Strategic trading in a dynamic noisy market. Dimitri Vayanos

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1 LSE Researc Online Article (refereed) Strategic trading in a dynamic noisy market Dimitri Vayanos LSE as developed LSE Researc Online so tat users may access researc output of te Scool. Copyrigt and Moral Rigts for te papers on tis site are retained by te individual autors and/or oter copyrigt owners. Users may download and/or print one copy of any article(s) in LSE Researc Online to facilitate teir private study or for non-commercial researc. You may not engage in furter distribution of te material or use it for any profit-making activities or any commercial gain. You may freely distribute te URL (ttp://eprints.lse.ac.uk) of te LSE Researc Online website. Cite tis version: Vayanos, D. (2001). Strategic trading in a dynamic noisy market [online]. London: LSE Researc Online. Available at: ttp://eprints.lse.ac.uk/arcive/ Tis is an electronic version of an Article publised in te Journal of finance 56 (1) pp Blackwell Publising. ttp:// ttp://eprints.lse.ac.uk Contact LSE Researc Online at: Library.Researconline@lse.ac.uk

2 Strategic Trading in a Dynamic Noisy Market Dimitri Vayanos Marc 10, 2003 ABSTRACT Tis paper studies a dynamic model of a financial market wit a strategic trader. In eac period te strategic trader receives a privately observed endowment in te stock. He trades wit competitive market makers to sare risk. Noise traders are present in te market. After receiving a stock endowment, te strategic trader is sown to reduce is risk exposure eiter by selling at a decreasing rate over time, or by selling and ten buying back some of te sares sold. Wen te time between trades is small, te strategic trader reveals te information regarding is endowment very quickly. MIT and NBER. I tank Anat Admati, Denis Gromb, Pete Kyle, Andy Lo, Lee-Bat Nelson, Paul Pfleiderer, Jose Sceinkman, Matt Spiegel, René Stulz, Rangarajan Sundaram, Jean Tirole, Jiang Wang, Ingrid Werner, Jeff Zwiebel, seminar participants at Cicago, Harvard, LSE, MIT, UCSD, and participants at te Western Finance Association, NBER market microstructure, and European Econometric Society conferences for very elpful comments. I am especially grateful to Avanidar Subramanyam for many valuable comments and suggestions. I tank Muamet Yildiz for excellent researc assistance.

3 Large traders, suc as dealers, mutual funds, and pension funds, play an important role in financial markets. 1 Many empirical studies sow tat tese agents trades ave a significant price impact. 2 Recent studies also sow tat large traders execute teir trades slowly, over several days, presumably to reduce teir price impact. 3 Tese studies raise a number of teoretical questions. For instance, wat dynamic strategies sould large traders employ to minimize teir price impact? In particular, ow quickly sould tey execute teir trades? How quickly does te price adjust to reflect te presence of a large trader? Given te importance of large traders, an analysis of teir dynamic strategies is relevant for understanding te daily and weekly beavior of returns, volume, and bid-ask spreads. It is also relevant for comparing trading mecanisms, in terms of liquidity provided to large traders, and information revealed in prices. An analysis of large traders dynamic strategies requires an assumption about teir motives to trade. Trading motives are generally divided into informational motives, arising from private information about asset payoffs, and allocational motives, suc as risk-saring, portfolio rebalancing, and liquidity. In a seminal paper, Kyle (1985) studies te dynamic strategy of a large trader wit informational motives (an insider ). Te insider is risk-neutral and trades wit risk-neutral market makers. Market makers agree to trade because tey cannot distinguis te insider from noise traders wo are also present in te market. Kyle sows tat te insider reveals is information slowly, until te time wen it is publicly announced. 4 If most large trades were motivated by information, large traders would significantly outperform te market. However, many empirical studies sow tat large traders do not significantly outperform, and may even underperform, te market. Moerover, tis performance is in spite of ig portfolio turnover. 5 Terefore, allocational motives must be important. Te dynamic strategies of large traders wit allocational motives ave received comparatively less attention. Te problem as some similarities wit te case of informational motives. For instance, a large trader wo wants to sell in order to edge a risky position, as private information tat e will sell and tat te price will fall. Tis is similar to an insider wo as private information tat te price will fall because of a negative earnings announcement. Te crucial difference, owever, is tat te time of te earnings announcement is exogenous, wile te time at wic te large trader sells is endogenous. Terefore, te large trader s speed of trade execution cannot be deduced from te insider s, since te latter 1

4 depends crucially on te time of te earnings announcement. Notice tat in te presence of allocational motives, te speed of trade execution determines two important aspects of te trading process. First, te speed of price adjustment, as in te presence of informational motives, and second, te allocative efficiency. If, for instance, te large trader sells slowly, optimal risk-saring is slowly acieved, and te trading process is not very efficient. In tis paper we study te dynamic strategy of a large trader wo trades to sare risk. 6 We consider a discrete-time, infinite-orizon, stationary economy wit a consumption good and two investment opportunities. Te first is a riskless bond and te second is a risky stock tat pays a random dividend in eac period. Te large trader is risk-averse and trades wit competitive market makers. We introduce a risk-saring motive troug a privately observed stock endowment tat te large trader receives in eac period. 7 For simplicity, we eliminate informational motives by assuming tat dividend information is public. To obtain te price impact in te absence of informational motives, we assume tat market makers are risk-averse. 8 In addition to te large trader and te market makers, small noise traders are present in te market. Since te large trader as a risk-saring motive, trade can take place even witout noise. We introduce noise because it is realistic to assume tat te large trader can conceal is trades to some extent. Our model is similar to Kyle (1985) in tat a large trader trades wit competitive market makers and noise traders. Te main differences between te two models are te following. First, in Kyle s model te large trader trades to exploit private information about asset payoffs. By contrast, in our model, information about asset payoffs is public and te large trader trades to sare risk. Second, in Kyle s model te large trader and te market makers are risk-neutral, wile in our model tey are risk-averse. We assume risk aversion so tat tere is scope for risk-saring. Tird, Kyle s model is non-stationary, since te large trader receives private information only at te beginning of te trading session. By contrast, our model is stationary, since te large trader receives a stock endowment in eac period. We assume stationarity so tat te model is tractable even wen traders are risk-averse. Our main results concern te dynamics of stock oldings and prices. We sow tat after an endowment sock, te large trader s stock oldings converge to a long-run limit, determined by optimal risk-saring between te large trader and te market makers. Moreover, tere are two patterns of convergence to te long-run limit: stock oldings can eiter decrease over time, or tey can decrease and ten increase. Te first convergence pattern is very intuitive. Stock oldings decrease over time as 2

5 te large trader sells to te market makers te fraction of is endowment corresponding to optimal risk-saring. We sow tat te first convergence pattern also as te following properties. First, te trading rate, defined as trade size (of te large trader s trades) over te size of subsequent trades, decreases over time. In particular, trade size decreases over time. Second, te price impact, defined as price cange over trade size, decreases over time. Te trading rate decreases over time as te information regarding te large trader s endowment gets reflected in te price. Wen te information is not reflected in te price, te large trader as an incentive to sell quickly in order to frontrun on tis information. Te second convergence pattern is somewat surprising. Te large trader sells to te market makers te fraction of is endowment corresponding to optimal risk-saring. He ten engages in a round-trip transaction, selling some sares only to buy tem back later. Tis pattern occurs wen tere is enoug noise, and wen te large trader is not very risk-averse relative to te market makers. Te large trader engages in te round-trip transaction because te market makers misinterpret te sale prior to tat transaction, i.e. te sale tat led to optimal risk-saring. Indeed, te market makers attribute te sale to te small traders, wo account for most order flow, and expect te large trader to absorb a fraction of te sale. Te large trader knows tat e initiated te sale and tat e will not absorb back a fraction. Terefore, e as private information tat te price will fall. He can trade on tat information, selling some sares and buying tem back wen te price falls. Te second convergence pattern suggests tat large traders strategies can be complicated and reminiscent of market manipulation. Te dynamics of stock oldings and prices take a very simple form in te continuoustime case, were te time between trades,, is small. Tey consist of two pases: a first sort pase, wose lengt goes to 0 wen goes to 0, and a second long pase. In te first pase, te large trader sells a fraction of is endowment (bounded away from 0) and te information regarding te endowment gets fully reflected in te price. In te second pase, optimal risk-saring is acieved. Te trading rate is tus very ig in te first pase and lower in te second pase. An important property of te dynamics for small, is tat te large trader reveals is information quickly, i.e. witin a time tat goes to 0 wen goes to 0. Tis is in contrast to te slow revelation of information in Kyle s (1985) model. 9 Te dynamics of stock oldings and prices depend on te noise and on traders risk aversion. We sow tat, as te noise increases, te large trader sells faster. However, te market makers and te large trader s risk aversion ave an ambiguous effect on te speed 3

6 of trade. Our results ave several empirical implications. In Section VI we derive tese implications, and relate our results to te empirical literature on large trades. Moreover, in Section VII we calibrate te model and study ow quickly te large trader trades for realistic parameter values. Tere is a small literature on te dynamic beavior of large traders wit allocational motives. Admati and Pfleiderer (1988) and Foster and Viswanatan (1990) allow large traders to optimize te timing of a single trade. Tey study weter suc timing decisions can explain te daily and weekly beavior of returns, volume, and bid-ask spreads. Seppi (1990) compares te upstairs market, were large traders execute teir trades as single blocks, to te downstairs market, were tey can trade over time. He derives conditions under wic large traders wit allocational motives prefer te upstairs market. Almgren and Criss (1999) and Bertsimas and Lo (1998) study te dynamic strategies of large traders wo ave a fixed time orizon to complete a trade and face an exogenous price reaction function. Vayanos (1999) studies te dynamic strategies of large traders wo trade to sare risk. Te main difference wit tis paper is tat tere is no noise. Te no noise assumption is somewat unrealistic, since it implies tat large traders cannot conceal teir trades. In tis paper we make te more realistic assumption tat tere is noise and sow tat in its presence, large traders strategies are very different. Indeed, in Vayanos (1999) large traders stock oldings decrease after an endowment sock and te trading rate is constant. By contrast, in tis paper stock oldings eiter decrease, in wic case te trading rate also decreases, or tey decrease and ten increase. Cao and Lyons (1999) study te dynamic strategies of large dealers wo sare risk in an inter-dealer market. Te main differences wit tis paper is tat tere is no noise and tat dealers can trade only for two periods. Te remainder of tis paper is organized as follows. In Section I we present te model. In Section II we determine a Nas equilibrium of te game between te market makers and te large trader. In Section III we study te dynamics of stock oldings and prices after te large trader receives an endowment sock, and in Section IV we study te dynamics in te continuous-time case. In Section V we examine ow te dynamics depend on te noise and on traders risk aversion. In Section VI we derive te empirical implications of our results, and in Section VII we calibrate te model. Finally, in Section VIII we conclude. All proofs are in te Appendix. 4

7 I. Te Model Time is continuous and goes from to. Trade takes place at times l, were l =.., 1, 0, 1,.., and > 0 is te time between trades. We refer to time l as period l. Tere is a consumption good and two investment opportunities. Te first investment opportunity is a riskless bond wit an exogenous, continuously compounded rate of return r. One unit of te consumption good invested in te bond in period l 1 returns e r units in period l. Te second investment opportunity is a risky stock tat pays a dividend d l in period l. We set d l = d l 1 + δ l, (1) and assume tat te dividend socks δ l are independent of eac oter and are normal wit mean 0 and variance σ 2. All traders learn δ l in period l, i.e. dividend information is public. Tere are tree types of traders: a large trader, market makers, and small noise traders. Te large trader is infinitely-lived and consumes c l in period l. 10 His utility over consumption is exponential wit coefficient of absolute risk aversion (CARA) α and discount rate β, i.e. exp( αc l β(l l 0 )). (2) l=l 0 In period l te large trader receives an endowment of ɛ l sares of te stock and d l 1 e r ɛ l units of te consumption good. Te consumption good endowment is te negative of te present value of expected dividends, d l /(1 e r ), times te stock endowment, ɛ l. 11 Te stock endowments ɛ l are independent of eac oter and of te dividend socks, and are normal wit mean 0 and variance σe. 2 Tey are private information to te large trader. Market makers are infinitely-lived and form a continuum wit measure 1. A market maker consumes c l in period l. His utility over consumption is exponential wit CARA α and discount rate β, i.e. exp( αc l β(l l 0 )). (3) l=l 0 Te assumption tat market makers form a continuum wit measure 1 means te following. First, a market maker maximizes equation (3) taking price as given. Market makers are tus competitive. Second, te market makers aggregate demand is derived by assigning eac market maker an index m in [0, 1] and integrating market makers demands over m. 5

8 Aggregate demand is tus te average of market makers demands. Te same is true for te market makers aggregate stock oldings, endowment, etc. 12 Small traders beavior is not derived from utility maximization. Small traders simply sell u l sares in period l. (Small traders buy if u l is negative.) Te sell orders u l are independent of eac oter, of te dividend socks, and of te large trader s endowments. Tey are normal wit mean 0 and variance σu. 2 Te sequence of events in period l is as follows. First, te large trader receives is endowments. Second, te market makers and te large trader learn te dividend sock. Tird, trade in te stock takes place. Fourt, te stock pays te dividend and fift, te market makers and te large trader consume. Trade is organized as follows. Te large trader and te small traders submit market orders and, simultaneously, te market makers submit demand functions. Te market-clearing price is ten determined and all trades take place at tis price. II. Equilibrium Determination In tis section we determine a stationary Nas equilibrium of te game between te market makers and te large trader. (A Nas equilibrium is stationary if traders equilibrium strategies are independent of time, olding te state variables constant.) In Section II.A we conjecture traders equilibrium strategies. Tese strategies form a Nas equilibrium if it is optimal for a trader to follow is strategy wen oter traders follow teirs. In Sections II.B and II.C we study traders optimization problems, and sow tat determining a stationary Nas equilibrium reduces to solving a system of non-linear equations. In Section II.D we solve te system for small order flow. We also derive an equation tat we use in Sections III and IV to explain te intuition beind our results. 13 A. Strategies We first introduce some notation. We denote by e l te large trader s stock oldings after trade in period l, and by s l te market makers expectation of e l, conditional on information available after trade in period l. We divide te large trader s stock oldings before trade in period l, e l 1 + ɛ l, into expected stock oldings, defined as te market makers expectation, s l 1, and into unexpected stock oldings. We denote by e l te market makers aggregate stock oldings after trade at period l. Wen all market makers follow teir equilibrium strategy, e l are also te stock oldings of eac market maker. To study te 6

9 market makers optimization problem, we will allow a market maker to deviate from is equilibrium strategy, but assume tat te oter market makers follow teirs. Since market makers form a continuum, te stock oldings of eac non deviating market maker will be e l. We will denote te stock oldings of te deviating market maker by e l + e l. Te conjectured period l equilibrium sell order for te large trader is (If x l is negative, it is a buy order.) x l = a e (e l 1 + ɛ l s l 1 ) + a s s l 1 a e e l 1. (4) Te sell order, x l, is linear in te large trader s unexpected stock oldings, e l 1 +ɛ l s l 1, te large trader s expected stock oldings, s l 1, and te market makers stock oldings, e l 1, all evaluated before trade in period l. 14 expect te coefficients a e, a s, and a e to be positive, i.e. te large trader sells wen is stock oldings are ig and te market makers stock oldings are low. We allow a e and a s to be different, i.e. te large trader can sell at different rates out of unexpected and expected stock oldings. We will sow tat te coefficients a e, a s, and a e are indeed positive and a e > a s. Te conjectured period l equilibrium demand for a market maker is ( ) x l (p l ) = B 1 e r d l p l A e e l 1 A s s l 1. (5) Demand, x l (p l ), is linear in te dividend rate, d l, te price, p l, te market makers stock oldings before trade in period l, e l 1, and te market makers expectation of te large trader s stock oldings before trade in period l, s l 1. Notice tat a unit increase in te dividend rate leaves demand unaffected only if te price increases by /(1 e r ). Tis is because /(1 e r ) is equal to te increase in te present value of expected dividends. Notice also tat traders conjectured strategies are stationary, since te coefficients a e, a s, a e, A e, A s, and B, are independent of l. Te price in period l is given by te market-clearing condition Using equation (5), we get p l = We x l (p l ) = x l + u l. (6) 1 e r d l A e B e l 1 A s B s l 1 1 B (x l + u l ). (7) B. Te Market Makers Optimization Problem A market maker maximizes te expectation of equation (3) assuming tat te oter market makers and te large trader follow teir equilibrium strategies. We formulate te market 7

10 maker s problem as a dynamic programming problem. Te state in period l is evaluated after trade takes place and before te stock pays te dividend. Tere are five state variables: te market maker s consumption good oldings tat we denote by M l, te dividend rate d l, te market maker s stock oldings e l + e l, te oter market makers stock oldings e l, and te market makers expectation s l, of te large trader s stock oldings. Tere are two control variables cosen between te state in period l 1 and te state in period l: te consumption c l 1, and te demand x l (p l ). Te dynamics of M l are given by te budget constraint M l = e r (M l 1 + d l 1 (e l 1 + e l 1 ) c l 1 ) p l x l (p l ). Since market makers form a continuum, te price p l in te budget constraint is given by equation (7), wic assumes tat all market makers follow teir equilibrium strategy. Given tat te large trader also follows is equilibrium strategy, equation (7) becomes p l = 1 e r d l A e B e l 1 A s B s l 1 1 B (a e(e l 1 + ɛ l s l 1 ) + u l ), (8) were A s = A s + a s and A e = A e a e. Te dynamics of d l are given by equation (1). Te dynamics of e l + e l are given by e l + e l = e l 1 + e l 1 + x l (p l ). (9) Te market maker s stock oldings after trade in period l are equal to is stock oldings after trade in period l 1, plus te sares tat e buys in period l. Similarly, te dynamics of e l are given by e l = e l 1 + x l + u l. (10) Te oter market makers stock oldings after trade in period l are equal to teir stock oldings after trade in period l 1, plus te order flow in period l. Given tat te large trader follows is equilibrium strategy, equation (10) becomes e l = (1 a e )e l 1 + a s s l 1 + a e (e l 1 + ɛ l s l 1 ) + u l. (11) We finally determine te dynamics of s l, te market makers expectation of te large trader s stock oldings e l. To determine tese dynamics, we first determine te dynamics of e l and ten use recursive filtering. Te dynamics of e l are given by e l = (e l 1 + ɛ l ) x l. (12) 8

11 Te large trader s stock oldings after trade in period l are equal to is stock oldings after trade in period l 1, plus is stock endowment in period l, minus is sell order in period l. Given tat te large trader follows is equilibrium strategy, equation (12) becomes e l = (e l 1 + ɛ l ) a e (e l 1 + ɛ l s l 1 ) a s s l 1 + a e e l 1. (13) We next use recursive filtering. We denote te information available to te market makers after trade in period l by I l. Te information I l consists of I l 1, te dividend rate d l (wic does not contain information on e l ), and te order flow x l + u l. To compute s l, te mean of e l conditional on I l, we proceed recursively. We first condition on I l 1 and ten compute te mean of e l conditional on te order flow x l + u l. Since all variables are normal, we can simply regress e l on x l + u l. (Te regression is conditional on I l 1.) To do te regression, we use equations (4) and (13). We also denote by Σ 2 e te variance of e l 1 conditional on I l Te regression implies tat were s l = E Il e l = (1 a s )s l 1 + a e e l 1 + g(x l + u l (a s s l 1 a e e l 1 )), (15) g = a e(1 a e )(Σ 2 e + σ 2 e) a 2 e(σ 2 e + σ 2 e) + σ 2 u. (16) Te mean of e l conditional on I l, is te sum of two terms. Te first term, (1 a s )s l 1 + a e e l 1, is te mean of e l conditional on I l 1. Te second term corresponds to te market makers surprise in period l. Te surprise is proportional to te difference between te order flow x l + u l, and te mean of te order flow conditional on I l 1, a s s l 1 a e e l 1. Te coefficient of proportionality g increases in (Σ 2 e + σ 2 e)/σ 2 u, wic is a measure of te relative order flow coming from te large and te small traders. Te dynamics of s l are given by equation (15). Since te large trader follows is equilibrium strategy, equation (15) becomes s l = (1 a s )s l 1 + a e e l 1 + g(a e (e l 1 + ɛ l s l 1 ) + u l ). (17) We conjecture tat te market maker s value function is V (M l, d l, e l, e l, s l ) = exp( α( 1 e r M l +d l (e l + e l )+F (Q, e l e l s l )+q)), (18) were F (Q, v) = (1/2)v t Qv for a matrix Q and a vector v, v t is te transpose of v, Q is a symmetric 3 3 matrix, and q is a constant. In Proposition 1 (in te Appendix) we 9

12 provide sufficient conditions for te demand in equation (5) to solve te market maker s optimization problem and for te function (18) to be te value function. Te conditions are on te coefficients A e, A s, B, a e, a s, a e, Q, and q. We explore te economic intuition beind tese conditions in Section II.D. C. Te Large Trader s Optimization Problem Te large trader maximizes te expectation of equation (2) assuming tat te market makers follow teir equilibrium strategy. We formulate te large trader s problem as a dynamic programming problem. Te state in period l is evaluated after trade takes place and before te stock pays te dividend. Tere are five state variables: te large trader s consumption good oldings tat we denote by M l, te dividend rate d l, te large trader s stock oldings e l, te market makers expectation s l, of te large trader s stock oldings, and te market makers stock oldings e l. Tere are two control variables cosen between te state in period l 1 and te state in period l: te consumption c l 1, and te market order x l. Te dynamics of M l are given by te budget constraint M l = e r (M l 1 + d l 1 e l 1 c l 1 ) d l 1 e r ɛ l + p l x l. (19) Te price p l in te budget constraint is given by equation (7). Te dynamics of d l, e l, s l, and e l are given by equations (1), (12), (15), and (10), respectively. We conjecture tat te large trader s value function is V (M l, d l, e l, s l, e l ) = exp( α( 1 e l s l e r M l + d l e l + F (Q, s l ) + q)). (20) e l were Q is a symmetric 3 3 matrix and q a constant. In Proposition 2 (in te Appendix) we provide sufficient conditions for te market order in equation (4) to solve te large trader s optimization problem, and for te function (20) to be te value function. Te conditions are on te coefficients A e, A s, B, a e, a s, a e, Q, and q. We explore te economic intuition beind tese conditions in te next section. D. Equilibrium for Small Order Flow Te optimization problems of Sections II.B and II.C produce a set of sufficient conditions for traders strategies to form a stationary Nas equilibrium. Te conditions are on A e, A s, B, a e, a s, and a e, te coefficients of traders strategies, and Q, q, Q, and q, te coefficients 10

13 of traders value functions. In te Appendix we combine tese conditions wit equations (14) and (16) of te market makers recursive filtering problem, and obtain a system, (S), of 20 non-linear equations. Te 20 unknowns are te six coefficients A e, A s, B, a e, a s, and a e, te 12 elements of te symmetric 3 3 matrices Q and Q, and te two coefficients g and Σ 2 e. In general, te system (S) is complicated and can only be solved numerically. Tis is because traders face bot fundamental risk (dividend risk) and price risk (risk coming from te effect of te oter traders on price). Te large trader, for instance, faces te risk of a negative dividend sock and te risk of selling at te same time as te small traders. Te cost of bearing dividend risk is exogenous and easy to compute. However, te cost of bearing price risk depends on traders strategies and is a non-linear function of A e, A s, B, a e, a s, a e, Q, and Q. Since dealing wit dividend risk is easy and wit price risk difficult, it is natural to study te case were price risk is small. Te large trader s problem does not cange qualitatively for small price risk. Indeed, te determinants of te large trader s strategy are price impact and te risk of olding a large position. Neiter goes to 0 as price risk gets small, since tere is dividend risk. Price risk is small wen order flow is small, i.e. wen te variances σe 2 and σu 2 of te endowment and te noise socks are small. To study te system (S) for small order flow, we set σu 2 = σ 2 uσe, 2 and assume tat σe 2 goes to 0 wile σ 2 u stays constant. We also set Σ 2 e = Σ 2 eσe. 2 For σe 2 = 0 te system (S) simplifies dramatically. In Teorem 1 we sow tat (S) collapses to a system of four non-linear equations in a e, a s, g, and Σ 2 e. Tis new system, (s), as a solution (obtained in closed-form for small in Section IV) and tus (S) as a solution. Te solution of (S) can be extended for small σe, 2 by te implicit function teorem. Teorem 1 Te system (S) as a solution for small σ 2 e. For σ 2 e = 0, te coefficients a e, a s, g, and Σ 2 e solve te system (s) of a e (1 (1 a e (1+g))e r )(1 a s (1+g))α (1 a e (1+g))(1 a s a e )(1 e r )α = 0, (21) a s 1 (1 a s (1 + g)(2 a s a e ))e r 1 (1 a s (1 + g))(1 a s a e )e r α + (α(1 a s) αa s )(1 e r ) = 0, (22) g = a e(1 a e )(Σ 2 e + ) a 2 e(σ 2 e + ) + σ 2 u, (23) 11

14 and Te coefficient a e is given by Σ 2 e = (1 a e) 2 (Σ 2 e + )σ 2 u a 2 e(σ 2 e + ) + σ 2 u. (24) a e α a s α = 0. (25) Te coefficients A e, A s, B, Q, and Q, are determined (in te Appendix) by solving systems of linear equations. Wen te variance σ 2 e of te endowment socks is zero, we can simplify te system (S) and explore te economic intuition beind its equations. We do not consider all te equations of (S), but only two equations tat we can derive by combining te equations of (S). (We derive te two equations in te Appendix.) Te first, market maker equation follows by combining te equations coming out of te market makers optimization problem. Te second, large trader equation similarly follows from te large trader s problem. Combining te market maker and large trader equations, we can obtain an equation tat we use in Sections III and IV to explain te intuition beind our results. Te market maker equation is p l + d l ασ2 2 e r 1 e r e l + E l p l+1 e r = 0. (26) Equation (26) states tat a market maker s marginal benefit of buying x sares in period l and selling tem in period l + 1, is 0. Te marginal benefit is te sum of four terms. Te first term represents te price p l tat te market maker pays at period l. Te second term represents te dividend d l tat te market maker receives in period l. Te tird term represents an inventory cost tat te market maker bears by olding a riskier position between periods l and l + 1. Tis inventory cost is proportional to te dividend risk σ 2, te market maker s CARA α, and te market maker s stock oldings e l. Finally, te fourt term represents te price tat te market maker expects to receive in period l+1, discounted in period l. (We denote by E l te expectation w.r.t. te market makers information, to distinguis it from te expectation w.r.t. te large trader s information.) Te large trader equation is + 1 B x l (ga s a e )(1 + g) A s a s B p l + d l ασ2 2 e r 1 e r e l + E l p l+1 e r l l+1 (1 a s (1 + g)) l l 2 E l x l e (l l)r = 0. (27) 12

15 Equation (27) states tat te large trader s marginal benefit of buying x sares in period l and selling tem in period l + 1, is 0. It is te counterpart of equation (26) wic sets te marginal benefit of a market maker to 0. Te large trader s marginal benefit is te sum of six terms. Te first four terms are as for te market maker. Te fift and sixt terms represent te impact of buying and selling sares on prices. Te fift term represents te impact on te period l price. It is te product of x l, te number of sares tat te large trader sells in equilibrium, times 1/B, te marginal price increase tat corresponds to te purcase of x sares. (Equation (7) implies tat a purcase of x sares increases te price by (1/B) x.) Te sixt term represents te impact on te price in periods l l + 1. For eac l, we multiply E l x l, te expected number of sares tat te large trader sells in equilibrium, by (ga s a e )(1 + g) A s a s B (1 a s(1 + g)) l l 2, (28) te marginal price increase tat corresponds to te purcase of x sares in period l and te sale of x sares in period l + 1. We ten discount in period l, and sum over l. Equation (28) implies tat te purcase and subsequent sale of x sares decreases te price in periods l l + 1 if ga s a e > 0 and increases te price oterwise. We first explain wy tis zero cumulative trade affects prices, and ten wy te effect depends on te sign of ga s a e. Te purcase and subsequent sale of x sares affects prices because it affects te market makers expectation of te large trader s stock oldings. Te reason wy it affects te market makers expectation, is te following. Te market makers revise teir expectation by comparing order flow to expected order flow. In period l, tey compare te purcase of x sares to 0. However, in period l + 1, tey compare te sale of x sares not to 0, but to te expectation tat te purcase in period l as created. Suppose, for instance, tat after a purcase, te market makers expect a purcase. Ten, te sale of x sares will send a stronger signal tan te purcase, and te market makers will increase teir expectation of te large trader s stock oldings. To determine weter te market makers expect a purcase after a purcase, we use equations (15) and (10). Tese equations imply tat a purcase of x sares in period l reduces s l by g x, e l by x, and expected (sell) order flow in period l + 1, a s s l a e e l, by (a s s l a e e l ) = (ga s a e ) x. Terefore, te market makers expect a purcase after a purcase if ga s a e > 0. Notice tat wen g is small, i.e. most order flow comes from te small traders, te market makers 13

16 expect a sale after a purcase. Tis is because tey attribute te purcase to te small traders, and expect te large trader to satisfy part of te demand, i.e. to sell some sares. Combining te market maker equation (26) and te large trader equation (27), we can obtain an equation tat we use to explain te intuition beind our subsequent results. Substituting te price p l from equation (26) into equation (27), we get σ 2 2 e r ) 1 e r (αe l αe l ) + (E l p l+1 E l p l+1 e r + 1 B x l (ga s a e )(1 + g) A s a s B l l+1 (1 a s (1 + g)) l l 2 E l x l e (l l)r = 0. (29) Te large trader s marginal benefit of buying sares in period l and selling tem in period l + 1, can be reduced to a sum of four terms. Te first term is te risk-saring term. Buying sares benefits te large trader wen is stock oldings e l, adjusted by is CARA α, are smaller tan te market makers stock oldings. Te second term is te frontrunning term. Suppose tat in equilibrium te large trader sells in period l + 1, and tat te sale is not expected by te market makers. In period l, te large trader as private information tat te price will fall, i.e. E l p l+1 E l p l+1 < 0. By buying sares, e foregoes frontrunning on tis information. Te tird term is te price impact term and represents te impact of buying sares on te period l price. Te fourt term represents te impact of buying and selling sares on te price in periods l l + 1. Since te impact is troug te market makers expectation, we refer to te fourt term as te belief manipulation term. III. Dynamics of Stock Holdings and Prices In tis section we study te dynamics of stock oldings and prices. To study tese dynamics, we first sow a result on te coefficients a e and a s of te large trader s equilibrium strategy. A. Te Coefficients a e and a s Te large trader s sell order in period l is given by equation (4) tat we reproduce below x l = a e (e l 1 + ɛ l s l 1 ) + a s s l 1 a e e l 1. Te coefficient a s is te rate at wic te large trader sells out of is expected stock oldings (conditional on te market makers information) s l 1. Te coefficient a e is similarly te rate at wic te large trader sells out of is unexpected stock oldings e l 1 + ɛ l s l 1. In Proposition 3 we compare te coefficients a e and a s for small order flow. Our numerical solutions confirm Proposition 3 for large order flow. 14

17 Proposition 3 For small σ 2 e, a e > a s. Proposition 3 states tat te large trader sells at a iger rate out of unexpected tan out of expected stock oldings. To explain te intuition beind tis result, we consider two extreme cases. First, te case were te large trader s stock oldings are expected, i.e. were tey are equal to te market makers expectation s l 1. Second, te case were te large trader s stock oldings are unexpected, i.e. were te market makers expectation is 0. We also suppose tat te large trader s stock oldings are ig relative to te market makers, so tat te large trader sells over time and te price falls. Wen stock oldings are unexpected, te large trader as private information tat te price will fall. By selling slowly, e bears te opportunity cost of not frontrunning on tis information. By contrast, wen stock oldings are expected, te large trader as no private information and bears no suc cost. Terefore, te large trader sells more slowly wen stock oldings are expected. In terms of equation (29), te frontrunning term is 0 wen stock oldings are expected and negative wen tey are unexpected. Terefore, te marginal benefit of buying in period l and selling in period l + 1, i.e. te marginal benefit of selling more slowly, is iger wen stock oldings are expected. B. Dynamics Te dynamics of stock oldings and prices are, a priori, complicated, because tey are generated by multiple endowment and noise socks. We can, owever, simplify te dynamics, using te linearity of te model. Indeed, because of linearity, te dynamics are simply te sum over all socks of te dynamics generated by eac sock. 16 We will study te dynamics generated by an endowment sock. Te dynamics generated by a noise sock ave a similar flavor. We normalize te endowment sock to 1, and assume tat it comes in period l, i.e. we set ɛ l = 1. To isolate te dynamics generated by tis sock, we set all oter endowment and noise socks to 0. In Proposition 4 we determine te dynamics of te large trader s stock oldings. Proposition 4 Te large trader s stock oldings before trade in period l l are e l 1 = a e a e (ga s a e ) + a s + a e (a e (1 + g) a s a e )(a s + a e ) (1 a s a e ) l l a e a s + (1 a e (1 + g)) l l. (30) a e (1 + g) a s a e 15

18 Proposition 4 implies tat stock oldings converge to te long-run limit a e /(a s + a e ), l and tat convergence takes place at a combination of te rates (1 a s a e ) l and (1 a e (1 + g)) l l. To prove Proposition 4, we determine te joint dynamics of te large trader s stock oldings, e l, te market makers expectation of te large trader s stock oldings, s l, and te market makers stock oldings, e l. Equations (13), (17), and (11), give e l, s l, and e l as linear functions of e l 1, s l 1, and e l 1. Te 3 3 matrix corresponding to tese equations as tree eigenvalues. Te first eigenvalue is 1, and corresponds to te long-run limit. Te oter two eigenvalues are 1 a s a e and 1 a e (1 + g) and correspond to te rates of convergence. Te dynamics of prices can be deduced from te dynamics of e l, s l, and e l. Equation (8) implies tat te price p l is a linear function of e l, s l, and e l. Terefore, te price converges to a long-run limit, and convergence takes place at a combination of te rates (1 a s a e ) l l and (1 a e (1 + g)) l l. We next study te long-run limits of stock oldings and prices, and ten te convergence to tese limits. Te long-run limit of te large trader s stock oldings is a e /(a s + a e ). Since te endowment sock is equal to 1, te long-run limit of te market makers stock oldings is a s /(a s +a e ). Terefore, in te long run, te endowment sock is divided between te large trader and te market makers according to te ratio a e /a s. Tis ratio is in fact te optimal risk-saring rule. Teorem 1 implies tat in te absence of price risk, te risk-saring rule coincides wit te standard risk-saring rule α/α. 17 Te long-run limit of te price can be deduced from te long-run limits of stock oldings and te market makers expectation of te large trader s stock oldings. In te proof of Proposition 4, we sow tat te latter is a e /(a s +a e ). Tis means tat, in te long run, te market makers form a correct expectation of te large trader s stock oldings. (As all results in tis section, tis result concerns te dynamics generated by one endowment sock. Because of te subsequent endowment and noise socks, te market makers never learn te large trader s stock oldings.) In Proposition 5 we sow some results on te convergence of stock oldings and prices to teir long-run limits. To state te proposition, we define two variables, te trading rate and te price impact. Te trading rate is te ratio of te large trader s sell order at a given period, to te sum of is sell orders at tat and subsequent periods. Te price impact is te ratio of minus te price cange at a given period, to te large trader s sell order at tat period. Formally, te trading rate in period l is x l / l l x l, and te price impact is (p l 1 p l )/x l. 16

19 Proposition 5 If ga s a e 0, te large trader s stock oldings decrease over time. Te trading rate and te price impact also decrease over time. If ga s a e < 0, te large trader s stock oldings decrease and ten increase over time. Proposition 5 implies tat tere are two patterns of convergence to te long-run limit: te large trader s stock oldings can decrease over time, or tey can decrease and ten increase. Moreover, wen stock oldings decrease over time, te trading rate and te price impact also decrease. 18 Te first convergence pattern is very intuitive. Stock oldings decrease over time, as te large trader sells to te market makers te fraction of is endowment corresponding to optimal risk-saring. Te trading rate decreases over time because, first, te large trader sells at a iger rate out of unexpected tan out of expected stock oldings (Proposition 3) and, second, te fraction of stock oldings tat are expected increases, as te market makers expectation becomes more accurate. Te price impact decreases over time because te fraction of te large trader s sell order tat is unexpected, and tat causes te price to cange, decreases. Figure 1 illustrates te first convergence pattern. 19 Te second convergence pattern is somewat surprising. Te large trader sells to te market makers te fraction of is endowment corresponding to optimal risk-saring. He ten engages in a round-trip transaction, selling some sares only to buy tem back later. Tis pattern occurs wen g is small, i.e. most order flow comes from te small traders, and wen a e /a s is large, i.e. te optimal risk-saring rule assigns many sares to te large trader. Te large trader engages in te round-trip transaction because te market makers misinterpret te sale prior to tat transaction, i.e. te sale tat led to optimal risk-saring. Indeed, te market makers attribute te sale to te small traders, wo account for most order flow, and expect te large trader to absorb a fraction of te sale. Te large trader knows tat e initiated te sale and tat e will not absorb back a fraction. Terefore, e as private information tat te price will fall. He can trade on tat information, selling some sares and buying tem back wen te price falls. Figure 2 illustrates te second convergence pattern. IV. Dynamics in te Continuous-Time Case In tis section we study te dynamics of stock oldings and prices in te continuous-time case, were te time between trades, is small. We consider te continuous-time case for 17

20 two reasons. First, because in real-world financial markets te time between trades can be very small. Second, because wen bot te order flow and te time between trades are small, we can determine te Nas equilibrium in closed-form. In Section IV.A we determine te closed-form solution and in Section IV.B we use te solution to study te dynamics of stock oldings and prices. A. Te Closed-Form Solution We obtain te closed-form solution wen bot te order flow and te time between trades are small. More precisely, we consider te Taylor expansion in of te coefficients A e, A s, B, a e, a s, a e, Q, Q, g, and Σ 2 e. For small σe, 2 i.e. for small order flow, we determine te order of te dominant term in te Taylor expansion, i.e. we determine weter te dominant term is of order, or, or 1, etc. For σe 2 = 0, we determine te dominant term in closed-form. In Proposition 6 we present te results for te coefficients a e, a s, a e, and g. We omit te oter coefficients, since we do not use tem in wat follows. Proposition 6 For small σ 2 e, a e = φ e + o( ), as = φ s + o(), a e = φ e + o(), and g = g 0 + o(1). For σ 2 e = 0, φ s is te positive root of g 0 = σ 2 1, u αr φ e = α(1 + g 0 ), 2(φ s ) 2 (1 + g 0 ) α α φ sr(2 + g 0 ) r 2 = 0, and φ e = φ s α/α. Proposition 6 implies tat for small order flow, a e is of order, a s and a e of order, and g of order 1. Our numerical solutions confirm tese results for large order flow. In Section IV.B we use tese results to study te dynamics of stock oldings and prices. Proposition 6 as also implications on ow te coefficients a e, a s, a e, and g depend on te exogenous parameters σ 2 u, α, and α, for σe 2 = 0. We explore tese implications in Section V, were we perform comparative statics. 18

21 Te result tat a e is of order and a s of order, is stronger tan te result a e > a s of Proposition 3. For small, te large trader sells at a muc iger rate, and not simply at a iger rate, out of unexpected tan out of expected stock oldings. Tis result turns out to be crucial for te dynamics of stock oldings and prices, so we explain te intuition beind it in te rest of tis section. We suppose tat before trade in period l, te large trader s stock oldings are equal to 1, and te market makers stock oldings to 0. Te large trader tus sells over time. Equation (29) states tat te marginal benefit of buying in period l and selling in period l + 1, i.e. te marginal benefit of selling more slowly, is te sum of te risk-saring term, te frontrunning term, te price impact term, and te belief manipulation term. Te risk-saring term is negative, since te large trader bears more dividend risk between periods l and l + 1. Since te variance of te dividend sock δ l+1 is σ 2, te risk-saring term is of order. Te price impact term is positive, since te price in period l increases. It is of te same order as te trade x l, of te large trader. Te belief manipulation term can be negative or positive, depending on weter te market makers expect a purcase or a sale after a purcase. For simplicity, we assume tat it is 0. Wen stock oldings are expected, i.e. wen s l 1 = 1, te frontrunning term is 0. Terefore, te price impact benefit of selling more slowly, as to equal te risk-saring cost. Tis means tat te trade x l is of order. Equation (4) implies tat x l = a s. Terefore, a s is of order. Wen stock oldings are unexpected, i.e. wen s l 1 = 0, te frontrunning term is negative. Tis is because te period l price does not reflect te large trader s future sales, and is iger tan te period l + 1 price. Substituting te price p l+1 from equation (7), we can write te frontrunning term as 1 B (E l x l+1 E l x l+1 ) e r. Te term in parenteses is te unexpected period l + 1 sell order, i.e. te sell order tat can be predicted by te large trader and not by te market makers. Te frontrunning term is simply te price impact of te unexpected sell order, discounted in period l. Notice tat te price impact term, (1/B)x l, is te negative of te frontrunning term, since it is te negative of te price impact of x l, te unexpected period l sell order. Te sum of te two terms is of order ) x l (E l x l+1 E l x l+1 e r. 19

22 Tis, rater tan x l, as to be of te same order as te risk-saring term, i.e. of order. Equations (4), (13), and (17) imply tat ) x l (E l x l+1 E l x l+1 e r = a e a e (e l s l )e r = a e (1 (1 a e (1 + g))e r ). Terefore, a e is of order. B. Dynamics As in Section III.B, we study te dynamics generated by one endowment sock. We first proceed euristically. To complement te euristic analysis, we ten study te dynamics in te limit wen goes to 0. In Proposition 6 we sowed tat a e, te selling rate out of unexpected stock oldings, is of order. Terefore, unexpected stock oldings get close to 0, teir long-run limit, witin a number of periods of order 1/, i.e. witin a time of order (1/ ) =. Witin tat time, total (expected plus unexpected) stock oldings get close to expected stock oldings. Tis appens bot because total stock oldings decrease, as te large trader sells a fraction of is endowment, and because expected stock oldings increase, as te market makers expectation becomes more accurate. Once total stock oldings get close to expected stock oldings, tey evolve more slowly. Indeed, in Proposition 6 we sowed tat a s, te selling rate out of expected stock oldings, is of order. Terefore, expected (and total) stock oldings get close to teir long-run limit, witin a number of periods of order 1/, i.e. witin a time of order (1/) = 1. Te above euristic analysis implies tat for small, te dynamics consist of two pases. Te first pase is sort, wit lengt of order. In tis pase, te large trader sells a fraction of is endowment, and te market makers form a correct expectation of te large trader s stock oldings. Te second pase is long. In tis pase, optimal risk-saring is acieved. Figure 3 illustrates te dynamics of te large trader s stock oldings for small. To complement te euristic analysis, we study te dynamics in te limit wen goes to 0. We assume tat te endowment sock comes at time t, and we determine te limit wen goes to 0, of te large trader s stock oldings at time t > t. We fix times t and t, rater tan periods l and l, because te time between two periods goes to 0 wen goes to 0. We determine te limit of te large trader s stock oldings in Proposition 7. Proposition 7 Te limit wen goes to 0, of te large trader s stock oldings at time t > t is φ e g 0 φ s φ e + φ s + φ e (1 + g 0 )(φ s + φ e ) e (φs+φ e)(t t). (31) 20

23 Proposition 7 implies tat in te limit wen goes to 0, stock oldings beave as follows. At time t tey are equal to 1, te endowment sock. Immediately after time t tey drop discontinuously to g 0 /(1 + g 0 ). Tey ten converge to te long-run limit φ e /(φ s + φ e ) at te rate e (φs+φ e)(t t). Te discontinuous drop corresponds to te first pase of te dynamics for small, and te exponential rate of convergence corresponds to te second pase. We sould empasize tat te discontinuous drop does not correspond to a single block trade, but to many small trades. Eac of tese trades is of order, and te trades are completed witin a time of order. An important property of te dynamics for small, is tat te large trader reveals is information quickly, i.e. witin a time tat goes to 0 wen goes to 0. Indeed, te market makers form a correct expectation of te large trader s stock oldings witin a time of order. Te fast revelation of information is in contrast to te slow revelation of information (witin a time tat does not go to 0 wen goes to 0) in Kyle s (1985) insider trading model. One reason wy te results are different may ave to do wit traders impatience. Our large trader is risk-averse and bears a cost wen olding a risky position. He is tus impatient to reduce is position. By contrast, Kyle s insider is risk-neutral and is not impatient to establis a position, as long as e does so before is information is publicly announced. 20 It is, owever, unlikely tat te difference in results is due to impatience. Indeed, we get fast revelation of information for all parameter values, including wen te risk aversion of te large trader is small. 21 A second reason wy te results are different may ave to do wit stationarity. Our model is stationary, since te large trader receives a stock endowment in eac period. By contrast, Kyle s model is non-stationary, since te insider receives private information only at te beginning of te trading session. Non-stationarity implies tat te fast revelation of information cannot be an equilibrium. Indeed, if te insider reveals is information quickly, market dept will increase over time and te insider will prefer to trade more slowly. To determine weter te difference in results is due to stationarity, one can examine a stationary model wit a risk-neutral insider wo receives private information over time. Suc a model is of independent interest. For instance, a result tat te insider reveals is information quickly would suggest tat markets are informationally efficient even in te presence of informational monopolists