A simple microstructural explanation of the concavity of price impact
AA simple microstructural explanation of the concavity of priceimpact
Sergey Nadtochiy *†‡§
First draft: January 6, 2020Current version: December 11, 2020
Abstract
This article provides a simple explanation of the asymptotic concavity of the price impact of a meta-ordervia the microstructural properties of the market. This explanation is made more precise by a model in whichthe local relationship between the order flow and the fundamental price (i.e. the local price impact) is linear,with a constant slope, which makes the model dynamically consistent. Nevertheless, the expected impact onmidprice from a large sequence of co-directional trades is nonlinear and asymptotically concave. The mainpractical conclusion of the proposed explanation is that, throughout a meta-order, the volumes at the best bidand ask prices change (on average) in favor of the executor. This conclusion, in turn, relies on two moreconcrete predictions, one of which can be tested, at least for large-tick stocks, using publicly available marketdata.
Keywords: concave price impact, ergodic diffusion, market microstructure, meta-order, microprice, VWAPstrategy.
The term price impact often refers to the fact that, on average, trades move the asset price in their direction.However, this general observation has several more specific interpretations, and, as pointed out in [25], it isimportant to differentiate between various types of price impact. First, one may consider the local impact:i.e., the expected price change as a function of the volume of a single trade. The latter is only relevant in themarkets where large trades occur often (e.g., OTC markets), but is not relevant, e.g., for the most popular stockexchanges. The second type of impact is the meta-order impact: i.e., the expected price change as a functionof the volume of a sequence of co-directed trades (such sequence is called a meta-order). This type of impactis more relevant for public exchanges, where most participants willing to buy or sell a large quantity of theasset split this quantity into smaller pieces (child orders) and submit each of them separately. Within the meta-order impact, one can distinguish two sub-types, depending on whether the relative rate of the execution, or its * The author thanks M. Mouyebe for conducting a numerical experiment whose results inspired this article. He thanks S. Jaimungal forproviding the market data used herein. The author also thanks J.-P. Bouchaud, P. Ustinov, K. Webster, and the anonymous referees, for theuseful discussions and comments that helped him improve this paper significantly. † Partial support from the NSF CAREER grant 1855309 is acknowledged. ‡ The data that support the findings of this study are available from the corresponding author upon reasonable request. § Address the correspondence to: Department of Applied Mathematics, Illinois Institute of Technology, 10 W. 32nd St., Chicago, IL60616 ([email protected]). a r X i v : . [ q -f i n . T R ] D ec otal duration, is fixed. This paper studies the meta-order impact for a fixed execution rate (also known as theexpected price trajectory).There exist various empirical studies that confirm the concavity of meta-order impact (see, e.g., [1], [3],[2], [26], [21], [4], [5]). They show that the expected price change as a function of traded volume is concave(see the left part of Figure 2), and some even claim a specific power (in particular, square-root) dependence ofthe impact on the volume. These empirical results motivated the search for a sound theoretical explanation ofthe concavity of price impact. The following four paragraphs describe the main types of explanations availableto date.The first explanation is rather heuristic and has not been documented in the academic literature (to theauthor’s best knowledge). It explains the concavity of price impact by the predictability of future prices andorder flow. To understand this argument, notice that the concavity of price impact is equivalent to the statementthat the marginal impact of the individual child orders decreases throughout a meta-order. Then, assuming, forexample, that the price dynamics have a mean-reverting component, one can argue that the latter will slow downthe deviation of the price from its initial level, causing smaller marginal impact of every subsequent trade. Similarly, if the market participants can predict (with some accuracy) the size of a meta-order, then, toward theend of its execution, they become less certain that the execution will continue, hence they limit their predatoryactions agains the executor (i.e., moving their limit orders or front running), causing smaller marginal impacttowards the end of the meta-order. One shortcoming of this type of arguments is, of course, is their relianceon the violation of the efficient market hypothesis, which states that prices are not predictable. Although,in practice, the latter holds only to a certain extent, typically, the inefficiencies in the market disappear overtime, once they become known and are of a significant magnitude. Therefore, given the prolonged history ofthe empirical studies in this area, it is hard to imagine that the predictability of future prices and order flowswould still exist on a level that would make a significant contribution to the price impact. Another downside ofthis explanation is the underlying assumption that the meta-order can be detected by other market participantsinstantaneously. Indeed, in practice, it takes some time for the predatory traders to detect a meta-order, implyingthat the very first few trades should cause smaller marginal impact than some of the subsequent ones, whichwould contradict the concavity of the price impact if the order flow predictability was the main driving forcefor the shape of the impact.Another type of explanation is based on the game-theoretic models in which a market-maker provides liq-uidity for the executor. The concavity is explained, for example, via the shape of market-maker’s preferences([10]) or via the distribution of the sizes of meta-orders ([8]). While such explanation is admissible for OTCmarkets, it is not clear whether the conclusions of such models extend to the order-driven markets (e.g., ex-changes), where multiple heterogeneous liquidity providers with varying inventories (as they trade with otherliquidity consumers) compete for clients’ orders.Additional evidence of the concavity (even square-root property) of price impact is obtained via the dimen-sional analysis in [19], [22]. Indeed, if one assumes that the impact depends only on a certain group of factors,then, after few additional invariance assumptions, one can deduce that the square-root is the only function thatproduces the right unit of measurement for the impact. This is a rather powerful method, but, unfortunately, itsheds very little light on the mechanism that generates the price impact, which is an important part of explainingthe phenomenon.The latent limit order book model (LLOB), described, e.g. in [25], [7], and the model based on Hawkes It is important to note that [5] both provides an empirical analysis of the price impact and proposes its explanation. However, the notionof price impact used in [5] is fundamentally different from the one used here and in other studies. Namely, [5] defines the price impactas the expected absolute value of the price change, which is of course dominated by the price volatility and is very different (qualitativelyand quantitatively) from the expected value of the price change, which is more commonly used as the notion of price impact (and the latternotion is adopted herein). Of course, this is only true if the initial price is on the right side of the level around which it mean-reverts. This is one of the flaws ofthis argument. proposed herein has several advantages. First, as shownin the remainder of this section, the main results of this paper can be stated in plain language, without appealingto any sophisticated technical arguments. In this sense, the proposed explanation is model-free. Nevertheless, aspecific modeling setting is described in Section 2, where, in particular, it is shown that the arguments presentedfurther in this section to explain the concavity of price impact can co-exist in a reasonable mathematical model,and where the relevant predictions are proven rigorously (this is where the technical arguments are needed).The model of Section 2 allows for multiple market participants, consuming and providing liquidity, whichmakes it well suited for order-driven markets. The setting of the model is “bottom-up”: i.e., its inputs haveclear economic meaning and can be measured from market data. The proposed explanation does not requirethe predictability of the future prices or of the oder flow and, hence, is consistent with the efficient markethypothesis. Finally, a significant advantage of the proposed explanation is that one of its two most importantpredictions (which directly imply the concavity of price impact) can be tested on real market data without theinformation about meta-orders (as the latter is notoriously difficult to obtain).Most of the remainder of this section is devoted to the description of the proposed explanation of theconcavity of price impact, and of the underlying assumptions, in plain language. The main assumption of this It is worth mentioning that, strictly speaking, the present model only explains asymptotic concavity: i.e., that the marginal impact atthe beginning of a long meta-order is higher than at its end. With a slight abuse of terminology, we refer to this property as concavitythroughout the paper. However, the numerical experiments reveal that, for a wide range of parameters’ values, the model produces globallyconcave price impact curves. X , which we refer to as the fundamental price (this terminologycomes from [12], [11], [13]), and which has the meaning of a signal predicting the direction of the next trade.Namely, between the jump times of the best bid and ask prices, the process X lies in the interval [ a, b ] (whichmay change after the best bid or ask price changes), and the closer it is to a (resp. b ) the more likely it is thatthe next trade will be a sell (resp. buy). In addition, we assume that the best bid (resp. ask) price changeswhen and only when X hits a (resp. b ). Another property that is required from X is that every trade makes apositive linear local impact on X , with the coefficient α > . For example, a buy trade of size δ changes X to X + αδ . Any process X that satisfies these properties is referred to as the fundamental price.Although the proposed explanation works for any fundamental price X , it is convenient to think of aconcrete example. Namely, measuring all prices in ticks and making the assumption that the bid-ask spread ofthe asset is always (or, in practice, almost always) equal to one (i.e., assuming that we study a large-tick asset),we conclude that the microprice X , defined as X t = P b + V bt V bt + V at , (1)where P b denotes the best bid price and V bt , V at denote the limit order volumes at the best bid and ask prices,respectively, satisfies all the properties of the fundamental price listed above. Indeed, the microprice alwaysstays in the interval [ P b , P b + 1] , by the above definition, and it is known to possess the desired predictivepower for the direction of the next trade (see, e.g., [24]). In addition, the large-tick property of the asset impliesthat the best ask (resp. bid) price changes when and only when X hits P b + 1 (resp. P b ). Finally, it is naturalto expect that trades have a positive impact on the microprice: e.g., a part of this impact is purely mechanical,as any buy trade decreases V a and any sell trade decreases V b .For the sake of concreteness, all the subsequent analysis is presented under the assumption of a large-tickasset and with the microprice playing the role of the fundamental price X . Then, in particular, it is clear thatthe best bid and the best ask prices are given by the roundings (cid:98) X (cid:99) and (cid:100) X (cid:101) , respectively, and that they changewhenever X hits an integer. Assuming that, between the trades, X follows a symmetric random walk, weconclude that its global dynamics must have a force, or a drift term, that pushes X away from the midprice.Indeed, if X is slightly above the midprice, the next trade is more likely to be a buy, which pushes X furtherup and makes the next buy even more likely, and so on. This can be viewed as the self-excitatory behavior ofthe fundamental price within the spread. For simplicity, let us focus on the dynamics of X between two nearestintegers – i.e., we consider X mod ( X modulo one). As explained above, the process X mod is a randomwalk (on the unit circle) with a drift that pushes it away from / . The stationary distribution of such a process must have a U-shaped density (see the top left part of Figure 1). Thus, before the execution of a meta-orderbegins, the distribution of the fundamental price modulo one has a U-shaped density. Next, let us analyzewhat happens toward the end of a meta-order. It is natural to assume that a meta-order, which is a sequence ofco-directed trades, introduces an additional drift term in X mod , of the same sign as the meta-order itself. Ifthis additional drift is constant and sufficiently large, it is easy to deduce that the stationary distribution of theresulting process is uniform (think, e.g., of a Brownian motion with a very large drift, run on a circle). It is clearthat the wings (i.e., the values at and ) of a U-shaped density are higher than the wings of a uniform density(the latter are equal to one). Interpolating heuristically between zero additional drift and the infinitely large This is similar in spirit to the model with uncertainty zones of [23], although the latter model only describes the last transaction price,as opposed to the pair of the best bid and ask prices. The assumption of linearity of local impact is no loss of generality, as we ultimately consider a regime where every individual tradehas infinitesimal size. It is important to notice that this property fails for the assets whose bid-ask spreads vary significantly. This is why the microprice isnot a good proxy for the fundamental price for such assets (see the also the discussion in Section 3.2). Recall that the stationary distribution, in particular, is the distribution of the value of the process at any fixed time, provided the processhas been running sufficiently long. X mod before a sufficiently long meta-order arehigher than the wings of its density at the end of this meta-order, for any positive execution rate (compare thetop left and the bottom right parts of Figure 1). Now, the phenomenon of “improving liquidity” becomes clear.Recalling that X is the microprice (see (1)), one easily sees that the lower wings of the stationary density of X mod imply smaller probability of observing low liquidity (i.e., low volume of limit orders) at the best bidor ask. It only remains to connect the wings of the stationary density of X mod to the price impact directly.To this end, notice that the expected impact on the midprice of a buy trade of size δ , submitted at time t = 0 , isgiven by E ( (cid:100) X mod αδ (cid:101) − , where X mod is a random variable whose density is given by the stationary density of X mod . As δ ↓ ,the leading order of the above expectation is given by the probability that X mod αδ > . The latter, inturn, is proportional to the wings of the stationary density of X mod . Thus, the expected marginal impact onthe midprice is proportional to the wings of the stationary distribution of the fundamental price modulo one. Asmentioned above, a meta-order introduces an additional drift term in the dynamics of the fundamental price,thus, switching the market into a different regime. In this new regime, X mod attains a new stationary density(provided the meta-order lasts long enough), whose wings are lower than the wings of the original stationarydensity. Repeating the above argument that connects the wings of the stationary density and the marginalimpact, we conclude that the marginal impact at the end of the meta-order is lower than at the beginning, whichimplies the (asymptotic) concavity of price impact.The above explanation needs an additional clarification, which leads to another assumption. Namely, theconclusion that the wings of the stationary distribution decrease during the meta-order relies on the assumptionthat the fundamental price obtains a constant drift during a meta-order, which is equivalent to the assumptionthat the meta-order is executed at a constant rate. However, a typical executor aims to hide her activity andtrade according to (a version of) the Volume Weighted Average Price (VWAP) strategy. The latter states that theexecution rate of a meta-order must be a constant fraction of the rate of the total traded volume in the market(see, e.g., [17], [6]). Thus, the drift that the fundamental price obtains during a meta-order is constant onlyif measured on the business clock – i.e., using the total traded volume instead of the actual time. Therefore,all processes in the above discussion should be run on the business clock, and the conclusions of the previousparagraph require the additional assumption that the execution strategy is similar to VWAP.Note that the explanation of the concavity of price impact presented above relies only on two predictions.The first one is the U-shape of the (global) stationary distribution of the fundamental price modulo one (run ona business clock). The second prediction is that, during a meta-order, the fundamental price (run on a businessclock) obtains an additional constant drift term. While it is impossible to test the second prediction without themeta-order data (although this prediction appears to be self-evident), the first prediction is verified using realmarket data in Section 3.2.The rest of the paper is organized as follows. The precise modeling assumptions and mathematical state-ments are given in Section 2. Sections 2.1 and 2.2 describe, respectively, the finite- and infinite-activity modelsfor X , in particular, giving the precise form of its drift in terms of the input parameters of the model. The finite-activity model is easier to understand intuitively, while the infinite-activity model allows for more tractablerepresentation of the price impact. Sections 2.3 and 2.4 are devoted to the definition and the computation ofthe price impact of a VWAP meta-order. These sections are complicated by the additional effort made in thispaper to avoid treating the executor of a meta-order as an exogenous entity and to use a realistic definition ofprice impact, consistent with the way it would be computed in practice. In particular, it is shown in Section 2.3that the proposed setting allows for multiple agents following VWAP-type strategies, with overlapping execu-tion intervals and with varying trade directions. The price impact is defined as the conditional expectation ofthe price change over the interval on which the first agent’s trades sum up to a given volume (the size of the5eta-order), given that all trades of this agent in this interval have the prescribed direction. The latter is doneto mimic the computation of price impact as a sample average of the price change over the execution intervals.Section 2.5 shows that the marginal price impact is proportional to the wings of the stationary distribution ofthe fundamental price modulo one, run on a business clock. Propositions 4 and 3 describe this connection,respectively, in the presence of a meta-order and in its absence. Section 2.6 proves that the aforementionedwings are lower in the presence of a meta-order (Theorem 1), thus, establishing the asymptotic concavity ofprice impact. The numerical and empirical analysis is presented in Section 3. The roots of the proposed model go back to the game-theoretic setting of [12], [11], [13]. However, the specificmodel proposed herein is natural enough and does not require any additional justification via equilibrium argu-ments. The core of the model is the assumption that the potential buyers and sellers arrive to the market oneby one, having their own reservation prices (i.e., the “fair” prices for the asset, in their view). If a reservationprice of an agent is above (below) the current best ask (bid) price, a sell (buy) trade occurs. The reservationprices are not independent across the potential buyers and sellers: they have a common component X and theidiosyncratic additive part, generated from a (symmetric around zero) distribution with c.d.f. F . It is shownin [13] that, for reasonable values of the model parameters, the agents providing liquidity via limit orders setthe equilibrium bid and ask prices exactly at (cid:98) X (cid:99) and (cid:100) X (cid:101) , respectively, thus, making the model consistentwith the setting described in Section 1. The details of the model are presented in the remainder of this section,with the main result (the asymptotic concavity of price impact) stated in Theorem 1. It is worth mentioningthat a part of this section contains the description of two models: with finite and infinite trading activity. Theformer is easier to interpret from the economic (or practical) point of view. The latter provides more tractableexpressions for the target quantities. We show that the two are consistent and, ultimately, focus our attentionon the infinite-activity model. The potential buyers/sellers arrive according to a Poisson process with jump times { S i } and intensity λ . Thereservation price of the i th buyer/seller is p S i = ˜ X S i − + ξ i , where ˜ X S i − := lim t ↑ S i ˜ X t , { ξ i } are i.i.d. random variables, with c.d.f. F , and ˜ X evolves according to ˜ X t = X + α (cid:88) S i ≤ t ∆ V S i + σ ( ˜ X t ) ˜ B t mod , with ˜ B being a Brownian motion independent of { S i , ξ i } , and with ∆ V S i := δ { ξ i ≥(cid:100) ˜ X Si − (cid:101)− ˜ X Si } − δ { ξ i ≤(cid:98) X Si − (cid:99)− X Si } . Throughout the rest of the paper, we make the following standing assumptions on F and σ .• F ∈ C (cid:15) ([ − , , for some (cid:15) ∈ (0 , , and F (cid:48) is symmetric around x = 0 in this range.• inf x ∈ [0 , ( F ( x −
1) + F ( − x )) > .• σ has period one and is symmetric around x = 1 / .6 inf x ∈ [0 , σ ( x ) > .• σ ∈ C (cid:15) ([0 , .In the above, we use the standard notation C n + (cid:15) ([ a, b ]) to denote the space of real-valued functions on [ a, b ] that are n times continuously differentiable, with (cid:15) -H¨older derivatives of order n .Denote by M a Poisson random measure with the compensator µ ( dt, dx ) = λdt ⊗ ( P ◦ ξ − i )( dx ) = λdt ⊗ dF ( x ) , independent of ˜ B . Then, the fundamental price and the order flow are described by the following system: ˜ X t = X + α ( N + t − N − t ) + σ ( ˜ X t ) ˜ B t ,N + t = δ (cid:82) t (cid:82) R { x ≥ β + ( ˜ X u − ) } M ( du, dx ) ,N − t = δ (cid:82) t (cid:82) R { x ≤ β − ( ˜ X u − ) } M ( du, dx ) , (2)where β + ( x ) := (cid:100) x (cid:101) − x, β − ( x ) := (cid:98) x (cid:99) − x. Note that N + t = (cid:88) S i ≤ t max(∆ V S i , , N − t = (cid:88) S i ≤ t max( − ∆ V S i , . The input to the model is ( α, σ, F, λ, δ ) . For analytic tractability, it is convenient to consider an infinite-activity limit of the model (2), as λ → ∞ . Inorder to avoid the explosion of total order flow, we need to assume that δ → , so that λδ → γ , with someconstant γ > . For simplicity, we assume that δ = γ/λ . Notice that dN t := d ( N + t − N − t ) = λδ (cid:16) F ( − β + ( ˜ X t )) − F ( β − ( ˜ X t )) (cid:17) dt + dZ t , where Z is a martingale. Then, (cid:104) Z (cid:105) t = (cid:90) t λδ (cid:16) F ( − β + ( ˜ X s )) + F ( β − ( ˜ X s )) (cid:17) ds ∼ δγ (cid:90) t (cid:16) F ( − β + ( ˜ X s )) + F ( β − ( ˜ X s )) (cid:17) ds → . Thus, we expect ˜ X to converge to X which is the solution of dX t = αγ (cid:0) F ( − β + ( X t )) − F ( β − ( X t )) (cid:1) dt + σ ( X t ) dB t , (3)equipped with the same initial condition as ˜ X . We do not make this statement precise, as we will in fact needthe convergence of time-changed processes, established in Lemma 1.It is important to notice that the drift of X in (3) is increasing on ( n, n + 1) , negative in ( n, n + 1 / , andpositive in ( n + 1 / , n + 1) , for any integer n . Indeed, for x ∈ ( n, n + 1) , the expression F ( − β + ( x )) − F ( β − ( x )) = F ( x − n − − F ( n − x ) is increasing in x , as the c.d.f. F is an increasing function, and the above expression is equal to zero at x = n + 1 / . These observations imply that the drift of X pushes it away from the midprice, consistent withthe conclusions of Section 1. 7 .3 Expected impact on midprice by a VWAP strategy In this subsection, we define the expected price impact of a VWAP execution strategy and find its convenientrepresentation (equation (13)) in the finite-activity model described above. It is assumed that the impact iscomputed from a sample of consecutive past trades of the agent of the total size L . Then, we define theexpected price impact in the infinite-activity model as a corresponding limit of the finite-activity impact (see(14)) and show that it has a natural interpretation in terms of the infinite-activity model itself (Proposition 1). Recall that the process ( N t = N + t − N − t ) represents the total order flow in the market. In the finite activitymodel (2), we have ˜ X t = X + αN t + (cid:90) t σ ( ˜ X u ) d ˜ B u , (4) N t = δ (cid:90) t (cid:90) R (cid:16) { x ≥ β + ( ˜ X u − ) } − { x ≤ β − ( ˜ X u − ) } (cid:17) M ( du, dx ) , (5)where ˜ B is a Brownian motion and M is an independent Poisson random measure (see, e.g., [14, Chapter II])with the compensator µ ( dt, dx ) = λdt ⊗ dF ( x ) . The total order flow is a sum of the order flows of K individual market participants (agents): N = K (cid:88) j =1 N j . (6)We assume that the agents follow VWAP-type strategies. Namely, the j th agent has the order flow N jt = δ (cid:90) t (cid:90) R (cid:16) { x ≥ β + ( ˜ X u − ) } + { x ≤ β − ( ˜ X u − ) } (cid:17) ζ j ( ˜ X u − , u ) M j ( du, dx ) , j = 1 , . . . , K, (7)where { M j } are independent Poisson random measures with the compensators µ j ( dt, dx ) = λ j dt ⊗ dF ( x ) , j = 1 , . . . , K, K (cid:88) j =1 λ j = λ, and ( ω, x, u ) (cid:55)→ ζ j ( x, u ) ∈ {± } is a random field, defined for all ( x, u ) ∈ R × R + , s.t.• { ζ j ( · , u ) } are independent across j = 1 , . . . , K , across u ≥ , and independent of ( { M j } , ˜ B ) ,• for each j = 1 , . . . , K and ( x, u ) ∈ R × R + , we have P ( ζ j ( x, u ) = 1) = F ( − β + ( x )) F ( − β + ( x )) + F ( β − ( x )) . (8)The random fields { ζ j } represent the heterogeneity of the agents in terms of their trading styles. Forexample, at any given moment in time, one agent may buy (i.e., ζ j = 1 ), while another one may sell (i.e., ζ j = − ). In general, we allow the agents’ trading styles to depend on the fundamental price and on their8diosyncratic sources of randomness. The assumptions we make on { ζ j } are not the most general, but theyensure that the model is consistent: i.e., the total order flow N = (cid:80) Kj =1 N j satisfies (5), with an appropriatelychosen Poisson random measure M having the prescribed compensator and being independent of ˜ B . The latterobservation is made precise in the appendix.To see why we call the agents’ strategies VWAP-type,notice that, in the present model, the total tradedvolume in the market at time t is given by ˜ V t = δ (cid:90) t (cid:90) R (cid:16) { x ≥ β + ( ˜ X u − ) } + { x ≤ β − ( ˜ X u − ) } (cid:17) M ( du, dx ) . Comparing the above with (7), we conclude that the j th agent in the proposed model trades with the rate thatis λ j /λ fraction of the overall rate, which makes it similar to VWAP. However, unlike the classical VWAPstrategy, where the trades are made in the same direction, our agents trade in different directions. It is alsoworth noting the difference between N (in (5)) and ˜ V given above: the former denotes the order flow process,to which the buy market orders contribute with the positive sign and the sell orders contribute with the negativesign, while ˜ V denotes the traded volume process, to which all trades contribute with the positive sign.Let us now assume that the first agent aims to compute the expected impact of a sequence of her buy trades(referred to as the execution interval) on the midprice. Recalling (7), we conclude that any such executioninterval can be characterized by the condition ζ = 1 . In order to compute this expected impact in practice, theagent (i) finds the past execution intervals in a sample of past L trades, (ii) records the changes in the quoted askprice over each interval, and (iii) computes the sample average of these changes. Mathematically, this processcorresponds to computing ˜ I L ( Q, δ, λ, λ ) := E (cid:16) (cid:100) ˜ X τ (cid:101) − (cid:100) ˜ X τ (cid:101) | ζ ( · , t ) = 1 , t ∈ ( τ , τ ] (cid:17) , where the constants Q > and L > denote, respectively, the (fixed) total amount of the asset purchasedby the first agent over each randomly selected execution interval and the size of the sample (measured intrades of the first agent) from which the execution intervals are selected. The random times τ and τ denote,respectively, the beginning and the end of a randomly chosen execution interval: τ := T ( ˜ V , η ) , τ := T ( ˜ V , ˜ V τ + Q ) , η ∼ U (0 , L ) , ˜ V t := δ (cid:90) t (cid:90) R (cid:16) { x ≥ β + ( ˜ X u − ) } + { x ≤ β − ( ˜ X u − ) } (cid:17) M ( du, dx ) , (9)where U (0 , L ) denotes the uniform distribution on [0 , L ] , we assume that η is independent of everything else,and we introduce the hitting time (or inverse) functional T ( Z, h ) := inf { t ≥ Z t > h } , for any process Z on [0 , ∞ ) and any level h ∈ R . Note that ˜ V defined above represents the first agent’s tradedvolume process. Thus, the above choice of τ corresponds to the assumption that each trade of the agent isequally likely to be the first trade of a VWAP meta-order.Next, we notice that (4)–(7) imply ˜ X t = X + αδ (cid:90) t (cid:90) R K (cid:88) j =1 (cid:16) { x ≥ β + ( ˜ X u − ) } + { x ≤ β − ( ˜ X u − ) } (cid:17) ζ j ( ˜ X u − , u ) M j ( du, dx ) + (cid:90) t σ ( ˜ X u ) d ˜ B u . (10) We only consider the execution intervals of buy trades, as the case of sell trades is analogous. ζ , we easily deduce that ˜ X τ is independentof the values of ζ on the random time interval ( τ , τ ] . Recall that τ − τ is a function of the path Z t :=˜ V τ + t − ˜ V τ for t ∈ [0 , T ( Z, Q )] : indeed, τ − τ = T ( ˜ V , ˜ V τ + Q ) − τ = T ( Z, Q ) . Thus, ˜ X τ is a function of the path of ( Z t , ˜ X τ + t ) for t ∈ [0 , T ( Z, Q )] . For convenience, we denote ˜ X τ =: g (cid:16) Z t , ˜ X τ + t , t ∈ [0 , T ( Z, Q )] (cid:17) . Using (10) and (9) again, we deduce that the conditional distribution of the path of ( Z t , ˜ X τ + t ) for t ∈ [0 , T ( Z, Q )] , given ˜ X τ = x and ζ j ( · , t ) = 1 for t ∈ ( τ , τ ] , is the same as the distribution of the path of ( ˜ N t ( x ) , ˜ Y t ( x ) , for t ∈ [0 , T ( ˜ N ( x ) , Q )] , where, ˜ N t ( x ) := δ (cid:90) t (cid:90) R (cid:16) { z ≥ β + ( ˜ Y u − ( x )) } + { z ≤ β − ( ˜ Y u − ( x )) } (cid:17) ˜ M ( du, dz ) , (11) ˜ Y t ( x ) := x + α ˜ N t ( x ) + αδ (cid:90) t (cid:90) R (cid:16) { z ≥ β + ( ˜ Y u − ( x )) } − { z ≤ β − ( ˜ Y u − ( x )) } (cid:17) ˆ M ( du, dz ) + (cid:90) t σ ( ˜ Y u ( x )) d ˜ W u , (12)for any x ∈ R , with ˜ W being a Brownian motion, and with ˜ M and ˆ M being independent Poisson randommeasures with the respective compensators ˜ ν ( dt, dx ) = λ dt ⊗ dF ( x ) , ˆ ν ( dt, dx ) = ( λ − λ ) dt ⊗ dF ( x ) . By possibly extending the probability space, we assume that ( ˜ Y ( x ) , ˜ N ( x )) are constructed on the same proba-bility space as ( ˜ X, ˜ V , η ) and are independent of the latter. The above observations imply ˜ I L ( Q, δ, λ, λ ) = E (cid:16) (cid:100) ˜ X τ (cid:101) − (cid:100) ˜ X τ (cid:101) | ζ ( · , t ) = 1 , t ∈ ( τ , τ ] (cid:17) = − E (cid:100) ˜ X τ (cid:101) + E (cid:16) E (cid:16) (cid:100) g ( Z t , ˜ X τ + t , t ∈ [0 , T ( Z, Q )]) (cid:101) | ˜ X τ , ζ ( · , t ) = 1 , t ∈ [ τ , τ ] (cid:17) | ζ ( · , t ) = 1 , t ∈ ( τ , τ ] (cid:17) = E (cid:20)(cid:16) E (cid:100) g ( ˜ N t ( x ) , ˜ Y t ( x ) , t ∈ [0 , T ( ˜ N ( x ) , Q )]) (cid:17) x = ˜ X τ (cid:101) | ζ ( · , t ) = 1 , t ∈ [ τ , τ ] (cid:21) − E (cid:100) ˜ X τ (cid:101) = E (cid:100) ˜ Y T ( ˜ N ( x ) ,Q ) ( ˜ X τ ) (cid:101) − E (cid:100) ˜ X τ (cid:101) , where we recalled the meaning of function g to obtain the las equality.Introducing the time-changed processes ¯ X v := ˜ X T ( ˜ V ,v ) , ¯ Y v ( x ) := ˜ Y T ( ˜ N ( x ) ,v ) ( x ) , we summarize the above result as ˜ I L ( Q, δ, λ, λ ) = E (cid:0) (cid:100) ¯ Y Q ( ¯ X η ) (cid:101) − (cid:100) ¯ X η (cid:101) (cid:1) , (13)where ( ˜ V , ˜ X ) are defined in (9), (10), and ( ˜ Y x , ˜ N x ) are defined in (11), (12).The process ˜ N represents the order flow of the first agent during her (buy) execution intervals. The in-terpretation of ˜ Y is also clear: it is a model for the fundamental price during the (buy) execution intervals ofthe first agent. Since the order flow is biased upwards in such intervals, the integrand in (11) has a sum oftwo (mutually exclusive) indicators, and the resulting nondecreasing process ˜ N creates an upward trend in thedynamics of ˜ Y . Note also that the processes ¯ X and ¯ Y ( x ) run on the business clock, hence, their indices aremeasured in traded volume. 10 .3.2 Infinite-activity model Note that the expected impact on midprice, given by (13), depends only on the distributions of ¯ X and ¯ Y ,defined in (13). Let us fix γ > and θ ∈ (0 , , and consider the limit of the expected impact: I L ( Q, θ ) := lim λ →∞ ˜ I L ( Q, γ/λ, λ, θλ ) , (14)provided it is well defined. The restriction δ = γ/λ is discussed at the beginning of Subsection 2.2. Thecondition λ = θλ is equivalent to the assumption that the market participation rate of the first agent is fixed aswe vary λ .It turns out that ¯ X converges (weakly) to ˆ X , given by ˆ X t = X + (cid:90) t ˆ µ ( θ, ˆ X u ) du + (cid:90) t ˆ σ ( θ, ˆ X u ) d ˆ B u , (15)with a Brownian motion ˆ B and with ˆ µ ( θ, y ) := α F ( − β + ( y )) − F ( β − ( y )) θ ( F ( − β + ( y )) + F ( β − ( y ))) , ˆ σ ( θ, y ) := σ ( y ) (cid:112) θγ ( F ( − β + ( y )) + F ( β − ( y ))) , while ¯ Y ( x ) converges (weakly) to ˆ Y ( x ) , given by ˆ Y t ( x ) = x + (cid:90) t ˆ µ ( θ, ˆ Y u ( x )) du + (cid:90) t ˆ σ ( θ, ˆ Y u ( x )) d ˆ W u , (16)with a Brownian motion ˆ W and with ˆ µ ( θ, y ) := α θF ( β − ( y )) + F ( − β + ( y )) − F ( β − ( y )) θ ( F ( − β + ( y )) + F ( β − ( y ))) . To make the above statement precise, we view ¯ X , ¯ Y ( x ) as random elements with values in the Skorokhodspace D ([0 , ∞ )) , and ˆ X , ˆ Y ( x ) as random elements with values in C ([0 , ∞ )) . Note also that the laws of theprocesses ˆ X and ˆ Y ( x ) are uniquely determined by (15) and (16), which can be easily seen by applying thescale function transformation and reducing these SDEs to the ones with no drifts and with Lipschitz diffusioncoefficients. Lemma 1. As λ → ∞ , for any x ∈ R , we have: P ◦ ¯ X − → P ◦ ˆ X − , P ◦ ( ¯ Y ( ¯ X η )) − → P ◦ ( ˆ Y ( ˆ X η )) − , where the convergence is in weak topology induced by the C ([0 , ∞ )) -seminorms.Proof: W.l.o.g., we only prove the convergence of ¯ X . Recall that the latter is a function of two processes, ˜ X and ˜ V . First, we prove the C -tightness of the joint law of ( ˜ X, ˜ V ) over λ → ∞ on a finite time interval [0 , T ] . Toprove the latter, it suffices to show (i) that the absolute values of the two processes are bounded in probability,uniformly over λ , and (ii) that ∀ ε > , lim ε (cid:48) → lim sup λ →∞ P (cid:32) sup t,s ∈ [0 ,T ] , | t − s |≤ ε (cid:48) | Z t − Z s | > ε (cid:33) = 0 , Z = ˜ X, ˜ V . Both (i) and (ii) follow from Chebyshev’s inequality and the estimate E sup u ∈ [ t,s ] | Z u − Z s | ≤ Cλ δ | t − s | , where C is a constant and we recall λ = θλ and δ = γ/λ .Next, we consider any limit point Λ (a probability measure on ( C ([0 , T ])) ) of the family { P ◦ ( ˜ X, ˜ V ) − } λ ,with the associated sequence { λ n → ∞} . In particular, ( ˜ X n , ˜ V ,n ) → ( ˇ X, ˇ V ) in weak topology induced bythe C -norm. Let us describe the dynamics of ( ˇ X, ˇ V ) . It is easy to see that, for any f ∈ C b ( R ) , E sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12) f ( ˇ V t ) − γθ (cid:90) t f (cid:48) ( ˇ V s ) (cid:0) F ( − β + ( ˇ X s )) + F ( β − ( ˇ X s )) (cid:1) ds (cid:12)(cid:12)(cid:12)(cid:12) = lim n →∞ E sup t ∈ [0 ,T ] (cid:12)(cid:12)(cid:12)(cid:12) f ( ˜ V ,nt ) − γθ (cid:90) t f (cid:48) ( ˜ V ,ns ) (cid:16) F ( − β + ( ˜ X ns )) + F ( β − ( ˜ X ns )) (cid:17) ds (cid:12)(cid:12)(cid:12)(cid:12) = 0 . Choosing an appropriate sequence of f approximating the identity, we deduce from the above that ˇ V t = γθ (cid:90) t (cid:0) F ( − β + ( ˇ X s )) + F ( β − ( ˇ X s )) (cid:1) ds, t ∈ [0 , T ] . (17)Similarly, for any < t < s ≤ T , f ∈ C b ( R ) , ≤ t < · · · < t k ≤ t , g ∈ C b ( R k ) , E (cid:20) g (cid:0) ˇ X t , ˇ V t , . . . , ˇ X t k , ˇ V t k (cid:1) (cid:18) f ( ˇ X s ) − f ( ˇ X t ) − αγ (cid:90) st f (cid:48) ( ˇ X u ) (cid:0) F ( − β + ( ˇ X u )) − F ( β − ( ˇ X u )) (cid:1) du (cid:19)(cid:21) = lim n →∞ E (cid:104) g (cid:16) ˜ X nt , ˜ V ,nt , . . . , ˜ X nt k , ˜ V ,nt k (cid:17) (cid:16) f ( ˜ X ns ) − f ( ˜ X nt ) − αγ (cid:90) st f (cid:48) ( ˜ X nu ) (cid:16) F ( − β + ( ˜ X nu )) − F ( β − ( ˜ X nu )) (cid:17) du (cid:19)(cid:21) = 0 . From the above, we deduce that the process M t := ˇ X t − γ (cid:82) t (cid:0) F ( − β + ( ˇ X u )) − F ( β − ( ˇ X u )) (cid:1) du , defined onthe canonical space ( C ([0 , T ])) , is a continuous martingale under Λ and, therefore, is given by a Brownianintegral. Using the test function, as in the above, we easily deduce that d (cid:104) M (cid:105) t = σ ( ˇ X t ) dt a.s. under Λ . Thuswe have shown that ˇ X can be written as ˇ X t = X + αγ (cid:90) t (cid:0) F ( − β + ( ˇ X u )) − F ( β − ( ˇ X u )) (cid:1) du + (cid:90) t σ ( ˇ X u ) d ˇ B u , t ∈ [0 , T ] , (18)where ˇ B is a Brownian motion under Λ . As the law of ( ˇ X, ˇ V ) is uniquely determined by (17) and (18), theconvergence of ( ˜ X n , ˜ V ,n ) holds along any sequence { λ n → ∞} .To conclude the proof, we notice that there exists ε > , s.t., Λ -a.s., ˇ V · ∈ K ε , with K ε := { f ∈ C ([0 , T ]) : f ( t ) − f ( s ) ≥ ε ( t − s ) , ∀ ≤ s < t ≤ T } . It is easy to see that the mapping ( f, g ) (cid:55)→ f ◦ g − is a continuousmapping from C ([0 , T ]) × K ε into C ([0 , T ]) . Thus, using the Skorokhod’s representation theorem and theportmanteau theorem, we conclude that, along any { λ n → ∞} , ¯ X n · := ˜ X n ( ˜ V ,n · ) − → ˇ X ( ˇ V · ) − =: ˆ X · , with the inverse being defined as a right-continuous function and with the convergence being in weak topologyinduced by the C -norm. Using (17) and (18), we easily show that ˆ X satisfies (15). Recalling that the solutionto (15) is unique in law, we complete the proof of the lemma.12 emark 1. The proof of Lemma 1 also shows that, in the infinite-activity model, during an execution intervalof the first agent, her order flow is given by θγ (cid:90) · (cid:0) F ( − β + ( Y u )) + F ( β − ( Y u )) (cid:1) du, where Y represents the dynamics of the fundamental price in such intervals (it is the limit of ˜ Y ). In addition,during any such interval, the total traded volume in the market is given by γ (cid:90) · (cid:0) F ( − β + ( Y u )) + F ( β − ( Y u )) (cid:1) du. (19) Thus, in the infinite-activity limit, the first agent still uses a VWAP strategy, with the participation rate θ . In view of Lemma 1, it is natural to expect that I L ( Q, γ, θ ) , given by (14), can be computed by replacing ( ¯ X, ¯ Y ) by ( ˆ X, ˆ Y ) in (13). Proposition 1.
For any
L > , Q > , and θ ∈ (0 , , the limit in (14) is well defined, and we have I L ( Q, θ ) = E (cid:16) (cid:100) ˆ Y Q ( ˆ X η ) (cid:101) − (cid:100) ˆ X η (cid:101) (cid:17) , (20) with η ∼ U (0 , L ) independent of ( ˆ X, ˆ Y ) .Proof: The proof follows from Lemma 1, the portmanteau theorem, and the fact that neither ˆ Y Q ( ˆ X η ) nor ˆ X η haveatoms.In the remainder of the paper, we stay in the setting of the infinite-activity model. In this subsection, we consider the limit of the expected price impact of a VWAP meta-order in the infinite-activity model, as the sample size L over which the impact is estimated increases to infinity (see (21)). Wethen express the resulting (infinite-activity and infinite-sample impact) through the stationary density of thefundamental price run on the business clock (Proposition 2).Recall that η ∼ U (0 , L ) , where L represents the length of the data sample from which the executionintervals are collected. As it is natural to estimate impact over a large sample, we consider L → ∞ and set I ( Q, θ ) := lim L →∞ I L ( Q, θ ) , (21)provided the limit is well defined. Not surprisingly, the large-sample expected impact on midprice turns outto be connected to the stationary distribution of the fundamental price. We begin with the following technicalresult. Lemma 2.
Let us fix an arbitrary θ > . Then, there exist unique stationary distributions of ˆ X mod and ˆ Y mod , with the densities ψ and χ , respectively. These densities are uniquely determined by the followingconditions: ˆ σ ( θ, · ) ψ ∈ C (cid:15) ([0 , , ∂ x (cid:0) ˆ σ ( θ, x ) ψ ( x ) (cid:1) − ∂ x (ˆ µ ( θ, x ) ψ ( x )) = 0 , x ∈ (0 , , (22) Note that the length of an individual execution interval (measured in trades of the first agent) may be much smaller than L : i.e., at thisstage, we do not make the assumption that the execution intervals are long. (0 + ) = ψ (1 − ) , (cid:90) ψ ( x ) = 1 , ˆ σ ( θ, · ) χ ∈ C (cid:15) ([0 , , ∂ x (cid:0) ˆ σ ( θ, x ) χ ( x ) (cid:1) − ∂ x (ˆ µ ( θ, x ) χ ( x )) = 0 , x ∈ (0 , , (23) χ (0 + ) = χ (1 − ) , (cid:90) χ ( x ) = 1 . Moreover, for any bounded Borel-measurable function G , we have, for any x ∈ R , lim T →∞ T (cid:90) T E G ( ˆ X t mod dt = lim T →∞ E G ( ˆ X T mod
1) = (cid:90) G ( z ) ψ ( z ) dz, lim T →∞ T (cid:90) T E G ( ˆ Y t ( x ) mod dt = lim T →∞ E G ( ˆ Y T ( x ) mod
1) = (cid:90) G ( z ) χ ( z ) dz. Proof:
W.l.o.g. we only consider the case of ˆ X mod . First, we notice that the assumptions on F and σ implythat ˆ µ i ( θ, · ) / ˆ σ ( θ, · ) ∈ C (cid:15) ([0 , , for i = 0 , . Then, for any c > , Theorem 6.5.3 in [18] yields theexistence and uniqueness of ˆ σ ψ ∈ C (cid:15) ([0 , satisfying the ODE in (22) with the boundary conditions ˆ σ ( θ, + ) ψ (0 + ) = ˆ σ ( θ, − ) ψ (1 − ) = c . Moreover, the maximum principle (or the Feynman-Kac formula)implies that ψ > . Hence, choosing c > appropriately, we can ensure that (cid:82) ψ ( x ) = 1 . Thus, we haveshown the existence and uniqueness of the solution to (22).By choosing an arbitrary f ∈ C ([0 , , satisfying f (0) = f (1) = 0 and f (cid:48) (0 + ) = f (cid:48) (1 − ) , applying Itˆo’sformula to f ◦ ( · mod X ) , integrating by parts, and using (ˆ σ ψ )(0 + ) = (ˆ σ ψ )(1 − ) , along with the ODE(22) and the dominated convergence, we show that ddt (cid:90) E f ( ˆ X t ( x ) mod ψ ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12) t =0 = 0 , (24)where ˆ X ( x ) is the solution to (15) with X = x . As follows from Theorem 5.4.20 and Remark 5.4.21 in [16], { ˆ X ( x ) } x ∈ R is a Markov family with the transition denoted by K ( x, A ) . Due to periodicity of the coefficients in(15), we have K ( x + n, A + n ) = K ( x, A ) . Then, it is easy to see that { ˆ X ( x ) mod } x ∈ [0 , is a Markov familywith the transition kernel (cid:80) ∞ n = −∞ K ( x, A + n ) . The Markov property and (24) imply that ψ is stationary.To show uniqueness of the stationary distribution of ˆ X mod , consider any such distribution and use thescale function transformation, along with the continuous differentiability and Gaussian estimates for the funda-mental solution of a linear (strictly) parabolic PDE with Lipschitz coefficients (see, e.g., [9]), to conclude thatthe stationary distribution has density ψ ∈ C ([0 , , s.t. ψ (0) = ψ (1) . Using Itˆo’s formula, we show that ψ isa weak solution to the ODE in (22) on (0 , , with the test functions in C ((0 , . Using the weak form of theODE (22), we improve the regularity and conclude that ˆ σ ( θ, · ) ψ ∈ C (cid:15) ((0 , , and, in turn, that the ODE(22) holds in classical sense. Thus, the first part of the proof yields uniqueness of the stationary distribution.Finally, to obtain the last statement of the lemma, it is a standard exercise to check that the families ofmeasures Q T ( dx ) := 1 T (cid:90) T P ( ˆ X t mod ∈ dx ) dt, ˜ Q T ( dx ) := P ( ˆ X T mod ∈ dx ) , x ∈ [0 , , parameterized by T ≥ , are tight and that each of their limit points (in weak topology), as T → ∞ , is astationary distribution of ˆ X mod . Since such distribution is unique, we obtain the statement of the lemma forbounded continuous G . As the stationary distribution has no atoms, this statement is extended to all boundedBorel-measurable G . 14 roposition 2. For any
Q > and θ > , the limit in (21) is well defined, and we have: I ( Q, θ ) = (cid:90) E (cid:16) (cid:100) ˆ Y Q ( x ) (cid:101) − (cid:17) ψ ( x ) dx, (25) where ψ is the density of the stationary distribution of ˆ X mod .Proof: Notice that, for any x ∈ R and any integer n , ˆ Y ( x + n ) = ˆ Y ( x ) . Then, using the independence of ˆ X , ˆ Y and η , and the uniform distribution of η , we have: I L ( Q ) = E (cid:16) (cid:100) ˆ Y Q ( ˆ X η ) (cid:101) − (cid:100) ˆ X η (cid:101) (cid:17) = E L (cid:90) L (cid:16) (cid:100) ˆ Y Q ( ˆ X s mod (cid:101) − (cid:100) ˆ X s mod (cid:101) (cid:17) ds = E L (cid:90) L G (cid:16) ˆ X s mod (cid:17) ds, where G ( x ) := E (cid:16) (cid:100) ˆ Y Q ( x ) (cid:101) − (cid:100) x (cid:101) (cid:17) . Using the ergodicity of X mod (see Lemma 2), E L (cid:90) L G (cid:16) ˆ X s mod (cid:17) ds → (cid:90) G ( x ) ψ ( x ) dx. In this subsection, we express the marginal expected price impact of a VWAP meta-order (i.e., the derivative ofthe expected price impact w.r.t. the size Q of the meta-order) through the wings of the stationary distributionof the fundamental price run on the business clock. This connection is one of the key steps in establishing thedesired concavity of the expected price impact, as described in Section 1. The target representation is derivedfor Q = 0 (Proposition 3) and for Q = ∞ (Proposition 4).First, we analyze the asymptotic behavior of ∂ Q I ( Q, θ ) as Q ↓ . Proposition 3.
For any θ > , lim Q ↓ ∂ Q I ( Q, θ ) = α ψ (1 − ) . Proof:
Since the drift and volatility of ˆ Y ( x ) are bounded and continuous and the volatility is bounded away fromzero, as Q → , we have, uniformly over x ∈ [0 , : E (cid:16) (cid:100) ˆ Y Q ( x ) (cid:101) − (cid:17) = (cid:16) P (cid:16) x + ˆ µ ( θ, x ) Q + ˆ σ ( θ, x ) (cid:112) Q ˆ W ≥ (cid:17) − P (cid:16) x + ˆ µ ( θ, x ) Q + ˆ σ ( θ, x ) (cid:112) Q ˆ W ≤ (cid:17)(cid:17) (1 + o (1))= (cid:18) Φ (cid:18) x − µ ( θ, x ) Q ˆ σ ( θ, x ) √ Q (cid:19) − Φ (cid:18) − x + ˆ µ ( x ) Q ˆ σ ( θ, x ) √ Q (cid:19)(cid:19) (1 + o (1)) , Φ is the standard normal c.d.f.. Then, using the continuity of ψ , the conditions ψ (0 + ) = ψ (1 − ) and ˆ σ ( θ, x ) = ˆ σ ( θ, − x ) , for x ∈ (0 , , as well as the mean value theorem and the dominated convergence, weobtain, as Q → : (cid:90) E (cid:16) (cid:100) ˆ Y Q ( x ) (cid:101) − (cid:17) ψ ( x ) dx = (cid:90) (cid:18) Φ (cid:18) x − µ ( θ, x ) Q ˆ σ ( θ, x ) √ Q (cid:19) ψ ( x ) − Φ (cid:18) x − − ˆ µ ( θ, − x ) Q ˆ σ ( θ, x ) √ Q (cid:19) ψ (1 − x ) (cid:19) dx (1 + o (1))= (cid:112) Q (cid:90) − / √ Q (cid:20) Φ (cid:18) x ˆ σ ( θ, x √ Q ) + ˆ µ ( θ, x √ Q ) √ Q ˆ σ ( θ, x √ Q ) (cid:19) ψ (1 + x (cid:112) Q ) − Φ (cid:18) x ˆ σ ( θ, x √ Q ) − ˆ µ ( θ, − x √ Q ) √ Q ˆ σ ( θ, x √ Q ) (cid:19) ψ ( − x (cid:112) Q ) (cid:21) dx (1 + o (1))= Qψ (1 − ) ˆ µ ( θ, − ) + ˆ µ ( θ, + )ˆ σ ( θ, − ) (cid:90) −∞ φ (cid:18) x ˆ σ ( θ, − ) (cid:19) dx (1 + o (1))= Q ψ (1 − )(ˆ µ ( θ, − ) + ˆ µ ( θ, + ))2 (1 + o (1)) = Qα ψ (1 − ) (1 + o (1)) . Similarly, we can analyze ∂ Q I ( Q, θ ) as Q → ∞ . Notice that, due to the Markov property of ˆ Y (see anal-ogous argument for the Markov property of ˆ X in the proof of Lemma 2) and the periodicity of the coefficientsin (16), we have: I ( Q + ∆ Q, θ ) − I ( Q, θ ) = lim L →∞ ( I L ( Q + ∆ Q, θ ) − I L ( Q, θ ))= lim L →∞ E (cid:16) (cid:100) ˆ Y Q +∆ Q ( ˆ X η ) (cid:101) − (cid:100) ˆ Y Q ( ˆ X η ) (cid:101) (cid:17) = lim L →∞ E (cid:16) (cid:100) ˆ Y ∆ Q ( R Q ( ˆ X η )) (cid:101) − (cid:100) R Q ( ˆ X η ) (cid:101) (cid:17) = lim L →∞ E (cid:16) (cid:100) ˆ Y ∆ Q ( Z ) (cid:101) − (cid:17) , where R ( x ) ∼ ˆ Y ( x ) mod and Z ∼ R Q ( ˆ X η ) mod are independent of ( ˆ X, ˆ Y ) . Repeating the proof ofProposition 2, we obtain lim L →∞ E (cid:16) (cid:100) ˆ Y ∆ Q ( Z ) (cid:101) − (cid:17) = (cid:90) E (cid:16) (cid:100) ˆ Y ∆ Q ( R Q ( x )) (cid:101) − (cid:17) ψ ( x ) dx. Applying the scale transformation to ˆ Y ( x ) , to eliminate the drift, it is easy to see that the density of ˆ Y t ( x ) ,denoted ¯ χ xt , can be written as ¯ χ xt ( y ) = Γ( t, x, y ) P ( y ) , t > , x, y ∈ R , where Γ( · , x, · ) ∈ C ε, ε , with some ε ∈ (0 , , is the fundamental solution of the parabolic PDE associatedwith the transformed ˆ Y x , and P is an exponentially bounded Lipschitz-continuous function whose derivativeis continuous everywhere except integers, where it has first order discontinuities. A direct computation showsthat ∂ t ¯ χ x − ∂ y (cid:0) ˆ σ ¯ χ x (cid:1) + ∂ y (ˆ µ ¯ χ x ) = 0 , (26)16here the equation holds globally in ( t, y ) ∈ (0 , ∞ ) × R in a weak sense and pointwise (with all derivativesbeing well defined) everywhere except (0 , ∞ ) × Z , with the left and the right limits being well defined at everyinteger y . Then, applying the Gaussian estimates for Γ , it is easy to see that, for any t > , the distribution of R t ( x ) ∼ ˆ Y t ( x ) mod has density χ xt ( y ) = (cid:88) n ∈ Z ¯ χ xt ( y + n ) , y ∈ [0 , (a similar argument was used in the proof of Lemma 2). Due to periodicity of the coefficients, we deduce that χ x satisfies (26) in the same sense as ¯ χ x .Repeating the proof of Proposition 3, we obtain, as ∆ Q → : E (cid:16) (cid:100) ˆ Y ∆ Q ( R Q ( x )) (cid:101) − (cid:17) = (cid:90) E (cid:16) (cid:100) ˆ Y ∆ Q ( y ) (cid:101) − (cid:17) χ xQ ( y ) dy = ∆ Qα χ xQ (1 − ) (1 + o (1)) , for every x ∈ (0 , . Thus, using the dominated convergence theorem, we conclude: ∂ Q I ( Q, θ ) = α (cid:90) χ xQ (1 − ) ψ ( x ) dx. (27)The following proposition describes the asymptotic behavior of ∂ Q I for large Q . Proposition 4.
For any γ, θ > , lim Q →∞ ∂ Q I ( Q, θ ) = α χ (1 − ) , with χ defined in Lemma 2.Proof: Using Ito’s formula, it is easy to see that u ( t, y ) = (cid:82) χ xt ( y ) ψ ( x ) dx is a weak solution to ∂ t u − ∂ y (cid:0) ˆ σ u (cid:1) + ∂ y (ˆ µ u ) = 0 , y ∈ (0 , , u ( t,
0) = u ( t,
1) = (cid:90) χ xt (1 − ) ψ ( x ) dx, u (0 , y ) = ψ ( y ) , (28)with u, ∂ y u ∈ L ([0 , T ] × [0 , . Recalling that χ x satisfies (26) along (0 , ∞ ) × { − } ) , we deduce that v ( t, y ) := u ( t, y ) − (cid:82) χ xt (1 − ) ψ ( x ) dx satisfies (28) with the same initial and with zero boundary conditions.Applying Theorem III.2.1 in [20] to v , we conclude that (cid:107) ∂ y v ( t, · ) = ∂ y u ( t, · ) (cid:107) L is bounded uniformly over t ≥ . This, in turn, yields uniform continuity of the family { u ( t, · ) } t ≥ . Since the weak limit of this family,as t → ∞ is χ (see Lemma 2), we conclude that it is also a strong limit in the uniform norm. This, along with(27), completes the proof of the proposition.Thus, we have shown that the marginal expected impact on the midprice at the beginning of an executionsequence is proportional to the stationary distribution of ˆ X mod at − . Similarly, we have shown that themarginal expected impact at the end of a sufficiently long execution sequence is proportional to the stationarydistribution of ˆ Y mod at − . In the next section, we show that, for small θ > , the former exceeds the latter,which proves the asymptotic concavity of the expected impact curve. In this subsection, we show the asymptotic concavity of the expected price impact of a VWAP meta-order:namely, for sufficiently small participation rates θ > , the marginal expected price impact at the beginning17f the meta-order (i.e., for Q ≈ ) is strictly larger than at its end (i.e., for Q ≈ ∞ ). This is done using therepresentation of the marginal expected price impact via the wings of the stationary distribution, established inthe previous subsection. The main result is summarized in Theorem 1.In view of Propositions 3 and 4, the asymptotic concavity of the expected price impact is equivalent to ψ (1 − ) > χ (1 − ) . To show the latter, we recall the ODEs (22) and (23) and, multiplying them by θ , we deducethe existence and uniqueness of function ( θ, x ) (cid:55)→ f ( θ, x ) , s.t. f ( θ, · ) ∈ C (cid:15) ([0 , and ∂ x f ( θ, x ) − ∂ x ((¯ µ ( x ) + θ ¯ µ ( x )) f ( θ, x )) = 0 , (29) f ( θ,
1) = f ( θ, , (cid:90) f ( θ, x )¯ σ ( x ) dx = 1 , (30)where ¯ σ ( y ) := √ θ ˆ σ ( y ) = σ ( y ) (cid:112) γ ( F ( y −
1) + F ( − y )) , ¯ µ ( y ) = αγ F ( y − − F ( − y ) σ ( y ) , ¯ µ ( y ) := 2 αγ F ( − y ) σ ( y ) , and we used β + ( y ) = 1 − y, β − ( y ) = − y, y ∈ (0 , . It is clear that ψ = f (0 , · ) / ¯ σ and χ = f ( θ, · ) / ¯ σ . Our goal is to show that, for small enough θ > , wehave f ( θ, < f (0 , . The next proposition establishes the desired result, but under two additional technicalassumptions. Assumption 1.
There exists a constant ρ ≥ , s.t. σ ( x ) = ρ (cid:112) γ ( F ( x −
1) + F ( − x ) , x ∈ (0 , . Note that all standing assumptions on σ are implied by the above assumption and the properties of F . Assumption 2.
The function F is log-concave in [ − , and F (cid:48) is nondecreasing in this range. Proposition 5.
For any x ∈ [0 , , there exists ∂ θ f ( · , x ) ∈ C ( R ) . Moreover, under Assumptions 1 and 2, wehave: ∂ θ f (0 , < .Proof: It is easy to see (e.g., using Feynman-Kac formula) that f ( θ, x ) is continuously differentiable in θ . Then,we differentiate (29) and (30) w.r.t. θ to obtain g xx − (¯ µ + θ ¯ µ ) g x − (¯ µ (cid:48) + θ ¯ µ (cid:48) ) g = ∂ x (¯ µ f (0 , · )) , g ( θ,
0) = g ( θ, , (31) (cid:90) g ( θ, x )¯ σ ( x ) dx = 0 , (32)for g ( θ, x ) := ∂ θ f ( θ, x ) .Next, we consider the case θ = 0 . The PDE (29) and the property ∂ x f (0 , /
2) = ¯ µ (1 /
2) = 0 (whichfollows from the fact that f (0 , · ) is symmetric around x = 1 / ) yield: ∂ x f (0 , x ) = 2¯ µ ( x ) f (0 , x ) , x ∈ (0 , . Note that the log-concavity of F can be ensured by requiring that the density of the distribution defined by F is log-concave. ∂ x (¯ µ f (0 , x )) = ¯ µ (cid:48) f (0 , x ) + ¯ µ ∂ x f (0 , x ) = (¯ µ (cid:48) + 2¯ µ ¯ µ ) f (0 , x )= γf (0 , x ) σ (cid:0) − αF (cid:48) ( − x ) σ − αF ( − x ) ∂ x σ + 4 αγ ( F ( x − − F ( − x )) F ( − x ) (cid:1) = 2 αγ f (0 , x ) σ ( − ρF (cid:48) ( − x ) F ( x − − ρF (cid:48) ( − x ) F ( − x ) − ρF ( − x ) F (cid:48) ( x − ρF ( − x ) F (cid:48) ( − x ) + 2 F ( x − F ( − x ) − F ( − x ) F ( − x )) ≤ αγ f (0 , x ) σ ( − F (cid:48) ( − x ) F ( x − − F ( − x ) F (cid:48) ( x −
1) + 2 F ( x − F ( − x ) − F ( − x ) F ( − x ))= 2 αγ f (0 , x ) σ (cid:18) − F (cid:48) ( − x ) F ( x −
1) + F ( − x ) F (cid:48) ( x − − F ( − x ) F (cid:48) ( x −
1) + 2 F ( − x ) (cid:90) x − − x F (cid:48) ( z ) dz (cid:19) . The right hand side of the above is clearly non-positive for x ∈ (0 , / . For x ∈ (1 / , its negativity isimplied by the monotonicity of F (cid:48) on R − and by the log-concavity of F : (log F ) (cid:48) ( x − < (log F ) (cid:48) ( − x ) ,F ( − x ) F (cid:48) ( x − < F (cid:48) ( − x ) F ( x − . Recall the ODE for g (0 , · ) : g xx − ¯ µ g x − ¯ µ (cid:48) g = ∂ x (¯ µ f (0 , · )) , g (0 ,
0) = g (0 , . (33)The rest of the proof follows from the maximum principle. Indeed, the ODE in (33) and the conditions ∂ x (¯ µ f (0 , · )) < , ¯ µ (cid:48) > imply that g (0 , · ) cannot have a strictly negative minimum in (0 , (otherwise, theODE cannot be satisfied at the minimum point of g (0 , · ) ). Then, if g (0 , ≥ , we conclude that g (0 , · ) ≥ ,which contradicts (32) (the case g (0 , · ) ≡ is easily excluded, since ¯ µ f (0 , · ) cannot be constant). Thus, weconclude that g (0 , < and complete the proof of the proposition.Thus, we have proved the main mathematical result of this paper. Theorem 1.
Under Assumptions 1 and 2, there exists ε > , s.t. lim Q ↓ ∂ Q I ( Q, θ ) > lim Q →∞ ∂ Q I ( Q, θ ) , for all θ ∈ (0 , ε ) . In this subsection, we present an analytically tractable model from the class described in the previous section.In addition to illustrating the tractability of the proposed setting, our goal herein is to compute numerically thestationary densities, the expected price impact of a VWAP meta-order, and the price resilience curve (Figures1–2). 19ssume, for simplicity, that ρ = 1 and that F is the c.d.f. of a uniform distribution on [ − a, a ] , for some a > . Then, all assumptions made in Section 2 are satisfied and, for x ∈ [0 , , F ( x ) = 12 a ( x + a ) , F ( x −
1) + F ( − x ) = 2 a − a , σ ( x ) = γ (2 a − a ¯ σ ( x ) = 1 , ¯ µ ( x ) = α a − x − , ¯ µ ( x ) := 2 α a − a − x ) . The ODE (29) becomes ∂ x f ( θ, x ) − α a − ∂ x ((2 x (1 − θ ) + 2 θa − f ( θ, x )) = 0 . To find the general solution of this ODE, we solve: u x − α a − x (1 − θ ) + 2 θa − u = C ,u ( x ) = g ( x ) exp (cid:18) α a − − θ ) (2 x (1 − θ ) + 2 θa − (cid:19) ,g (cid:48) ( x ) = C exp (cid:18) − α a − − θ ) (2 x (1 − θ ) + 2 θa − (cid:19) ,g ( x ) = C (cid:90) x −∞ exp (cid:32) − α (1 − θ )2 a − (cid:18) y + 2 θa − − θ ) (cid:19) (cid:33) dy + C = C Φ (cid:32) (cid:114) α (1 − θ )2 a − (cid:18) x + 2 θa − − θ ) (cid:19)(cid:33) + C ,u ( x ) = exp (cid:18) α a − − θ ) (2 x (1 − θ ) + 2 θa − (cid:19) (cid:34) C Φ (cid:32) (cid:114) α (1 − θ )2 a − (cid:18) x + 2 θa − − θ ) (cid:19)(cid:33) + C (cid:35) . The boundary conditions yield exp (cid:18) α a − − θ ) (2 θa − (cid:19) (cid:34) C Φ (cid:32) (cid:114) α (1 − θ )2 a − θa − − θ ) (cid:33) + C (cid:35) = exp (cid:18) α a − − θ ) (2(1 − θ ) + 2 θa − (cid:19) (cid:34) C Φ (cid:32) (cid:114) α (1 − θ )2 a − (cid:18) θa − − θ ) (cid:19)(cid:33) + C (cid:35) ,C = C exp (cid:16) α (1 − θ +2 θa ) a − − θ ) (cid:17) Φ (cid:18) (cid:113) α (1 − θ )2 a − (cid:16) θa − − θ ) (cid:17)(cid:19) − exp (cid:16) α (2 θa − a − − θ ) (cid:17) Φ (cid:18) (cid:113) α (1 − θ )2 a − θa − − θ ) (cid:19) exp (cid:16) α (2 θa − a − − θ ) (cid:17) − exp (cid:16) α (1 − θ +2 θa ) a − − θ ) (cid:17) . The restriction a > in order to satisfy the assumptions on F stated at the beginning of Section 2.1. The main purpose of thistechnical assumption is to ensure that the rates of arrival of the buy an sell orders do not vanish. f ( θ, x ) = u ( x ) (cid:82) u ( y ) dy ,u ( x ) = exp (cid:18) α (2 x (1 − θ ) + 2 θa − a − − θ ) (cid:19) (cid:34) exp (cid:18) α (1 − θ + 2 θa ) a − − θ ) (cid:19) Φ (cid:32)(cid:114) α (1 − θ )2 a − − θ + 2 θa − θ (cid:33) − exp (cid:18) α (2 θa − a − − θ ) (cid:19) Φ (cid:32)(cid:114) α (1 − θ )2 a − θa − − θ (cid:33) + (cid:18) exp (cid:18) α (2 θa − a − − θ ) (cid:19) − exp (cid:18) α (1 − θ + 2 θa ) a − − θ ) (cid:19)(cid:19) Φ (cid:32) (cid:114) α (1 − θ )2 a − (cid:18) x + 2 θa − − θ ) (cid:19)(cid:33)(cid:35) . Figure 1 describes the shape of the stationary density f ( θ, · ) , for various values of θ . We can see that, aspredicted by the theoretical results (recall Proposition 5), the value of the density at the boundary decreases as θ increases. Moreover, we see that the stationary density becomes more skewed toward the left, as θ increases.The latter indicates that, during the execution of a meta-order, the fundamental price is more likely to be closerto the best bid than to the best ask price, which is consistent with the phenomenon of “improving liquidity”discussed in Section 1. Indeed, if we interpret the fundamental price as the microprice, the fact that it is closerto the best bid price means that the volume of limit orders at the best ask is higher than the volume at the bestbid, which implies better liquidity for the executor (assuming that the total volume at the best bid and ask doesnot change on average).Next, for a fixed participation rate θ ∈ (0 , , we compute the expected price impact as a function ofexecuted volume Q (i.e., the expected price trajectory). Recall the formula (25), I ( Q, θ ) = (cid:90) E (cid:100) ˆ Y xQ (cid:101) f (0 , x ) dx − , where, in the present case, ˆ Y xt = x + (cid:90) t ˆ µ ( θ, ˆ Y xu ) du + 1 √ θ ˆ W t , ˆ µ ( θ, y ) = αθ (2 a −
1) (2 θa − y mod − θ )) . The PDE for u Q,θ ( t, x ) := E (cid:100) ˆ Y xQ − t (cid:101) is given by u Q,θt + ˆ µ ( θ, x ) u Q,θx + 12 θ u
Q,θxx = 0 , t ∈ [0 , Q ) , x ∈ R , u Q,θ ( Q, x ) = (cid:100) x (cid:101) . We use the explicit Euler scheme to approximate u and compute the impact curve I ( · , θ ) by approximatingnumerically the integral I ( Q, θ ) = (cid:90) u Q,θ (0 , x ) f (0 , x ) dx − . The result of this computation is shown in the left part of Figure 2.Finally, we address the question of price resilience, which measures the expected trajectory of the midpriceafter a meta-order has been executed. We assume that the execution of the meta-order lasted long enough,so that the process describing the conditional distribution of the fundamental price run on the business time21f the executor, ˆ Y , had entered into its stationary regime before the execution was over. Mathematically, thelatter means that, after the execution, the fundamental price run on the business time of the market follows theprocess ˇ Y xt = x + (cid:90) t ˇ µ ( ˇ Y xu ) du + ˇ W t , where ˇ W is a Brownian motion, and ˇ µ ( y ) := α a − y mod − . Hence, the price resilience is defined as R ( ¯ V , θ ) := (cid:90) E (cid:100) ˇ Y ¯ V (cid:101) f ( θ, x ) dx − , where ¯ V represents the total traded volume in the market. The right part of Figure 2 shows R ( ¯ V , θ ) as afunction of ¯ V . Note that, since θ = 0 . , the range of values of ¯ V , in the right part of Figure 2, is chosen tomatch the range of values of Q , in the left part: indeed, the execution of a meta-order of size Q via a VWAPstrategy with participation rate θ = 0 . will terminate when the total traded volume becomes ¯ V = Q/θ = 5 Q .It is clear from the right part of Figure 2 that the price resilience is convex and that the expected midprice doesnot decay to its initial level, which is consistent with the existing theoretical and empirical findings. Remark 2.
Figure 2 indicates that the expected price impact of a VWAP meta-order in the proposed model isasymptotically linear, for large Q , hence, the concavity becomes less pronounced for large order sizes. This isindeed the case, and it follows from the existence of lim Q →∞ ∂ Q I ( Q, θ ) , provided the latter is strictly positive.In principle, thisr limit may be zero, but this is excluded by the assumptions on F and σ made at the beginningof Section 2.1. Remark 3.
It is important to note that the choice of the model parameters in the above example was dictatedpurely by the desire to obtain analytic tractability. In this work, we do not address the important questionof what model parameters are consistent with the empirically observed dynamics of microprice and orderflow, which requires a separate investigation. Once the realistic parameters are found, one can determine themagnitude of the predicted change in the wings of the stationary distribution, during a VWAP meta-order, forrealistic participation rates, to see whether the concavity of the expected price impact predicted by the modelis significant for practical purposes.
The heuristic argument described in Section 1 shows that the concavity of the price impact can be derived fromtwo predictions. The first prediction is that the stationary distribution of the fundamental price modulo ticksize, run on the business clock, is U-shaped. And the second prediction states that, during the execution of aVWAP meta-order, the fundamental price (run on the business clock) obtains an additional constant drift in thedirection of the order. Although one cannot test empirically the second assumption without having access tothe information about meta-orders, it does not seem that this assumption requires any additional justificationbeyond common sense. Therefore, in this subsection, we test the first assumption, using only publicly availabledata, without any information about the meta-orders themselves.The experiment presented here uses the data from NASDAQ exchange obtained via the ITCH protocol,which provides information about every event in the limit order book. An event may be an execution, anaddition, or a cancelation, of a limit order, and the associated prices and volumes are either specified directly22r can be recovered from prior events (see [6] for a more detailed description of the ITCH protocol). Usingthis data, one can reverse-engineer the trade volumes of the volumes of limit orders at the first few levels ofthe limit order book, at the time of every event. The time interval we use covers November 3–7 of 2014 andincludes the tickers CSCO, INTC, MSFT, VOD, LBTYK, LVNTAFollowing the discussion in Section 1, we interpret the fundamental price X as the microprice (measuredin ticks) and, in turn, X mod as the limit order imbalance: X t mod ≈ V bt V bt + V at , where V b and V a are the limit order volumes at the best bid and ask respectively, and t is restricted to the timestamps of the events recorded in the database. Our goal is to estimate the stationary density f (0 , · ) of ˆ X mod ,where ˆ X stands for the fundamental price X run on the business clock: ˆ X t = X ( V · ) − ( t ) , where V t is the total traded volume by time t . Due to the ergodicity of ˆ X mod (see Lemma 2), for any ≤ a < b ≤ , we have (cid:90) ba f (0 , y ) dy = lim T →∞ T (cid:90) T [ a,b ] ( ˆ X t ) dt = lim T →∞ T (cid:90) ( V · ) − ( T )0 [ a,b ] ( X t ) dV t = lim T →∞ V T (cid:90) T [ a,b ] ( X t ) dV t . Splitting the interval [0 , into intervals J , . . . , J of length . , we obtain the following natural approxi-mation f (0 , x ) ≈ V T (cid:90) T J k ( X t ) dV t , x ∈ J k , k = 1 , . . . , . (34)In order to implement (34), we need to make an important decision on how to interpret the integral in itsright hand side. Recall that the theoretical connection between the marginal expected price impact and the tailsof the stationary distribution of ˆ X mod , given by Propositions 3 and 4, is established in the infinite-activityregime, in which V · is continuous and, hence, there is no ambiguity in how to understand the integral in theright hand side of (34). The infinite-activity assumption is important for the theoretical results of this work,as it allows us to ignore the sizes of market orders (more precisely, the joint distribution of these sizes andthe fundamental price) in deriving the expression for the marginal expected impact. On the other hand, thisassumption is clearly an abstraction, as the actual market orders do not have infinitesimal sizes and, as a result,the process V · changes by jumps only. Since the process X · also jumps at the jump times of V · , we need tochoose an appropriate (natural) interpretation of the integral in (34). The following interpretations are usedherein.1. To explain the first interpretation (and the resulting estimator), let us recall that the purpose of our esti-mation of the stationary density is to connect it to the expected marginal impact of a VWAP executor.This connection is based on the observation that, by the nature of the VWAP strategy, the executor’strades are mimicking the evolution of the total traded volume process V . Hence, the total fraction ofmarket orders executed while the fundamental price is in the interval J k is equal to the fraction of theexecutor’s trades submitted while X is in J k . It remains to notice that the latter fraction, for J k close tozero or one, determines the expected marginal impact of the executor. Thus, a natural interpretation ofthe right hand side of (34) should provide a reasonable approximation to the fraction of executor’s trades The author thanks S. Jaimungal for providing this data. X is in J k . Notice that the discontinuity of V · makes it impossible to follow the VWAPstrategy perfectly: indeed, the VWAP executor cannot submit her market orders exactly at the sametime as the ones submitted by other traders. Assuming that the executor copies a fraction of every marketorder submitted by other traders (with the size of the fraction determined by her participation rate) andsubmits her copy right before the associated market order of another trader with probability w , and rightafter it with probability − w (independent of everything else), we arrive at the estimator f (0 , x ) ≈ (cid:80) t i ≤ T ∆ V t i (cid:88) t i ≤ T (cid:32) w J k (cid:32) V bt i − V bt i − + V at i − (cid:33) + (1 − w ) J k (cid:32) V bt i + V bt i + + V at i + (cid:33)(cid:33) ∆ V t i ,x ∈ J k , k = 1 , . . . , , (35)where { t i } are the arrival times of the market orders, and V a/bt i − and V a/bt i + denote, respectively, the limitorder volumes right before and right after the i th market order. We refer to (35) as the “weighted es-timator”. Note that, from a numerical point of view, this estimator corresponds to approximating theintegrand in (34), at the jump times of the integrator, via a weighted average of its values to the left andto the right of the jump time. From a financial point of view, the size of w measures how well and howfast the executor can predict the size of the next market order. An executor who is unable or unwillingto predict the individual sizes of external market orders, or who is not fast enough to submit her orderahead of the external one, will have w = 0 . For w = 0 . , we call (35) the “equally weighted estimator”.2. The next estimator is based on the assumption that the VWAP executor does not try to predict the sizes ofindividual market orders submitted by other traders, and that she aims to reproduce the desired fractionof the total traded volume by executing trades of a fixed size each at the (dynamically changing) rate atwhich other market orders arrive. In this case, (34) changes to f (0 , x ) ≈ (cid:80) t i ≤ T (cid:90) T J k ( X t ) d (cid:88) t i ≤ T [0 ,t ] ( t i ) , x ∈ J k , k = 1 , . . . , . (36)Assuming, as in item 1 above, that the executor aims to submit each trade at the same time as a marketorder from another trader arrives, and that she is slightly early with probability w , and slightly late withprobability − w , we obtain the estimator f (0 , x ) ≈ (cid:80) t i ≤ T (cid:88) t i ≤ T (cid:32) w J k (cid:32) V bt i − V bt i − + V at i − (cid:33) + (1 − w ) J k (cid:32) V bt i + V bt i + + V at i + (cid:33)(cid:33) ,x ∈ J k , k = 1 , . . . , , (37)which we call the “weighted uniform estimator”. For w = 0 . , we call (37) the “equally weighteduniform estimator”.3. The last estimator proposed herein is based on interpreting the integral in the right hand side of (34) asa limit of the integrals with continuous integrators (in the spirit of the infinite-activity model studied inthe preceding sections). Namely, for each discontinuity time t i of V · and for any ε > small enough, sothat V · is constant in [ t i − ε, t i ) and in ( t i , t i + ε ] , we replace (cid:90) t i + εt i − ε J k ( X t ) dV t There is, of course, a more fundamental reason why it is impossible to follow the VWAP perfectly – it is because the executor doesnot possess a perfect predictive power for the future traded volume. However, herein, we focus on a different reason. Of course, this strategy would not correspond to a perfect VWAP, but, as mentioned before, the latter is impossible anyway. R ik := (cid:90) ∆ V ti J k (cid:32) V bt i − V bt i − + V at i − − v (cid:33) dv or (cid:90) ∆ V ti J k (cid:32) V bt i − − vV bt i − + V at i − − v (cid:33) dv, (38)depending on whether t i is the time of a buy or a sell order, respectively. Note that this approximationis based on the assumption that the execution of the i th market order takes ε units of time. However,since the values of the integrals in (38) do not depend on ε , one can always make this assumption,provided ε is smaller than the resolution at which the events are recorded in the database. As the integralof J k ( X t ) dV t over the compliment of (cid:83) i [ t i − ε, t i + ε ] is zero, the above determines the value of theassociated estimator f (0 , x ) ≈ (cid:80) t i ≤ T ∆ V t i R ik , x ∈ J k , k = 1 , . . . , , (39)which we refer to as the “continuous estimator”.In order to implement the estimators (35), (37) and (39) in a manner that is consistent with the theoreticalpart of the paper, one needs to restrict the sample of available trades. First of all, only the trades submitted dur-ing the NASDAQ normal trading hours (9:30am–4pm Eastern Standard Time) are considered in this empiricalanalysis. Second, in order to stay as close as possible to the assumption of a constant one-tick bid-ask spread,we further restrict the universe of market orders to those that arrive when the bid-ask spread is equal to one tickand that do not execute any limit orders located in the higher levels of the limit order book (i.e., higher thanthe first level, measured right before the market order arrives). In addition, if the i th market order is a buy andif the bid-ask spread becomes larger than one tick right after this order is executed, we set V at i + := 0 , and weproceed symmetrically for the sell orders. This restriction ensures that, in the business time intervals includedin our analysis, the desired relationship between the fundamental price and the bid and ask prices is preserved,and that the financial rationale described in the first item of the above list remains valid.The estimated stationary density f (0 , · ) is presented in Figures 3–8, for each estimator and each ticker inour sample. Figures 3–5 show the estimated stationary density over one week of data. They clearly indicate theU-shape property of the density for most tickers, except LBTYK and LVNTA. The reason that the theoreticallypredicted U-shape property fails for the latter tickers stems from the fact that they are not large-tick stocks:Table 1 confirms that the percentage of traded volume that occurs while the spread is equal to a single tick issignificantly smaller for LBTYK and LVNTA than for the other tickers. Indeed, for a small-tick stock, themicroprice does not satisfy the properties of a fundamental price stated in the introduction: a change in the bestbid or ask price may not correspond to the microprice crossing the best bid or ask.Figure 6 indicates that the U-shape property of the stationary density can also be observed on a single-dayestimate. However, the presence of daily trends distorts the symmetry of this distribution. Averaging overmultiple days helps restore this symmetry.Figures 7–8 show that the U-shape property of the weighted estimate is not very robust w.r.t. the weight w : on our sample, it is observed for w ≤ . (the smaller is w the stronger is the U-pattern) but disappears,and even reverses, as w approaches one. On the other hand, as shown in Figures 7–8, the weighted uniform The integrals in (38) can be computed in a closed but somewhat cumbersome form. One may also be tempted to explain the lack of the U-shape property of the stationary densities of LBTYK and LVNTA by therelatively low traded volume of these stocks, which may negatively affect the quality of the estimates. However, the traded volume ofLBTYK is very close to that of VOD, and the latter ticker has U-shaped stationary density. We thank the anonymous referee who pointed it out. w . Overall, theempirical analysis supports the prediction of a U-shaped stationary density.To conclude this section, let us explain why it is very challenging to test empirically the second theoreticalprediction of this work – that the wings of the fundamental price distribution decrease during a meta-order –without using the meta-order data itself. One may be tempted to treat the time periods with pronounced trendsin the order flow as being analogous to the execution intervals of meta-orders. If this was a fair analogy, then,in principle, one could estimate the stationary price distribution over such intervals and compare its wings tothe stationary distribution estimated over the entire sample. However, there is a subtlety that makes this tasksignificantly more complicated. The theoretical conclusion of Section 2.6 that the wings of the fundamentalprice distribution decrease during a meta-order is not just a consequence of the fact that the price obtains a driftof a constant sign during a meta-order. Namely, for the conclusion of Section 2.6 to hold it is also importantthat the VWAP execution strategy does not alter the relationship between the fundamental price and the arrivalrate of the market orders. Indeed, a VWAP executor partially hides her activity by submitting her marketorders at the same rate as other market participants. As a result, during the VWAP meta-order, the overallarrival rate of the market orders remains the same (symmetric U-shaped) function of the fundamental price (see(19)) – only the balance between the buy and sell market orders changes during the execution. The latter canalso be interpreted as the assumption that, on average, the executor’s activity remains largely undetected by theother traders, so that they do not change their behavior and, in particular, the relationship between the limitorder imbalance and the arrival rate of the market orders remains unchanged. In general, there is no reasonwhy this property would hold during a typical period of positive or negative trend in the order flow, whichmeans that the distribution of the microprice in such time periods may not have the same properties as duringa VWAP meta-order. Indeed, the top left panel of Figure 6 shows the stationary distribution of the micropricemodulo tick size on a day with a positive trend in the order flow. It is clear that this distribution has more massconcentrated in [0 . , than in [0 , . . On the other hand, Figure 1 shows that the opposite is expected to occurduring a buy VWAP meta-order. Thus, in order to test the second theoretical prediction of this work, one needsto find the time intervals during which the order flow has a significant trend, while the relationship between themicroprice and the arrival rate of the market orders remains the same as in the overall sample. Needless to say,it is very challenging to detect such intervals, and it presents an interesting topic for further investigation. In this appendix, we show that there exists a Poisson random measure (PRM) M with the compensator µ ( dt, dx ) = λdt ⊗ dF ( x ) , such that (cid:90) t (cid:90) R (cid:16) { x ≥ β + ( ˜ X u − ) } − { x ≤ β − ( ˜ X u − ) } (cid:17) M ( du, dx ) (40) = (cid:90) t (cid:90) R K (cid:88) j =1 (cid:16) { x ≥ β + ( ˜ X u − ) } + { x ≤ β − ( ˜ X u − ) } (cid:17) ζ j ( ˜ X u − , u ) M j ( du, dx ) , where { M j , ζ j } are described in (7)–(8) and ˜ X t = X + αδ (cid:90) t (cid:90) R K (cid:88) j =1 (cid:16) { x ≥ β + ( ˜ X u − ) } + { x ≤ β − ( ˜ X u − ) } (cid:17) ζ j ( ˜ X u − , u ) M j ( du, dx ) + (cid:90) t σ ( ˜ X u ) d ˜ B u . Mathematically, it means that, on the business clock, the drift of the price process that is due to the meta-order is constant. j and introduce the random function g ( u, x ) := x ( β − ( ˜ X u − ) ,β + ( ˜ X u − )) ( x ) + η + ( u ) { } ( ζ j ( ˜ X u − , u )) R \ ( β − ( ˜ X u − ) ,β + ( ˜ X u − )) ( x )+ η − ( u ) {− } ( ζ j ( ˜ X u − , u )) R \ ( β − ( ˜ X u − ) ,β + ( ˜ X u − )) ( x ) , where η + u and η − u are random variables independent of everything else and distributed according to dF ( x ) F ( − β + ( ˜ X u − )) [ β + ( ˜ X u − ) , ∞ ) ( x ) , dF ( x ) F ( β − ( ˜ X u − )) ( −∞ ,β − ( ˜ X u − )] ( x ) , respectively. Let us show that the random measure ˜ M j ( du, dx ) := M j ( du, dx ) ◦ g ( u, · ) − is a PRM with the same compensator as M j . Indeed, for any predictable random function Φ (w.r.t. F which isthe completion of the natural filtration of ( ˜ M j , ˜ B ) ), we have E (cid:18)(cid:90) ts (cid:90) R Φ( u, x ) ˜ M j ( du, dx ) | F s (cid:19) = E (cid:18)(cid:90) ts (cid:90) R Φ( u, g ( u, x )) M j ( du, dx ) | F s (cid:19) = E (cid:20) E (cid:18)(cid:90) ts (cid:90) R Φ( u, g ( u, x )) M j ( du, dx ) | ζ j , η + , η − , F { M j } , ˜ Bs (cid:19) | F s (cid:21) = E (cid:20) E (cid:18)(cid:90) ts (cid:90) R (cid:16) Φ( u, x ) ( β − ( ˜ X u − ) ,β + ( ˜ X u − )) ( x ) + Φ( u, e + ( u )) { } ( f ( ˜ X u − , u )) R \ ( β − ( ˜ X u − ) ,β + ( ˜ X u − )) ( x )+Φ( u, e − ( u )) {− } ( f ( ˜ X u − , u )) R \ ( β − ( ˜ X u − ) ,β + ( ˜ X u − )) ( x ) (cid:17) λ j du dF ( x ) | F { M j } , ˜ Bs (cid:17) f = ζ j , e + = η + , e − = η − | F s (cid:21) = E (cid:34)(cid:90) ts E (cid:32)(cid:90) β + ( ˜ X u − ) β − ( ˜ X u − ) Φ( u, x ) dF ( x ) + Φ( u, η + ( u )) { } ( ζ j ( ˜ X u − , u ))(1 − F ( β + ( ˜ X u − )) + F ( β − ( ˜ X u − )))+Φ( u, η − ( u )) {− } ( ζ j ( ˜ X u − , u ))(1 − F ( β + ( ˜ X u − )) + F ( β − ( ˜ X u − ))) | ζ j [0 ,u ) , η +[0 ,u ) , η − [0 ,u ) , F { M j } , ˜ Bu (cid:17) λ j du | F s (cid:105) = E (cid:34)(cid:90) ts (cid:32)(cid:90) β + ( ˜ X u − ) β − ( ˜ X u − ) Φ( u, x ) dF ( x ) + (cid:90) ∞ β + ( ˜ X u − ) Φ( u, x ) dF ( x ) + (cid:90) β + ( ˜ X u − ) −∞ Φ( u, x ) dF ( x ) (cid:33) λ j du | F s (cid:35) = E (cid:20)(cid:90) ts (cid:90) E Φ( u, x ) λ j du dF ( x ) | F s (cid:21) . 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Market Microstructure and Liquidity , 1(2),2015.Table 1: Total traded volume (in number of shares) and the fraction of this volume that is used in the estimate(i.e., of the volume due to the trades that arrive while the bid-ask spread is at one tick and do not eat into thehigher levels of the book), for the time period November 3-7, 2014.CSCO INTC MSFT VOD LBTYK LVNTATraded volume 20,848,074 25,212,394 29,949,217 3,955,416 4,602,998 1,047,510% of traded volume used 87.25 85.23 87.03 92.09 64.49 42.7329igure 1: Stationary density f ( θ, · ) , for θ = 0 (top left), θ = 0 . (top right), θ = 0 . (bottom left), and θ = 0 . .Parameters used are: F ( x ) = [ − a,a ] ( x ) / (2 a ) , a = 1 . , ρ = 1 , α = 10 .Figure 2: Left: the expected impact on midprice I ( Q, θ ) (measured in the number of ticks) as a function ofexecuted volume Q . Right: the price resilience R ( ¯ V , θ ) as a function of total traded volume ¯ V . Parametersused are: F ( x ) = [ − a,a ] ( x ) / (2 a ) , a = 1 . , ρ = 1 , α = 10 , θ = 0 . .30igure 3: Estimated stationary density f (0 , · ) for CSCO (left) and INTC (right) over November 3–7, 2014. Thegraphs show the equally weighted estimate (blue line), the equally weighted uniform estimate (orange line),and the continuous estimate (green line).Figure 4: Estimated stationary density f (0 , · ) for MSFT (left) and VOD (right) over November 3–7, 2014. Thegraphs show the equally weighted estimate (blue line), the equally weighted uniform estimate (orange line),and the continuous estimate (green line). 31igure 5: Estimated stationary density f (0 , · ) for LBTYK (left) and LVNTA (right) over November 3–7, 2014.The graphs show the equally weighted estimate (blue line), the equally weighted uniform estimate (orangeline), and the continuous estimate (green line). 32igure 6: Estimated stationary density f (0 , · ) for MSFT over: Nov 3, 2014 (top left), Nov 4, 2014 (top right),Nov 5, 2014 (middle left), Nov 6, 2014 (middle right), Nov 7, 2014 (bottom left), Nov 3-7, 2014 (bottomright). The graphs show the equally weighted estimate (blue line), the equally weighted uniform estimate(orange line), and the continuous estimate (green line).33igure 7: Estimated stationary density f (0 , · ) for CSCO (left) and INTC (right) over November 3–7, 2014. Thegraphs show the weighted estimate with w = 1 (blue line), the weighted uniform estimate with w = 1 (orangeline), and the continuous estimate (green line).Figure 8: Estimated stationary density f (0 , · ) for MSFT (left) and VOD (right) over November 3–7, 2014. Thegraphs show the weighted estimate with w = 1 (blue line), the weighted uniform estimate with w = 1= 1