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7 The aggregate position of the noise traders at t is denoted by u t, which is an exogenous Gaussian process given by u t = u + t du τ, with du t = σ v db u t, () where B u t is a Brownian motion independent from B v t. The aggregate position of the informed trader at t is denoted by x t. The informed trader is risk-neutral and chooses x t to maximize expected utility at t = given by U [ 1 ] = E (v 1 p t+ dt ) dx t [ 1 ] = E (v 1 p t dp t ) dx t, (3) where p t+ dt = p t + dp t is the price at which the order dx t is executed. 8 Prior to the liquidation date, t = 1, the price equals the conditional expectation of v 1 given the information available until that point. The market maker does not observe the individual orders of the informed trader or the noise traders, but only the aggregate order flow y t = u t + x t. (4) More generally than Back and Pedersen (1998), we assume that the market maker also learns about the asset value. At t + dt, the market maker receives a noisy signal of the innovation in asset value: dz t = dv t + de t, with de t = σ e db e t, (5) where Bt e is a Brownian motion independent from all the others. We denote by Ω t the market maker s information set right before trading at t, and the fundamental value conditional on this information set by q t = E [v 1 Ω t ]. (6) We also denote by Ω y t the market maker s information set right after she observes the aggregate order flow at t. Because she is competitive and risk-neutral, the price at which 8 Because the optimal trading strategy of the informed trader might have a stochastic component, we cannot set E( dp t dx t ) = as, e.g., in the Kyle (1985) model. 7

10 In the following we refer to the first term of trading strategy (11) as the level trading as it consists in trading on the difference between the level of the asset value and the price level. It is the same as in Kyle (1985), Back, Cao, and Willard (). The second term of (11) is new and we call it news trading as it consists in trading on innovation of the asset value, i.e., on news. We define the following constants: a = σ u σ v, b = σ e σ v, c = Σ. (1) σv Theorem 1. In the benchmark model, there is a unique linear equilibrium. It is described by the equations (8), (9) and (11), where the coefficients are given by β B t = ( 1 a 1/ c + b ) 1/, (13) 1 t c 1 + b γt B =, (14) λ B t = 1 ( c + b ) 1/, (15) a 1/ 1 + b µ B 1 t = 1 + b. (16) In the fast model, there is a unique linear equilibrium. It is described by equations (8), (1), (11), where the coefficients are given by β F t = 1 1 t a 1/ cf 1/ ( ) 1 + f + b + bf f, (17) γt F = a 1/ f 1/, (18) λ F t = 1 1 a 1/ f 1/ ( + b + bf), (19) µ F 1 + f t = + b + bf, () ρ F t = 1 a 1/ f 1/ 1 + f, (1) and f is the unique root in (, 1) of the cubic equation ( (1+c)b b ) f 3 + ( b(b+)(c+1) (b+1) ) f + (b+) ( (b+)(c+1) 1 ) f 1 =. () 1

16 Proposition 7. In the benchmark model, the cumulative price impact is CPI B t,t+τ = k B 1 [ 1 ( 1 τ ) ] λ B β B, (33) 1 t while in the fast model it is CPI F t,t+τ = k F + k F 1 [ 1 ( 1 τ ) ] (λ F µ F ρ F )β F, (34) 1 t where k B 1 = β B Σ, (35) k F = γ F ((λ F µ F ρ F )γ F + µ F )σ v, (36) k F 1 = β F Σ + γ F (1 (λ F µ F ρ F )γ F µ F )σ v. (37) One can see from the formulas that in the benchmark model the cumulative price impact starts from near-zero values when τ is very small, while in the fast model it starts from a positive value, k F. This is because in the benchmark model there is no news trading, and the price impact takes time to build up, as we have also seen in Figure 1. By contrast, in the fast model the news trading component has an immediate and large price impact, but additionally it has a level trading component that also builds up price impact over time. The cumulative price impact is thus a measure that allows to test whether a particular high frequency trader engages in news trading or not. 16

17 A Proofs A.1 Proof of Proposition 1 Benchmark model The expected flow of profit of the informed trader is E [dπ t ] = E [(v 1 p t+dt )dx t ] = ( β t Σ t + γ t σ v λ t γ t σ v) dt, (38) where Σ t = E [(v t p t ) ]. 1 The problem of the informed trader is to maximize [ 1 ] E dπ t (39) subject to the law of motion of Σ t. We have Σ t+dt = E[(v t+dt p t+dt ) ] = E[(v t + dv t p t µ t (dv t + de t ) λ t (β t (v t p t )dt + γ t dv t + du t )) ] = Σ t + (1 λ t γ t µ t ) σ vdt + µ t σ edt + λ t σ udt λ t β t Σ t dt, therefore the law of motion of Σ t is given by Σ t = λ t β t Σ t + (1 λ t γ t µ t ) σ v + µ t σ e + λ t σ u. (4) Substituting (4) into (38) and (39) the informed trader problem becomes max E (β t,γ t) [ ( 1 (β t Σ t + (1 λ tγ t µ t ) σv + µ t σe + λ t σu λ t β t λ t β t ) + γ t σ v λ t γ t σ v Integrated by parts and following the same reasoning as in Kyle (1985) we obtain that Σ 1 =, that λ t is a constant, and that γ t =. Now, the pricing rule. Equations (6) and (9) imply that µ t is the regression coefficient 1 It would be equivalent to write the trading strategy as β t (v t q t+dt )dt + γ t dv t since the difference with the actual trading strategy would be of the order dt 3/. Similarly, one could define Σ t using q t+dt or p t+dt instead of p t. ) dt ] 17

18 of dz t on v 1 conditional on Ω y t dt (i.e., Ω t {dz t }): µ t = Cov(v 1, dz t Ω y t dt ) V ar(dz t Ω y t dt ) = Cov(v 1 + dv τ, dv t + de t Ω y t dt ) V ar(dv t + de t Ω y t dt ) = σ v. σv + σe Equations (7) and (8) imply that λ t is the regression coefficient of dy t on v 1 conditional on Ω t : λ t = Cov(v 1, dy t Ω t ) V ar(dy t Ω t ) = Cov(v t Ω t, β t (v t p t )dt + du t Ω t ) V ar(β t (v t p t )dt + du t Ω t ) = β tσ t. σu Since λ t is constant, so is β t Σ t. Equation (4) then implies that Σ t is constant. Therefore Σ t = (1 t)σ and β t = β /(1 t). Finally, we integrate (4) between and 1 and substituting for λ and µ we obtain β and λ. Fast model law of motion of Σ t. We have Now, the informed trader maximizes (39) where (38) and subject to the Σ t+dt = E[(v t+dt q t+dt ) ] = E[(v t + dv t q t dq t (λ t µ t ρ t )dy t µ t dz t ) ] = E[(v t + dv t q t (λ t µ t ρ t )(β t (v t q t )dt + γ t dv t + du t ) µ t (dv t + de t )) ] = Σ t + (1 (λ t µ t ρ t )γ t µ t ) σvdt + (λ t µ t ρ t ) σudt + µ t σedt (λ t µ t ρ t )β t Σ t dt, therefore Σ t = (λ t µ t ρ t )β t Σ t + (1 (λ t µ t ρ t )γ t µ t ) σ v + (λ t µ t ρ t ) σ u + µ t σ e. (41) Proceeding as above we obtain Σ 1 =, λ t is a constant, and γ t = µ t λ t + µ t ρ t. Equations (7) and (8) imply that λ t is the regression coefficient of dy t on v 1 condi- 18

19 tional on Ω t : λ t = Cov(v 1, dy t Ω t ) V ar(dy t Ω t ) = Cov(v 1, β t (v t p t )dt + γ t dv t + du t Ω t ) V ar(β t (v t p t )dt + γ t dv t + du t Ω t ) = β tσ t + γ t σv. γt σv + σu ρ t is the regression coefficient of dz t on dy t conditional on Ω t : ρ t = Cov(dz t, dy t Ω t ) V ar(dy t Ω t ) = Cov(dv t + de t, β t (v t p t )dt + γ t dv t + du t Ω t ) V ar(β t (v t p t )dt + γ t dv t + du t Ω t ) = γ t σv. γt σv + σu Equations (6) and (1) imply that µ t is the regression coefficient of v 1 on dz t ρ t dy t conditional on Ω t : µ t = Cov(v 1, dz t ρ t dy t Ω t ) V ar(dz t ρ t dy t Ω t ) = Cov(v 1, dv t + de t ρ t β t (v t p t )dt ρ t γ t dv t ρ t du t Ω t ) V ar(dv t + de t ρ t β t (v t p t )dt ρ t γ t dv t ρ t du t Ω t ) = ρ tβ t Σ t + (1 ρ t γ t )σv (1 ρ t γ t ) σv + ρ t σu + σe = γ t σvβ t Σ t + σuσ v + σe(γ t σv + σu). σ 4 u σ v +γ t σ4 v σ u γ t σ v +σ u 19

20 B Old notes Start with some notation in the fast model: ν = λ ρµ. B.1 Proof of Proposition 7 Preliminary calculations We first compute Cov(v t p t, v t+τ p t+τ ) = Cov(v t p t, v t p t ) = Σ t τ τ Cov(v t p t, dp t+s ) ds Cov(v t p t, νβ t+s (v t+s p t+s )) ds Differentiate with respect to τ: which rewrites as τ Cov(v t p t, v t+τ p t+τ ) = νβ t+τ Cov(v t p t, v t+τ p t+τ ) τ log Cov(v 1 t p t, v t+τ p t+τ ) = νβ t+τ = νβ 1 t τ = νβ τ log(1 t τ) We integrate between and t: log Cov(v t p t, v t+τ p t+τ ) log Cov(v t p t, v t p t ) = νβ log( 1 t τ 1 t Using Cov(v t p t, v t p t ) = Σ t we get We also compute Cov(v t p t, v t+τ p t+τ ) = Σ t ( 1 t τ 1 t Cov(dv t, v t+τ p t+τ ) = Cov(dv t, dv t dp t ) = (1 νγ µ)σ vdt τ τ ) νβ Cov(dv t, dp t+s ) ds νβ t+s Cov(dv t, v t+s p t+s ) ds )

21 Proceeding as above we obtain ( ) νβ 1 t τ dt Cov(dv t, v t+τ p t+τ ) = (1 νγ µ)σ v 1 t Covariance informed trades with following price changes instant price change (i.e., quote update): Covariance with next Cov(dx t, dp t +) = Cov(µdv t, (ν λ)γdv t + µdv t ) = γ((ν λ)γ + µ)σ vdt Covariance with following price changes: Cov(dx t, dp t+τ = νβ t+τ (β t Cov(v t p t, v t+τ p t+τ )dt + γcov(dv t, v t+τ p t+τ )) dτ ( ) ( ) νβ = νβ t+τ βt Σ t + γ(1 νγ µ)σv 1 t τ dtdτ 1 t ( ) ( ) νβ 1 = νβ t βt Σ t + γ(1 νγ µ)σv 1 t τ dtdτ 1 t Therefore: ( ) τ ( ) νβ 1 1 t s Cov(ẋ t, p t+τ p t+ ) = γ((ν λ)γ + µ)σv + νβ t βt Σ t + γ(1 νγ µ)σv ds 1 t = γ((ν λ)γ + µ)σv + ( β t Σ t + γ(1 νγ µ)σv) [ ( ) ] νβ 1 t τ 1 1 t B. Informed trader strategy and inventories Autocorrelation of informed trades in the slow market Corr(dx t, dx t+τ ) = while in the fast market it is equal to zero. ( 1 τ ) λβ 1/ 1 t This is because the covariance of informed trades is of the order dt in both markets while the variance is of the order dt in slow market and dt in fast market. The variance of informed trader inventories at t = 1 in slow market V ar(x 1 ) = β λ Σ (4) 1

22 and in the fast market V ar(x 1 ) = β ν Σ + γ(1 νγ/ µ) σv (43) ν Numerically it seems that V ar(x 1 ) is always larger in the fast model, but not by much (intuition = much larger volume but no auto-correlation of order flow). Figure : Informed trader inventory Green: X t in slow market Blue: X t in fast market (Σ = σ v = σ u = σ e = 1)

23 .8 Figure 3: Price and fundamental value Red: v t Green: p t in slow market Blue: p t in fast market B.3 Comparison benchmark / fast model B.3.1 Proof of equation (43) Variance of inventories X T = T dx t: V ar(x T ) = = T t= T T V ar(dx t ) + t= γ σ v dt t= T T t T t τ= Cov(dx t, dx t+τ ) + (β t β t+τ Cov(v t p t, v t+τ p t+τ ) dt + γβ t+τ Cov(dv t, v t+τ p t+τ )) dτ dt ( ) T t = γ σvt + βt Σ t + γ(1 νγ µ)σv (1 t) νβ β t+τ (1 t τ) νβ dτ dt t= T τ= ( ) = γ σvt + βt Σ t + γ(1 νγ µ)σv (1 t) νβ 1 t= ν ((1 t)νβ (1 T ) νβ ) dt T = γ σvt + ( β Σ + γ(1 νγ µ)σ ν v) ( T 1 T ) νβ 1 (1 (1 T )νβ 1 ) We have V ar(x 1 ) = β ν Σ + γ(1 νγ/ µ) σv ν 3

24 B.4 Discrete time fast model Here instead of differential notation, d dstuff, we should use stuff notation. At each date t = 1,..., T : (i) the informed trader observes dv t and the MM observes dz t 1 = dv t 1 + de t 1, de t 1 N (, σe) (except at date 1 for the MM) (ii) the MM quotes bid = ask = Q t (iii) the informed trader submits dx t and liquidity traders du t N (, σu) (iv) the MM observes the order flow dy t = dx t + du t and sets the price P t We look for a linear equilibrium: dx t = β t (v t Q t ) + γ t dv t P t = Q t + λ 1,t dy t Q t+1 = Q t + λ,t dy t + l t dz t MM zero profit conditions (t = 1,..., T ) P t = E[v t Ω t ] Ω t {dy 1, dz 1,...dy t 1, dz t 1, dy t } Q t+1 = E[v t Ω z t ] Ω z t {dy 1, dz 1,...dy t 1, dz t 1, dy t, dz t } We also denote Σ t = V [v t Ω t ] Σ z t = V [v t Ω z t ] Boundary conditions: Q 1 =, Σ z = 4

25 Lemma 1. λ 1,t = λ,t = β t Σ z t 1 + (β t + γ t )σ v β t Σ z t 1 + (β t + γ t ) σ v + σ u β t Σ z t 1(σv + σe) + (β t + γ t )σvσ e ( ) β t Σ z t 1 + (β t + γ t ) σv + σu σ e + ( ) βt Σ z t 1 + σu σ v l t = σvσ u β t γ t Σ z t 1σv ( ) β t Σ z t 1 + (β t + γ t ) σv + σu σ e + ( ) βt Σ z t 1 + σu σ v Σ t = Σ z t 1 + σ v β t Σ z t 1 + β t (β t + γ t )Σ z t 1σ v + (β t + γ t ) σ 4 v β t Σ z t 1 + (β t + γ t ) σ v + σ u Σ z t = Σ z t 1 + σ v β t (Σ z t 1) (σv + σe) + βt Σ z t 1σv 4 + σvσ 4 u + (β t + γ t ) σvσ 4 e + β t (β t + γ t )Σ z t 1σvσ e ( ) β t Σ z t 1 + (β t + γ t ) σv + σu σ e + ( ) βt Σ z t 1 + σu σ v The value function of the informed trader at time t-(iii), t = 1,..., T, is π t = α 1,t 1 (v t Q t ) + α,t 1 dv t + α 3,t 1 (v t Q t )dv t + δ t 1 = max dx E [ dx(v t P t ) + α 1,t (v t+1 Q t+1 ) + α,t dv t+1 + α 3,t (v t+1 Q t+1 )dv t+1 + δ t v t, dv t ] and α 1,T = α,t = α 3,T = δ T = Lemma. β t = 1 α 1,t λ,t (λ 1,t α 1,t λ,t) γ t = α 1,t λ,t l t λ 1,t α 1,t λ,t α 1,t 1 = β t (1 λ 1,t β t ) + α 1,t (1 λ,t β t ) α,t 1 = λ 1,t γ t + α 1,t (λ,t γ t + l t ) α 3,t 1 = β t λ 1,t γ t + γ t (1 λ 1,t β t ) α 1,t (1 λ,t β t )(λ,t γ t + l t ) δ t 1 = α 1,t [ σ v + λ,tσ u + l t σ e] + α,t σ v + α 3,t σ v + δ t Substituting for β t and γ t in the expression of α 1,t 1 we obtain: α 1,t 1 = 1 4α t(λ,t λ 1,t ) 4(λ 1,t α 1,t λ,t) In the limit dt, we obtain α 1,t λ,t 1/, γ t l t /(λ 1,t λ,t ) 5

26 Proof of Lemma 1 Let s start with P t : Prior: v t 1 Q t N dv t Signal:, Σz t 1 σv dy t = β t (v t 1 Q t ) + (β t + γ t )dv t + du t Posterior: v t 1 Q t dv t dy t N µ 1 µ, σ 1 ρσ 1 σ ρσ 1 σ σ where µ 1 µ = Cov t 1 Q t dv t = 1 β t Σ z t 1 + (β t + γ t ) σ v + σ u, dy t V ar(dy t ) 1 dy t β tσ z t 1 (β t + γ t )σ v dy t and σ 1 ρσ 1 σ ρσ 1 σ σ, dy t = V ar v t 1 Q t Cov v t 1 Q t V ar(dy t ) 1 Cov v t 1 Q t dv t dv t dv t = Σz t 1 1 βt Σ z t 1 β t (β t + γ t )Σ z t 1σv σv βt Σ z t 1 + (β t + γ t ) σv + σu β t (β t + γ t )Σ z t 1σv (β t + γ t ) σv 4, dy t Therefore E[v t Ω t ] = β t Σ z t 1 + (β t + γ t )σv dy βt Σ z t 1 + (β t + γ t ) σv + σu t and V [v t Ω t ] = Σ z t 1 + σ v β t Σ z t 1 + β t (β t + γ t )Σ z t 1σ v + (β t + γ t ) σ 4 v β t Σ z t 1 + (β t + γ t ) σ v + σ u Now let s do Q t+1 : 6

27 Prior: v t 1 Q t N, Σz t 1 dv t σ v Signal: dy t = β t (v t 1 Q t ) + (β t + γ t )dv t + du t dz t = dv t + de t Posterior: v t 1 Q t dv t dy t, dz t N µ 1 µ, σ 1 ρσ 1 σ ρσ 1 σ σ = where µ 1 = Cov v t 1 Q t, dy t V ar dy t dy t µ dv t dz t dz t dz t 1 ( ) β t Σ z t 1 + (β t + γ t ) σv + σu σ e + ( ) β tσ z t 1(σv + σe)dy t β t (β t + γ t )Σ z t 1σvdz t βt Σ z t 1 + σu σ v (β t + γ t )σvσ edy t + (βt Σ z t 1σv + σvσ u)dz t 1 and σ 1 ρσ 1 σ = V ar v t 1 Q t ρσ 1 σ σ dv t 1 Cov v t 1 Q t, dy t V ar dy t Cov v t 1 Q t, dy t dv t dz t dz t dv t dz t Therefore E[v t Ω z t ] = ( ) βt Σ z t 1(σv + σe) + (β t + γ t )σvσ e dyt + ( ) σvσ u β t γ t Σ z t 1σv dzt ( ) β t Σ z t 1 + (β t + γ t ) σv + σu σ e + ( ) βt Σ z t 1 + σu σ v and V [v t Ω z t ] = Σ z t 1 + σv β t (Σ z t 1) (σv + σe) + βt Σ z t 1σv 4 + σvσ 4 u + (β t + γ t ) σvσ 4 e + β t (β t + γ t )Σ z t 1σvσ e ( ) β t Σ z t 1 + (β t + γ t ) σv + σu σ e + ( ) βt Σ z t 1 + σu σ v 7

28 Proof of Lemma π t = max dx = max dx E [dx(v t Q t λ 1,t dx λ 1,t du t ) + α 1,t (v t Q t + dv t+1 λ,t dx λ,t du t l t dv t l t de t ) + α,t dv t+1 + α 3,t (v t Q t + dv t+1 λ,t dx λ,t du t l t dv t l t de t )dv t+1 + δ t ] dx(v t Q t λ 1,t dx) + α 1,t ( (vt Q t λ,t dx l t dv t ) + σ v + λ,tσ u + l t σ e + α,t σ v + α 3,t σ v + δ t ) FOC: dx = 1 α 1,tλ,t (λ 1,t α 1,t λ,t) (v t Q t ) + α 1,tλ,t l t dv λ 1,t α 1,t λ t,t Therefore π t = [β t (v t Q t ) + γ t dv t ] [(1 λ 1,t β t )(v t Q t ) λ 1,t γ t dv t ] + α 1,t [(1 λ,t β t )(v t Q t ) (λ,t γ t + l t )dv t ] +α 1,t [ σ v + λ,tσ u + l t σ e] + α,t σ v + α 3,t σ v + δ t = [ β t (1 λ 1,t β t ) + α 1,t (1 λ,t β t ) ] (v t Q t ) References + [ λ 1,t γ t + α 1,t (λ,t γ t + l t ) ] dv t + [ β t λ 1,t γ t + γ t (1 λ 1,t β t ) α 1,t (1 λ,t β t )(λ,t γ t + l t )] (v t Q t )dv t +α 1,t [ σ v + λ,tσ u + l t σ e] + α,t σ v + α 3,t σ v + δ t [1] Back, Kerry, Henry Cao, and Gregory Willard (): Imperfect Competition among Informed Traders, Journal of Finance, 55, [] Back, Kerry, and Hal Pedersen (1998): Long-Lived Information and Intraday Patterns, Journal of Financial Markets, 1, [3] Biais, Bruno, Thierry Foucault, and Sophie Moinas (11): Equilibrium High Frequency Trading, Working Paper. [4] Brogaard, Jonathan (1): High Frequency Trading and Its Impact on Market Quality, Working Paper. 8

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