Agent-based models for latent liquidity and concave price impact
AAgent-based models for latent liquidity and concave price impact
Iacopo Mastromatteo, Bence T´oth, and Jean-Philippe Bouchaud
Capital Fund Management, 23-25, Rue de l’Universit´e 75007 Paris, France
We revisit the “ ε -intelligence” model of T´oth et al. (2011), that was proposed as a minimalframework to understand the square-root dependence of the impact of meta-orders on volume infinancial markets. The basic idea is that most of the daily liquidity is “latent” and furthermorevanishes linearly around the current price, as a consequence of the diffusion of the price itself.However, the numerical implementation of T´oth et al. (2011) was criticised as being unrealistic,in particular because all the “intelligence” was conferred to market orders, while limit orders werepassive and random. In this work, we study various alternative specifications of the model, forexample allowing limit orders to react to the order flow, or changing the execution protocols. Byand large, our study lends strong support to the idea that the square-root impact law is a verygeneric and robust property that requires very few ingredients to be valid. We also show that thetransition from super-diffusion to sub-diffusion reported in T´oth et al. (2011) is in fact a cross-over,but that the original model can be slightly altered in order to give rise to a genuine phase transition,which is of interest on its own. We finally propose a general theoretical framework to understandhow a non-linear impact may appear even in the limit where the bias in the order flow is vanishinglysmall. I. INTRODUCTION
Understanding price impact is arguably one of themost important problems in financial economics. From atheoretical standpoint, price impact is the transmissionbelt that allows private information to be reflected byprices. But by the same token, it is also the very mech-anism by which prices can be distorted, or even crash,under the influence of uninformed trades and/or fire-saledeleveraging. Price impact is also a cost for trading firms– in fact the dominant one when assets under manage-ment become substantial. As a rough order of magnitude,impact costs for individual stocks are ten times largerthan fixed costs for a firm that trades a mere 1% of the av-erage daily volume. Now, the simplest guess is that priceimpact should be linear, i.e. proportional to the (signed)volume of a transaction. This is in fact the assumptionof the seminal microstructure model proposed by Kyle in1985 [1]. This paper has had a profound influence on thefield, with over 6000 citations as of mid-2013. A linearimpact model is at the core of many different studies,concerning for example optimal execution strategies, liq-uidity estimators, agent based models, volatility models,etc.Quite surprisingly, however, the last 15 years have wit-nessed mounting empirical evidence invalidating the sim-ple linear impact framework, suggesting instead a sub-linear, square-root like growth of impact with volume,often dubbed the “square-root impact law”. One shouldhowever carefully distinguish at this point different def-initions of “price impact” that lead to very different de-pendence in volume. For example, the average impact of Empirical papers consistently measuring an impact curve closeto square root date back to 1997 [3], see also [2, 4, 5] and Refs.therein. More recent results again support the same law, see[6–8] and Figs. 1 and 2 below. a single market order is found to be a strongly concavefunction of the volume, with a significant dependence onthe microstructure (tick size, order priority, etc.). Theconcavity is primarily the result of a conditioning effect:the size of a market order very rarely exceeds the totalvolume of limit orders that sit at the opposite best price.Therefore, large market orders match large outstandingvolumes, and result in small price changes.The square-root impact law that we have been refer-ring to above is both more relevant and more universal.It concerns the average impact of a “meta-order” of size Q , which is a sequence of orders in the same directionfrom the same investor, incrementally executed in themarket using either market or limit orders, that sum upto a certain quantity Q . This definition is more relevantbecause trades are usually much too large to be executedin a single shot, but must rather be fragmented in (many)small orders that are executed progressively. The impactof a meta-order is also surprisingly universal: the square-root law has been reported by many different groups, andseems to hold for completely different markets (equities,futures, FX, etc.), epochs (from the mid-nineties, whenliquidity was provided by market makers, to the presentday electronic markets), microstructure (small ticks vs.large ticks), market participants and underlying tradingstrategies (fundamental, technical, etc.) and style of ex-ecution (using limit or market orders – see Fig. 1; withhigh or low participation rate, etc.). In all these cases,the average relative price difference I between the firstand the last trade of a meta-order of volume Q is welldescribed by the following law: I = Y σ D (cid:18) QV D (cid:19) δ , (1)where δ is an exponent in the range 0.4 – 0.7, Y is a co-efficient of order unity and σ D , V D are respectively dailyvolatility and daily traded volume [2, 4, 5], see also [6–8]. The proprietary data of CFM published in [2] corre- a r X i v : . [ q -f i n . T R ] D ec sponds to nearly 500,000 meta-orders between June 2007and December 2010 on a variety of futures contracts, andleads to δ ≈ . δ ≈ . Q ranging from a few 10 − to afew percent of the average daily liquidity. Newer data upto the end of 2012 leave these estimates unchanged. Asan illustrative example, we plot in Fig. 1 the impact curveobtained for a specific futures contract (GBP), while inFig. 2 we represent the impact exponents estimated ona set of representative futures contracts. CFM’s data onindividual stocks is also compatible with δ ≈ . − . P r i c e c h a n g e / D a il yv o l a t ili t y Executed volume / Daily volumeTransient impact of CFM trades on GBP − − − − Lag (trades)
All ordersLimit orders x . FIG. 1. Impact of CFM trades on the GBP futures mar-ket, obtained by averaging over 3 × meta-orders executedduring the period 2008–2012. The full lines with symbols inthe main plot correspond to two styles of execution of themeta-order (either with a mix of limit and market orders, orexclusively with limit orders). The average impact in the twocases appears to be the same. The soft dashed lines plottedfor comparison show power-laws with exponents 0.5 and 1.The thick dashed line indicates the result of a power-law fit,with exponent δ = 0 .
44. In the inset we plot the intra-daysign autocorrelation function for the same product averagedover all the trades of year 2012, exhibiting a power-law decaywith exponent γ ≈ . This finding is in our opinion truly remarkable, on sev-eral counts. First, because it is so universal (across mar-kets and execution strategies) and stable over time, it isindeed tempting to call it a “law” akin to physical laws.There are in our experience not that many stable em-pirical relations in financial markets, and this is one ofthe most compelling that we have encountered. Second,a square-root dependence is totally counter-intuitive, atleast at first sight, because it is non-additive . In other I m p a c t e x p o n e n t δ Persistence exponent γ AUDBRENT CDCRUDEDAX DJEURFTSE GBPNSDQFIG. 2. Plot of the impact exponent δ against the persistenceexponent γ characterizing the autocorrelation of trade signsfor a representative set of futures contracts. The impact ex-ponent δ has been computed by using a dataset consisting ofapproximately 10 meta-orders, executed during the period2008-2012. Buy orders and sell orders lead to very similarresults. The exponent γ is calculated by using intra-day datafor the year 2012, consisting of roughly 10 trades per day percontract (market-wide). Note that there not seem to be anysignificant correlation between δ and γ , in disagreement withthe model of [9], which predicts δ = γ (dashed line). words, a square-root law entails that the last Q/ ∼
40% of the first Q/
2! Butsurely if the last Q/ Q/
2, impact should become additive again. Thismeans that the only possibility for such a strange be-haviour to hold is that there must exist some memory inthe market that extends over a time scale longer than thetypical time needed to execute a meta-order. The secondingredient needed to explain the concavity of the square-root impact is that the last Q/ Q/
2. In other words, after hav-ing executed the first half of the meta-order, the liquidityopposing further moves must somehow increase. Still, itis quite a quandary to understand how such non-lineareffects can appear, even when the bias in the order flowis vanishingly small.In a previous publication [2], we proposed a minimalmodel based on the above two intuitive ingredients (mem-ory and liquidity increase), in order to rationalize the uni-versal square-root dependence of the impact. Our argu-ment relied on the existence of slow “latent” order book,i.e. orders to buy/sell that are not necessarily placed inthe visible order book but that only reveal themselvesas the transaction price moves closer their limit price.We showed, using both analytical arguments and numer-ical simulations of an artificial market, that the liquidityprofile is V-shaped, with a minimum around the currentprice and a linear growth as one moves away from thatprice. This explains why the resistance to further movesincreases with the executed volume, and provides a sim-ple explanation – borne out by numerical simulations –for the square-root impact [2]. By the same token, a van-ishing expected volume available around the mid-priceleads to very small trades having anomalously large im-pact, as indeed reflected by the singular behaviour ofthe square-root function near the origin. This has ledus to the notion of an inherently critical liquidity in fi-nancial markets. The presence of a liquidity funnel lo-calized around the mid-price is in fact a feature whichis expected in any micro-structural model encompass-ing the notions of ordered prices and market clearing,and emerges even in highly stylized “reaction-diffusion”models such as [10, 11]. Hence we expect our predictionsconcerning price impact to be rather general and robustwith respect to the precise specification of the model.Still, there is a large degree of arbitrariness in thechoices that we made to construct a statistically efficientartificial market where prices behave as random walks,and some questions have been raised about the general-ity of our results, as well as on some more subtle pointsthat were not fully clarified in [2]. In particular, ourinitial model conferred all the “intelligence” to liquid-ity takers, whereas liquidity providers were entirely pas-sive and acting randomly. In real markets, however, weknow that statistical efficiency is the result of a complex“tug-of-war” between liquidity takers – who create trendsthrough the fragmentation of their trades – and liquidityproviders – who attempt to benefit from the correlatedflow of market orders by actively increasing the liquidityopposing the flow (see the discussion in [12, 13]).The aim of the present paper is to revisit and ex-tend the model and the arguments of Ref. [2]. Westart by providing more precise numerical results aboutthe phase transition, observed in [2], between a super-diffusive (trending) market and a sub-diffusive (mean-reverting) market as the parameters of the model arevaried. This model is actually extremely interesting inits own right, as an example of a random walk in anadaptive environment, for which very few exact math-ematical results are available. We then study variousaspects of the impact of a meta-order, in particular howthe execution style affects the shape of the impact andhow the impact decays after the last trade of the meta-order. We broadly confirm the conclusions of [2], thatsuch an “ ε -intelligence” model is indeed sufficient to re-produce a concave impact function, as long as the exe-cution of the meta-order takes place on time scales much In particular by our friends and colleagues D. Farmer, J. Gatheral& F. Lillo. shorter than the renewal time of the latent order book.We show that different execution styles (i.e. aggressivemarket orders or passive limit orders) hardly affect theshape of the impact function, which demonstrates thatthe universality of impact concavity is a consequence ofthe coarse-grained properties of the supply function, butnot of the details of the microstructure.We then introduce a variation of the original settingof the model, by giving a more symmetric role to marketand limit orders for ensuring price diffusion. Limit ordersnow adapt to the market order flow and explicitly act asbuffers against further price moves. This specificationallows us to reproduce the long range correlation of thesign of limit orders found in market data, which exactlymirrors the long range correlation of the sign of marketorders. In the original version of the model [2], these limitorder correlations are totally absent and the confiningrole of the order book is purely mechanical. We find thatthe impact function is again concave in this setting, andlooks even closer to empirical data.We end the paper by providing a general theoreticalframework in which all the results discussed so far canbe qualitatively understood. We sketch some analyticalcalculations that might allow one to go beyond the nu-merical simulations and compute explicitly the variousproperties of the model (super- or sub-diffusion proper-ties, temporal dependence of the impact function, etc.).The completion of this program is however left for futurestudies.In section II we introduce the model and discuss itsproperties in the absence of meta-orders, while in sectionIII we show that a concave impact function is indeedobserved for different execution protocols. Section IVpresents the generalization of the model in which marketand limit orders are allowed to interact, while in sectionV we construct a theoretical framework in which all theresults discussed so far can be qualitatively understood.Finally, we draw our conclusions in section VI.
II. A DYNAMICAL MODEL FOR “LATENT”LIQUIDITYA. The building-blocks of the original model
The basic assumption underlying the arguments putforward in [2] is the existence of a slowly evolving latent order book storing the volume that market participantswould be willing to trade at any given price p . This latentorder book is where the “true” liquidity of the marketlies, at variance with the real order book where only avery small fraction of this liquidity is revealed, and thatevolves on very fast time scales. In particular, marketmaking/high frequency trading contributes heavily to thelatter but only very thinly to the former. This hypothesis is motivated by market data, whichdemonstrates that only a very small fraction of the vol-ume daily traded on a market is instantly available in theorder book [12]. The vast majority of the daily tradedvolume in fact progressively reveals itself as trading pro-ceeds: liquidity is a dynamical process, see [12–14],and for an early study carrying a similar message, [15].Clearly, traders tend to hide their intentions as long asthey can, as they have no incentive in giving away privateinformation too soon by adding orders to the real orderbook. In fact, the actual decision to trade at a certainprice p in the future could itself be “latent”: think forexample of a mean-reversion algorithm that would decideto sell if the price ever went up by a certain quantity. Weimagine that the volume in the latent order book mate-rializes in the real order book with a probability thatincreases sharply when the distance between the tradedprice and the limit price decreases. In particular, we pos-tulate below that the latent order book and the real orderbook coincide at the best quotes.The model for the evolution of the latent order bookis inspired from “zero-intelligence” models for the realorder book [16, 17], but with some additional featuresthat allow the price to be diffusive [2]. In such setting,the latent order book is modeled as a discrete price gridpopulated by orders of two species (buy or sell) of variablesize. Sell orders sit on the left ( bid ) side of the book,while buy orders populate its right ( ask ) side. Such abook is described by specifying for both sides and foreach price level how much volume is instantly availablefor trading, while its time evolution is dictated by threetypes of stochastic processes: • Depositions : An investor who becomes poten-tially interested in buying or selling shares at aprice p places a (virtual) limit order of a unit vol-ume at that price level. We suppose that theselimit orders arrive at rate λ per unit price, whichfor simplicity we assume to be uniform along theprice line. • Cancellations : Traders might remove orderswhich were present in the latent order book. Weassume any order to have the same probability perunit time ν to be canceled. • Trades : A buy (sell) market order might hit thebook, resulting in a trade which reduces the vol-ume available on the best ask (bid). Obviously, ifthe volume on a given price level is completely con-sumed, a price change is instantly triggered. We It is actually worth noticing that the “square-root” impact lawhas not been much affected by the development of high-frequencytrading; this is yet another strong argument in favor of the latentliquidity models. assume that market orders follow a Poisson pro-cess and denote by µ the rate at which this typeof orders arrive in the market. We neglect here thewell known activity clustering in financial markets[18], but this effect is, we believe, irrelevant for thepresent problem. The signs of these market orders,on the other hand, have long-range correlations, seebelow. The statistics of the volume of each marketorder will turn out to play an important role, asdiscussed in the next subsection.Notice that the choice µ = 1 corresponds to measuringtime in units of market orders. An estimate of µ onstock markets leads to µ = 0 . −
10 s − . In the followingwe will often refer to market order time by using thesymbol t , as opposed to τ which will label real time.Without loss of generality, we will also take the tick size w (i.e. the spacing of the price grid) to be 10 − (in, say,$).In the present version of the model, we assume the de-position rate of limit orders to be independent of the sideof the book and the cancellation rate to be independentof the sign of the order (but see section IV below). Thesign of market orders, on the other hand, is determinedby a non-trivial process, such as to generate long-rangecorrelations, in agreement with empirical findings [14].More precisely, the sign (cid:15) t of market order number t haszero mean (in the absence of a meta-order that wouldlead to a locally biased flow), (cid:104) (cid:15) t (cid:105) = 0, but is character-ized by a power-law decaying autocorrelation function: (cid:104) (cid:15) t (cid:15) t (cid:48) (cid:105) = g t − t (cid:48) ∝ | t − t (cid:48) | − γ with γ < This choice ismotivated by empirical evidence showing long range cor-relation in the market order flow, favoring an exponent γ ≈ . γ ≈ . V m.o. is an increasingfunction of the prevailing volume at the best V best . Weproposed in [2] to set V m.o. = max ( (cid:98) f V best (cid:99) , (cid:98) . . . (cid:99) means taking the integer part and f is a random Specifically, we consider the order generation prescription ofRef. [19]: we use a power-law distribution p ( L ) ∼ L − γ − for thedurations L of trends of buy or sell market orders. By taking thesign of each trend to be positive or negative with equal probabil-ity we can obtain an autocorrelation function (cid:104) (cid:15) t (cid:15) t (cid:48) (cid:105) ∼ | t − t (cid:48) | − γ .While the short-time properties of the model might depend onthis choice, its long-range behavior should be independent of thedetails of the order generation mechanism as long as the asymp-totic properties are the same. variable in [0 , P ( f ) given by: P ( f ) = ζ (1 − f ) ζ − . (2)In this way it is possible to tune the aggressivity of mar-ket orders through a single parameter ζ which allows oneto interpolate between the case where each market orderhas a unit volume ( ζ = ∞ ) and the case where each mar-ket order completely exhausts the volume at the oppositebest ( ζ = 0). Intuitively, large values of ζ (i.e. small vol-umes for each trade) decrease the impact of each trade,and therefore the volatility of the market (for a relateddiscussion, see [22]).The cancellation rate defines a time scale τ ν = ν − which is of crucial importance for the model, since this isthe memory time of the market. For times much largerthan τ ν , all limit orders have been canceled and replacedelsewhere, so that no memory of the initial (latent) orderbook can remain. Now, as we emphasized above, a con-cave (non-additive) impact law can only appear if somekind of memory is present. Therefore, we will study thedynamics of the system in a regime where times are smallcompared to τ ν . From a mathematical point of view, rig-orous statements about the diffusive nature of the price,and the non-additive nature of the impact, can only beachieved in the limit where ν/µ →
0, i.e. in marketswhere the latent liquidity profile changes on a time scalevery much longer than the inverse trading frequency. Al-though τ ν is very hard to estimate directly using marketdata, it is reasonable to think that trading decisionsonly change when the transaction price changes by a fewpercent, which leads to τ ν ∼ a few days in stocks andfutures markets. Hence, we expect the ratio ν/µ to beindeed very small, on the order of 10 − , in these markets. B. Super-diffusion vs. sub-diffusion of prices
We first investigate the statistics of price changes in ourartificial model, in particular the variogram D ( t ) definedas: D ( t ) = (cid:104) ( p ( t + t ) − p ( t )) (cid:105) , (3)where the averaging is either over t for a single trajec-tory (as with real empirical data), or over different tra-jectories for a given t – but in both cases t must bechosen (cid:29) τ ν in order to be in the stationary state. Auseful quantity is σ t = D ( t ) /t (the so-called “signatureplot”), which can be seen as a measure of the squaredvolatility on time scale t . For a purely diffusive process(e.g. the Brownian motion), σ t is strictly independent Remember again that ν is not the cancel rate in the real (visible)order book, which is extremely high, 10 s − or so, but the cancelrate of trading intentions in the latent order book, which aremuch slower. of t . A “sub-diffusive” process is such that σ t is a de-creasing function of t , signalling mean-reversion, whereasa “super-diffusive” process is such that σ t is an increas-ing function of t , signalling trends. A simple exampleis provided by the fractional Brownian motion, which issuch that D ( t ) ∝ t H , where H is the Hurst exponentof the process. The usual Brownian case corresponds to H = 1 / H > / H < /
2) is tantamount tosuper- (resp. sub-) diffusion. It may also happen thatthe process becomes diffusive at long times, i.e., σ t tendsto a finite, non-zero value σ ∞ when t → ∞ .The above dynamical liquidity model contains an in-gredient that favors super-diffusion (the long range cor-related nature of the order flow), and an ingredient thatfavors sub-diffusion (the long memory time of the orderbook itself). Let us be more explicit. When the memorytime of the order book τ ν is very short, the autocorre-lation of the price changes is dominated by the autocor-relation of the order flow. It is easy to show that fora power-law autocorrelation with exponent γ , as definedabove, the Hurst exponent of the price change is givenby: H = 12 , when γ > H = 1 − γ > , when γ < , (4)with logarithmic corrections for γ = 1. The same resultwould in fact hold for an arbitrary memory time τ ν , butwhen market orders always consume the whole volumeavailable at the best quote (i.e. when ζ → uncorrelated order flow (i.e. γ → ∞ ) in a quicklyevolving environment ( τ ν ∼ µ − ) the price process isobviously diffusive.However, the same situation of a totally uncorrelatedorder flow γ → ∞ , but now with a very slowly evolvingorder book τ ν (cid:29) µ − , turns out to be far from trivial,and leads to a strongly sub-diffusive dynamics. Intu-itively, this strong mean-reverting behaviour is explainedby the following argument: imagine that the price hasbeen drifting upwards for a while. The buy side of thebook, below the current price, has had little time to refillyet, whereas the sell side of the book is full and creates abarrier resisting further increases. Subsequent sell mar-ket orders will therefore have a larger impact than buyorders, pushing the price back down.When ζ → γ → ∞ , then H = 1 / This is actually the reason why the early zero-intelligence modelsof [16] were unsatisfactory as models of true prices. As can beseen in Fig. 11 in that paper, the price is indeed strongly sub-diffusive within these models. The need for a correlated orderflow to counter-balance this effect was in fact duly noted in theconclusion of that paper. ζ → ∞ where the executed volume is unity, simulationsshow that the price motion is actually confined , i.e.: σ t ≈ σ ∞ − c √ t , (5)in the double limit t → ∞ , ν → νt →
0, andwhere c is a constant depending upon the values of µ and λ . This result can be intuitively understoodafter realizing that σ ∞ is proportional to the stationaryvalue of the spread: in this regime the price bouncesindefinitely in the region of the bid-ask spread, whereasthe volumes of price levels outside that zone growlinearly in time, leading to a trapping effect.When ζ is finite (neither zero nor infinite), an analogywith diffusion-reaction models, studied in [10, 11, 23],suggests H = 1 / τ ν (cid:29) µ − . Within thatframework, buy and sell orders are described as parti-cles of type A and B diffusing along the infinite priceline. The market clearing condition is then modeled byassuming that buy and sell orders annihilate each otheras soon as they meet ( A + B → ∅ ), the interface betweenthe A -rich region and the B -rich region defines the posi-tion of the mid-point. This diffusion-annihilation prob-lem has been studied in details, and previous numericalresults suggest an exponent H = 1 / τ ν (cid:29) µ − , and found a result compati-ble with H = 1 / ζ , as predicted in the diffusion-annihilationmodel [25]. Note that in the order book model consid-ered here, orders do not “diffuse” directly, so the mappingto a diffusion-annihilation problem, if correct, does notseem to be trivial. We have not been able to construct aconvincing argument that would show that the two mod-els are in the same universality class, and the similaritiesbetween the two could be misleading.
C. Diffusive prices and market efficiency
From a financial point of view, both super-diffusionand sub-diffusion lead to arbitrage opportunities, i.e.strategies that try to profit from the trends or mean-reversion patterns that exist when H (cid:54) = 1 /
2. The tradingrules that emerge must be such that simple strategies arenot profitable, i.e. prices are close to random walks with H ≈ /
2, a property often called “statistical efficiency”. Within the original setting of our model, and much in A power-law fit of the results however leads to H ≈ . Whether or not this random walk behaviour is indicating themarkets are “efficient” in the sense that prices reflect fundamen-tal values is another matter. We believe that while the former the spirit of [12, 20], we have two parameters γ and ζ toplay with, that allow us to tune the relative strength oftrending effects induced by market orders and the mean-reversion forces induced by limit orders. Note that since γ is empirically found to be smaller than unity, marketsmust clearly operate in the regime τ ν (cid:29) µ − , otherwise asuper-diffusive behaviour with H = 1 − γ/ γ, ζ plane, separating a phase where the price issuper-diffusive (below that line) from a phase where it issub-diffusive (above that line). The artificial market istherefore found to be only “viable” (i.e. efficient) alonga certain line ζ c ( γ ) in parameter space. Numerically, thelocation of this critical line is approximately determinedby comparing σ t for t = 10 µ − and t = 10 µ − [2]. Thisleads to the result shown in Fig. 3, with new data.Mathematically, the claim of [2], is the following: forany γ <
1, there is a critical value ζ c ( γ ), such that thebehaviour of the price in the intermediate asymptotics regime µ − (cid:28) t (cid:28) τ ν evolves from being super-diffusive( σ t ∝ t H with H > /
2) for ζ < ζ c ( γ ) to sub-diffusive( σ t ∝ t H with H < /
2) for ζ > ζ c ( γ ). We have redoneextensive numerical simulations of the original model,and our conclusion is different. Although there is indeeda value of ζ for which the price is approximately diffu-sive for long enough times to be of practical significance,we find that the super- → sub-diffusion transition is infact only a cross-over. Running careful simulations forlong enough times (while still in the intermediate regime µ − (cid:28) t (cid:28) τ ν ), we see that the effect of long-range cor-relations in the signs of the trades always dominates atlong times, and lead to an asymptotic Hurst exponent H = 1 − γ/ γ < super- to sub-diffusiontransition truly exists in a mathematical sense. For thispurpose, we slightly modify the rule that sets the size ofmarket orders as follows: given a specific sign for a mar-ket order, its volume is set to be a power ψ of the volumeon the opposite best V best : V m.o. = max( (cid:98) V ψ best (cid:99) ,
1) (6)with ψ ∈ [0 , V best ≥ V m.o. . Clearly, larger val-ues of ψ correspond to more aggressive orders, so that property is indeed obeyed, the mechanisms that lead to statisti-cal efficiency have little to do with the activity of fundamentalarbitrageurs. See [14] for an extended discussion of this point. − − γ O r d e r agg r e ss i v i t y ζ Diffusivity estimator log (cid:16) σ ( t =1000) σ ( t =10) 101000 (cid:17) − − γ O r d e r agg r e ss i v i t y ψ . . . . . . . . . . . . . . Left ) Phase diagram for the model in the regime µ = 0 . − , λw = 5 × − s − , ν = 10 − s − . The diffusive natureof the model is assessed by considering the quantity S = log (cid:16) D ( t =10 ) D ( t =10 ) (cid:17) , so that the perfectly diffusive regime diffusioncorresponds to S = 0 (thick black line). ( Right ) The same quantity is plotted for the modified model in which the orderconsumption is controlled by the ψ exponent for the same set of parameters. In both cases, for any value of γ one can find acritical ζ beyond which the behavior of the model passes from super-diffusion to (apparent) sub-diffusion. for ψ = 0 one recovers unit execution, and the price isconfined (see Eq. (5) above). If ψ = 1, on the other hand,price is trivially super-diffusive when γ <
1. But when ψ <
1, the volume eaten by market orders is (asymptot-ically) a very small fraction of the volume at best, sug-gesting that the confining effect of the book might endup dominating the dynamics for small enough ψ . Thisis what we find numerically – see Figs. 3 and 4. Moreprecisely, we now find that the Hurst exponent H of theprice process is a continuously varying function of γ and ψ , monotonically increasing from H ( γ, ψ = 0) = 0 to H ( γ, ψ = 1) = min(1 − γ/ , / γ <
1, thereis therefore a critical value ψ c ( γ ) such that market effi-ciency is strictly recovered for µ − (cid:28) τ (cid:28) τ ν . For τ (cid:29) τ ν and γ < D. The volume distribution at the best
Finally, we have studied the volume distribution at thebest bid (or the best ask). The results are summarizedin Fig. 5, where we plot the empirical distribution thatwe obtain for various values of γ and ζ . The volumedistribution is very broad, and decays as a power lawwith a universal exponent (independent of γ and ζ ) witha value close to − .
5. One in fact finds a peak in theprobability of very large volumes, associated with pricelevels which have never been hit yet by market orders,that have therefore an average volume equal to λw/ν ,where w is the tick size. We can therefore conclude thatthe broad distribution of volumes is induced by the largedepth of the book, which forces the price process to visitprice bins with volumes ranging from values close to zero(found in the region of the spread) to values around themaximum depth (for yet unexplored price regions).We find that the distribution of queue durations is alsobroad, which is in our case due to the persistence of themarket order flow. Intuitively, a bid queue can survivea long period of time if a long stream of orders hits theask queue, but as soon as a sequence of orders of theappropriate sign is started, the bid is emptied after a fi-nite number of orders (typically of the order of log λ/ν ).However, we have not found a way to derive the appar-ently universal value − / . × − . × − . × − . × − . × − t Diffusion in ζ -model, γ = 0 . D i ff u s i o n c o e ffi c i e n t σ t = D ( t ) / t − − − − − − t Diffusion in ψ -model, γ = 0 . ζ = 0 . ζ = 0 . ζ = 0 . ζ = 1 ζ = 1 . ζ = 2 ψ = 0 . ψ = 0 . ψ = 0 . ψ = 0 . ψ = 0 . ψ = 0 . Left ) Signature plots σ t for the parameter choice µ = 0 . − , λw = 5 × − s − , ν = 10 − s − , γ = 0 . ζ . Decreasing values of ζ lead to more strongly diffusive behavior. Note however that the long-time behaviour isalways super-diffusive in this case. ( Right ) Signature plots for the modified model with power law volume consumption, forvarious values of the ψ exponent. The parameters set is the same one adopted in the left plot, except for the value of γ = 0 . ψ = 0 and ψ = 1. Note that in this case, the long time behaviour issub-diffusive when ψ (cid:46) .
8, super-diffusive for larger values of ψ and exactly diffusive for ψ = ψ c ≈ . III. THE CONCAVE IMPACT OFMETA-ORDERS
We now revisit the numerical results of [2] for the im-pact of meta-orders, and study more precisely the de-pendence of this impact on the style and “aggressivity”of the execution schedule.
A. Execution with market orders
We first define more precisely how meta-orders are in-troduced in the model, on top of the previously defined“background” order flow that builds an unbiased, diffu-sive price time series. Our choice is to introduce an extraagent (the “trader”) into the market, which buys (with-out loss of generality) Q shares within the time interval[0 , T ], by executing market orders at a fixed time rate µφ (the case of limit order execution is discussed later). Theflow of market orders is now biased, with (cid:104) (cid:15) t (cid:105) = ϕ , where ϕ is an increasing function of φ usually called participa-tion rate . The relation between the participation rate ϕ and the frequency φ depends in general on the order flowof both the trader and the background, and reduces to ϕ = φ φ when on average the trader and the rest of themarket submit individual market orders of the same vol-ume. After time T , the meta-order ends, and the marketorder flow immediately reverts to its unperturbed state.We first keep the original setting of T´oth et al. [2] andwork in a region of the plane γ, ζ , and for a range of time scales such that the price is approximately diffusive. (Infact, a concave impact can also be observed even in thesuper- and sub-diffusive phases.)We allow the trader to submit orders with a volumeextracted by a different distribution with respect to theone used by the rest of the market, in order to modeldifferent execution styles for the submission of the meta-order. Accordingly, we introduce a quantity ζ (cid:48) determin-ing the volume consumption of the trader: analogouslyto the case of ζ , we suppose that the fraction of volume f (cid:48) consumed by any of the market orders submitted by thetrader is distributed according to p ( f (cid:48) ) = ζ (cid:48) (1 − f (cid:48) ) ζ (cid:48) − .This implies that for all values of ζ (cid:48) (cid:54) = ∞ , whenever theexecuted quantity Q is fixed, the execution time T fluc-tuates according to the actual liquidity conditions of themarket. Conversely, choosing a submission protocol withfixed T requires Q to fluctuate. Only in the unit exe-cution case ζ (cid:48) = ∞ it is possible to fix at once Q and T .Because of the bias in the order flow, the average pricechange (cid:104) p T − p (cid:105) between the start and the end of themeta-order is no longer zero. The questions we want toask are:1. Is the dependence of the initial impact I = (cid:104) p T − p (cid:105) on Q concave and how does it depend on thefrequency φ ?2. Does the impact depend on the execution style (pa-rameterized here by ζ (cid:48) )?3. What happens to the price at large times after the − − − − − − γ = 0 . ζ ∈ [0 . , . F r e q u e n c y [ Sh a r e s ] − − − − − − ζ = 1 γ ∈ [0 . , . ζ (left plot) and of γ (right plot). The curves on the left plot are almost exactly superposed, while the ones on the right plot share approximativelythe same slope. The grey points on the right plot correspond to the case of independent market orders. The lines plotted forcomparison show a power-law decay with exponent − .
5. The parameters used to obtain this figures correspond to the onesused to generate Fig. 3, so that the maximum depth is given by λw/ν = 5 × . meta-order is over (i.e., what is the permanent partof the impact (cid:104) p ∞ − p (cid:105) )?We investigated market impact for several values ofthe ζ (cid:48) parameterizing the volume taken with each mar-ket order by the trader. We have considered executionvolumes in the range Q = [0 , φ have been studied in the range [0 . , Q and of φ that has been considered, wehave simulated 3 × realizations of the submission pro-cess and computed the average price change during andafter each execution. In all the cases under investigation,the impact I T was found to be a concave function of thevolume Q : I T ∝ Q δ δ < , (7)confirming the results reported in [2]. For example, weplot in Fig. 6 the results obtained for the case γ = 0 . ζ (cid:48) = ζ = 0 .
95. The dependence of the impactexponent δ on ϕ is shown in the right panel of Fig. 6,for different values of ζ (cid:48) . Note that the impact is signif-icantly less concave in the case of unit order execution ζ (cid:48) = ∞ (see right panel of Fig. 6), whereas δ is foundto be compatible with empirical data, i.e. δ ≈ . − . ζ (cid:48) , includ-ing the case ζ (cid:48) = 0, corresponding to “greedy” execution.We also notice that for low participation ratio the impactexponent drops significantly. This is due to a condition-ing effect which favors slow execution of buy meta-ordersfor negative price trajectories and a fast execution in thecase of positive price trajectory, causing a bias effect atlow φ , as better elucidated in section III C. In order to check the robustness of the results withrespect to the specification of the model, we have simu-lated the execution of a meta-order also for the ψ -modeldescribed above, in which the volume consumed by mar-ket orders is a power of the volume available at the best.We have chosen the same submission schedule as for the ε -intelligence market, and tested the same range of ex-ecuted volumes and participation ratios. Even in thiscase, we find quantitatively very similar results with re-spect to the previous case, as reported in the bottompanel of Fig. 6.The relaxation of impact after the end of a meta-orderis a particularly important topic, which has attractedconsiderable attention recently. Farmer et al. [9] arguethat a ‘fair price’ mechanism should by at play, such thatthe impact of a meta-order reverts at long times to a valueprecisely equal to the average price at which the meta-order was executed (see also [27]). This seems to be con-firmed by the empirical data analysed in [7, 8]; however,such an analysis is quite tricky, as it involves some degreeof arbitrariness in the choice of the timescale for the re-laxation of price after the end of the meta-order. Indeed,we have not been able to confirm this result on CFM’sproprietary trades. Even within our synthetic marketframework, the long time behaviour of the impact is quitenoisy. Our data suggests that the impact decays to a fi-nite value, which seems to be higher than the ‘fair price’benchmark, although we cannot exclude a slow decay toa smaller value. More specifically, we find that perma-nent and transient component of the impact obey twodifferent scalings: while the transient component of the0 P r i c e c h a n g e P r i c e c h a n g e φφ = 0 . φ = 0 . φ = 0 . φ = 1 φ = 3 . φ = 10 ζ ′ = ∞ ζ ′ = ζζ ′ = 0 φ = 0 . φ = 0 . φ = 0 . φ = 1 φ = 3 . φ = 10 ψ ′ = ψ FIG. 6. (
Top ) Temporary impact for the execution of a metaorder in the case γ = 0 . ζ (cid:48) = ζ = 0 .
95, for the set of parameters µ = 0 . − , λw = 5 × − s − , ν = 10 − s − . The right plot shows the fitted exponent for the impact function under thisparticular execution schedule (solid line), compared with the ones corresponding to different execution protocols (dashed lines).Note that except for unit execution where concavity is weaker, the value of the impact exponent δ is compatible with empiricaldata. ( Bottom ) Temporary impact for the modified model in which the ψ parameter controls the order consumption mechanism.We considered the case ψ (cid:48) = ψ = 0 .
75 and γ = 0 . µ = 0 . − , λw = 5 × − s − , ν = 10 − s − . The right plot shows the fitted impact exponent. The results that weobtain for this model are very close to the ones reported above for the ε -intelligence model. The soft dashed lines in the topand bottom left panel are plotted for reference, and indicate the scalings I ∝ Q / and I ∝ Q . impact is described by a concave law, its permanent com-ponent is linear, and hence dominates the total impactfor long enough trades. This behavior can be understoodon the basis of the arguments that will be presented insection V, where we show how the linear component ofthe impact is initially hidden by a concave transient ef-fect due to the partial adaptation of the order book tothe modified order flux. Overall, we confirm again theresults presented in [2], see their Fig. 5 (right). We doalso confirm that the initial part of the decay, just afterthe meta-order is completed, is very steep, of the type: I T + t − I T ∝ − t θ , with θ <
1. Fig. 7 shows four typicaldecay curves for market impact for different participationrates.
B. Plasticity of the order book
The shape of the latent order book plays an importantrole in determining the properties of the model, namelythe diffusion behaviour and the price impact function dis-cussed above. This will be substantiated more preciselyin the last section of this paper (see Eqs. (18) and (23)).The stationary shape of the latent order book when nometa-order is present is represented in Fig. 8, for a choiceof parameters such that the price dynamics is approxi-mately diffusive ( γ = 0 . ζ = 0 .
95, while µ = 0 . − , λw = 5 × − s − , ν = 10 − s − ). More generally, wealways find that the average book volume is an increasingfunction of the price level p − p (where p is the currentprice). The book profile ρ ( p ) increases from ρ ( p ) = 0at the mid-price, to the asymptotic value ρ ( ±∞ ) = λ/ν .The size of the “liquidity hole” around p is determined1 P r i c e c h a n g e Market order time t
600 900 1500 φ = 1 φ = 0 . φ = 0 . φ = 0 . T = 300 µ − and stochastic executedquantity Q for different participation rates. We have used γ = 0 . ζ = ζ (cid:48) = 0 . µ = 0 . − , λ w = 5 × − s − , ν =10 − s − , while the simulation consisted of 3 × realizationsof the submission process. Even though the impact is concaveat small times, it finally crosses over to a linear (in time)regime. The relaxation part is represented in semi-logarithmicscale. by the price scale p (cid:63) ∝ (cid:113) µ λ ν . Therefore, the small can-cellation limit corresponds to the limit of large latentvolume λ/ν → ∞ and large liquidity hole p (cid:63) → ∞ .As shown in [2, 28], the dynamics of the average shapeof the order book can be approximately described, in thediffusive regime, by the equation ∂ (cid:104) ρ ( p, t ) (cid:105) ∂t = D ∂ (cid:104) ρ ( p, t ) (cid:105) ∂p − ν (cid:104) ρ ( p, t ) (cid:105) + λ , (8)where D is the price diffusion constant (or volatilitysquared). This implies that in the stationary state onehas: (cid:104) ρ ∞ ( p ) (cid:105) = λν (cid:16) − e − p/p (cid:63) (cid:17) , (9)where p (cid:63) = (cid:113) D ν . This prediction is compared in Fig. 8against simulated data, with no free fitting parameter.It is interesting to study how the shape of the orderbook is progressively deformed by the order flow, andhow this determines the impact of further trades. Wetherefore also show in Fig. 8 the asymptotic shape ofthe latent book after a long buy meta-order is executed.One can clearly see how the bid side and the ask side ofthe book become asymmetric , while the bid-ask spreadremains substantially unchanged (inset of Fig. 8). The volume on the ask (sell) side of the book increases, lead-ing to a smaller expected price impact for trades in thesame direction, and therefore a concave impact. Con-versely, the volume on the bid (buy) side of the bookdecreases, which implies that the impact of a sell ordercoming at the end of a buy meta-order will be substan-tially larger than both the impact of an additional buyorder and the average impact of buy/sell orders in equi-librium. Hence, impact is expected to relax when a buymeta-order stops, because a subsequent balanced orderflow will have an average negative impact on the price. V o l u m e [ s h a r e s ] PriceBook plasticity0510152025 0 0.1 0.2
Unperturbed volumePerturbed ask volumePerturbed bid volumePhenomenological model
FIG. 8. Average shape of the book at equilibrium before(thick solid line) and after (dashed lines) a long buy meta-order executed at a frequency φ = 0 . ζ (cid:48) = ∞ . Weconsidered the same parameters as in Fig. 6, except for alarger value of ν = 10 − µ . The soft dashed line indicatesthe prediction of Eq. (8) for the unperturbed state of thebook. The plot shows that after a long buy meta-order theask levels are on average more populated than the bid ones.The presence of an offset between bid and ask volume in theperturbed case also evidences that the bulk of the book canstore information about the past order flow. C. Execution with limit orders
The execution of a large meta-order in a real marketusually involves a (sometimes large) fraction of limit or-ders. Interestingly, empirical data indicate that even inthis case the impact function is a concave function of thevolume, quite comparable to the impact of market orderexecution (see Fig. 1). This requires any reliable modelof market impact to predict concave impact function re-gardless of the execution protocol (via market, limit or-ders, or both). This is again an indication that marketimpact is a “coarse-grained” effect that depends on the2true liquidity and not on market microstructure. This iswhy we want to ascertain that the same is true withinour numerical model and study the impact of a buy meta-order executed through limit orders only.In fact, modelling limit order execution is more com-plex than describing market order execution, as the for-mer involves several possible choices: at which price levelshould the orders be submitted? How should the aver-age lifetime of the orders sitting in the book be fixed?What is the volume deposed with each submission? Ourchoice is to consider for simplicity a stylized executionstrategy for limit orders, mimicking as closely as possiblethe ζ -execution strategy described above for the case ofmarket order execution. We expect our results not tobe strongly sensitive to the precise specifications of theexecution protocol. Accordingly, we have introduced ontop of the unperturbed flow of the ε -intelligence modelan extra agent submitting limit orders at the best bid fora volume equal to V l.o. = max( (cid:98) f V best (cid:99) , , (10)with a constant, deterministic fraction f of the volumeof the best bid. As in the market order case, we stud-ied depositions occurring at a rate µφ . The execution isthen interrupted as soon as the cumulated volume exe-cuted exceeded a target volume Q . We have consideredthe same background as in the market order submissioncase (i.e., a market approximately diffusive with γ = 0 . ζ = 0 .
95) and averaged our results over 3 × real-izations of the submission process. We have also assumedthat orders are never canceled from the latent order book,implying that, although the orders might disappear fromthe real order book, they are reinserted as soon as thebook moves close to their original price level.The remarkable result is that even with limit-orderexecution we still measure concave impact curves. Forintermediate participation rates ( φ from 0.1 to 1), wealso found price changes similar to the ones measuredfor a market order execution (see Fig. 9) . This resultindicates that within the present framework market im-pact should be regarded as a property stemming from themechanism with which the market clears volume imbal-ances, rather than a feature depending upon the detailsabout how such imbalances are created. In this respect,the concave shape of market impact should be regardedas a universal feature reflecting the regime of critical liq-uidity provision in which our synthetic market operates. At small execution frequencies ( φ < . Q for a buy meta-order executed throughmarket orders favors negative price trajectories, as a trader with ζ (cid:48) (cid:54) = ∞ tends to wait for the price to uptrend in order to exe-cute large volumes at the ask. The inverse effect is measured forthe (buy) limit order execution case, as a negative price swing isrequired in order to clear orders sitting at the bid. This favorspositive price trajectories, thus biasing the price change up. P r i c e c h a n g e Executed volumeTransient impact for limit execution . . . . φ Impact exponent φ = 0 . φ = 0 . φ = 0 . φ = 1FIG. 9. Temporary impact induced by a meta-order executedthrough the deposition of limit orders. We have consideredsubmissions of volumes equal to a fraction f = 1 / µ = 0 . − , λw =5 × − s − , ν = 10 − s − . As in Fig. 6, soft dashed linesindicate the reference scalings I ∝ Q / and I ∝ Q . IV. AN ALTERNATIVE MODEL:“STIMULATED LIQUIDITY REFILL”
The results presented above pertain to the originalmodel of T´oth et al. [2], and broadly confirm the mainfinding of that paper, i.e. that the impact of meta-ordersis indeed concave, and in quantitative agreement withempirical data. However, this model has been criticizedon the basis that price efficiency is ensured by tuningthe ζ parameter that governs the statistics of market or-ders only, while assuming limit orders to be essentiallyrandom and passive. Surely one expects that price effi-ciency actually results from a subtle “tit-for-tat” balancebetween the flow of market orders and the correspondingcounterflow of limit orders – see e.g. [13, 14, 29, 30] forempirical evidence. Although the aim of [2] was to showthat one could build a simple latent order book modelthat is consistent both with price diffusion and with aconcave impact, we agree that moving closer to reality isindeed necessary to make the story more compelling.To that effect, we now present a model where priceefficiency is maintained through a “stimulated liquidityrefill” mechanism, whereby market orders attract a liq-uidity counterflow. More precisely, we posit that after amarket order of sign (cid:15) , the probability for the next limitorder to be on the ask (+) or on the bid ( − ) side of the3order book is biased as: P ± ( (cid:15) ) = 1 ± α(cid:15) , (11)where α ∈ [0 ,
1] is a new parameter describing the limitorder flow reaction to market orders. The statistics of themarket order flow is still captured by the two parametersdefined above: 0 < γ < ζ describesthe aggressivity of market orders (i.e. the fraction of theopposite volume against which they execute).The model studied in the previous sections correspondsto α = 0, i.e. a complete decoupling between market or-der flow and liquidity provision. For α >
0, on the otherhand, more volume is on average placed on the ask sideof the book after a buy market order, and vice-versa forsell market orders. This prescription for limit order depo-sition reproduces the empirically known correlation be-tween the signs of market orders and limit orders [29], andthe long range correlation of the limit order flow [21, 30].Notice that even though limit orders are described by ashort memory process, the induced correlation betweenlimit orders is effectively long range due to their interac-tion with market orders.Now, choosing a value of ζ such that for α = 0 themarket is super-diffusive, one can study how increasingvalues of α progressively decreases the positive autocor-relations induced by the market orders, and eventuallyleads to an approximately diffusive price for a “critical”value of α . This is summarized in Fig. 10, where a phasediagram analogous to the one of Fig. 3 is shown in theplane γ, α for the specific case ζ = 0 .
4, with a crossovervalue α c ( γ ) for which the market is approximately effi-cient.Finally, we have simulated the execution of meta-orders within the present specification of the model andcalculated the corresponding price impact. We again ob-tain a strongly concave shape of the impact as a functionof the size of the meta-order, I T ∝ Q δ , as shown in theinset of Fig. 11. Note that the value of the exponent δ ≈ . − . more concave than in the original model, with an exponent δ ≈ . − . δ ≈ . − . − . − − . . . (cid:16) σ ( t =1000) σ ( t =10) 101000 (cid:17) . . . . . γ . . . . L i m i t o r d e r i m b a l a n c e p a r a m e t e r α FIG. 10. Phase diagram for the model with correlated limitorders in the plane γ, α . We used the parameters ζ = 0 . µ = 0 . − , λw = 5 × − s − , ν = 10 − s − . The diffusionbehaviour is estimated as in Fig. 3. We again find, withinthis setting, a crossover line that separates sub- and super-diffusion regimes. exponent δ : we will show that the more prompt is theresponse, the more concave is the behavior of the impactfunction. V. A GENERAL FRAMEWORK FORMARKOVIAN ORDER BOOKS
We now outline a general theoretical framework inwhich the results presented in the above sections can bequalitatively justified. In order to do this, we construct afaithful representation of our synthetic market allowingus to relate the main properties of the price process tothe ones of the order book. In particular we will be ableto relate the dynamic properties of the price (e.g., marketimpact and diffusivity) to the dynamic properties of thefirst levels of the order book. We will show that the onlyhypothesis needed to do this is that the evolution of theorder book is Markovian. Such an hypothesis is triviallyfulfilled by our synthetic market (the state of the orderbook after an event only depends on its shape before theevent). Notice that, even though the order book itselfstores no memory about its past, the long memory of thesign process (cid:15) t induces indirectly long-ranged correlationson the state of the book.For the sake of clarity, we will first present the resultspredicted by the propagator model [20], which is the sim-4 P r i c e c h a n g e Executed volumeTransient impact for ζ -execution mechanism . . . φ Impact exponent φ = 0 . φ = 0 . φ = 0 . φ = 1 φ = 3 . φ = 10FIG. 11. Temporary impact for the execution of a meta-orderin a model with correlated limit orders. We have considered α = 0 . γ = 0 . ζ (cid:48) = ζ = 0 .
4, corresponding to an ap-proximately diffusive price dynamics. The choice of the otherparameters is µ = 0 . − , λw = 5 × − s − , ν = 10 − s − .As in Fig. 6, we use soft dashed lines in order to display thereference scalings I ∝ Q / and I ∝ Q . Impact is stronglyconcave, with a weak dependence of the impact exponent onthe participation rate. The inset shows the dependence of theimpact exponent δ upon the participation rate φ . plest setting where one can address the issues of marketefficiency and anomalous impact. In fact, even though the predictions of the propagator model and of the above ε -intelligence model are quite different, the structure ofthe two models are in fact closely related. The anal-ogy between the equations governing price diffusion andmarket impact for the two family of models will turn outto be useful in understanding some general properties ofMarkovian order book models. A. Linear models of price impact
The propagator model assumes that the relation be-tween sign of trades and price changes can be describedby a linear relation of the form [14, 20] p t = p + t − (cid:88) t (cid:48) =0 G t − t (cid:48) (cid:15) t (cid:48) , (12)where p t represents the mid-price after trade t and (cid:15) t isa stochastic term denoting the sign of the market ordernumber t . The propagator G t − t (cid:48) describes how a tradeexecuted at time t (cid:48) influences the price at a subsequenttime t (i.e., it is assumed that G t − t (cid:48) = 0 for t < t (cid:48) ). Thisimplies that the properties of the price p t are completelyspecified by the properties of the market order flux (cid:15) t .In order to describe the scenario of our synthetic market,we will take (cid:104) (cid:15) t (cid:105) = ϕ (13) (cid:104) (cid:15) t (cid:15) t (cid:48) (cid:105) − (cid:104) (cid:15) t (cid:105)(cid:104) (cid:15) t (cid:48) (cid:105) = (1 − ϕ ) g t − t (cid:48) . (14)This allows us to model the long range correlation ofmarket orders through the term g t − t (cid:48) ∼ | t − t (cid:48) | − γ togetherwith the execution of a meta-order with a participationrate ϕ . The mean and variance can be easily calculated,and result in: (cid:104) p t − p (cid:105) = ϕ t − (cid:88) t (cid:48) =0 G t − t (cid:48) (15) (cid:104) ( p t − p ) (cid:105) − (cid:104) ( p t − p ) (cid:105) = (1 − ϕ ) t − (cid:88) t (cid:48) ,t (cid:48)(cid:48) =0 G t − t (cid:48) G t − t (cid:48)(cid:48) g t (cid:48) − t (cid:48)(cid:48) . (16)Eq. (15) expresses the impact of a meta-order executedat the participation rate ϕ , while Eq. (16) determinesthe diffusion properties of the model. In particular theprocess is diffusive if and only if the sum in Eq. (16) islinear in t . This condition fixes the large time behaviorof the propagator, relating its shape to the correlation ofthe order flow: as shown in [20] the propagator model isdiffusive if and only if G t ∼ t − β with β = − γ (see alsothe discussion in [14]). It will also be convenient to decompose the marketimpact as (cid:104) p t − p (cid:105) = ϕ G ∞ t + ϕ t − (cid:88) t (cid:48) =0 ( G t − t (cid:48) − G ∞ ) , (17)in order to distinguish the linear contribution to the im-pact from the decaying one, which we will call transient .Two scenarios for the market impact are compatible withthis framework:5 • The transient impact term is integrable . Thesecond term then converges to a finite constant andthe total impact is dominated at large times by thelinear term of equation (17). • The transient impact term is not integrable .The total impact at large times is a sum of lin-ear and transient contributions. In particular if theterm G ∞ is zero, only the transient component sur-vives.Since the diffusivity of prices requires G t ∼ t − β , onehas G ∞ = 0. It results that at large times (cid:104) p t − p (cid:105) ∼ ϕ t γ . Inserting Q = ϕt , one then finds for the impact: I ∼ Q γ ϕ − γ . Realistic values of γ (e.g. γ ≈ .
5) thenimply that for this model the impact exponent is δ ≈ .
75, with a weak ϕ . dependence of the impact on theparticipation rate, but at odds with the empirical resultspresented in [2, 4, 5] where δ is closer to 0 . slowly adapts tothe presence of a bias in the order flow. Notice in factthat the minimal ingredients in order to have anomalousimpact are (i) the fact that the bias ϕ is reduced andeventually absorbed ( (cid:104) ∆ p t (cid:105) = ϕ G t − → t large)and (ii) the fact that it takes a long time to do it (i.e.the propagator G t is non-integrable). From this pointof view, anomalous impact stems from market efficiencyand from the presence of a long range correlated orderflow. B. A Markovian model for the latent order book
The model described in the above section is aneffective one, which does not take into account the factthat in a double auction market the price moves aredetermined by the local condition of the order book. Theeffect of a market order indeed depends upon the volumeat the best, whereas limit orders and cancellations canchange prices even in absence of trades. Hence, if thedescription provided by the propagator model is at leastqualitatively correct, it means that the kernel G t − t (cid:48) hasto be thought of as an effective quantity incorporating aremarkable amount of information about the order bookstructure, rather than a fundamental property of themodel.A more accurate description of the price process, tak-ing into account the one dimensional structure in whichthe price diffuses, can be formulated in the referenceframe of the last execution price after trade number t (which we denote by (cid:96) t ). The best bid price just beforetrade t + 1 is b t and the best ask price just before trade t + 1 is a t . Note that if the book does not evolve betweentrade number t and just before trade number t + 1, theneither a t or b t is equal to (cid:96) t : the former case is when (cid:15) t = + and the latter when (cid:15) t = − . If between the twotrades some activity has taken place in the book, then (cid:96) t is in general not equal to either of them. With thisconvention, we show in appendix A that for a Markovianevolution of the book one has: (cid:104) ∆ (cid:96) t (cid:105) ϕ = (cid:104) π t (cid:105) ϕ + ϕ (cid:104) s t (cid:105) ϕ ∆ (cid:96) t := (cid:96) t +1 − (cid:96) t , (18)where the average (cid:104) . . . (cid:105) ϕ is over all possible evolutions ofthe order book, and the mid-price π t and the half-spread s t are given by: π t = a t + b t s t = a t − b t . (20)Note that because prices are counted from (cid:96) t , π t tends tobe positive after a sell and negative after a buy.Eq. (18) is analogous to Eq. (15) for the impact. It ex-presses the fact that under a constant bias the imbalanceparameter ϕ is linearly coupled to the spread, while theaverage asymmetry of the book is captured by the term π t , which is zero by symmetry when ϕ = 0. One expectsthat in the presence of a positive bias, trades are morelikely to happen at the ask, and therefore (cid:104) π t (cid:105) ϕ is neg-ative and partly compensates the second term. In orderfor the model to describe strictly anomalous impact, oneshould impose the “slow absorption” condition, analo-gous to the one of the previous section, which in thiscontext would read: (cid:104) π t (cid:105) ϕ + ϕ (cid:104) s t (cid:105) ϕ ∼ t − β , (21)with β > (cid:104) π ∞ (cid:105) ϕ + ϕ (cid:104) s ∞ (cid:105) ϕ = 0 . (22)These conditions express the fact that in order for the im-pact to be truly anomalous, the book should accumulatean asymmetry large enough to exactly compensate thespread term which is linearly coupled to the bias. Other-wise, the impact must be linear in ϕ and no asymptoticconcavity can be present. The speed at which such asym-metry forms should then control the impact exponent: aquickly adapting book would give an integrable contribu-tion ( β >
1) to the average price change, and a boundedimpact, whereas the presence of long memory in the or-der book would generate a non-trivial impact exponent δ = 1 − β .The diffusion properties of the model (correspondingto Eq. (16) in the propagator model) can be obtainedfrom the autocorrelation function:6 (cid:104) ∆ (cid:96) t ∆ (cid:96) (cid:105) ϕ − (cid:104) ∆ (cid:96) t (cid:105) ϕ (cid:104) ∆ (cid:96) (cid:105) ϕ = s (1 − ϕ ) (cid:18) (cid:104) π t (cid:105) ϕ, + − (cid:104) π t (cid:105) ϕ, − (cid:19) + s (1 − ϕ ) (cid:18) (cid:104) s t (cid:105) ϕ, + + (cid:104) s t (cid:105) ϕ, − (cid:19) g t (23)where the symbol (cid:104)·(cid:105) ϕ,(cid:15) indicates an average conditionalto the sign of the first trade (cid:15) and s denotes the sta-tionary value of the half-spread. Again, the details aboutthe derivation of these equations can be found in Ap-pendix A. Notice that these results are independent ofthe particular choice of dynamics for the book; the infor-mation about the time evolution of the book (or equiv-alently, the information about what happens in betweenmarket orders) is fully encoded in the averages (cid:104) . . . (cid:105) ϕ and (cid:104) . . . (cid:105) ϕ,(cid:15) . C. Comparison between the two models and openproblems
Unlike for the propagator model, we now have two de-grees of freedom to determine the average price change:the bias ϕ is coupled to the quantity (cid:104) s t (cid:105) (equivalently,each unbalanced buy trade pushes price up by half of thespread), while the mid-price (cid:104) π t (cid:105) is independent of thebias, and accounts for the average pressure due to thebook shape. In particular if more volume is available onthe ask side, (cid:104) π t (cid:105) is negative because large price changesare more likely to occur on the bid side, where the den-sity of orders is smaller. This is consistent with what wehave numerically shown in section III: in our syntheticmarket a bias in the order flow is partially compensated,while the book slowly relaxes to a perturbed stationaryvalue in which more volume sits on the ask side of thebook. In the numerical model, the slow relaxation of thebook takes place in the regime µ − (cid:28) τ (cid:28) τ ν , withinwhich an anomalous response can indeed build up.Another important difference between the propagatormodel and the ε -intelligence model concerns the impactexponent. In the propagator model, imposing that pricesare diffusive uniquely fixes the long time behavior of G t − t (cid:48) in terms of the correlation of the flow, as recalledabove. This then immediately fixes the impact exponentas δ = (1 + γ ) /
2. This is the case because G t − t (cid:48) controlsboth the mean and the variance of the price process. Un-der our general order book model, instead, we now dealwith two independent quantities subject to two differentboundary conditions ( (cid:104) π t (cid:105) ϕ,(cid:15) and (cid:104) s t (cid:105) ϕ,(cid:15) ). This givesmore flexibility to the model, which means that there isno longer a unique relation between the correlation ex-ponent γ and the impact exponent δ , which will dependupon the details of the order book dynamics.It is interesting to notice that within our framework theobject manipulated in order to study market impact isthe average price change (cid:104) ∆ (cid:96) t (cid:105) ϕ , rather than the impactitself. In this language the concavity of market impactcorresponds to the fact that once a bias is added to theflow of orders, (cid:104) ∆ (cid:96) t (cid:105) ϕ will progressively decrease such as to compensate (partially or completely) the bias. Inother words, this point of view suggests the idea that theexpected price change (cid:104) ∆ (cid:96) t (cid:105) ϕ is a Lyapunouv function ofthe order book dynamics, as required by the assumptionof market (statistical) efficiency: once some predictabil-ity (the bias in the order flow) is introduced, one expectsthat the dynamics of the market will remove arbitrageopportunities . Simulation results support this inter-pretation, as demonstrated by Fig. 12, where we showthat after a transient following the beginning of the meta-order, the average price change relaxes to a non-zero sta-tionary value (as the impact has a residual permanentcomponent). Once the meta-order is over, another tran-sient follows and finally at large times the average pricechange reverts back to zero. As a final note, we remarkthat the ε -intelligence market presented in section II ful-fills all the hypotheses required for the theoretical frame-work presented in this section to hold. In other words, weshould in principle be able to provide an exact descriptionof our synthetic market. However, completing this pro-gram is technically difficult and goes beyond the scope ofthis paper. Whereas Eq. (23) relates the evolution of av-erage price at time t + 1 with the expected best quotes attime t , one would need to solve an infinite set of relationsbetween the price gap at level n and time t + 1 with thegap at level n + 1 and time t . This would correspond incontinuous approximation to a partial differential equa-tion for the dynamics of order book. We hope to comeback to this problem in a future work. This can be rigorously shown in other market models such as theMinority Game setting of [31]. However in the setting of the Mi-nority Game the expected price change is reduced by the actionof market participants which by construction adapt in order toremove predictability from the market. In this context the rea-son for such decrease is purely mechanical, because sequence oforders on one side of the market tend to hit progressively highervolumes. The fact that the ε -intelligence agents choose how much liquidityto take according to the volume on the best does not introduce adependence on the book of the process (cid:15) t : although the volumetaken is conditioned to the book shape, the sign of the orders isnot. Another necessary condition for the validity of this descrip-tion is the absence of trades through (i.e., trades hitting multipleprice levels), which are absent by construction in an ε -intelligencemarket. The only caveat which should be considered is that oneneeds ζ (cid:48) = ζ : the market orders associated with the meta-orderand with the environment should consume liquidity in the sameway if one wants to describe the modified flow through Eqs. (13)and (14). − . − . . . . tT = 3000 µ − Average price change during a meta-order of variable size − . − . . . . tT = 300 µ − φ = 0 . φ = 0 . φ = 0 . φ = 0 . φ = 0 . φ = 0 . FIG. 12. Average price change during and after the execution of a meta-order for the same choice of parameters as in Fig. 7with T = 3000 µ − (left plot) and with T = 300 µ − (right plot) . The area under the curve represents the price impact of themeta-order. The figures show the transition to the asymptotic linear regime occurring for extremely long orders ( T ∼ µ − ),as opposed to the initial transient regime which causes concave impact. Also notice that the dynamics always tends to decreasethe average price change, as it is expected in a market in which arbitrage opportunities are reduced as a consequancy of tradingitself. VI. CONCLUSIONS
The aim of this work was to revisit and clarify the“ ε -intelligence” model of T´oth et al. [2], that was pro-posed as a minimal framework to understand the surpris-ing non-additive, square-root dependence of the impactof meta-orders in financial markets. The basic mecha-nism, substantiated by the analytical and numerical re-sults of [2], is that most of the daily liquidity is “la-tent” and furthermore vanishes linearly around the cur-rent price, as a consequence of the diffusion of the priceitself, which depletes the nearby liquidity. Still, the nu-merical implementation of this idea requires several extraspecifications that are to some extent arbitrary, and therobustness of the scenario of T´oth et al. needed to beascertained.Our conclusions broadly support the universality ofthe results reported in [2], which is in itself importantsince the square-root dependence of the impact is em-pirically found to be independent of the market, epoch,microstructure, execution style, etc. It would be con-ceptually difficult to understand this universality if thetheoretical results depended crucially on the microscopicspecification of the model. This is in fact, in our opinion,one of the difficulty of the “fair price” scenario of Farmeret al. [9] which crucially depends, among other things,on the shape of the order-flow autocorrelation function(see for example Fig. 2, which shows that the prediction δ = γ is not supported by our data on futures).We have proposed and studied a variant of the model of T´oth et al. where market efficiency does not rely en-tirely on the statistics of market orders (as in the orig-inal version [2]), but rather comes from the interactionbetween the flow of market order and the “tit-for-tat”reaction of limit orders, which tend to replenish the sideof the book that is under pressure. This specification isfar more realistic than the original one, and in fact allowsus to account for the long-range correlation of the sign ofboth market and limit orders, an empirical fact that wasnot reproduced in [2]. We find that the impact is evenmore concave in volume than in the original version ofthe model, and in fact closer to empirical results.We have also investigated different execution protocols,in particular one where the meta-order is executed usinglimit orders only , with the same qualitative result. Thisis an important finding, because there is a commonlyheld view that the impact of market orders is fundamen-tally different from that of limit orders. Although this iscorrect at the level of single trades [30], empirical resultssuggest that the impact of meta-orders depend very littleon the ratio of market to limit orders used for execution(see Fig. 1).Finally, we have shown that the transition from super-diffusion to sub-diffusion reported in [2] is in fact a cross-over that depends on the time scale over which the diffu-sion behaviour is probed. Although not hugely relevantfor practical applications, this issue is of some impor-tance from a theoretical point of view, as our model is anexample of a random walk in an adaptive environment,for which very few mathematical results are available.8We have shown how the original model can be slightlyaltered in order to give rise (at least numerically) to agenuine phase transition between a super-diffusive anda sub-diffusive phase, such that purely diffusive motionis only realized on a co-dimension one sub-space of theparameters. It would be very interesting to obtain ana-lytical results on this transition, on the time-dependentshape of the latent order book and on the impact of meta-orders within these simple models of markets. We haveprovided in the last section of this paper some ingredientsthat may enable one to achieve this program, within theframework of Markovian latent order books. As a generalresult, we have shown that anomalous, non-additive be-haviour of impact requires that the liquidity buffer adaptsslowly to the order flow [31], in such a way that theasymptotic price change induced by a meta-order van-ishes. This suggests that expected price change acts asa Lyapunouv function of the order book dynamics, andas such is deeply related to market efficiency: once somepredictability (the bias in the order flow) is introduced,one should expect that the dynamics of the market willact to remove arbitrage opportunities. At least concep-tually, this is close to the arguments put forth in [9, 27],although both the detailed ingredients and the conclu-sions differ in the two approaches. These ideas, and theirprecise relation with the “microstructure invariance” ofKyle & Obizhaeva [6], would be well worth elucidatingfurther.By and large, our study lends strong support to theidea that the square-root impact law is a very genericand robust property, and requires very few ingredientsto be valid. It is expected to hold in any market, pro-vided the correlation time of the latent liquidity is muchlonger than the inter-transaction time. We believe thatthe impact on the implied volatility of option markets, forexample, will show a similar concavity. This would sup-port the use of this square-root impact law to discountthe expected cost of liquidation from the mark-to-marketvalue of positions, as advocated in [32]. ACKNOWLEDGMENTS
We warmly thank our collaborators J. De Lataillade,C. Deremble, Z. Eisler, J. Kockelkoren and Y. Lemp´eri`erefor many very useful discussions. We have also benefitted from interesting comments and suggestions by R. Beni-chou, X. Brokmann, J. Donier, D. Farmer, J. Gatheral,A. Kyle, C. Lehalle, F. Lillo, M. Potters and H. Wael-broeck.
Appendix A: The Markovian book model: mean andcorrelations
In section V B we presented an order book model con-structed in the moving reference frame of the last exe-cution price (cid:96) t (where t labels trade time), and linkedits properties to the ones of the sign process (cid:15) t . Herewe want to show that Eqs. (18) and (23) describing theevolution of the last execution price can be derived byusing basic properties of a Markovian order book, whichis defined as the one in which the state right before thetrade t + 1, denoted by ρ (cid:15)t +1 , depends just upon the statebefore trade number t , ρ (cid:15)t , and upon the sign of the t -thtrade (cid:15) t . This condition can equivalently be written as p ( ρ (cid:15)t +1 | ρ (cid:15) , . . . , ρ (cid:15)t , (cid:15) , . . . , (cid:15) t ) = p ( ρ (cid:15)t +1 | ρ (cid:15)t , (cid:15) t ) . (A1)In order to derive a master equation for p ( ρ (cid:15)t | ρ (cid:15) , (cid:15) ), oneneeds to specify the statistics for the process (cid:15) t , whichwe assume to be defined by the relations p ( (cid:15) t ) = 1 + ϕ(cid:15) t p ( (cid:15) t , (cid:15) ) = ϕ (cid:18) (cid:15) t (cid:19) (cid:18) (cid:15) (cid:19) (A3)+ ϕ (1 − ϕ ) (cid:18) (cid:15) t (cid:19) (cid:18) (cid:19) + (1 − ϕ ) ϕ (cid:18) (cid:19) (cid:18) (cid:15) (cid:19) + (1 − ϕ ) (cid:18) g t (cid:15) t (cid:15) (cid:19) . This describes a market in which a fraction ϕ of tradersis submitting orders of fixed, positive sign, while the re-maining 1 − ϕ are correlated among themselves (althoughuncorrelated with the buyers). These are exactly the con-ditions specifying a meta-order submission process underthe ε -intelligence model (as described in section III). Inthis regime, one can derive a master equation for theevolution of the book via p ( ρ (cid:15)t +1 | ρ (cid:15) , (cid:15) ) = (cid:88) ρ (cid:15)t ,(cid:15) t p ( ρ (cid:15)t | ρ (cid:15) , (cid:15) ) p ( ρ (cid:15)t +1 | ρ (cid:15)t , (cid:15) t ) p ( (cid:15) t | (cid:15) ) (A4)= (cid:88) ρ (cid:15)t p ( ρ (cid:15)t | ρ (cid:15) , (cid:15) ) (cid:20) p ( ρ (cid:15)t +1 | ρ (cid:15)t , +) + p ( ρ (cid:15)t +1 | ρ (cid:15)t , − )2 + p ( ρ (cid:15)t +1 | ρ (cid:15)t , +) − p ( ρ (cid:15)t +1 | ρ (cid:15)t , − )2 (cid:18) ϕ + g t (1 − ϕ ) (cid:15) ϕ(cid:15) (cid:19)(cid:21) , which is an evolution equation for the probability ofobserving a book in a specific configuration at a given instant of time, given a starting condition ρ (cid:15) and a9sign (cid:15) for the first trade. The master equation for theunconditional probability p ( ρ (cid:15)t +1 | ρ (cid:15) ) can be analogouslyobtained (either by summing Eq. (A5) by (cid:15) with theappropriate weights p ( (cid:15) ), or by directly using Eq. (A4)in order to obtain the evolution equation). In any case,the equation for the unconditional evolution correspondsto (A5) with the substitution g t = 0. Notice that inEq. (A5) the effect of market orders has been integratedout, and is kept into account through the terms pro-portional to ϕ and g t . Interestingly, even though themarket order process can have long memory, Eq. (A5)only couples subsequent times.By defining the mid-price π t and the half-spread s t , one can use Eq. (A5) to calculate the evolution oftheir averages. In particular Eq. (18) can be obtained bymultiplying the master equation for the unconditionalaverage by ∆ (cid:96) t and summing over the book configura-tions ρ (cid:15)t +1 . In order to finally recover Eq. (18), one hasthen to use the fact that (in absence of trade through)it holds (cid:88) ρ (cid:15)t +1 ∆ (cid:96) t p ( ρ (cid:15)t +1 | ρ (cid:15)t , +) = a ( ρ + t ) = a t (A5) (cid:88) ρ (cid:15)t +1 ∆ (cid:96) t p ( ρ (cid:15)t +1 | ρ (cid:15)t , − ) = b ( ρ − t ) = b t , (A6)where a ( ρ + t ) and b ( ρ − t ) are functions of the book state ρ (cid:15) expressing the ask and the bid price. Notice that arelation analogous to (18) can be derived in terms of con-ditional averages, so that it is also possible to write (cid:104) ∆ (cid:96) t (cid:105) ϕ = (cid:18) (cid:104) π t (cid:105) ϕ, + + (cid:104) π t (cid:105) ϕ, − (cid:19) + ϕ (cid:18) (cid:104) π t (cid:105) ϕ, + − (cid:104) π t (cid:105) ϕ, − (cid:104) s t (cid:105) ϕ, + + (cid:104) s t (cid:105) ϕ, − (cid:19) , which relates the conditional and unconditional values ofthe averages. In particular by matching term by termEqs. (18) and (A7) one obtains (cid:104) π t (cid:105) ϕ = (cid:104) π t (cid:105) ϕ, + + (cid:104) π t (cid:105) ϕ, − (cid:104) s t (cid:105) ϕ,(cid:15) = (cid:104) s t (cid:105) ϕ, − (cid:15) (A8) (cid:104) π t (cid:105) ϕ − (cid:104) π t (cid:105) ϕ,(cid:15) = − (cid:15) ( (cid:104) s t (cid:105) ϕ − (cid:104) s t (cid:105) ϕ,(cid:15) ) . (A9)The expression of the autocorrelation function for thisprocess is slightly more involved to derive: one can obtainEq. (23) by considering a symmetric initial condition forthe book and by writing (cid:104) ∆ (cid:96) t ∆ (cid:96) (cid:105) ϕ = s (cid:88) (cid:15) (cid:15) (cid:18) ϕ (cid:15) (cid:19) (cid:104) ∆ (cid:96) t (cid:105) ϕ,(cid:15) . (A10)After using the master equation (A5) in order to obtainthe relation (cid:104) ∆ (cid:96) t (cid:105) ϕ,(cid:15) = (cid:104) π t (cid:105) ϕ,(cid:15) + (cid:18) ϕ + g t (1 − ϕ ) (cid:15) ϕ(cid:15) (cid:19) (cid:104) s t (cid:105) ϕ,(cid:15) , (A11)Eq. (23) can then be recovered. [1] Kyle, A. (1985). 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