ T s T f time (days) Figre 4: The trading volme nder nlimited (green) and limited cognition (ble), when σ =.3. In the figre, T ψ denotes the argment maximm of Q. That the price wold be higher withot cognition limits sonds intitive. Unbonded cognition enables traders to continosly allocate the asset to those who vale it the most. Sch an efficient allocation cold be expected to raise the price, and this is indeed what happens when (21) holds. Bt, as stated in Proposition 5, there are cases where the price can be higher when cognition is limited than when it is nbonded. The intition is as follows. Arond time zero, low valation traders are marginal. With limited cognition, low valation traders have a higher expected marginal tility becase they take into accont the possibility that they may have switched to high valation. Conseqently, they demand more assets, which tends to psh p prices. This effect is stronger when low valation traders are marginal for a longer period, that is, when the shock is more severe (s close to one) and when their tility flow is not too concave (σ close to zero). 3.4 Robstness to recrring liqidity shocks So far we focsed on a simple model generating a deterministic eqilibrim price process. In this sbsection, we show that or qalitative reslts are pheld in a more general model, with stochastic shocks and stochastic prices. Ths, or baseline one shock analysis can be viewed, at least qalitatively, as the implse response of a more general model with recrring shocks. In the spirit of Dffie, Gârlean, and Pedersen (27) we assme that aggregate liqidity shocks occr at Poisson arrival times with intensity κ>. As in the baseline model, when 23

24 ashockhits,allinvestorsswitchtothelow valationstateandrecoverlateratindependent exponential times with intensity γ. We also assme that, at the time of a shock, all traders withdraw from the market and halt trading, to re-design their holding plans. Traders resme trading and revise their holding plan at the first sbseqent jmp of their information process. 23 We focs on stationary eqilibria in which the price is stochastic, bt can be written as fnction of one random variable: the time elapsed since the last aggregate shock. In Section IV, page 18 of the spplementary appendix, we show that the strctre of the eqilibrim remains qalitatively the same as in or baseline model. In particlar, the market clearing condition is the same as in Lemma 2. Also, a trader s objective fnction is t e (r+ρ+κ)( t) E t [v(θ,q t, )] + κw (q t, ) ξ q t, d where ξ rp ṗ κ(p p ), and the time index, t or, denote the time elapsed since the last liqidity shock. This is similar to the vale fnction in Lemma 1, bt with two changes. First, the definition of the opportnity cost of holding asset is modified so as to reflect the possibility that a new shock might hit the market with intensity κ, bringing down the price to p.second,thetilityflow is agmented by κw (q t, ), where W (q) is the vale of holding q nits of the asset at the time of a new aggregate liqidity shock, ntil the first sbseqent information event. In Section IV, page 18 of the spplementary appendix, we solve for W (q), compte the optimal holdings and, sbstitting them in the market clearing condition, obtain the following proposition: Proposition 6. With recrrent random aggregate liqidity shocks, there exists an eqilibrim in which i) the price jmps each time the market is hit by a shock, ii) in between jmps the price is the fnction of the time elapsed since the last shock which is the niqe bonded soltion of the ODE: (r + κ)p ṗ =1 + κ r δκ(r + ρ + κ) (r + ρ)(r + ρ + κ + γ) Qσ h, δκ2 (r + κ)(r + ρ + κ) r(r + ρ)(r + ρ + κ + γ) e (r+κ)z e (r+ρ+κ)z Q σ h,z dz, 23 This is in line with evidence from the flash crash of May 6 th, 21: the Secrity and Exchanges Commission reports that atomated trading systems pased in reaction to the sdden price decline in order to allow traders and risk managers to flly assess the risks before trading was resmed. See online. 24

25 where Q h, is defined in eqation (IV.6), page 21 of the spplement. In the spplement we provide varios nmerical examples based on or analytical characterization. We consider varios vales of κ, setσ =.3, and otherwise keep the same parameter vales as in Table 1. In the example where aggregate liqidity shocks occr every qarter on average (κ =4)wefindthatholdingpoliciesarehmp shapedandthatthepriceisincreasing, jst as in or baseline model. However, since traders anticipate the occrrence of ftre liqidity shocks, the price level is abot 9.5% lower. 4 Implementation with realistic market instrments So far, or characterization of eqilibrim has been cast in abstract terms, sch as holding plans and market clearing. We now stdy how these holding plans can be implemented with realistic market instrments. In so doing, we focs on electronic order driven markets. Sch venes are the major trading mechanism for stocks arond the world (e.g., in the US NASDAQ and the NYSE, and in Erope Eronext, the London Stock Exchange and the Detsche Börse.) In these markets, traders can place limit sell (resp. by) orders reqesting exection at prices at least as large (resp. low) as their limit price. These orders are stored in the book, ntil they are exected, cancelled or modified. Traders can also place market orders, reqesting immediate exection. A limit sell order standing in the book is exected, at its limit price, when hit by an incoming by order (either a market order or a limit by order with a higher price limit), if there are no nexected sell orders in the book at lower prices, or at the same price bt placed at an earlier point in time. The case of a limit by order is symmetric. 24 There are mltiple possible implementations of or eqilibrim. For instance, there is a trivial implementation, where all traders desiring to change their holdings continosly sbmit market orders or limit orders at the crrent eqilibrim price. While this has the advantage of simplicity, this implementation is at odds with important stylized facts; for example it leads to an empty limit order book. To narrow down the set of possible implementations while giving rise to realistic dynamics, we restrict or attention to the case where market participants alter their trading strategies only when their information process jmps. This is qite natral in or framework, and it follows if, between two jmps of their information process, traders cognitive 24 Note also that a limit order to sell and a limit order to by placed at the same price cannot both remain together in the limit order book: as soon as offers cross each other they are exected, in accordance with price and time priority rles. 25

26 resorces are devoted to the complex task of assessing the liqidity stats of their instittion, which leaves little opportnity to alter trading plans. 4.1 Implementing the eqilibrim with limit orders and algorithms First we assme that, when their information process jmps, in addition to market and limit orders, traders can place trading algorithms. The latter are compter programs feeding orders to the market over time, in response to pre-specified ftre changes in market variables. 25 Note that both limit orders and algorithms enable trades to happen withot direct hman intervention while the trader is engaged in information collection and processing. 26 Lastly, in keeping with or limited cognition assmption, we do not allow algorithms to condition their orders on changes in the liqidity stats of the instittion occrring between jmps of its information process. It is straightforward to implement the holding plan of high valation traders. Before T f,as soon as their information process reveals that their instittion has recovered from the shock, they place a limit order to by at the crrent price, which is immediately exected. 27 At that point in time, since they now have their optimal holdings, they cancel any limit order they wold have previosly placed in the book. The case of low valation traders is more intricate. Indeed, one mst bear in mind that the eqilibrim price given in Proposition 1 is strictly increasing over (,T f ). This implies that any limit order to by sbmitted at time t is either immediately exected (if the limit price is greater than p t )orneverexected(ifthelimitpriceislowerthanp t ). Conseqently, if a trader only places (limit or market) orders when her information process jmps, she cannot implement increasing holding plans. In contrast, when the eqilibrim holding plan is decreasing, it can be implemented by placing at time t an immediately exected market order to sell, as well as limit orders to sell at price p >p t which will be exected later. Now, Proposition 3 states that eqilibrim holdings are decreasing if and only if condition (21) holds. This leads to the following Proposition: 25 Becase we have no aggregate ncertainty, market-level variables (price, volme, qote, etc.) are in or model deterministic fnctions of time. Ths, conditioning on time is enogh to make the algorithm also depend on the state of the market. In a more general model with aggregate ncertainty, one wold need to explicitly allow algorithms to depend on market level variables at time. 26 This is in line with the view of Harris (23), who arges that Limit orders represent absent traders [enabling them] to participate in the markets while they attend to bsiness elsewhere. 27 Eqivalently, they cold place a market order to by. Assming they place a limit order to by at p t pins down the market clearing price. 26

27 Proposition 7. The eqilibrim characterized in Proposition 1 can be implemented by traders placing market and limit orders (only) when their information process jmps if and only if (21) holds. If (21) does not hold, asset holding plans are hmp shaped. As arged above, in order to implement the increasing branch of the hmp, traders cannot se limit by orders: instead, they mst rely on algorithms. For concreteness, consider the example illstrated in Figre 5. Interpret t 1 and t 2 as the time of two consective jmps of the information process of a trader, both revealing low valation. At t 2 the trader s asset holdings ndergo a discrete downward jmp, from q t1,t 2 to q t2,t 2.Thenthey continosly increase, reach a flat, and finally decrease again. To implement this holding plan, the trader places a market sell order at time t 2,whichimplementstheinitialdownwardjmp in her optimal holdings. She also programs her algorithm to place a seqence of market by orders after t 2, which implement the increasing part of her holding plan. Finally, to implement the smoothly decreasing part of her holding plan, she ses a schedle of limit sell orders. As shown on Figre 5, the limit sell orders placed at time t 2 start execting before those placed at time t 1. Since the eqilibrim price process is increasing, this means that these orders are placed at lower prices than the previos ones. The figre also shows that the holding plan set at time t 2 declines less steeply than its t 1 conterpart. This reflects that the qantity offered in the book at corresponding prices is lower for the plan set at t 2 than for the plan set at t 1. Ths, to implement the new holding plan, at t 2 the trader cancels some of the orders placed at t 1, replacing them with more aggressive orders. Finally note that, at t 2,thetraderalso modifies the trading algorithm generating the prchases necessary to implement the increasing part of her holding plan. This can be interpreted in terms of hman intervention resetting the parameters of the trading algorithm. This discssion is smmarized in or next proposition. Proposition 8. Consider a trader observing low valation at time t<t f. If (21) does not hold sch trader can implement the optimal holding plan arising in Proposition 1 by placing market and limit sell orders at t and programming her trading algorithm to trigger market by orders at times >t. Retrning to Figre 5, now interpret t 1 and t 2 as the times at which the information processes of two traders jmp, each time revealing low valation. The figre illstrates that the late trader starts selling before the early trader implying that the limit sell orders of the late trader are 27

28 1 q t1,.8 q t2, t 1 t 2 T f time (days) Figre 5: The holding plans of an early low valation trader and of a late low valation trader, when σ =.3 placed at lower prices than the limit sell orders of the early trader, i.e., there is nderctting. 28 To see why this is optimal, compare the time expected valations of the time t 1 trader and the time t 2 trader, assming that for both there has been no new information event by time. For the trader who observed low valation at time t 1 the probability that her valation is high now is: 1 e γ( t 1).Fortheothertraderitis1 e γ( t 2),whichislowersincet 2 >t 1. Hence the expected valation of the time t 1 trader is greater than that of the time t 2 trader. This is why the time t 1 trader sells later. 29 Taking stock of the above reslts, we now describe the overall market dynamics prevailing when (21) does not hold and Q > 1forsome. Thereareforsccessivephasesinthemarket, as illstrated in Figre 6. There exists a time T 1 <T f sch that, from time to T 1,lowvalationtradersplace limit sell orders at lower and lower prices, i.e., there is nderctting. These limit orders accmlate in the book, withot immediate exection. Correspondingly, the best ask decreases and the depth on the ask side of the book increases. Dring this period, low valation traders also place market sell orders, which are exected against limit by orders placed by high valation traders and algorithms. 28 This is in line with the empirical reslts of Biais, Hillion, and Spatt (1995), Griffiths, Smith, Trnbll, and White (2) and Elll, Holden, Jain, and Jennings (27). 29 This goes along the same lines as the intition why low valation traders placing trading plans at t will initially sell and then by from agents placing their trading plans later. 28

29 Denoting by T ψ (T 1,T f )thetimeatwhichq achieves its maximm, dring [T 1,T ψ ] the best ask qote remains constant and the depth on the ask side of the book declines. Indeed, dring this period, low valation traders stop nderctting the best qote, while high valation traders cancel their limit orders. Dring this second phase of the market, there are still no exections at the best ask, and trades are initiated by low valation traders placing market sell orders, exected against the limit by orders placed by high valation traders. The third phase is between T ψ and T f. Dring this period, high valation traders hit the limit sell orders otstanding in the book with market orders to by (or marketable limit orders). Correspondingly, the depth on the ask side contines to decline, and the best ask price increases. 3 Finally, after time T f, the market has recovered from the shock, and the price remains constant at 1/r. The algorithmic trading strategies generated by or model are in line with stylized facts and empirical findings. That algorithms progressively bild p an increasing position via sccessive by orders can be interpreted in terms of order splitting. That they by progressively as the price deterministically trends pwards can be interpreted in terms of short term momentm trading. That after the liqidity shock traders bild p long positions which they will eventally nwind is in line with the findings of Brogaard (21) that algorithms do not withdraw after large price drops and benefit from price reversals. Or theoretical reslts are also in line with the empirical findings of Hendershott and Riordan (21) that trading algorithms provide liqidity when it is scarce and rewarded. Indeed, the strategies followed by or algo traders (who by initially while simltaneosly placing limit orders to sell to be exected later) is a form of market making, similar to that arising in Grossman and Miller (1988). Note however that, while in Grossman and Miller some market participants are exogenosly assmed to be liqidity providers and other liqidity consmers, in or model all participants are identical 3 Note that there is no bid ask bonce in this market. Until time T ψ all trades are exected at the limit price of by orders stemming from traders who jst realized their valation was high. After T ψ all trades are exected against the best limit order to sell otstanding in the book. Note also that the tick size is zero, so that prices are not restricted to a discrete grid. Ths there is a seqence of exections at smoothly increasingly high bid prices, followed by a seqence of exections at smoothly increasing ask prices. Ths in eqilibrim there is positive serial correlation in sccessive order types, in line with the empirical findings of Biais, Hillion, and Spatt (1995), Griffiths, Smith, Trnbll, and White (2) and Elll, Holden, Jain, and Jennings (27). 29

30 Panel A: Price and best ask in the book price best ask T 1 T ψ T f Panel B: Stock of limit sell orders in the book 1.5 T f time (days) Figre 6: Price dynamics (Panel A) and limit order book activity (Panel B), when σ =.3. Spplementary appendix VIII.18 provides the formlas leading to Panel B. ex ante, yet they play different roles in the market becase of differences in the realizations of their information and valation processes. 4.2 Eqilibrim withot trading algorithms What happens if trading algorithms are not available and traders can only place limit and market orders at the time of information events? Sppose for now that the eqilibrim price is increasing (which will trn ot to be the case in eqilibrim). As mentioned in Section 4.1 if traders place orders only when their information process jmps, this precldes them from implementing increasing holding plans. Instead, they can implement only decreasing holding plans. Therefore, if condition (21) does not hold, the eqilibrim will differ from that arising in Proposition 1. Instead, the eqilibrim is as in the next proposition. Proposition 9. If s>σ/(1 + σ) and traders can place only limit and market orders when their information process jmps, there exists an eqilibrim in which the price path is strictly increasing over (,T f ), and eqal to 1/r over [T f, ). High-valation traders, and low valation traders after T f, follow the same asset holding plan as in Proposition 1. For low valation traders before T f, the optimal holding plan q t, is continos in (t, ) and strictly less than 1. 3

31 Moreover, there exists T φ (,T f ) and a strictly decreasing fnction φ : (,T φ ] R +, sch that: If t (,T φ ], then q t, is constant for [t, φ t ], and strictly decreasing for (φ t,t f ). If t (T φ,t f ], then q t, is strictly decreasing for (t, T f ). While the proposition describes the eqilibrim in abstract terms, in the the appendix we provide closed form analytical soltions for all relevant eqilibrim objects. Also, one may wonder whether the eqilibrim in Proposition 9 is niqe. We provide a partial answer to this qestion in the spplementary appendix to this paper. We show that the eqilibrim of Proposition 9 is niqe in the class of Markov eqilibria, i.e., eqilibria where traders find it optimal to choose holding plans that depend only on the information-event time (the crrent aggregate state) and their crrent tility stats (their crrent idiosyncratic state). 31 The intition for Proposition 9 is as follows. When s>σ/(1+σ), time t <T φ low valation traders wold choose hmp-shaped holding plans if their holdings were not constrained to be decreasing (the solid crve in Figre 7). Faced with the constraint of choosing a decreasing holding plan, they iron the increasing part of the hmp-shaped plans (the dashed crve in Figre 7). To implement the holding plans of this proposition, time t low valation traders place market sell orders, as well as schedles of limit sell orders, which begin execting at time φ t. Proposition 9 also implies that, similarly to the case where traders can se algorithms, there is nderctting in eqilibrim: since φ t is strictly decreasing, sccessive traders place limit sell orders at lower and lower prices. 32 Note that, both in Proposition 1 and Proposition 9, the time at which the price flly recovers from the liqidity shock is T f, the time at which the residal spply S (defined in (12)) reaches. This is becase S is a fnction of the total qantity of the asset broght to the market and of the nit demand of high valation traders, both of which are naffected by whether or not low valation traders can se algorithms. The next proposition offers frther insights into the comparison of the price paths arising in Proposition 9 and Proposition Importantly, or proof makes no apriorimonotonicity restriction on the price path. Instead, we consider general and possibly non-monotonic price paths. We then show, via elementary optimality and market clearing considerations, that the preference dynamics and the focs on Markov eqilibria imply that the price path is continos, strictly increasing ntil time T f, and flat after time T f. 32 Note that in Proposition 9 we obtain nderctting for all (s, σ) sch that σ/(1 + σ) <s. In Proposition 1, by contrast, we obtain nderctting for the smaller set of parameters sch that the maximm of Q is greater than 1. 31

32 1.2 1 nconstrained plan ironed ot plan q t, t φ t T f time (days) Figre 7: Holding plans of algorithmic verss limit order traders, when σ =.3, given the eqilibrim price of Proposition 9. Proposition 1. When s>σ/(1 + σ): The price arising in Proposition 1 is strictly lower than its conterpart in Proposition 9 arond time zero for ρ close to. The price arising in Proposition 1 is strictly higher than its conterpart in Proposition 9 between T φ and φ ; The price is identical in Proposition 1 and in Proposition 9 after time φ. This Proposition shows that, shortly after the liqidity shock, the price can be lower when instittions se trading algorithms. 33 The intition is as follows. When they cannot se algorithms, traders know they will not be able to by back before their next information event. Hence they initially sell less, which redces the selling pressre on the price. Conseqently the price can be higher than when traders are able to se algorithms. An otside observer may conclde that banning the se of algorithms cold be socially beneficial, since it wold alleviate the initial price pressre created by the liqidity shock. However, inferring aggregate welfare implications from price effects is misleading. Indeed, Proposition 2 implies that the eqilibrim arising with algorithms Pareto dominates that arising when traders can place only limit and market orders when their information process jmps. 33 While we are able to establish this reslt only analytically for small ρ, or nmerical calclations sggests that it can hold for larger vale of ρ. In particlar, it does hold for the vale ρ = 25 that we have chosen for the nmerical calclations presented in this paper as well as in or spplementary appendix. 32

33 Panel A: Volme.15.1 with algorithms with limit orders.5 T ψ T f Panel B: Stock of limit sell orders in the book 1 with algorithms with limit orders.5 T f time (days) Figre 8: Trading volme (pper panel) and volme of limit orders otstanding in the book (lower panel), when σ =.3 Figre 8 plots the trading volme (pper panel) and the volme of limit orders otstanding in the book (lower panel) in Propositions 1 and 9. Trading volme is higher when instittions are able to se algorithms. This reflects the additional trading volme generated by rond trip trades. Qalitatively, the stock of limit orders in the book has the same shape in Propositions 1 and 9. The book is filled progressively as limit orders to sell are placed by low valation traders. Then, cancellations and exections lead to a decrease in the stock of orders in the book. Dring the early phase, the amont of limit orders otstanding is higher in Proposition 1 than in Proposition 9. Indeed, with algorithms low valation traders by after their information event, this indces them to place more limit order to sell, in order to nwind this position towards the end of the price recovery process. 5 Conclsion This paper stdies the reaction of traders and markets to liqidity shocks nder cognition limits. We model the aggregate liqidity shock as a transient decline in the valation of the asset by all participants. While instittions recover from the shock at random times, traders with limited cognition observe the stats of their instittion only when their own information process 33

34 jmps. We interpret this delay as the time it takes traders to collect and process information abot positions, conterparties, risk exposre and compliance. We characterize eqilibrim prices and asset holding plans nder two alternative market strctres: when traders can se algorithms to atomatically change their holding plan while investigating the liqidity stats of their instittion, and when they can place only market and limit orders when their information process jmps. Or analysis sggests that, to an otside observer, algorithms may seem to be price destabilizing. We show however that they play a socially sefl role by facilitating market making: the eqilibrim prevailing when traders can se algorithms always Pareto dominates that prevailing if traders can only place limit orders. This reflect the absence of externalities, as the asset holding constraints imposed by cognition limits do not depend on other traders actions. Note however that in or analysis there is no adverse selection. In a more general model with changes in common vales and where some slow traders cold not se algorithms, while fast traders cold, information asymmetries cold arise. 34 In this context, algorithmic trading cold inflict negative externalities on slow traders, and eqilibrim might no longer be optimal. Biais, Focalt, and Moinas (21a) analyze these isses in a one period model. Becase of their static setp, however, they are nable to consider rich dynamic order placement policies sch as those arising in the present model. An interesting, bt challenging, avene of frther research wold be to extend the present model to the asymmetric information case. In doing so one cold take stock of the insights of Goettler, Parlor, and Rajan (29) and Pagnotta (21) who stdy dynamic order placement nder asymmetric information. 34 Consistent with the view that algorithmic traders have sperior information Hendershott and Riordan (21) and Brogaard (21) find empirically that algorithmic trades have greater permanent price impact than slow trades and lead price discovery. 34

35 A Proofs This appendix provides the proofs omitted in the main body of the paper. To keep the exposition concise, some intermediate reslts and calclations are gathered in or spplementary appendix, Biais, Hombert, and Weill (21b). A.1 Proof of Lemma 1 Let s begin by deriving a convenient expression for the intertemporal cost of bying and selling assets. For this we let τ <τ 1 <τ 2 <...denote the seqence of information events. For acconting prposes, we can always assme that, at her n-th information event, the trader sells all of her assets, q τn 1,τ n, and prchases a new initial holding q τn,τ n. Ths, the expected inter-temporal cost of the holding plan can be written: E p τ1 e rτ1 q,τ1 + e rτn p τn q τn,τ n + n=1 τn+1 τ n p dq τn,e r e rτn+1 p τn+1 q τn,τ n+1 Given that p is continos and piecewise continosly differentiable, and that q τn, has bonded variations, we can integrate by part (see Theorem in Carter and Van Brnt, 2), keeping in mind that d/d(e r p )= e r ξ. This leads to: E p τ1 e rτ1 q,τ1 + n=1 τn+1 =constant + E e r ξ q τ, d. τ n e r ξ q τn, d = p s + E n= τn+1 τ n. e r ξ q τn, d In the above, the first eqality follows by adding and sbtracting q, = s, and by noting that q, is constant; the second eqality follows by sing or τ notation for the last contact time before. With the above reslt in mind, we find that, p to a constant, we can rewrite the intertemporal payoff net of cost as: E e v(θ r,q τ,) ξ q τ, d. (A.1) Now for any t, the probability that τ t is the probability that N N t =, which is eqal to e ρ( t). Ths, the distribtion of τ has an atom of mass e ρ at t =, and then the density ρe ρ( t) for t (,]. Hence, after applying Baye s rle, (A.1) rewrites as: e e r ρ E v(θ,q, ) ξ q, τ = + ρe ρ( t) E v(θ,q t, ) ξ q t, τ = t dt d. (A.2) To simplify this expression, we rely on the following lemma, proved in Section VIII.1, page 44 in the spplementary appendix. Lemma A.1. E [v(θ,q t, ) ξ q t, τ = t] =E [v(θ,q t, ) ξ q t, ] for all t. This lemma is clearly tre for t =sinceq, = s for all, and since the information event process is independent from the type process. In Section VIII.1 in the spplementary appendix, we show that it also holds for t>. Intitively, this is becase of two facts. First, as noted above, the information event process is independent from the type process. Second, {τ = t} = {N t N t = 1 and N N t =} only depends on increments of the information process at and after t, which are independent from the trader s information one instant before t, and hence independent from the predictable process q t,. Relying on Lemma A.1, eqation 35

36 (A.2) becomes: = e (r+ρ) E [v(θ,s) ξ s] d + e r ρe ρ( t) E [v(θ,q t, ) ξ q t, ] dt d e (r+ρ) E [v(θ,s) ξ s] d + E e rt e (r+ρ)( t) E t [v (θ,q t, ) ξ q t, ] d ρdt, where E t [ ] refers to the expectation conditional pon F t and the second line follows from changing the order of integration and applying the law of iterated expectations. t A.2 Preliminary reslts abot Q We begin with preliminary reslts that we se repeatedly in this appendix. For the first preliminary reslt, let Ψ(Q) inf{ψ :(1 µ hψ ) 1/σ Q 1}, and ψ Ψ(Q ). We have: Lemma A.2 (Preliminary reslts abot Q ). Eqation (17) has a niqe soltion, Q. Moreover, Q < (1 µ h ) 1/σ and Q is continosly differentiable with Q = eρ (s µ h ) e ρ (1 µ h ) 1+1/σ Q. (A.3) ψ e ρt (1 µ ht ) 1+1/σ dt In the proof of this Lemma, in Section VIII.2 page 44 of the spplementary appendix, we first show existence, niqeness, and the ineqality Q < (1 µ h ) 1/σ sing the monotonicity and continity of eqation (17) with respect to Q. We then show that the soltion is continosly differentiable sing the Implicit Fnction Theorem. For the second preliminary reslt, consider eqation (17) after removing the min operator in the integral: ρe ρ( t) (1 µ ht ) 1+1/σ Q dt = S Q eρt (s µ ht ) dt eρt (1 µ ht ) 1+1/σ dt. (A.4) Now, whenever Q 1, it is clear that Q also solves eqation (17). Given that the soltion of (17) is niqe it follows that Q = Q. Conversely if Q = Q, sbtracting (17) from (A.4) shows that: ρe ρ( t) (1 µ ht ) (1 µ ht ) 1/σ Q min{(1 µ ht ) 1/σ Q, 1} dt =. Since the integrand is positive, this can only be tre if (1 µ ht ) 1/σ Q =min{(1 µ ht ) 1/σ Q, 1} for almost all t. Letting t deliversq 1. Taken together, we obtain: Lemma A.3 (A sefl eqivalence). Q 1 if and only if Q = Q. The next Lemma, proved in Section VIII.3 page 46 of the spplementary appendix, provides basic properties of Q : Lemma A.4 (Preliminary reslts abot Q ). The fnction Q is continos, satisfies Q + = s and Q Tf =. It is strictly decreasing over (,T f ] if s σ/(1 + σ) and hmp-shaped otherwise. A.3 Proof of Proposition 1 The asset holding plans are determined according to Lemmas 4-6. The asset holding plan of high-valation traders are niqely determined for (,T f ], and are indeterminate for >T f. The holding plan of lowvalation traders are niqely determined, as Lemma A.2 shows that eqation (17) has a niqe soltion. Now, trning to the price, the definition of Q implies that the price solves rp =1 δ(1 µ h )Q σ +ṗ for <T f. For T f, the fact that high-valation traders are indifferent between any asset holdings in [, 1] implies that 36

37 rp =1+ṗ. Bt the price is bonded and positive, so it follows that p =1/r. Since the price is continos at T f, this provides a niqe candidate eqilibrim price path. Clearly this candidate is C 1 over (,T f ) and (T f, ). To show that it is continosly differentiable at T f note that, given Q Tf = and p Tf =1/r, ODE (18) implies that ṗ T f =. Obviosly, since the price is constant after T f, ṗ T + f = as well. We conclde that ṗ is continos at = T f as well. Next, we show that the candidate eqilibrim ths constrcted is indeed an eqilibrim. For this recall from Lemma A.2 that <Q < (1 µ h ) 1/σ, which immediately implies that < 1 rp +ṗ < 1 for <T f. It follows that high-valation traders find it optimal to hold one nit. Now one can directly verify that, for <T f, the problem of low-valation traders is solved by q t, =min{(1 µ ht ) 1/σ Q, 1}. For T f, 1 rp +ṗ = and so the problem of high-valation traders is solved by any q t, [, 1], while the the problem of low-valation traders is clearly solved by q t, =. The asset market clears at all dates by constrction. We already know that the price is eqal to 1/r for t>t f so the last thing to show is that it is strictly increasing for t<t f. Letting (1 µ h ) 1/σ Q for T f, and = for T f. In Section VIII.4, page 47 in the spplementary appendix, we show that: Lemma A.5. The fnction is strictly decreasing over (,T f ]. Now, in terms of,thepricewrites: p = e r(y ) 1 δ σ y dy = e rz 1 δ σ z+ dz, after the change of variable y = z. Since is strictly decreasing over (,T f ), and constant over [T f, ), it clearly follows from the above formla that p is strictly increasing over (,T f ). A.4 Proof of Proposition 2 Let s begin with a preliminary remark. By definition, any feasible allocation q satisfies the market clearing condition E [ q τ,] =s. Taken together with the fact that ξ = rp ṗ is deterministic, this implies: = E e r q τ,ξ d e r E [q τ,] ξ d = e r sξ d = p s. With this in mind, consider the eqilibrim asset holding plan of Proposition 1, q, and sppose it does not solve the planning problem. Then there is a feasible asset holding plan q that achieves a strictly higher vale of the objective (2). Bt, by (A.6), this asset holding plan has the same cost as the eqilibrim asset holding plan. Sbtracting the inter-temporal cost (A.6) from the inter-temporal tility (A.1), we obtain that V (q ) >V(q), which contradicts individal optimality. (A.5) (A.6) A.5 Proof of Proposition 3 Taken together, Lemma A.3 and Lemma A.4 immediately imply that Lemma A.6. The fnction Q satisfies Q + = s, Q Tf =.Ifs σ/(1 + σ), then it is strictly decreasing over (,T f ].Ifs>σ/(1 + σ) and Q 1 for all (,T f ], Q is hmp-shaped over (,T f ]. The only case that is not covered by the Lemma is when s>σ/(1 + σ) and Q > 1 for some (,T f ]. In this case, note that for small and close to T f, we have that Q < 1. Given that Q is hmp-shaped, it follows that the eqation Q = 1 has two soltions, <T 1 <T 2 <T f. For (,T 1 ](resp. [T 2,T f ]), 37

38 Q 1 and is increasing (resp. decreasing), and ths Lemma A.3 implies that Q = Q and increasing (resp. decreasing) as well. In particlar, since we know from Lemma A.2 that Q is continosly differentiable, we have that Q T 1 > and Q T 2 <. Ths, Q changes sign at least once in (T 1,T 2 ). To conclde, in Section VIII.5 we establish: Lemma A.7. The derivative Q changes sign only once in (T 1,T 2 ). A.6 Proof of Proposition 4 In Section VIII.6, page 5 in the spplementary appendix, we prove the following asymptotic reslts: Lemma A.8. As ρ goes to infinity: T f (ρ) T s (A.7) s µ h Q (ρ) = (1 µ h ) 1 γ 1+ 1 s µh o, [,T 1+1/σ ρ (1 µ h ) 1/σ s ] (A.8) σ 1 µ h ρ T ψ (ρ) = arg max Q s µ h (ρ) arg max. (A.9) [,T f (ρ)] [,T s] (1 µ h ) 1+1/σ With this in mind we can stdy the asymptotic behavior of price, allocation, and volme. A.6.1 Asymptotic price Proposition 1 shows that the price at all times t [T f, ) is eqal to 1/r. Bt we know from Lemma A.8 that T f (ρ) T s as ρ. Clearly, this implies that, for all t (T s, ), as ρ the price converges towards 1/r, its nlimited cognition conterpart. Integrating the ODEs of Proposition 1 shows that, at all times t [,T s ], the price is eqal to: p t = Tf (ρ) t e r( t) (1 δ(1 µ h )Q σ (ρ)) d + e r(t f (ρ) t). r Bt Lemma A.8 shows that (1 µ h )Q σ (ρ) converges point wise towards [(s µ h )/(1 µ h )] σ. Moreover, we know that (1 µ h )Q σ [, 1]. It ths follows from dominated convergence that p t converges to its nlimited cognition conterpart. A.6.2 Asymptotic distribtion of asset holdings Let s start with some T s. With nlimited cognition, traders whose valation is low at time hold (s µ h )/(1 µ h ), and traders whose valation is high hold one share. With limited cognition, the time cross sectional distribtion of asset holdings across low valation traders is, by or sal law of large nmbers argment, the time distribtion of asset holding generated by the holding plan of a representative trader, conditional on θ =. In Section VIII.7, page 52 in the spplementary appendix, we establish: Lemma A.9. For all ε>, asρ : Proba qτ, s µ h >ε 1 µ h θ = Proba qτ, 1 >ε θ = h. (A.1) (A.11) 38

39 A.6.3 Asymptotic volme Basic formlas. As above, let T ψ denote the arg max of the fnction Q. For any time <T ψ and some time interval [, + d], the only traders who sell are those who have an information event dring this time interval, and who find ot that they have a low valation. Ths, trading volme dring [, + d] can be compted as the volme of assets sold by these traders, as follows. Jst before their information event, low valation traders hold on average: E [q τ, θ = ] =e ρ s + ρe ρ( t) q,t, dt, (A.12) since a fraction e ρ of them have had no information event and still hold s nit of the asset and a density ρe ρ( t) of them had their last information event at time t and hold q,t,. Instantaneos trading volme is then: V = ρ(1 µ h ) E [q τ, θ = ] q,,, (A.13) where ρ(1 µ h ) is the measre of low valations investors having an information event, the term in large parentheses is the average size of low valation traders sell orders, and q,, is their asset holding right after the information event. For any time (T ψ,t f ) and some time interval [, +d], the only traders who by are those who have an information event dring this time interval, and who find ot that they have a high valation. Trading volme dring [, + d] can be compted as the volme of assets prchased by these traders: V = ρµ h 1 E [q τ, θ = h], (A.14) where the time average asset holdings of high-valation investors is fond by plgging (A.12) into the market clearing condition: µ h E [q τ, θ = h]+(1 µ h )E [q τ, θ = ] =s. For >T f, the trading volme is not zero since high valation traders contine to by from the low valation traders having an information event: V = ρ(1 µ h )E [q τ, θ = ]. (A.15) Taking the ρ limit. We first note that q,, (ρ) =min{(1 µ h ) 1/σ Q, 1} =(1 µ h ) 1/σ Q (ρ). Next, we need to calclate an approximation for: E [q τ, θ = ] =se ρ + min{(1 µ ht ) 1/σ Q (ρ), 1}ρe ρ( t) dt. For this we follow the same calclations leading to eqation (VIII.12) in the proof of Lemma A.8, page 51 in the spplementary appendix, bt with f(t, ρ) =min{(1 µ ht ) 1/σ Q (ρ), 1}. This gives: E [q τ, θ = ] =se ρ + ρe ρ( t) f(t, ρ) dt = f(, ρ) 1 ρ f t(, ρ)+o =(1 µ h ) 1/σ Q (ρ)+ 1 γ s µ h 1 + o = s µ h + γ ρ σ 1 µ h ρ 1 µ h ρ 1 ρ 1 s 1 µ h + o 1, ρ where the second line follows from plgging eqation (A.8) into the first line. Sbstitting this expression into eqations (A.13) and (A.14), we find after some straightforward maniplation that, when ρ goes to infinity, 39

40 V γ(s µ h )/σ for <T ψ ( ), V γ(1 s) for (lim T ψ ( ),T s ), and V for >T s. The trading volme in the Walrasian eqilibrim is eqal to the measre of low valation instittions who become high-valation instittions: γ(1 µ h ), times the amont of asset they by at that time: 1 (s µ h )/(1 µ h ). Ths the trading volme is γ(1 s). To conclde the proof, note that after taking derivatives of Q ( ) withrespectto, it follows that Q T ψ ( ) ( ) = s µ ht ψ ( ) σ =1 s which implies in trn that γ(s µ h )/σ > γ(1 s) for <T ψ ( ). A.7 Proof of Proposition 5 A.7.1 First point: when (21) holds Given the ODEs satisfied by the price path in both markets, it sffices to show that, for all <T s, σ s (1 µ h )Q σ µh >. 1 µ h Besides, when s σ/(σ + 1), it follows from Lemma A.3 and Lemma A.4 that: Q = Q = eρt (s µ ht ) dt eρt (1 µ ht ) 1+1/σ dt. Plgging the above and rearranging, we are left with showing that: F =(s µ h ) e ρt (1 µ ht ) 1+1/σ dt (1 µ h ) 1+1/σ e ρt (s µ ht ) <. Bt we know from the proof of Lemma A.4, eqation (VIII.5), page 46 in the spplementary appendix, that F has the same sign as Q, which we know is negative at all >sinces σ/(σ + 1). A.7.2 Second point: when s is close to 1 and σ is close to The price at time is eqal to: p = + e r ξ d, where we make the dependence of s and σ explicit. With perfect cognition, ξ =1 δ((s µ h )/(s µ h )) σ = 1 δ (1 (1 s)e γ ) σ for <T s, and ξ = 1 for >T s. Therefore p =1/s δi(s), where: I(s) Ts e r (1 (1 s)e γ ) σ d, where we make explicit the dependence on s. Similarly, with limited cognition, the price at time is eqal to p =1/s δj(s), where: J(s) Tf e r (1 µ h )Q σ d. We begin with a Lemma proved in Section VIII.8.1, page 53 in the spplementary appendix: 4

41 Lemma A.1. When s goes to 1, bothi(s) and J(s) go to 1/r. Therefore, p goes to (1 δ)/r both with perfect cognition and with limited cognition. Besides, with perfect cognition: 1 p (s )=(1 δ)/r + δ I (s) ds, s and with limited cognition: 1 p (s )=(1 δ)/r + δ J (s) ds. s The next two lemmas compare I (s) and J (s) for s in the neighborhood of 1 when σ is not too large. The first Lemma is proved in Section VIII.8.2, page 54 in the spplementary appendix: Lemma A.11. When s goes to 1: I (s) constant if r>γ, I (s) Γ 1 (σ) log((1 s) 1 ) if r = γ, I (s) Γ 2 (σ)(1 s) 1+r/γ if r<γ, where the constant terms Γ 1 (σ) and Γ 2 (σ) go to when σ. In this Lemma and all what follows f(s) g(s) means that f(s)/g(s) 1whens 1. The second Lemma is proved in Section VIII.8.3, page 55 in the spplementary appendix: Lemma A.12. Assme γ + γ/σ ρ>. There exists a fnction J (s) J (s) sch that, when s goes to 1: J (s) + if r>γ, J (s) Γ 3 (σ) log((1 s) 1 ) if r = γ, J (s) Γ 4 (σ)(1 s) 1+r/γ if r<γ, where the constant terms Γ 3 (σ) and Γ 4 (σ) go to strictly positive limits when σ. Lemmas A.11 and A.12 imply that, if σ is close to, then J (s) >I (s) for s in the left-neighborhood of 1. The second point of the proposition then follows. A.8 Proofs of Proposition 9 A.8.1 The candidate eqilibrim Before formlating or gess, we need the following Lemma, proved in in Section VIII.9, page 59 in the spplementary appendix: Lemma A.13. Sppose s>σ/(1 + σ) and let T φ > be the niqe soltion of: µ htφ σ = 1+ 1 s 1. σ Then T φ <T s and, for all t [,T φ ) there exists a niqe φ t (T φ,t f ) sch that (A.16) φt t e ρ (1 µ h ) 1+1/σ (s µ ht ) (1 µ ht ) 1+1/σ (s µ h ) d =. (A.17) 41

42 In addition t φ t is continosly differentiable, strictly decreasing over [,T φ ],andlim t Tφ φ t = T φ. For concision, we directly state below the analytical formla defining the candidate eqilibrim (a heristic constrctive proof is available from the athors pon reqest). The price path. The price path is continos and solves the following ODEs. When t (,T φ ): s µht rp t ṗ t = ξ t =1 δ 1 µ ht When t (T φ,φ ): σ + δ d dt s µ ht (1 µ ht ) 1+1/σ σ φt e (r+ρ)( t) (1 µ h ) d, t (A.18) rp t ṗ t = ξ t =1 δ 1 µ ht 1 µ hφ 1 t σ s µhφ 1 t. 1 µ hφ 1 t (A.19) When t (φ,t f ): rp t ṗ t = ξ t =1 δ(1 µ ht )Q σ t. (A.2) where Q t is defined in eqation (A.4). Lastly, when t T f, p t =1/r. Clearly, together with the continity conditions at the bondaries of each intervals (T φ, φ, and T f ), the above ODEs niqely define the price path. High valation traders holding plans. The asset holding plans of high valation traders are as follows. For t [,T f ), the time t high valation trader holds q t, = 1 for all t. For t T f,thetime t high valation trader holds q t, = s/µ ht for all t. Since these asset holding plan are constant, we gess that they are implemented by sbmitting market orders at time t. Low valation traders holding plans. The asset holding plans of low valation traders are as follows. For t (,T φ ], the time t low valation trader holds: q t, = s µ ht for all [t, φ t ] (A.21) 1 µ ht 1/σ 1 =(1 µ ht ) 1/σ ξ Q for all [φ t,t f ), where Q (A.22) δ(1 µ h ) = for all T f, (A.23) and where ξ is defined in eqations (A.18) and (A.19). For t [T φ,t f ) q t, =(1 µ ht ) 1/σ Q for all [t, T f ) (A.24) = for all T f. (A.25) Lastly, for t T f and t, q t, =. We gess that, in order to implement her holding plan at the information event time t, the trader sbmits market orders and a schedle of limit sell orders. Note that, after plgging in the definition of ξ φt given in eqation (A.19), one sees that the above defined asset holding plan is continos at = φ t.sinceq Tf = Q Tf =, we also have continity at = T f. Lastly in Lemma A.16 below, we will show that when t [,T φ )(t [T φ,t f )) and [φ t,t f ]( [t, T f ]), the holding plan q t, is strictly decreasing. The verification proof is organized as follows. Section A.8.2 provides preliminary properties of eqilibrim objects. Section A.8.3 proves the optimality of the candidate asset holding plans. Section A.8.4 concldes by showing that the market clears at all dates. 42

43 A.8.2 Preliminary reslts We begin by noting that, by direct inspection of or gess, the candidate eqilibrim coincides with the eqilibrim of Proposition 1 at all times t>φ. Next, a reslt we se repeatedly is: Lemma A.14. For t [,T φ ], d s µ ht dt (1 µ ht ) 1+1/σ, with an eqality only if t = T φ. The proof is in Section VIII.1, page 61 in the spplementary appendix. Next, we prove in Section VIII.11, page 61 in the spplementary appendix: Lemma A.15. For t (,T f ), rp t ṗ t = ξ t (, 1). This Lemma ensres that Q, in eqation (A.22), is well defined, and is also helpfl to establish the optimality of trading strategies. To show that asset holding plans are decreasing when φ t,wewillneedthe following Lemma, proved in Section VIII.12, page 62 in the spplementary appendix: Lemma A.16. Q is continos over (,T f ), strictly increasing over (,T φ ), and strictly decreasing over (T φ,t f ). Eqipped with the above Lemma, we show in Section VIII.13, page 63 in the spplementary appendix, another crcial property of the price path: Lemma A.17. The candidate eqilibrim price, p t, is continosly differentiable and strictly increasing over [,T f ]. A.8.3 Asset holding plans are optimal Exection times of limit orders. The first step is to verify that the candidate asset holding plans can be implemented sing market and limit orders only, as explained in Section A.8.1. To that end, we begin by deriving the exection times associated with alternative limit orders to by or sell. Limit orders to by or sell sbmitted at t T f. Consider the case of limit sell orders (the case of limit by order is symmetric). In the candidate eqilibrim, the price is constant for all t T f. Clearly, this means that a limit order to sell at the ask price a>p Tf is never exected. By the price priority rle, a limit order to sell at price a<p Tf is exected immediately. We can always ignore sch limit orders becase they are clearly dominated by a market order to sell, which is also exected immediately, bt has a strictly higher exection price, p Tf. Lastly, we arge that a limit order to sell at price a = p Tf sbmitted at time t T f is exected immediately, at time t. First we know from the price priority rle that this order is exected at or after time t. Bt note that, at all times >t T f, the limit order book mst be empty at price p Tf : otherwise, by the time priority rle, the earliest sbmitted limit order at price p Tf wold be exected at time, which wold contradict the fact that asset holding plans are constant after T f. Ths, by any time >t, all limit orders to sell at price p Tf, and in particlar those sbmitted at time t, have been exected. Since this is tre for all >t, this implies that a limit order to sell at price p Tf is exected immediately at time t, and is ths eqivalent to a market order. Limit orders to sell sbmitted at t<t f. Lemma A.17 showed that the price path is strictly increasing over [,T f ]. Therefore, the price priority rle implies that: a limit order to sell at price a>p Tf is never exected; a limit order to sell at price a [p t,p Tf ) is exected when the price reaches a, i.e. at the time sch that p = a; a limit order to sell at price a = p Tf is exected at or after time T f. Bt, as noted in the previos paragraph, the limit order book is empty at all times >T f, and the same reasoning as above shows that this limit order mst be exected at or before T f. Taken together, these two remarks imply that a limit order to sell at price a = p Tf, sbmitted at time t<t f, is exected exactly at time T f. As above, we can ignore limit orders to sell at price a<p t. By the price priority rle, sch limit orders are exected immediately at price a<p t, and therefore are clearly dominated by a market order to sell at price p t. Limit orders to by sbmitted at t<t f. By the price priority rle: a limit order to by at price b p t is never exected; limit order to by at price b>p t is exected immediately at price b, and is ths clearly dominated by market order to by at price p t. 43

44 A re-statement of the trader s problem. Based on the above, we show that the trader s problem redces to maximizing her tility from one information event to the next: Lemma A.18. In the candidate eqilibrim, an asset holding plan q t, solves the trader s problem if and only if it maximizes t e (r+ρ)( t) E t [v(θ,q t, )] ξ q t, d (A.26) for almost all (t, ω) R Ω,and sbject to the constraint that q t, is decreasing for [t, t T f ],and constant for t T f. For the if part, note that the exection times of Section A.8.3 imply that, pon an information event at time t: a trader cannot sbmit a limit order to by exected at time >t; if t<t f, a trader can sbmit a limit order to sell at any time (t, T f ]; a trader cannot sbmit a limit order to sell at any time >T f. Therefore, in the candidate eqilibrim, a trader s asset holding plan q t, can never increase, can decrease in an arbitrary fashion over [t, t T f ], and mst stay constant after t T f. This means that the vale of the time t asset holding plan, q t,, is less than the maximm vale of the maximization problem in Lemma A.18. Therefore, an asset holding plan solves the trader s problem if it solves the program of Lemma A.18 for almost all (t, ω) R Ω. To prove the only if part, we proceed by contrapositive. Sppose that q t, does not maximize (A.26) for some positive measre set of R Ω. Then, consider the following change of holding plan: for all times and events in that set, switch to a plan that achieves a higher vale in the objective (A.26), and keep yor original holding plan the same otherwise. To see that the change of holding plan is feasible, note that the exection times of Section A.8.3 imply that, if the change of holding plan reqires modifying orders sbmitted at some information event t 1, all of these modifications can be ndone at any sbseqent information event t 2 >t 1. Indeed a limit sell order at price p (p,p Tf ] is exected at time regardless of its sbmission time t 1 <. Ths, if sch an order is modified at t 1 and is still nfilled at time t 2,thent 2 <and the modification can be ndone. Other kind of orders (limit sell orders at different prices, limit by orders, or market orders) are either immediately exected at information event t 1, or never exected, so any modification at t 1 can be ndone with market orders at t 2. Taken together, this shows that the change of holding plan is feasible and, clearly, it increases the vale of the trader s objective. Optimality of high valation traders asset holding plans. By constrction, the candidate asset holding plan is constant for all given t, so it can be implemented by only sbmitting market orders at each information event. To show that the candidate asset holding plan is optimal, it sffices to show that it maximizes a high valation trader s tility flow at each time. To see that this is indeed the case, recall from Lemma A.15 that ξ (, 1) for (,T f ), and that by constrction we have that ξ = 1 for >T f. Therefore, the flow tility v(h, q) ξ q =min{q, 1} ξ q, is maximized by q =1if<T f, and by any q [, 1] if T f. In particlar, q t, = 1 and, as long as t T f >T s, q t, = s/µ ht 1 maximizes the flow tility for T f. Ths, the candidate asset holding plan is optimal for high valation traders. Optimality of low valation traders asset holding plans. After time T f, the price is constant and eqal to 1/r, so the opportnity cost is ξ = 1. Given that v q (, ) = v(h, ) = 1 and v q (, q) < 1 for q>, this immediately implies that q = is the asset holding maximizing the flow payoff, E [v q (θ,q) θ t = ] ξ q, for any t<. Clearly, given the exection time of Section A.8.3, a low valation trader can always implement this zero asset holding for all T f. Indeed, if t<t f, sch trader simply needs to sbmit a limit order to sell all her remaining assets at price p Tf. If t>t f, sch trader simply needs to sbmit a market order to sell all her assets at time t. We conclde that we can restrict attention to holding plans sch that q t, = for all 44

45 T f. Plgging this restriction into the objective (A.26), we find that the low valation trader s problem at time t<t f redces to choosing q t, in order to maximize Tf t e (r+ρ)( t) M(, q t, ) d (A.27) where M(, q) E [v(θ,q) θ t = ] ξ q =(1 ξ ) q δ 1 µ h 1 µ ht q 1+σ 1+σ, (A.28) sbject to the constraint that q t, is decreasing over [t, T f ]. We now verify that or candidate asset holding plan solves this optimization problem. First, we note that, by Lemma A.16, q t, is, as reqired, decreasing over [t, T f ]. Second, we show below that two sfficient conditions for optimality are (i) (ii) Tf t Tf t (r+ρ)( t) M e q (, q t,)q t, d =, (r+ρ)( t) M e q (, q t,) q t, d for any decreasing fnction q t,, where the partial derivative is well defined becase, since q t, is decreasing, q t, q t,t =(s µ ht )/(1 µ ht ) < 1. To see why condition (i) and (ii) are sfficient, consider another decreasing holding plan q t,.wehave Tf t e (r+ρ)( t) [M(, q t, ) M(, q t, )] d Tf t (r+ρ)( t) M e q (, q t,)(q t, q t, ) d, where the first ineqality follows from the concavity of q M(, q), and the second ineqality follows from (i) and (ii). The proof that condition (i) and (ii) hold follows from algebraic maniplations that we gather in Section VIII.14. A.8.4 The market clears at all times For all T f, high valation traders who had at least one information event hold q t, = 1. Plgging this into eqation (8) and rearranging, this leads to the market clearing condition: ρe ρ( t) (1 µ ht )E [q t, θ t = ] dt = ρe ρ( t) (s µ ht ) dt. (A.29) Market clearing at (,T φ ). Then, for all t, it follows from eqation (A.21) that low valation traders hold q t, =(s µ ht )/(1 µ ht ) and so clearly the market clearing condition (A.29) holds. Market clearing at (T φ,φ ). Then, it follows from eqation (A.21) and (A.22) that low valation traders hold q t, =(s µ ht )/(1 µ ht )ift φ 1, and (1 µ ht ) 1/σ Q if t [φ 1,]. Ths, the left hand side of (A.29) writes: φ 1 ρe ρ( t) (1 µ ht ) s µ ht dt + ρe ρ( t) (1 µ ht ) 1+1/σ Q dt. 1 µ ht φ 1 (A.3) Plgging ξ, as defined in eqation (A.19), into the definition of Q, as defined in eqation (A.22), we obtain: Q = s µ hφ 1 (1 µ hφ 1 )1+1/σ 45

46 Next, sing the implicit eqation (A.17) defining φ 1, we obtain that: Q = φ 1 φ 1 ρe ρ( t) (s µ ht ) dt ρe ρ( t) (1 µ ht ) 1+1/σ dt. Plgging in (A.3) shows that the market clearing condition (A.29) holds. Market clearing at (φ,t f ). First note that by eqation (A.2), we have Q = Q,whereQ is defined in eqation (A.4). Then, the demand from low-valation investors is by definition of Q, ρe ρ( t) (1 µ ht ) 1+1/σ Q dt = ρe ρ( t) (s µ ht ) dt Market clearing at >T f. The market clears by constrction of the holding plans of high and low valation investors after time T f. A.9 Proof of Proposition 1 For this proof the sperscript AT E (algorithmic trading eqilibrim) refers to the eqilibrim objects of Proposition 1, while the sperscript LOE (limit order eqilibrim) refers to the eqilibrim object of Proposition 9. First point: p AT E = p LOE for φ. By constrction p AT E = p LOE for (φ,t f ). This immediately imply that p AT E Second point: p AT E >p LOE Lemma A.19. Let (T φ,φ ). If Q AT E q AT E t, q LOE t, = p LOE for all φ. =1/r for T f, and ξ AT E for (T φ,φ ). To obtain the reslt, we prove the following Lemma. Q LOE for all t (,) with a strict eqality t (,φ 1 = ξ LOE, then time t low valation traders asset holdings satisfy ). This Lemma is proved in Section VIII.15. Since high valation traders hold 1 nit of the asset in both the ATE and the LOE, it cannot be that both markets clear at a date (T φ,φ ) sch that Q AT E Q LOE. Ths, Q AT E <Q LOE for all (T φ,φ ). From the definition of Q AT E,wehaveξ AT E =1 (1 µ h )Q AT E σ. From (A.23), in the LOE we have ξ LOE =1 (1 µ h )Q LOE σ. Ths ξ AT E >ξ LOE for (T φ,φ ), and ξ AT E = ξ LOE for φ. This clearly implies that p AT E >p LOE for (T φ,φ ). Third point: p AT E <p LOE for ρ close to zero. We start by establishing a preliminary reslt. Since all the eqilibrim variables are defined as roots of continosly differentiable fnctions that are well-defined in ρ =, with non-zero partial derivatives, we can apply the Implicit Fnction Theorem to show that: Lemma A.2. Every eqilibrim object has a well-defined limit when ρ and is continos in ρ over ρ [, + ). Moreover, those limits satisfy the same eqations as in the case ρ> after letting ρ =. The proof is in Section VIII.16, page 68 in the spplementary appendix. Denoting all these limits with a hat, ˆ, we show that Lemma A.21. The limiting prices at time zero satisfy ˆp AT E < ˆp LOE. And the third point of Proposition 1 follows by continity. 46

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