News Trading and Speed

Transcription

1 Preliminary and Incomplete Please do not quote! News Trading and Speed Thierry Foucault Johan Hombert Ioanid Roşu April 1, 1 Abstract Adverse selection occurs in financial markets because certain investors have either (a) more precise information, or (b) superior speed in accessing or exploiting information. To disentangle the effects of precision and speed on market performance, we compare two models in which a dealer and a more precisely informed trader continuously receive news about the value of an asset. In the first model the trader and the dealer are equally fast, while in the second model the trader receives the news one instant before the dealer. Compared with the first model, in the second model: (1) the optimal strategy of the informed trader exhibits a large and volatile component, called news trading; () the fraction of trading volume due to the informed investor increases from near zero to a large value; (3) liquidity decreases; (4) short-term price changes are more correlated with asset value changes; (5) the fraction of price volatility due to trading (quote updates) increases (decreases); nevertheless, (6) price volatility and (7) price informativeness stay the same. Our results suggest that the speed component of adverse selection is necessary to explain certain empirical regularities from the world of high frequency trading. Keywords: Insider trading, Kyle model, noise trading, trading volume, algorithmic trading, informed volatility, price impact. We thank Pete Kyle, Stefano Lovo, and Dimitri Vayanos for valuable comments. HEC Paris, foucault@hec.fr. HEC Paris, hombert@hec.fr. HEC Paris, rosu@hec.fr. 1

2 1 Introduction The recent advent of high frequency trading (HFT) in financial markets has raised numerous questions about the role of high frequency traders and their strategies. 1 Because of the proprietary nature of HFT and its extraordinary speed, it is difficult to characterize HFT strategies in general. Nevertheless, there is increasing evidence that at least some high frequency traders exploit very quick access to public information in an attempt to trade before everyone else. For example, in their online advertisement for real-time data processing tools, Dow Jones states: Timing is everything and to make lucrative, well-timed trades, institutional and electronic traders need accurate real-time news available, including company financials, earnings, economic indicators, taxation and regulation shifts. Dow Jones is the leader in providing high-frequency trading professionals with elementized news and ultra low-latency news feeds for algorithmic trading. 3 Clearly, trading more quickly than other market participants leads to adverse selection: at least for a short while, fast traders are better informed and trade quickly to exploit their speed advantage. 4 In general, adverse selection occurs because some investors have either (a) more precise information, or (b) superior speed in accessing or exploiting information. Traditionally, the market microstructure literature, e.g., Kyle (1985), has mainly focused on the first type of adverse selection. In contrast, the speed component of adverse selection has received little attention. Our paper focuses on this second type of adverse selection. To separate the role of precision and speed, we consider two models of trading under asymmetric information. In both models, a risk-neutral informed trader and a competitive dealer (or market maker) continuously learn about the value of an asset. 5 In both 1 In many markets around the world, high frequency trading currently accounts for a majority of trading volume. Hendershott, Jones, and Menkveld (11) report that in 9 as much as 73% of trading volume in the United States was due to high frequency trading. A similar result is obtained by Brogaard (11) for NASDAQ, and Chaboud, Chiquoine, Hjalmarsson, and Vega (9) for various Foreign Exchange markets. High frequency trading has been questioned espectially after the U.S. Flash Crash on May 6, 1. See, e.g., Kirilenko, Kyle, Samadi, and Tuzun (11). SEC (1) attributes the following characteristics to HFT: (1) the use of extraordinarily highspeed and sophisticated computer programs for generating, routing, and executing orders; () use of co-location services and individual data feeds offered by exchanges and others to minimize network and other types of latencies. 3 See 4 Hendershott and Riordan (11) find that on NASDAQ the marketable orders of high frequency traders have a significant information advantage and are correlated with future price changes. 5 The setup is similar to that of Back and Pedersen (1998), except that in our model the dealer also

3 models, the informed trader receives a more precise stream of news than that received by the dealer. The only difference lies in the timing of access to the stream of news. In the first model, the benchmark model, the informed trader and the dealer are equally fast. In the second model, the fast model, the informed trader receives the news one instant before the dealer. We show that even an infinitesimal speed advantage leads to large differences in the predictions of the two models. We further argue that the fast model is better suited to describe the world of high frequency trading. For example, consider the recent increase in trading volume observed in various exchanges around that world, which in part has been attributed to the rise of HFT. At high frequencies, traditional models such as Kyle (1985), or extensions such as our benchmark model have difficulty in generating a large trading volume of investors with superior information. To see, this, consider Figure 1. As it is apparent from the plot, the fast model can account for a significant participation rate of informed trading at higher frequencies, while the informed trader in the benchmark model is essentially invisible at high frequencies. 6 Thus, accounting for adverse selection due to speed is important if we want to explain the large observed trading volume due to HFT. Figure 1: Informed trading volume at various frequencies. The figure plots the fraction of trading volume due to the informed trader at various frequencies (second, minute, hour, day, month) in (a) the benchmark model, marked with ; and (b) the fast model, marked with. The parameters used are σ u = σ v = σ e = Σ = 1 (see Theorem 1). The liquidation date t = 1 corresponds to 1 calendar years Second Minute Hour Day Month receives news about the asset value. 6 In our benchmark model, as in Kyle (1985), there is a single informed trader. We have checked that the pattern shown in the figure can be obtained in models with multiple informed traders, such as Back, Cao, and Willard (). 3

4 Why would a small speed advantage for the informed trader translate into such a large different in outcomes? For this, we need to understand the difference in optimal strategies of the informed trader in the two models. In principle, when the asset value changes over time, there are two components of the optimal strategy: (1) Level Trading (or the low-frequency, drift, or deterministic component). This is a multiple of the difference between the asset value and the price, and changes slowly over time. Also, it is proportional to the time interval between two trades, thus it is small relative to the other component. () News Trading (or the high-frequency, volatility, or stochastic component). This is a multiple of the new signal, i.e., the innovation in asset value, and changes every instant. This component is relatively much larger than the level trading component. With no asymmetry in speed, the informed trader in the benchmark model does not have any incentive to trade on the news innovation, and trades only on the level of the asset value: the price impact of news trading would otherwise be too high. By contrast, in the fast model the informed trader also engages in news trading, in anticipation of a price move in the next instant due to the incorporation of news by the public. We call this the front running effect, except that it is legal since it is based on publicly available information. In addition, there is a substitution effect, due to the fact that in aggregate the price informativeness must be the same in the two models. 7 Thus, e.g., in the fast model, the informed trader is less aggressive in level trading, to compensate for the large news trading. Once we understand the optimal strategy of the informed trader, all the comparisons between the benchmark model and the fast model are driven by the front running effect and the substitution effect. To begin with, trading volume is higher in the fast model: in addition to the noise trading which is assumed the same in the two models, there is the large news trading component from the informed trader (the level component is too small to matter at high frequencies). As observed in Figure 1, the fraction of trading 7 As in the Kyle (1985) model, in both the benchmark and the fast model, the informed trader releases information at a constant rate, and such that at the end there is no money left on the table. 4

5 volume due to the informed trader is much larger at high frequencies, due to the large news trading component. Liquidity is smaller in the fast model: besides the usual adverse selection coming from the superior precision of the informed trader, front running generates additional adverse selection. This effect is mitigated by the substitution effect: the informed trader is less aggressive in level trading, and thus generates a smaller adverse selection due to signal precision. As mentioned above, price informativeness the same in the two models. Nevertheless, at high frequencies there is a sense in which the fast model has more informative prices: because the news trading component, trades are more correlated with current innovations in asset value, and therefore price changes are also more correlated with innovations in asset value. Total price informativeness is the same because of the substitution effect: in the fast model, the intensity of level trading is smaller. Price volatility is the same in the two models, due to the substitution effect. Note that price volatility arises from both trading and quote updates, since the dealer also learns about the asset value and changes quotes. In the fast model, the contribution of trades to price volatility is larger, because of the volatile news trading component of informed trading. In the benchmark model, the informed order flow is auto-correlated: there is only the level trading component, which changes direction only very slowly over time. In the fast model, the informed order flow has zero auto-correlation: at high frequencies, news trading dominates level trading, and the innovations in asset value are uncorrelated. Our results suggest that the fast model is better suited than the benchmark model to describe the strategies of high frequency traders: Brogaard (11) observes that their order flow is indeed volatile, and there is little evidence of auto-correlation. One consequence of our model is that, when trading takes place at high frequencies, one must interpret with caution the results of vector autoregressions such as those of Hendershott and Riordan (9). These regressions are based on a positive correlation between the informed order flow and short term price changes, which in our fast model are essentially zero at high frequencies, because of the large noise coming from news trading. To overcome these problems, we define the cumulative price impact as the 5

6 covariance between the net informed trade per unit of time and the subsequent price change at various time horizons. We show that the cumulative price impact is well defined even in the presence of high frequency noise, and can distinguish empirically between the fast model and the benchmark model. Finally, we can interpret some current regulatory proposals as attempts to eliminate the speed advantage of high frequency traders. In our framework, this translates into going from the fast model equilibrium into the benchmark model equilibrium. Since noise traders are not modeled here, we cannot discuss welfare in our framework, but we can study the potential effect of regulation on various measures on market performance. From the discussion above, we see that eliminating the speed advantage of the informed trader does not change price informativeness or price volatility, but: (i) reduces trading volume; (ii) reduces overall adverse selection, and thus increases market liquidity. 1.1 Literature Review To Be Developed For some recent theoretical articles on speed, see Biais, Foucault, and Moinas (11), or Pagnotta and Philippon (11). Model Trading occurs over the time interval [, 1]. The risk-free rate is taken to be zero. During [, 1], a single informed trader ( he ) and uninformed noise traders submit market orders to a competitive market maker ( she ), who sets the price at which the trading takes place. There is a risky asset with liquidation value v 1 at time 1. The informed trader learns about v 1 over time, and the expectation of v 1 conditional on his information available until time t follows a Gaussian process given by v t = v + t dv τ, with dv t = σ v db v t, (1) where v is normally distributed with mean and variance Σ, and Bt v is a Brownian motion. We refer to v t as the asset value or the fundamental value, and to dv t as the innovation in asset value, or the news. Thus, the informed trader observes v at time and, at each time t + dt [, 1] observes dv t. 6

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

8 trading takes place at t is given by p t = E [v 1 Ω y t ]. (7) We consider two different models: the benchmark model and the fast model, which differ according to the timing of information arrival and trading. The sequence of events in each model is summarized in Table 1. Table 1: Timing in the benchmark model and in the fast model Benchmark Model Fast Model 1. Informed trader observes dv t 1. Informed trader observes dv t. Market maker observes dz t = dv t + de t. Trading at t + dt 3. Trading at t + dt 3. Market maker observes dz t = dv t + de t In the benchmark model, the order of events during the time interval [t, t + dt] is as follows. The informed trader observes dv t and the market maker receives the signal dz t. The market maker sets the quotes based on the information set Ω t = {z τ } τ t {y τ } τ<t. Then, the informed trader submits a market order dx t and noise traders also submit their order du t. The information set of the market maker when she sets the execution price is Ω t = {z τ } τ t {y τ } τ t. In the fast model, the informed trader receives information before the market maker. First, the informed trader observes dv t and submits the market order dx t along with the noise traders orders du t. The market maker posts quotes based on Ω t = {z τ } τ<t {y τ } τ<t and the execution price is conditional on the information set Ω t = {z τ } τ<t {y τ } τ t. After trading has taken place, the market maker receives the signal dz t and updates the quotes. 3 Equilibrium The equilibrium concept is similar to Kyle (1985) or Back and Pedersen (1998). We are looking for equilibria in which the pricing rule is linear, i.e., the order flow at time t is 8

9 executed at price p t+ dt = q t+ dt + λ t dy t. (8) The bid-ask spread is zero and q t+ dt is the mid-quote (equal to the bid and the ask). The price impact coefficient (the Kyle lambda ) is a measure of the liquidity of the market. A large λ t implies that market orders move the price by a lot, i.e., the market is illiquid. In the benchmark model, the market maker receives her signal and updates the quotes before trading takes place. Therefore, we look for a quote updating rule of the form q t+ dt = p t + µ t dz t. (9) In the fast model, the market maker receives her signal and therefore updates the quotes only after trading has taken place. Thus, the updated quotes correspond to the next trading time, t + dt. We look for a quote updating rule of the form 9 q t+ dt = p t+dt + µ t ( dz t ρ t dy t ). (1) In Equation (1) the market maker updates the quotes based on her signal as well as on the order flow. The reason is that the order flow contains information about dv t and thus about dz t. At equilibrium, the term ρ t dy t will be equal to the expected value of dz t conditional on Ω y t, the market maker s information after trading and before she receives signal dz t, and the market maker will update her beliefs based on the unexpected part of dz t. We assume that the strategy of the informed trader is of the form dx t = β t (v t p t ) dt + γ t dv t. (11) We solve for the optimal β t and γ t in this class of strategies. In the Appendix, we show that in the discrete time version of both models, in a linear pricing equilibrium, the (unconstrained) optimal strategy is linear and of the same form as (11). 9 Note that, in the benchmark model p t is a classic Itô process but q t is not, while in the fast model q t is a classic Itô process and p t is not. In the proofs of our results, we only make use of the process p t in the benchmark model and of q t in the fast model. 9

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

11 In both models, when a, b, c +, the equilibrium converges to the unique linear equilibrium in the continuous time version of Kyle (1985). To give some intuition for this result, note that the strategy of the informed trader in the benchmark model and the in fast model are, respectively, dx B t = β B t (v t p t ) dt, dx F t = β F t (v t p t ) dt + γ F t dv t. Both strategies have a level trading component. In both models, the informed trader receives more precise signals than the market maker. Indeed, the informed knows perfectly v t while the market maker s best forecast is p t. Therefore, it is optimal for the informed trader to trade on the forecast error of the market maker v t p t. This forecast error is slowly moving, because its change is of the order of ( dv t dp t ) dt, which at high frequencies is negligible. The informed trader trades smoothly on the forecast error, in the sense that the level trading component is of the order dt. The key difference between the two models is that only in the fast model the informed trader s optimal strategy has a news trading component. The reason is that in the benchmark model, when the trader submits the order dx t, all the signals { dv τ } τ t that he has received are given the same weight in the optimal strategy. By contrast, in the fast model the marker maker has not incorporated the signal dz t = dv t + de t in the price yet. Therefore, it is optimal for the informed trader to trade aggressively on dv t before the market maker receives information dz t. The news trading component is very volatile since it is an innovation. It also generates a much larger order flow than the level trading component, since it is of order dt 1/. The following comparative result can be proved using directly the formulas from Theorem 1. Proposition 1. In the context of Theorem 1, for all values of the parameters we have 11

12 the following inequalities: β F < β B (3) λ F > λ B (4) µ F < µ B. (5) In the benchmark equilibrium, β increases in σ v, σ u, σ e, and decreases in Σ ; λ increases in σ v, σ e, Σ, and decreases in σ u ; µ increases in σ v, decreases in σ e, and is constant in σ u, Σ. In the fast equilibrium, β increases in σ v, σ u, σ e, and decreases in Σ ; γ increases in σ v, σ u, and decreases in σ e, Σ ; λ increases in σ v, σ e, Σ, and decreases in σ u ; µ increases in σ v, decreases in σ e, Σ, and is constant in σ u ; ρ increases in σ v, and decreases in σ u, σ e, Σ. 4 Empirical implications 4.1 Informed trading volume We define the fraction of trading volume coming from the informed trader as ITR t = Var( dx t ) Var( du t ) + Var( dx t ). (6) Proposition. In the benchmark model ITR t = at all times. In the fast model ITR t = γ σ v σ u+γ σ v (, 1). In the benchmark model, the informed trader s optimal strategy has only a level trading component. The level trading component consists in a drift in the asset holding x t. This generates a trading volume that is an order of magnitude smaller than the trading volume generated by the noise traders. Formally, informed trading volume is of the order dt while noise trading volume is of the order dt 1/. By contrast, in the fast 1

13 model, the informed trader s optimal strategy includes a news trading component. The news trading component is volatile (i.e., stochastic), which generates a trading volume that is of the same order of magnitude as the noise trading volume. Informed trading volume is exactly zero in the benchmark model because we work in continuous time. In the discrete time version of the model, informed trading volume is non zero but it converges quickly to zero as the trading frequency increases. In Figure 1 we show that in the benchmark model the trading volume is already very small when trading takes place at the weekly frequency. By contrast, in the fast model, informed trading volume converges to a limit between zero and one when the trading frequency becomes very large. 4. Liquidity We already proved the following result in Proposition 1: Corollary 1. Liquidity is lower in the fast model than in the benchmark model, i.e., λ F > λ B. The market is less liquid in the fast model since there is more adverse selection than in the benchmark model. Indeed, the informed trader has more precise information in both models, and, in the fast model only, the informed is also faster. This generates a second source of adverse selection, coming from news trading. Note that there is a mitigating effect: adverse selection coming from level trading is smaller in the fast model. This is implied by Proposition 1, which shows that βt F < βt B. To understand the intuition behind this result, we need to discuss price informativeness, which shows that there is a substitution effect between level trading (β) and news trading (γ). 4.3 Price informativeness We define the instantaneous covariance between changes in the price and in the fundamental value of the asset as ICov( dp t, dv t ) = lim Cov(p t+ δt δt 13 p t δt, v t+ δt v t δt ). (7)

14 Proposition 3. ICov(dp t, dv t ) is larger in the fast model than in the benchmark model. The increased covariance in the fast market comes from the news trading component, which is perfectly correlated with the change in asset value. Formally, the instantaneous covariance in the benchmark model only comes from the correlation of value changes with quote updates: ICov B ( dp t, dv t ) = Cov(µ B t dz t, dv t ) = µ B σ vdt, (8) where price changes due to trading are negligible. By contrast, in the fast model, value changes are correlated with both quotes updates and trading: ICov F (dp t, dv t ) = Cov(λ F t γt F dv t +µ F t (dz t ρ F t γt F dv t ), dv t ) = ((λ F µ F ρ F )γ F +µ F )σ v dt. (9) Although µ B > µ F, i.e., the covariance of value changes with quote updates is higher in the benchmark, the covariance with total price changes is higher in the fast model. One might believe that prices changes more correlated with value changes should lead to better price informativeness. This is however not the case. Define price informativeness as Σ t = E [ (v t p t ) ]. (3) Proposition 4. Σ B t = Σ F t at all times. While new information (about the innovation dv t ) is incorporated more quickly into the price in the fast model thanks to the news trading component, older information (about dv τ, τ < t) is incorporated less quickly because level trading in less intense in the fast model, i.e., βt F < βt B. There is a substitution between news trading and level trading, which leaves price informativeness unchanged when one moves from the benchmark to the fast model. The intuition for the substitution effect is that the informed trader competes with himself when using his information advantage. Trading more on news now eats the profit from trading on the levels in the future. Therefore, when news trading increases in the fast model, level trading has to decrease. 14

15 4.4 Price variance decomposition Following Hasbrouck (1991a, 1991b) we define trade informativeness as the fraction of price volatility due to trading: TI t = V ar(λ t dy t ) V ar( dp t ). (31) Proposition 5. Trade informativeness is higher in the fast model than in the benchmark model. More information that is incorporated through trading in the fast model. This is because the informed trader can act on news before the market marker revises the quotes. Therefore, trading is more intense and quote revision is less intense, which increases the Hasbrouck measure of trade informativeness. In terms of price informativeness, as stated in Proposition 4, the net effect of more informative trades and less informative quote revisions is exactly zero. In terms of price volatility, the net effect of more aggressive trades and less aggressive quote revisions is also zero. Proposition 6. Price volatility is the same in the fast model and in the benchmark model. The reason is that, in an efficient market, price changes are not autocorrelation. Therefore, the short-term price volatility is always equal to the long-term price volatility Σ + σ v. 4.5 Cumulative price impact We define the cumulative price impact of informed trade dx t as CPI t,t+τ ( ) dxt = Cov dt, p t+τ q t+ dt. (3) q t+ dt is the mid-quote when the order dx t is submitted to the market and p t+τ is the price after τ > periods of time. 15

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

29 [5] Chaboud, Alain, Benjamin Chiquoine, Erik Hjalmarsson, and Clara Vega (9): Rise of the Machines: Algorithmic Trading in the Foreign Exchange Market, Working Paper, Board of Governors of the Federal Reserve System. [6] Hasbrouck, Joel (1991a): Measuring the Information Content of Stock Trades, Journal of Finance, 46, [7] Hasbrouck, Joel (1991b): The Summary Informativeness of Stock Trades: An Econometric Analysis, Review of Financial Studies, 4, [8] Hendershott, Terrence, Charles Jones, and Albert Menkveld (11): Does Algorithmic Trading Improve Liquidity?, Journal of Finance, 66, [9] Hendershott, Terrence, and Ryan Riordan (1): Algorithmic Trading and Information, Working Paper. [1] Hendershott, Terrence, and Ryan Riordan (11): High Frequency Trading and Price Discovery, Working Paper. [11] Kirilenko, Andrei, Albert Kyle, Mehrdad Samadi, and Tugkan Tuzun (11): The Flash Crash: The Impact of High Frequency Trading on an Electronic Market, Working Paper. [1] Kyle, Albert (1985): Continuous Auctions and Insider Trading, Econometrica, 53, [13] Pagnotta, Emiliano, and Thomas Philippon (11): Competing on Speed, Working Paper. [14] SEC (1): Concept Release on Equity Market Structure, Release No ; File No. S