Trading and Liquidity with Limited Cognition


 Darrell Wiggins
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1 Trading and Liqidity with Limited Cognition Brno Biais, Johan Hombert, and PierreOlivier 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, limitorders, 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 DaphineNYSEEronext 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 PierreOlivier Weill from the spport of the National Science Fondation, grant SES Tolose School of Economics (CNRSCRM, IDEI), HEC Paris, University of California Los Angeles and NBER,
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 highvalation at the first jmp in this process. Unconstrained efficiency wold reqire that lowvalation instittions sell to highvalation 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 infinitelylived 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 nonstorable 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 lefthand side of (4) is the instittion s marginal tility flow over [t, t + dt]. The righthand 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 intertemporal 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 timet information abot her instittion: the history of her informationevent 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 (Wellbehaved 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 resell 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 intertemporal 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 intertemporal 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 crosssectional average asset holding be eqal to s, thepercapitaassetspply. 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 crosssectional 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 intertemporal problem redces to pointwise 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 marketclearing 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 firstorder 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 marketclearing 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 oneeqationinonenknown 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 46. 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 marketclearing 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 MasColell, 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 hmpshaped. 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
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